Theorem mdsymlem5 5780

Description: Lemma for mdsym 5784.

Hypotheses

Ref	Expression
mdsymlem1.1	⊢ A ∈ Cℋ
mdsymlem1.2	⊢ B ∈ Cℋ
mdsymlem1.3	⊢ C = (A ∨ℋ p)

Assertion

Ref	Expression
mdsymlem5	⊢ ((q ∈ Atoms ∧ r ∈ Atoms) → (¬ q = p → ((p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B)) → (((c ∈ Cℋ ∧ A ⊆ c) ∧ p ∈ Atoms) → (p ⊆ c → p ⊆ ((c ∩ B) ∨ℋ A))))))

Distinct variable group(s):   q,r,C   c,p,q,r,A   B,c,p,q,r

Proof of Theorem mdsymlem5

Step	Hyp	Ref	Expression
1		atnem0 5766	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((q ∈ Atoms ∧ p ∈ Atoms) → (¬ q = p ↔ (q ∩ p) = 0ℋ))
2	1	anbi2d 468	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((q ∈ Atoms ∧ p ∈ Atoms) → ((p ⊆ (q ∨ℋ r) ∧ ¬ q = p) ↔ (p ⊆ (q ∨ℋ r) ∧ (q ∩ p) = 0ℋ)))
3	2	3adant3 599	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((q ∈ Atoms ∧ p ∈ Atoms ∧ r ∈ Atoms) → ((p ⊆ (q ∨ℋ r) ∧ ¬ q = p) ↔ (p ⊆ (q ∨ℋ r) ∧ (q ∩ p) = 0ℋ)))
4		atexch 5769	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((q ∈ Cℋ ∧ p ∈ Atoms ∧ r ∈ Atoms) → ((p ⊆ (q ∨ℋ r) ∧ (q ∩ p) = 0ℋ) → r ⊆ (q ∨ℋ p)))
5		atelch 5742	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (q ∈ Atoms → q ∈ Cℋ )
6	4, 5	syl3an1 619	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((q ∈ Atoms ∧ p ∈ Atoms ∧ r ∈ Atoms) → ((p ⊆ (q ∨ℋ r) ∧ (q ∩ p) = 0ℋ) → r ⊆ (q ∨ℋ p)))
7	3, 6	sylbid 178	. . . . . . . . . . . . . . . . . . 19 ⊢ ((q ∈ Atoms ∧ p ∈ Atoms ∧ r ∈ Atoms) → ((p ⊆ (q ∨ℋ r) ∧ ¬ q = p) → r ⊆ (q ∨ℋ p)))
8	7	exp3a 292	. . . . . . . . . . . . . . . . . 18 ⊢ ((q ∈ Atoms ∧ p ∈ Atoms ∧ r ∈ Atoms) → (p ⊆ (q ∨ℋ r) → (¬ q = p → r ⊆ (q ∨ℋ p))))
9	8	3com23 616	. . . . . . . . . . . . . . . . 17 ⊢ ((q ∈ Atoms ∧ r ∈ Atoms ∧ p ∈ Atoms) → (p ⊆ (q ∨ℋ r) → (¬ q = p → r ⊆ (q ∨ℋ p))))
10	9	3expa 612	. . . . . . . . . . . . . . . 16 ⊢ (((q ∈ Atoms ∧ r ∈ Atoms) ∧ p ∈ Atoms) → (p ⊆ (q ∨ℋ r) → (¬ q = p → r ⊆ (q ∨ℋ p))))
11	10	adantrl 311	. . . . . . . . . . . . . . 15 ⊢ (((q ∈ Atoms ∧ r ∈ Atoms) ∧ (c ∈ Cℋ ∧ p ∈ Atoms)) → (p ⊆ (q ∨ℋ r) → (¬ q = p → r ⊆ (q ∨ℋ p))))
12	11	adantrd 308	. . . . . . . . . . . . . 14 ⊢ (((q ∈ Atoms ∧ r ∈ Atoms) ∧ (c ∈ Cℋ ∧ p ∈ Atoms)) → ((p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B)) → (¬ q = p → r ⊆ (q ∨ℋ p))))
13	12	imp32 281	. . . . . . . . . . . . 13 ⊢ ((((q ∈ Atoms ∧ r ∈ Atoms) ∧ (c ∈ Cℋ ∧ p ∈ Atoms)) ∧ ((p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B)) ∧ ¬ q = p)) → r ⊆ (q ∨ℋ p))
14	13	adantrl 311	. . . . . . . . . . . 12 ⊢ ((((q ∈ Atoms ∧ r ∈ Atoms) ∧ (c ∈ Cℋ ∧ p ∈ Atoms)) ∧ (A ⊆ c ∧ ((p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B)) ∧ ¬ q = p))) → r ⊆ (q ∨ℋ p))
15	14	adantrr 312	. . . . . . . . . . 11 ⊢ ((((q ∈ Atoms ∧ r ∈ Atoms) ∧ (c ∈ Cℋ ∧ p ∈ Atoms)) ∧ ((A ⊆ c ∧ ((p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B)) ∧ ¬ q = p)) ∧ p ⊆ c)) → r ⊆ (q ∨ℋ p))
16		sstr 1511	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((q ⊆ A ∧ A ⊆ (c ∨ℋ A)) → q ⊆ (c ∨ℋ A))
17		mdsymlem1.1	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ A ∈ Cℋ
18		chub2t 5425	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((A ∈ Cℋ ∧ c ∈ Cℋ ) → A ⊆ (c ∨ℋ A))
19	17, 18	mpan 518	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (c ∈ Cℋ → A ⊆ (c ∨ℋ A))
20	16, 19	sylan2 346	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((q ⊆ A ∧ c ∈ Cℋ ) → q ⊆ (c ∨ℋ A))
21		sstr 1511	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((p ⊆ c ∧ c ⊆ (c ∨ℋ A)) → p ⊆ (c ∨ℋ A))
22		chub1t 5424	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((c ∈ Cℋ ∧ A ∈ Cℋ ) → c ⊆ (c ∨ℋ A))
23	17, 22	mpan2 519	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (c ∈ Cℋ → c ⊆ (c ∨ℋ A))
24	21, 23	sylan2 346	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((p ⊆ c ∧ c ∈ Cℋ ) → p ⊆ (c ∨ℋ A))
25	20, 24	anim12i 268	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((q ⊆ A ∧ c ∈ Cℋ ) ∧ (p ⊆ c ∧ c ∈ Cℋ )) → (q ⊆ (c ∨ℋ A) ∧ p ⊆ (c ∨ℋ A)))
26	25	anandirs 395	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((q ⊆ A ∧ p ⊆ c) ∧ c ∈ Cℋ ) → (q ⊆ (c ∨ℋ A) ∧ p ⊆ (c ∨ℋ A)))
27	26	ancoms 334	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((c ∈ Cℋ ∧ (q ⊆ A ∧ p ⊆ c)) → (q ⊆ (c ∨ℋ A) ∧ p ⊆ (c ∨ℋ A)))
28	27	adantll 309	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((q ∈ Cℋ ∧ p ∈ Cℋ ) ∧ c ∈ Cℋ ) ∧ (q ⊆ A ∧ p ⊆ c)) → (q ⊆ (c ∨ℋ A) ∧ p ⊆ (c ∨ℋ A)))
29		chlubt 5426	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((q ∈ Cℋ ∧ p ∈ Cℋ ∧ (c ∨ℋ A) ∈ Cℋ ) → ((q ⊆ (c ∨ℋ A) ∧ p ⊆ (c ∨ℋ A)) ↔ (q ∨ℋ p) ⊆ (c ∨ℋ A)))
30		chjclt 5330	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((c ∈ Cℋ ∧ A ∈ Cℋ ) → (c ∨ℋ A) ∈ Cℋ )
31	17, 30	mpan2 519	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (c ∈ Cℋ → (c ∨ℋ A) ∈ Cℋ )
32	29, 31	syl3an3 621	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((q ∈ Cℋ ∧ p ∈ Cℋ ∧ c ∈ Cℋ ) → ((q ⊆ (c ∨ℋ A) ∧ p ⊆ (c ∨ℋ A)) ↔ (q ∨ℋ p) ⊆ (c ∨ℋ A)))
33	32	3expa 612	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((q ∈ Cℋ ∧ p ∈ Cℋ ) ∧ c ∈ Cℋ ) → ((q ⊆ (c ∨ℋ A) ∧ p ⊆ (c ∨ℋ A)) ↔ (q ∨ℋ p) ⊆ (c ∨ℋ A)))
34	33	adantr 306	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((q ∈ Cℋ ∧ p ∈ Cℋ ) ∧ c ∈ Cℋ ) ∧ (q ⊆ A ∧ p ⊆ c)) → ((q ⊆ (c ∨ℋ A) ∧ p ⊆ (c ∨ℋ A)) ↔ (q ∨ℋ p) ⊆ (c ∨ℋ A)))
35	28, 34	mpbid 170	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((q ∈ Cℋ ∧ p ∈ Cℋ ) ∧ c ∈ Cℋ ) ∧ (q ⊆ A ∧ p ⊆ c)) → (q ∨ℋ p) ⊆ (c ∨ℋ A))
36	35	adantrl 311	. . . . . . . . . . . . . . . . . 18 ⊢ ((((q ∈ Cℋ ∧ p ∈ Cℋ ) ∧ c ∈ Cℋ ) ∧ (A ⊆ c ∧ (q ⊆ A ∧ p ⊆ c))) → (q ∨ℋ p) ⊆ (c ∨ℋ A))
37		chlejb2t 5430	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((A ∈ Cℋ ∧ c ∈ Cℋ ) → (A ⊆ c ↔ (c ∨ℋ A) = c))
38	17, 37	mpan 518	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (c ∈ Cℋ → (A ⊆ c ↔ (c ∨ℋ A) = c))
39	38	biimpa 324	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((c ∈ Cℋ ∧ A ⊆ c) → (c ∨ℋ A) = c)
40	39	adantll 309	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((q ∈ Cℋ ∧ p ∈ Cℋ ) ∧ c ∈ Cℋ ) ∧ A ⊆ c) → (c ∨ℋ A) = c)
41	40	adantrr 312	. . . . . . . . . . . . . . . . . 18 ⊢ ((((q ∈ Cℋ ∧ p ∈ Cℋ ) ∧ c ∈ Cℋ ) ∧ (A ⊆ c ∧ (q ⊆ A ∧ p ⊆ c))) → (c ∨ℋ A) = c)
42	36, 41	sseqtrd 1536	. . . . . . . . . . . . . . . . 17 ⊢ ((((q ∈ Cℋ ∧ p ∈ Cℋ ) ∧ c ∈ Cℋ ) ∧ (A ⊆ c ∧ (q ⊆ A ∧ p ⊆ c))) → (q ∨ℋ p) ⊆ c)
43	42	exp45 303	. . . . . . . . . . . . . . . 16 ⊢ (((q ∈ Cℋ ∧ p ∈ Cℋ ) ∧ c ∈ Cℋ ) → (A ⊆ c → (q ⊆ A → (p ⊆ c → (q ∨ℋ p) ⊆ c))))
44	43	anasss 337	. . . . . . . . . . . . . . 15 ⊢ ((q ∈ Cℋ ∧ (p ∈ Cℋ ∧ c ∈ Cℋ )) → (A ⊆ c → (q ⊆ A → (p ⊆ c → (q ∨ℋ p) ⊆ c))))
45		atelch 5742	. . . . . . . . . . . . . . . . 17 ⊢ (p ∈ Atoms → p ∈ Cℋ )
46	45	anim1i 269	. . . . . . . . . . . . . . . 16 ⊢ ((p ∈ Atoms ∧ c ∈ Cℋ ) → (p ∈ Cℋ ∧ c ∈ Cℋ ))
47	46	ancoms 334	. . . . . . . . . . . . . . 15 ⊢ ((c ∈ Cℋ ∧ p ∈ Atoms) → (p ∈ Cℋ ∧ c ∈ Cℋ ))
48	44, 5, 47	syl2an 349	. . . . . . . . . . . . . 14 ⊢ ((q ∈ Atoms ∧ (c ∈ Cℋ ∧ p ∈ Atoms)) → (A ⊆ c → (q ⊆ A → (p ⊆ c → (q ∨ℋ p) ⊆ c))))
49	48	adantlr 310	. . . . . . . . . . . . 13 ⊢ (((q ∈ Atoms ∧ r ∈ Atoms) ∧ (c ∈ Cℋ ∧ p ∈ Atoms)) → (A ⊆ c → (q ⊆ A → (p ⊆ c → (q ∨ℋ p) ⊆ c))))
50		pm3.26 256	. . . . . . . . . . . . . 14 ⊢ ((q ⊆ A ∧ r ⊆ B) → q ⊆ A)
51	50	ad2antlr 321	. . . . . . . . . . . . 13 ⊢ (((p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B)) ∧ ¬ q = p) → q ⊆ A)
52	49, 51	syl7 24	. . . . . . . . . . . 12 ⊢ (((q ∈ Atoms ∧ r ∈ Atoms) ∧ (c ∈ Cℋ ∧ p ∈ Atoms)) → (A ⊆ c → (((p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B)) ∧ ¬ q = p) → (p ⊆ c → (q ∨ℋ p) ⊆ c))))
53	52	imp44 289	. . . . . . . . . . 11 ⊢ ((((q ∈ Atoms ∧ r ∈ Atoms) ∧ (c ∈ Cℋ ∧ p ∈ Atoms)) ∧ ((A ⊆ c ∧ ((p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B)) ∧ ¬ q = p)) ∧ p ⊆ c)) → (q ∨ℋ p) ⊆ c)
54	15, 53	sstrd 1513	. . . . . . . . . 10 ⊢ ((((q ∈ Atoms ∧ r ∈ Atoms) ∧ (c ∈ Cℋ ∧ p ∈ Atoms)) ∧ ((A ⊆ c ∧ ((p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B)) ∧ ¬ q = p)) ∧ p ⊆ c)) → r ⊆ c)
55		pm3.27 260	. . . . . . . . . . . . 13 ⊢ ((q ⊆ A ∧ r ⊆ B) → r ⊆ B)
56	55	ad2antlr 321	. . . . . . . . . . . 12 ⊢ (((p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B)) ∧ ¬ q = p) → r ⊆ B)
57	56	ad2antlr 321	. . . . . . . . . . 11 ⊢ (((A ⊆ c ∧ ((p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B)) ∧ ¬ q = p)) ∧ p ⊆ c) → r ⊆ B)
58	57	adantl 305	. . . . . . . . . 10 ⊢ ((((q ∈ Atoms ∧ r ∈ Atoms) ∧ (c ∈ Cℋ ∧ p ∈ Atoms)) ∧ ((A ⊆ c ∧ ((p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B)) ∧ ¬ q = p)) ∧ p ⊆ c)) → r ⊆ B)
59	54, 58	jca 236	. . . . . . . . 9 ⊢ ((((q ∈ Atoms ∧ r ∈ Atoms) ∧ (c ∈ Cℋ ∧ p ∈ Atoms)) ∧ ((A ⊆ c ∧ ((p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B)) ∧ ¬ q = p)) ∧ p ⊆ c)) → (r ⊆ c ∧ r ⊆ B))
60		ssin 1659	. . . . . . . . 9 ⊢ ((r ⊆ c ∧ r ⊆ B) ↔ r ⊆ (c ∩ B))
61	59, 60	sylib 173	. . . . . . . 8 ⊢ ((((q ∈ Atoms ∧ r ∈ Atoms) ∧ (c ∈ Cℋ ∧ p ∈ Atoms)) ∧ ((A ⊆ c ∧ ((p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B)) ∧ ¬ q = p)) ∧ p ⊆ c)) → r ⊆ (c ∩ B))
62		chlej1t 5427	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((r ∈ Cℋ ∧ (c ∩ B) ∈ Cℋ ∧ q ∈ Cℋ ) → (r ⊆ (c ∩ B) → (r ∨ℋ q) ⊆ ((c ∩ B) ∨ℋ q)))
63		mdsymlem1.2	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ B ∈ Cℋ
64		chinclt 5416	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((c ∈ Cℋ ∧ B ∈ Cℋ ) → (c ∩ B) ∈ Cℋ )
65	63, 64	mpan2 519	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (c ∈ Cℋ → (c ∩ B) ∈ Cℋ )
66	62, 65	syl3an2 620	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((r ∈ Cℋ ∧ c ∈ Cℋ ∧ q ∈ Cℋ ) → (r ⊆ (c ∩ B) → (r ∨ℋ q) ⊆ ((c ∩ B) ∨ℋ q)))
67	66	3comr 618	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((q ∈ Cℋ ∧ r ∈ Cℋ ∧ c ∈ Cℋ ) → (r ⊆ (c ∩ B) → (r ∨ℋ q) ⊆ ((c ∩ B) ∨ℋ q)))
68	67	3expa 612	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((q ∈ Cℋ ∧ r ∈ Cℋ ) ∧ c ∈ Cℋ ) → (r ⊆ (c ∩ B) → (r ∨ℋ q) ⊆ ((c ∩ B) ∨ℋ q)))
69	68	adantr 306	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((q ∈ Cℋ ∧ r ∈ Cℋ ) ∧ c ∈ Cℋ ) ∧ q ⊆ A) → (r ⊆ (c ∩ B) → (r ∨ℋ q) ⊆ ((c ∩ B) ∨ℋ q)))
70		sstr2 1510	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((r ∨ℋ q) ⊆ ((c ∩ B) ∨ℋ q) → (((c ∩ B) ∨ℋ q) ⊆ ((c ∩ B) ∨ℋ A) → (r ∨ℋ q) ⊆ ((c ∩ B) ∨ℋ A)))
71		chlej2t 5428	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((q ∈ Cℋ ∧ A ∈ Cℋ ∧ (c ∩ B) ∈ Cℋ ) → (q ⊆ A → ((c ∩ B) ∨ℋ q) ⊆ ((c ∩ B) ∨ℋ A)))
72	17, 71	mp3an2 640	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((q ∈ Cℋ ∧ (c ∩ B) ∈ Cℋ ) → (q ⊆ A → ((c ∩ B) ∨ℋ q) ⊆ ((c ∩ B) ∨ℋ A)))
73	72, 65	sylan2 346	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((q ∈ Cℋ ∧ c ∈ Cℋ ) → (q ⊆ A → ((c ∩ B) ∨ℋ q) ⊆ ((c ∩ B) ∨ℋ A)))
74	73	adantlr 310	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((q ∈ Cℋ ∧ r ∈ Cℋ ) ∧ c ∈ Cℋ ) → (q ⊆ A → ((c ∩ B) ∨ℋ q) ⊆ ((c ∩ B) ∨ℋ A)))
75	74	imp 277	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((q ∈ Cℋ ∧ r ∈ Cℋ ) ∧ c ∈ Cℋ ) ∧ q ⊆ A) → ((c ∩ B) ∨ℋ q) ⊆ ((c ∩ B) ∨ℋ A))
76	70, 75	syl5 22	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((r ∨ℋ q) ⊆ ((c ∩ B) ∨ℋ q) → ((((q ∈ Cℋ ∧ r ∈ Cℋ ) ∧ c ∈ Cℋ ) ∧ q ⊆ A) → (r ∨ℋ q) ⊆ ((c ∩ B) ∨ℋ A)))
77	76	com12 13	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((q ∈ Cℋ ∧ r ∈ Cℋ ) ∧ c ∈ Cℋ ) ∧ q ⊆ A) → ((r ∨ℋ q) ⊆ ((c ∩ B) ∨ℋ q) → (r ∨ℋ q) ⊆ ((c ∩ B) ∨ℋ A)))
78		chjcomt 5423	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((q ∈ Cℋ ∧ r ∈ Cℋ ) → (q ∨ℋ r) = (r ∨ℋ q))
79	78	ad2antll 320	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((q ∈ Cℋ ∧ r ∈ Cℋ ) ∧ c ∈ Cℋ ) ∧ q ⊆ A) → (q ∨ℋ r) = (r ∨ℋ q))
80	79	sseq1d 1527	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((q ∈ Cℋ ∧ r ∈ Cℋ ) ∧ c ∈ Cℋ ) ∧ q ⊆ A) → ((q ∨ℋ r) ⊆ ((c ∩ B) ∨ℋ A) ↔ (r ∨ℋ q) ⊆ ((c ∩ B) ∨ℋ A)))
81	77, 80	sylibrd 179	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((q ∈ Cℋ ∧ r ∈ Cℋ ) ∧ c ∈ Cℋ ) ∧ q ⊆ A) → ((r ∨ℋ q) ⊆ ((c ∩ B) ∨ℋ q) → (q ∨ℋ r) ⊆ ((c ∩ B) ∨ℋ A)))
82	69, 81	syld 27	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((q ∈ Cℋ ∧ r ∈ Cℋ ) ∧ c ∈ Cℋ ) ∧ q ⊆ A) → (r ⊆ (c ∩ B) → (q ∨ℋ r) ⊆ ((c ∩ B) ∨ℋ A)))
83	82	adantrl 311	. . . . . . . . . . . . . . . . . 18 ⊢ ((((q ∈ Cℋ ∧ r ∈ Cℋ ) ∧ c ∈ Cℋ ) ∧ (p ⊆ (q ∨ℋ r) ∧ q ⊆ A)) → (r ⊆ (c ∩ B) → (q ∨ℋ r) ⊆ ((c ∩ B) ∨ℋ A)))
84		sstr2 1510	. . . . . . . . . . . . . . . . . . 19 ⊢ (p ⊆ (q ∨ℋ r) → ((q ∨ℋ r) ⊆ ((c ∩ B) ∨ℋ A) → p ⊆ ((c ∩ B) ∨ℋ A)))
85	84	ad2antrl 322	. . . . . . . . . . . . . . . . . 18 ⊢ ((((q ∈ Cℋ ∧ r ∈ Cℋ ) ∧ c ∈ Cℋ ) ∧ (p ⊆ (q ∨ℋ r) ∧ q ⊆ A)) → ((q ∨ℋ r) ⊆ ((c ∩ B) ∨ℋ A) → p ⊆ ((c ∩ B) ∨ℋ A)))
86	83, 85	syld 27	. . . . . . . . . . . . . . . . 17 ⊢ ((((q ∈ Cℋ ∧ r ∈ Cℋ ) ∧ c ∈ Cℋ ) ∧ (p ⊆ (q ∨ℋ r) ∧ q ⊆ A)) → (r ⊆ (c ∩ B) → p ⊆ ((c ∩ B) ∨ℋ A)))
87	86	exp32 294	. . . . . . . . . . . . . . . 16 ⊢ (((q ∈ Cℋ ∧ r ∈ Cℋ ) ∧ c ∈ Cℋ ) → (p ⊆ (q ∨ℋ r) → (q ⊆ A → (r ⊆ (c ∩ B) → p ⊆ ((c ∩ B) ∨ℋ A)))))
88		atelch 5742	. . . . . . . . . . . . . . . . 17 ⊢ (r ∈ Atoms → r ∈ Cℋ )
89	5, 88	anim12i 268	. . . . . . . . . . . . . . . 16 ⊢ ((q ∈ Atoms ∧ r ∈ Atoms) → (q ∈ Cℋ ∧ r ∈ Cℋ ))
90	87, 89	sylan 343	. . . . . . . . . . . . . . 15 ⊢ (((q ∈ Atoms ∧ r ∈ Atoms) ∧ c ∈ Cℋ ) → (p ⊆ (q ∨ℋ r) → (q ⊆ A → (r ⊆ (c ∩ B) → p ⊆ ((c ∩ B) ∨ℋ A)))))
91	90	adantrr 312	. . . . . . . . . . . . . 14 ⊢ (((q ∈ Atoms ∧ r ∈ Atoms) ∧ (c ∈ Cℋ ∧ p ∈ Atoms)) → (p ⊆ (q ∨ℋ r) → (q ⊆ A → (r ⊆ (c ∩ B) → p ⊆ ((c ∩ B) ∨ℋ A)))))
92	91	imp31 280	. . . . . . . . . . . . 13 ⊢ (((((q ∈ Atoms ∧ r ∈ Atoms) ∧ (c ∈ Cℋ ∧ p ∈ Atoms)) ∧ p ⊆ (q ∨ℋ r)) ∧ q ⊆ A) → (r ⊆ (c ∩ B) → p ⊆ ((c ∩ B) ∨ℋ A)))
93	92	adantrr 312	. . . . . . . . . . . 12 ⊢ (((((q ∈ Atoms ∧ r ∈ Atoms) ∧ (c ∈ Cℋ ∧ p ∈ Atoms)) ∧ p ⊆ (q ∨ℋ r)) ∧ (q ⊆ A ∧ r ⊆ B)) → (r ⊆ (c ∩ B) → p ⊆ ((c ∩ B) ∨ℋ A)))
94	93	anasss 337	. . . . . . . . . . 11 ⊢ ((((q ∈ Atoms ∧ r ∈ Atoms) ∧ (c ∈ Cℋ ∧ p ∈ Atoms)) ∧ (p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B))) → (r ⊆ (c ∩ B) → p ⊆ ((c ∩ B) ∨ℋ A)))
95	94	adantrr 312	. . . . . . . . . 10 ⊢ ((((q ∈ Atoms ∧ r ∈ Atoms) ∧ (c ∈ Cℋ ∧ p ∈ Atoms)) ∧ ((p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B)) ∧ ¬ q = p)) → (r ⊆ (c ∩ B) → p ⊆ ((c ∩ B) ∨ℋ A)))
96	95	adantrl 311	. . . . . . . . 9 ⊢ ((((q ∈ Atoms ∧ r ∈ Atoms) ∧ (c ∈ Cℋ ∧ p ∈ Atoms)) ∧ (A ⊆ c ∧ ((p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B)) ∧ ¬ q = p))) → (r ⊆ (c ∩ B) → p ⊆ ((c ∩ B) ∨ℋ A)))
97	96	adantrr 312	. . . . . . . 8 ⊢ ((((q ∈ Atoms ∧ r ∈ Atoms) ∧ (c ∈ Cℋ ∧ p ∈ Atoms)) ∧ ((A ⊆ c ∧ ((p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B)) ∧ ¬ q = p)) ∧ p ⊆ c)) → (r ⊆ (c ∩ B) → p ⊆ ((c ∩ B) ∨ℋ A)))
98	61, 97	mpd 46	. . . . . . 7 ⊢ ((((q ∈ Atoms ∧ r ∈ Atoms) ∧ (c ∈ Cℋ ∧ p ∈ Atoms)) ∧ ((A ⊆ c ∧ ((p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B)) ∧ ¬ q = p)) ∧ p ⊆ c)) → p ⊆ ((c ∩ B) ∨ℋ A))
99	98	exp32 294	. . . . . 6 ⊢ (((q ∈ Atoms ∧ r ∈ Atoms) ∧ (c ∈ Cℋ ∧ p ∈ Atoms)) → ((A ⊆ c ∧ ((p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B)) ∧ ¬ q = p)) → (p ⊆ c → p ⊆ ((c ∩ B) ∨ℋ A))))
100	99	exp4d 298	. . . . 5 ⊢ (((q ∈ Atoms ∧ r ∈ Atoms) ∧ (c ∈ Cℋ ∧ p ∈ Atoms)) → (A ⊆ c → ((p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B)) → (¬ q = p → (p ⊆ c → p ⊆ ((c ∩ B) ∨ℋ A))))))
101	100	exp32 294	. . . 4 ⊢ ((q ∈ Atoms ∧ r ∈ Atoms) → (c ∈ Cℋ → (p ∈ Atoms → (A ⊆ c → ((p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B)) → (¬ q = p → (p ⊆ c → p ⊆ ((c ∩ B) ∨ℋ A))))))))
102	101	com34 36	. . 3 ⊢ ((q ∈ Atoms ∧ r ∈ Atoms) → (c ∈ Cℋ → (A ⊆ c → (p ∈ Atoms → ((p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B)) → (¬ q = p → (p ⊆ c → p ⊆ ((c ∩ B) ∨ℋ A))))))))
103	102	imp4c 284	. 2 ⊢ ((q ∈ Atoms ∧ r ∈ Atoms) → (((c ∈ Cℋ ∧ A ⊆ c) ∧ p ∈ Atoms) → ((p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B)) → (¬ q = p → (p ⊆ c → p ⊆ ((c ∩ B) ∨ℋ A))))))
104	103	com24 37	1 ⊢ ((q ∈ Atoms ∧ r ∈ Atoms) → (¬ q = p → ((p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B)) → (((c ∈ Cℋ ∧ A ⊆ c) ∧ p ∈ Atoms) → (p ⊆ c → p ⊆ ((c ∩ B) ∨ℋ A))))))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   ∧ w3a 581   = weq 797   = wceq 1091   ∈ wcel 1092   ∩ cin 1486   ⊆ wss 1487  (class class class)co 3001   C_ℋ cch 4968   ∨_ℋ chj 4972  0_ℋc0h 4974  Atomscat 4980

This theorem is referenced by:  mdsymlem6 5781

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079  ax-ac 1080  ax-hilex 4983  ax-hvaddcl 4984  ax-hvcom 4985  ax-hvass 4986  ax-hvzercl 4987  ax-hvaddid 4988  ax-hvmulcl 4989  ax-hvmulid 4991  ax-hvmulass 4992  ax-hvdistr1 4993  ax-hvdistr2 4994  ax-hvmulzer 4995  ax-hicl 5043  ax-his1 5045  ax-his2 5046  ax-his3 5047  ax-his4 5048  ax-hcompl 5113

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-sup 2154  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1st 3087  df-2nd 3088  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-plpr 3958  df-mpr 3959  df-enr 3960  df-nr 3961  df-plr 3962  df-mr 3963  df-ltr 3964  df-0r 3965  df-1r 3966  df-m1r 3967  df-c 4034  df-0 4035  df-1 4036  df-i 4037  df-r 4038  df-plus 4039  df-mul 4040  df-lt 4041  df-sub 4133  df-neg 4135  df-div 4216  df-le 4277  df-n 4423  df-2 4462  df-3 4463  df-4 4464  df-n0 4535  df-z 4564  df-seq 4661  df-exp 4676  df-sqr 4728  df-re 4790  df-im 4791  df-cj 4792  df-abs 4793  df-clim 4876  df-hvsub 4996  df-hnorm 5074  df-cauchy 5102  df-hlim 5107  df-sh 5114  df-ch 5127  df-oc 5156  df-ch0 5157  df-pj 5244  df-shsum 5275  df-span 5276  df-chj 5277  df-chsup 5278  df-cv 5712  df-at 5737

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