Theorem mdsymlem3 5778

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
mdsymlem3	⊢ ((((p ∈ Atoms ∧ ¬ (B ∩ C) ⊆ A) ∧ p ⊆ (A ∨ℋ B)) ∧ ¬ A = 0ℋ) → ∃r ∈ Atoms ∃q ∈ Atoms (p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B)))

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

Proof of Theorem mdsymlem3

Step	Hyp	Ref	Expression
1		atelch 5742	. . . . 5 ⊢ (p ∈ Atoms → p ∈ Cℋ )
2		mdsymlem1.3	. . . . . . . 8 ⊢ C = (A ∨ℋ p)
3	2	a1i 7	. . . . . . 7 ⊢ (p ∈ Cℋ → C = (A ∨ℋ p))
4		mdsymlem1.1	. . . . . . . 8 ⊢ A ∈ Cℋ
5		chjclt 5330	. . . . . . . 8 ⊢ ((A ∈ Cℋ ∧ p ∈ Cℋ ) → (A ∨ℋ p) ∈ Cℋ )
6	4, 5	mpan 518	. . . . . . 7 ⊢ (p ∈ Cℋ → (A ∨ℋ p) ∈ Cℋ )
7	3, 6	eqeltrd 1163	. . . . . 6 ⊢ (p ∈ Cℋ → C ∈ Cℋ )
8		mdsymlem1.2	. . . . . . 7 ⊢ B ∈ Cℋ
9		chinclt 5416	. . . . . . 7 ⊢ ((B ∈ Cℋ ∧ C ∈ Cℋ ) → (B ∩ C) ∈ Cℋ )
10	8, 9	mpan 518	. . . . . 6 ⊢ (C ∈ Cℋ → (B ∩ C) ∈ Cℋ )
11	7, 10	syl 12	. . . . 5 ⊢ (p ∈ Cℋ → (B ∩ C) ∈ Cℋ )
12		chrelat2t 5761	. . . . . 6 ⊢ (((B ∩ C) ∈ Cℋ ∧ A ∈ Cℋ ) → (¬ (B ∩ C) ⊆ A ↔ ∃r ∈ Atoms (r ⊆ (B ∩ C) ∧ ¬ r ⊆ A)))
13	4, 12	mpan2 519	. . . . 5 ⊢ ((B ∩ C) ∈ Cℋ → (¬ (B ∩ C) ⊆ A ↔ ∃r ∈ Atoms (r ⊆ (B ∩ C) ∧ ¬ r ⊆ A)))
14	1, 11, 13	3syl 21	. . . 4 ⊢ (p ∈ Atoms → (¬ (B ∩ C) ⊆ A ↔ ∃r ∈ Atoms (r ⊆ (B ∩ C) ∧ ¬ r ⊆ A)))
15	14	biimpa 324	. . 3 ⊢ ((p ∈ Atoms ∧ ¬ (B ∩ C) ⊆ A) → ∃r ∈ Atoms (r ⊆ (B ∩ C) ∧ ¬ r ⊆ A))
16	15	ad2antll 320	. 2 ⊢ ((((p ∈ Atoms ∧ ¬ (B ∩ C) ⊆ A) ∧ p ⊆ (A ∨ℋ B)) ∧ ¬ A = 0ℋ) → ∃r ∈ Atoms (r ⊆ (B ∩ C) ∧ ¬ r ⊆ A))
17	4	atcvat4 5775	. . . . . . . . . . . . . . 15 ⊢ ((r ∈ Atoms ∧ p ∈ Atoms) → ((¬ A = 0ℋ ∧ r ⊆ (A ∨ℋ p)) → ∃q ∈ Atoms (q ⊆ A ∧ r ⊆ (p ∨ℋ q))))
18	17	exp4b 296	. . . . . . . . . . . . . 14 ⊢ (r ∈ Atoms → (p ∈ Atoms → (¬ A = 0ℋ → (r ⊆ (A ∨ℋ p) → ∃q ∈ Atoms (q ⊆ A ∧ r ⊆ (p ∨ℋ q))))))
19	18	com34 36	. . . . . . . . . . . . 13 ⊢ (r ∈ Atoms → (p ∈ Atoms → (r ⊆ (A ∨ℋ p) → (¬ A = 0ℋ → ∃q ∈ Atoms (q ⊆ A ∧ r ⊆ (p ∨ℋ q))))))
20	19	com23 32	. . . . . . . . . . . 12 ⊢ (r ∈ Atoms → (r ⊆ (A ∨ℋ p) → (p ∈ Atoms → (¬ A = 0ℋ → ∃q ∈ Atoms (q ⊆ A ∧ r ⊆ (p ∨ℋ q))))))
21	20	imp4b 283	. . . . . . . . . . 11 ⊢ ((r ∈ Atoms ∧ r ⊆ (A ∨ℋ p)) → ((p ∈ Atoms ∧ ¬ A = 0ℋ) → ∃q ∈ Atoms (q ⊆ A ∧ r ⊆ (p ∨ℋ q))))
22		ssin 1659	. . . . . . . . . . . 12 ⊢ ((r ⊆ B ∧ r ⊆ C) ↔ r ⊆ (B ∩ C))
23	2	sseq2i 1525	. . . . . . . . . . . . . 14 ⊢ (r ⊆ C ↔ r ⊆ (A ∨ℋ p))
24	23	biimp 133	. . . . . . . . . . . . 13 ⊢ (r ⊆ C → r ⊆ (A ∨ℋ p))
25	24	adantl 305	. . . . . . . . . . . 12 ⊢ ((r ⊆ B ∧ r ⊆ C) → r ⊆ (A ∨ℋ p))
26	22, 25	sylbir 176	. . . . . . . . . . 11 ⊢ (r ⊆ (B ∩ C) → r ⊆ (A ∨ℋ p))
27	21, 26	sylan2 346	. . . . . . . . . 10 ⊢ ((r ∈ Atoms ∧ r ⊆ (B ∩ C)) → ((p ∈ Atoms ∧ ¬ A = 0ℋ) → ∃q ∈ Atoms (q ⊆ A ∧ r ⊆ (p ∨ℋ q))))
28	27	adantrr 312	. . . . . . . . 9 ⊢ ((r ∈ Atoms ∧ (r ⊆ (B ∩ C) ∧ ¬ r ⊆ A)) → ((p ∈ Atoms ∧ ¬ A = 0ℋ) → ∃q ∈ Atoms (q ⊆ A ∧ r ⊆ (p ∨ℋ q))))
29	28	com12 13	. . . . . . . 8 ⊢ ((p ∈ Atoms ∧ ¬ A = 0ℋ) → ((r ∈ Atoms ∧ (r ⊆ (B ∩ C) ∧ ¬ r ⊆ A)) → ∃q ∈ Atoms (q ⊆ A ∧ r ⊆ (p ∨ℋ q))))
30	29	adantlr 310	. . . . . . 7 ⊢ (((p ∈ Atoms ∧ ¬ (B ∩ C) ⊆ A) ∧ ¬ A = 0ℋ) → ((r ∈ Atoms ∧ (r ⊆ (B ∩ C) ∧ ¬ r ⊆ A)) → ∃q ∈ Atoms (q ⊆ A ∧ r ⊆ (p ∨ℋ q))))
31	30	adantlr 310	. . . . . 6 ⊢ ((((p ∈ Atoms ∧ ¬ (B ∩ C) ⊆ A) ∧ p ⊆ (A ∨ℋ B)) ∧ ¬ A = 0ℋ) → ((r ∈ Atoms ∧ (r ⊆ (B ∩ C) ∧ ¬ r ⊆ A)) → ∃q ∈ Atoms (q ⊆ A ∧ r ⊆ (p ∨ℋ q))))
32	31	imp 277	. . . . 5 ⊢ (((((p ∈ Atoms ∧ ¬ (B ∩ C) ⊆ A) ∧ p ⊆ (A ∨ℋ B)) ∧ ¬ A = 0ℋ) ∧ (r ∈ Atoms ∧ (r ⊆ (B ∩ C) ∧ ¬ r ⊆ A))) → ∃q ∈ Atoms (q ⊆ A ∧ r ⊆ (p ∨ℋ q)))
33		atnem0 5766	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((q ∈ Atoms ∧ r ∈ Atoms) → (¬ q = r ↔ (q ∩ r) = 0ℋ))
34	33	ancoms 334	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((r ∈ Atoms ∧ q ∈ Atoms) → (¬ q = r ↔ (q ∩ r) = 0ℋ))
35		sseq1 1521	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (q = r → (q ⊆ A ↔ r ⊆ A))
36	35	biimpcd 137	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (q ⊆ A → (q = r → r ⊆ A))
37	36	con3d 87	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (q ⊆ A → (¬ r ⊆ A → ¬ q = r))
38	37	imp 277	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((q ⊆ A ∧ ¬ r ⊆ A) → ¬ q = r)
39	38	adantrl 311	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((q ⊆ A ∧ (r ⊆ (B ∩ C) ∧ ¬ r ⊆ A)) → ¬ q = r)
40	34, 39	syl5bi 183	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((r ∈ Atoms ∧ q ∈ Atoms) → ((q ⊆ A ∧ (r ⊆ (B ∩ C) ∧ ¬ r ⊆ A)) → (q ∩ r) = 0ℋ))
41	40	adantll 309	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((p ∈ Atoms ∧ r ∈ Atoms) ∧ q ∈ Atoms) → ((q ⊆ A ∧ (r ⊆ (B ∩ C) ∧ ¬ r ⊆ A)) → (q ∩ r) = 0ℋ))
42	41	adantr 306	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((p ∈ Atoms ∧ r ∈ Atoms) ∧ q ∈ Atoms) ∧ r ⊆ (p ∨ℋ q)) → ((q ⊆ A ∧ (r ⊆ (B ∩ C) ∧ ¬ r ⊆ A)) → (q ∩ r) = 0ℋ))
43		chjcomt 5423	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((p ∈ Cℋ ∧ q ∈ Cℋ ) → (p ∨ℋ q) = (q ∨ℋ p))
44		atelch 5742	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (q ∈ Atoms → q ∈ Cℋ )
45	43, 1, 44	syl2an 349	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((p ∈ Atoms ∧ q ∈ Atoms) → (p ∨ℋ q) = (q ∨ℋ p))
46	45	adantlr 310	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((p ∈ Atoms ∧ r ∈ Atoms) ∧ q ∈ Atoms) → (p ∨ℋ q) = (q ∨ℋ p))
47	46	sseq2d 1528	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((p ∈ Atoms ∧ r ∈ Atoms) ∧ q ∈ Atoms) → (r ⊆ (p ∨ℋ q) ↔ r ⊆ (q ∨ℋ p)))
48		atexch 5769	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((q ∈ Cℋ ∧ r ∈ Atoms ∧ p ∈ Atoms) → ((r ⊆ (q ∨ℋ p) ∧ (q ∩ r) = 0ℋ) → p ⊆ (q ∨ℋ r)))
49	48, 44	syl3an1 619	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((q ∈ Atoms ∧ r ∈ Atoms ∧ p ∈ Atoms) → ((r ⊆ (q ∨ℋ p) ∧ (q ∩ r) = 0ℋ) → p ⊆ (q ∨ℋ r)))
50	49	3com13 615	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((p ∈ Atoms ∧ r ∈ Atoms ∧ q ∈ Atoms) → ((r ⊆ (q ∨ℋ p) ∧ (q ∩ r) = 0ℋ) → p ⊆ (q ∨ℋ r)))
51	50	3expa 612	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((p ∈ Atoms ∧ r ∈ Atoms) ∧ q ∈ Atoms) → ((r ⊆ (q ∨ℋ p) ∧ (q ∩ r) = 0ℋ) → p ⊆ (q ∨ℋ r)))
52	51	exp3a 292	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((p ∈ Atoms ∧ r ∈ Atoms) ∧ q ∈ Atoms) → (r ⊆ (q ∨ℋ p) → ((q ∩ r) = 0ℋ → p ⊆ (q ∨ℋ r))))
53	47, 52	sylbid 178	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((p ∈ Atoms ∧ r ∈ Atoms) ∧ q ∈ Atoms) → (r ⊆ (p ∨ℋ q) → ((q ∩ r) = 0ℋ → p ⊆ (q ∨ℋ r))))
54	53	imp 277	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((p ∈ Atoms ∧ r ∈ Atoms) ∧ q ∈ Atoms) ∧ r ⊆ (p ∨ℋ q)) → ((q ∩ r) = 0ℋ → p ⊆ (q ∨ℋ r)))
55	42, 54	syld 27	. . . . . . . . . . . . . . . . . 18 ⊢ ((((p ∈ Atoms ∧ r ∈ Atoms) ∧ q ∈ Atoms) ∧ r ⊆ (p ∨ℋ q)) → ((q ⊆ A ∧ (r ⊆ (B ∩ C) ∧ ¬ r ⊆ A)) → p ⊆ (q ∨ℋ r)))
56	55	exp3a 292	. . . . . . . . . . . . . . . . 17 ⊢ ((((p ∈ Atoms ∧ r ∈ Atoms) ∧ q ∈ Atoms) ∧ r ⊆ (p ∨ℋ q)) → (q ⊆ A → ((r ⊆ (B ∩ C) ∧ ¬ r ⊆ A) → p ⊆ (q ∨ℋ r))))
57	56	exp31 293	. . . . . . . . . . . . . . . 16 ⊢ ((p ∈ Atoms ∧ r ∈ Atoms) → (q ∈ Atoms → (r ⊆ (p ∨ℋ q) → (q ⊆ A → ((r ⊆ (B ∩ C) ∧ ¬ r ⊆ A) → p ⊆ (q ∨ℋ r))))))
58	57	com24 37	. . . . . . . . . . . . . . 15 ⊢ ((p ∈ Atoms ∧ r ∈ Atoms) → (q ⊆ A → (r ⊆ (p ∨ℋ q) → (q ∈ Atoms → ((r ⊆ (B ∩ C) ∧ ¬ r ⊆ A) → p ⊆ (q ∨ℋ r))))))
59	58	imp3a 279	. . . . . . . . . . . . . 14 ⊢ ((p ∈ Atoms ∧ r ∈ Atoms) → ((q ⊆ A ∧ r ⊆ (p ∨ℋ q)) → (q ∈ Atoms → ((r ⊆ (B ∩ C) ∧ ¬ r ⊆ A) → p ⊆ (q ∨ℋ r)))))
60	59	com24 37	. . . . . . . . . . . . 13 ⊢ ((p ∈ Atoms ∧ r ∈ Atoms) → ((r ⊆ (B ∩ C) ∧ ¬ r ⊆ A) → (q ∈ Atoms → ((q ⊆ A ∧ r ⊆ (p ∨ℋ q)) → p ⊆ (q ∨ℋ r)))))
61	60	imp4b 283	. . . . . . . . . . . 12 ⊢ (((p ∈ Atoms ∧ r ∈ Atoms) ∧ (r ⊆ (B ∩ C) ∧ ¬ r ⊆ A)) → ((q ∈ Atoms ∧ (q ⊆ A ∧ r ⊆ (p ∨ℋ q))) → p ⊆ (q ∨ℋ r)))
62	61	anasss 337	. . . . . . . . . . 11 ⊢ ((p ∈ Atoms ∧ (r ∈ Atoms ∧ (r ⊆ (B ∩ C) ∧ ¬ r ⊆ A))) → ((q ∈ Atoms ∧ (q ⊆ A ∧ r ⊆ (p ∨ℋ q))) → p ⊆ (q ∨ℋ r)))
63		pm3.26 256	. . . . . . . . . . . . . 14 ⊢ ((q ⊆ A ∧ r ⊆ (p ∨ℋ q)) → q ⊆ A)
64	63	adantl 305	. . . . . . . . . . . . 13 ⊢ ((q ∈ Atoms ∧ (q ⊆ A ∧ r ⊆ (p ∨ℋ q))) → q ⊆ A)
65	64	a1i 7	. . . . . . . . . . . 12 ⊢ ((p ∈ Atoms ∧ (r ∈ Atoms ∧ (r ⊆ (B ∩ C) ∧ ¬ r ⊆ A))) → ((q ∈ Atoms ∧ (q ⊆ A ∧ r ⊆ (p ∨ℋ q))) → q ⊆ A))
66		pm3.26 256	. . . . . . . . . . . . . . . 16 ⊢ ((r ⊆ B ∧ r ⊆ C) → r ⊆ B)
67	22, 66	sylbir 176	. . . . . . . . . . . . . . 15 ⊢ (r ⊆ (B ∩ C) → r ⊆ B)
68	67	ad2antrl 322	. . . . . . . . . . . . . 14 ⊢ ((r ∈ Atoms ∧ (r ⊆ (B ∩ C) ∧ ¬ r ⊆ A)) → r ⊆ B)
69	68	adantl 305	. . . . . . . . . . . . 13 ⊢ ((p ∈ Atoms ∧ (r ∈ Atoms ∧ (r ⊆ (B ∩ C) ∧ ¬ r ⊆ A))) → r ⊆ B)
70	69	a1d 14	. . . . . . . . . . . 12 ⊢ ((p ∈ Atoms ∧ (r ∈ Atoms ∧ (r ⊆ (B ∩ C) ∧ ¬ r ⊆ A))) → ((q ∈ Atoms ∧ (q ⊆ A ∧ r ⊆ (p ∨ℋ q))) → r ⊆ B))
71	65, 70	jcad 455	. . . . . . . . . . 11 ⊢ ((p ∈ Atoms ∧ (r ∈ Atoms ∧ (r ⊆ (B ∩ C) ∧ ¬ r ⊆ A))) → ((q ∈ Atoms ∧ (q ⊆ A ∧ r ⊆ (p ∨ℋ q))) → (q ⊆ A ∧ r ⊆ B)))
72	62, 71	jcad 455	. . . . . . . . . 10 ⊢ ((p ∈ Atoms ∧ (r ∈ Atoms ∧ (r ⊆ (B ∩ C) ∧ ¬ r ⊆ A))) → ((q ∈ Atoms ∧ (q ⊆ A ∧ r ⊆ (p ∨ℋ q))) → (p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B))))
73	72	exp3a 292	. . . . . . . . 9 ⊢ ((p ∈ Atoms ∧ (r ∈ Atoms ∧ (r ⊆ (B ∩ C) ∧ ¬ r ⊆ A))) → (q ∈ Atoms → ((q ⊆ A ∧ r ⊆ (p ∨ℋ q)) → (p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B)))))
74	73	adantlr 310	. . . . . . . 8 ⊢ (((p ∈ Atoms ∧ ¬ (B ∩ C) ⊆ A) ∧ (r ∈ Atoms ∧ (r ⊆ (B ∩ C) ∧ ¬ r ⊆ A))) → (q ∈ Atoms → ((q ⊆ A ∧ r ⊆ (p ∨ℋ q)) → (p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B)))))
75	74	adantlr 310	. . . . . . 7 ⊢ ((((p ∈ Atoms ∧ ¬ (B ∩ C) ⊆ A) ∧ p ⊆ (A ∨ℋ B)) ∧ (r ∈ Atoms ∧ (r ⊆ (B ∩ C) ∧ ¬ r ⊆ A))) → (q ∈ Atoms → ((q ⊆ A ∧ r ⊆ (p ∨ℋ q)) → (p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B)))))
76	75	adantlr 310	. . . . . 6 ⊢ (((((p ∈ Atoms ∧ ¬ (B ∩ C) ⊆ A) ∧ p ⊆ (A ∨ℋ B)) ∧ ¬ A = 0ℋ) ∧ (r ∈ Atoms ∧ (r ⊆ (B ∩ C) ∧ ¬ r ⊆ A))) → (q ∈ Atoms → ((q ⊆ A ∧ r ⊆ (p ∨ℋ q)) → (p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B)))))
77	76	r19.22dv 1278	. . . . 5 ⊢ (((((p ∈ Atoms ∧ ¬ (B ∩ C) ⊆ A) ∧ p ⊆ (A ∨ℋ B)) ∧ ¬ A = 0ℋ) ∧ (r ∈ Atoms ∧ (r ⊆ (B ∩ C) ∧ ¬ r ⊆ A))) → (∃q ∈ Atoms (q ⊆ A ∧ r ⊆ (p ∨ℋ q)) → ∃q ∈ Atoms (p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B))))
78	32, 77	mpd 46	. . . 4 ⊢ (((((p ∈ Atoms ∧ ¬ (B ∩ C) ⊆ A) ∧ p ⊆ (A ∨ℋ B)) ∧ ¬ A = 0ℋ) ∧ (r ∈ Atoms ∧ (r ⊆ (B ∩ C) ∧ ¬ r ⊆ A))) → ∃q ∈ Atoms (p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B)))
79	78	exp32 294	. . 3 ⊢ ((((p ∈ Atoms ∧ ¬ (B ∩ C) ⊆ A) ∧ p ⊆ (A ∨ℋ B)) ∧ ¬ A = 0ℋ) → (r ∈ Atoms → ((r ⊆ (B ∩ C) ∧ ¬ r ⊆ A) → ∃q ∈ Atoms (p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B)))))
80	79	r19.22dv 1278	. 2 ⊢ ((((p ∈ Atoms ∧ ¬ (B ∩ C) ⊆ A) ∧ p ⊆ (A ∨ℋ B)) ∧ ¬ A = 0ℋ) → (∃r ∈ Atoms (r ⊆ (B ∩ C) ∧ ¬ r ⊆ A) → ∃r ∈ Atoms ∃q ∈ Atoms (p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B))))
81	16, 80	mpd 46	1 ⊢ ((((p ∈ Atoms ∧ ¬ (B ∩ C) ⊆ A) ∧ p ⊆ (A ∨ℋ B)) ∧ ¬ A = 0ℋ) → ∃r ∈ Atoms ∃q ∈ Atoms (p ⊆ (q ∨ℋ r) ∧ (q ⊆ A ∧ r ⊆ B)))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   = weq 797   = wceq 1091   ∈ wcel 1092  ∃wrex 1202   ∩ 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:  mdsymlem4 5779

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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