Theorem atcvat3 5774

Description: A condition implying that a certain lattice element is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68.

Hypothesis

Ref	Expression
atcvat3.1	⊢ A ∈ Cℋ

Assertion

Ref	Expression
atcvat3	⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → (((¬ B = C ∧ ¬ C ⊆ A) ∧ B ⊆ (A ∨ℋ C)) → (A ∩ (B ∨ℋ C)) ∈ Atoms))

Proof of Theorem atcvat3

Step	Hyp	Ref	Expression
1		atcvat3.1	. . . . . . . . . . . 12 ⊢ A ∈ Cℋ
2		chcv1t 5751	. . . . . . . . . . . 12 ⊢ ((A ∈ Cℋ ∧ C ∈ Atoms) → (¬ C ⊆ A ↔ A ⋖ (A ∨ℋ C)))
3	1, 2	mpan 518	. . . . . . . . . . 11 ⊢ (C ∈ Atoms → (¬ C ⊆ A ↔ A ⋖ (A ∨ℋ C)))
4	3	biimpa 324	. . . . . . . . . 10 ⊢ ((C ∈ Atoms ∧ ¬ C ⊆ A) → A ⋖ (A ∨ℋ C))
5	4	adantll 309	. . . . . . . . 9 ⊢ (((B ∈ Atoms ∧ C ∈ Atoms) ∧ ¬ C ⊆ A) → A ⋖ (A ∨ℋ C))
6	5	adantrr 312	. . . . . . . 8 ⊢ (((B ∈ Atoms ∧ C ∈ Atoms) ∧ (¬ C ⊆ A ∧ B ⊆ (A ∨ℋ C))) → A ⋖ (A ∨ℋ C))
7		chjcomt 5423	. . . . . . . . . . . . . . . . 17 ⊢ ((B ∈ Cℋ ∧ C ∈ Cℋ ) → (B ∨ℋ C) = (C ∨ℋ B))
8	7	opreq2d 3013	. . . . . . . . . . . . . . . 16 ⊢ ((B ∈ Cℋ ∧ C ∈ Cℋ ) → (A ∨ℋ (B ∨ℋ C)) = (A ∨ℋ (C ∨ℋ B)))
9		chjasst 5446	. . . . . . . . . . . . . . . . . 18 ⊢ ((A ∈ Cℋ ∧ C ∈ Cℋ ∧ B ∈ Cℋ ) → ((A ∨ℋ C) ∨ℋ B) = (A ∨ℋ (C ∨ℋ B)))
10	1, 9	mp3an1 639	. . . . . . . . . . . . . . . . 17 ⊢ ((C ∈ Cℋ ∧ B ∈ Cℋ ) → ((A ∨ℋ C) ∨ℋ B) = (A ∨ℋ (C ∨ℋ B)))
11	10	ancoms 334	. . . . . . . . . . . . . . . 16 ⊢ ((B ∈ Cℋ ∧ C ∈ Cℋ ) → ((A ∨ℋ C) ∨ℋ B) = (A ∨ℋ (C ∨ℋ B)))
12	8, 11	eqtr4d 1131	. . . . . . . . . . . . . . 15 ⊢ ((B ∈ Cℋ ∧ C ∈ Cℋ ) → (A ∨ℋ (B ∨ℋ C)) = ((A ∨ℋ C) ∨ℋ B))
13	12	adantr 306	. . . . . . . . . . . . . 14 ⊢ (((B ∈ Cℋ ∧ C ∈ Cℋ ) ∧ B ⊆ (A ∨ℋ C)) → (A ∨ℋ (B ∨ℋ C)) = ((A ∨ℋ C) ∨ℋ B))
14		chlej2t 5428	. . . . . . . . . . . . . . . 16 ⊢ ((B ∈ Cℋ ∧ (A ∨ℋ C) ∈ Cℋ ∧ (A ∨ℋ C) ∈ Cℋ ) → (B ⊆ (A ∨ℋ C) → ((A ∨ℋ C) ∨ℋ B) ⊆ ((A ∨ℋ C) ∨ℋ (A ∨ℋ C))))
15		pm3.26 256	. . . . . . . . . . . . . . . 16 ⊢ ((B ∈ Cℋ ∧ C ∈ Cℋ ) → B ∈ Cℋ )
16		chjclt 5330	. . . . . . . . . . . . . . . . . 18 ⊢ ((A ∈ Cℋ ∧ C ∈ Cℋ ) → (A ∨ℋ C) ∈ Cℋ )
17	1, 16	mpan 518	. . . . . . . . . . . . . . . . 17 ⊢ (C ∈ Cℋ → (A ∨ℋ C) ∈ Cℋ )
18	17	adantl 305	. . . . . . . . . . . . . . . 16 ⊢ ((B ∈ Cℋ ∧ C ∈ Cℋ ) → (A ∨ℋ C) ∈ Cℋ )
19	14, 15, 18, 18	syl3anc 629	. . . . . . . . . . . . . . 15 ⊢ ((B ∈ Cℋ ∧ C ∈ Cℋ ) → (B ⊆ (A ∨ℋ C) → ((A ∨ℋ C) ∨ℋ B) ⊆ ((A ∨ℋ C) ∨ℋ (A ∨ℋ C))))
20	19	imp 277	. . . . . . . . . . . . . 14 ⊢ (((B ∈ Cℋ ∧ C ∈ Cℋ ) ∧ B ⊆ (A ∨ℋ C)) → ((A ∨ℋ C) ∨ℋ B) ⊆ ((A ∨ℋ C) ∨ℋ (A ∨ℋ C)))
21	13, 20	eqsstrd 1534	. . . . . . . . . . . . 13 ⊢ (((B ∈ Cℋ ∧ C ∈ Cℋ ) ∧ B ⊆ (A ∨ℋ C)) → (A ∨ℋ (B ∨ℋ C)) ⊆ ((A ∨ℋ C) ∨ℋ (A ∨ℋ C)))
22		chjidmt 5436	. . . . . . . . . . . . . . 15 ⊢ ((A ∨ℋ C) ∈ Cℋ → ((A ∨ℋ C) ∨ℋ (A ∨ℋ C)) = (A ∨ℋ C))
23	17, 22	syl 12	. . . . . . . . . . . . . 14 ⊢ (C ∈ Cℋ → ((A ∨ℋ C) ∨ℋ (A ∨ℋ C)) = (A ∨ℋ C))
24	23	ad2antlr 321	. . . . . . . . . . . . 13 ⊢ (((B ∈ Cℋ ∧ C ∈ Cℋ ) ∧ B ⊆ (A ∨ℋ C)) → ((A ∨ℋ C) ∨ℋ (A ∨ℋ C)) = (A ∨ℋ C))
25	21, 24	sseqtrd 1536	. . . . . . . . . . . 12 ⊢ (((B ∈ Cℋ ∧ C ∈ Cℋ ) ∧ B ⊆ (A ∨ℋ C)) → (A ∨ℋ (B ∨ℋ C)) ⊆ (A ∨ℋ C))
26		chlej2t 5428	. . . . . . . . . . . . . . 15 ⊢ ((C ∈ Cℋ ∧ (B ∨ℋ C) ∈ Cℋ ∧ A ∈ Cℋ ) → (C ⊆ (B ∨ℋ C) → (A ∨ℋ C) ⊆ (A ∨ℋ (B ∨ℋ C))))
27	1, 26	mp3an3 641	. . . . . . . . . . . . . 14 ⊢ ((C ∈ Cℋ ∧ (B ∨ℋ C) ∈ Cℋ ) → (C ⊆ (B ∨ℋ C) → (A ∨ℋ C) ⊆ (A ∨ℋ (B ∨ℋ C))))
28		pm3.27 260	. . . . . . . . . . . . . . 15 ⊢ ((B ∈ Cℋ ∧ C ∈ Cℋ ) → C ∈ Cℋ )
29		chjclt 5330	. . . . . . . . . . . . . . 15 ⊢ ((B ∈ Cℋ ∧ C ∈ Cℋ ) → (B ∨ℋ C) ∈ Cℋ )
30	28, 29	jca 236	. . . . . . . . . . . . . 14 ⊢ ((B ∈ Cℋ ∧ C ∈ Cℋ ) → (C ∈ Cℋ ∧ (B ∨ℋ C) ∈ Cℋ ))
31		chub2t 5425	. . . . . . . . . . . . . . 15 ⊢ ((C ∈ Cℋ ∧ B ∈ Cℋ ) → C ⊆ (B ∨ℋ C))
32	31	ancoms 334	. . . . . . . . . . . . . 14 ⊢ ((B ∈ Cℋ ∧ C ∈ Cℋ ) → C ⊆ (B ∨ℋ C))
33	27, 30, 32	sylc 62	. . . . . . . . . . . . 13 ⊢ ((B ∈ Cℋ ∧ C ∈ Cℋ ) → (A ∨ℋ C) ⊆ (A ∨ℋ (B ∨ℋ C)))
34	33	adantr 306	. . . . . . . . . . . 12 ⊢ (((B ∈ Cℋ ∧ C ∈ Cℋ ) ∧ B ⊆ (A ∨ℋ C)) → (A ∨ℋ C) ⊆ (A ∨ℋ (B ∨ℋ C)))
35	25, 34	eqssd 1518	. . . . . . . . . . 11 ⊢ (((B ∈ Cℋ ∧ C ∈ Cℋ ) ∧ B ⊆ (A ∨ℋ C)) → (A ∨ℋ (B ∨ℋ C)) = (A ∨ℋ C))
36		atelch 5742	. . . . . . . . . . . 12 ⊢ (B ∈ Atoms → B ∈ Cℋ )
37		atelch 5742	. . . . . . . . . . . 12 ⊢ (C ∈ Atoms → C ∈ Cℋ )
38	36, 37	anim12i 268	. . . . . . . . . . 11 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → (B ∈ Cℋ ∧ C ∈ Cℋ ))
39	35, 38	sylan 343	. . . . . . . . . 10 ⊢ (((B ∈ Atoms ∧ C ∈ Atoms) ∧ B ⊆ (A ∨ℋ C)) → (A ∨ℋ (B ∨ℋ C)) = (A ∨ℋ C))
40	39	breq2d 2072	. . . . . . . . 9 ⊢ (((B ∈ Atoms ∧ C ∈ Atoms) ∧ B ⊆ (A ∨ℋ C)) → (A ⋖ (A ∨ℋ (B ∨ℋ C)) ↔ A ⋖ (A ∨ℋ C)))
41	40	adantrl 311	. . . . . . . 8 ⊢ (((B ∈ Atoms ∧ C ∈ Atoms) ∧ (¬ C ⊆ A ∧ B ⊆ (A ∨ℋ C))) → (A ⋖ (A ∨ℋ (B ∨ℋ C)) ↔ A ⋖ (A ∨ℋ C)))
42	6, 41	mpbird 171	. . . . . . 7 ⊢ (((B ∈ Atoms ∧ C ∈ Atoms) ∧ (¬ C ⊆ A ∧ B ⊆ (A ∨ℋ C))) → A ⋖ (A ∨ℋ (B ∨ℋ C)))
43	42	exp 291	. . . . . 6 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → ((¬ C ⊆ A ∧ B ⊆ (A ∨ℋ C)) → A ⋖ (A ∨ℋ (B ∨ℋ C))))
44	29, 1	jctil 240	. . . . . . . 8 ⊢ ((B ∈ Cℋ ∧ C ∈ Cℋ ) → (A ∈ Cℋ ∧ (B ∨ℋ C) ∈ Cℋ ))
45	44, 36, 37	syl2an 349	. . . . . . 7 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → (A ∈ Cℋ ∧ (B ∨ℋ C) ∈ Cℋ ))
46		cvexcht 5763	. . . . . . 7 ⊢ ((A ∈ Cℋ ∧ (B ∨ℋ C) ∈ Cℋ ) → ((A ∩ (B ∨ℋ C)) ⋖ (B ∨ℋ C) ↔ A ⋖ (A ∨ℋ (B ∨ℋ C))))
47	45, 46	syl 12	. . . . . 6 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → ((A ∩ (B ∨ℋ C)) ⋖ (B ∨ℋ C) ↔ A ⋖ (A ∨ℋ (B ∨ℋ C))))
48	43, 47	sylibrd 179	. . . . 5 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → ((¬ C ⊆ A ∧ B ⊆ (A ∨ℋ C)) → (A ∩ (B ∨ℋ C)) ⋖ (B ∨ℋ C)))
49	48	adantr 306	. . . 4 ⊢ (((B ∈ Atoms ∧ C ∈ Atoms) ∧ ¬ B = C) → ((¬ C ⊆ A ∧ B ⊆ (A ∨ℋ C)) → (A ∩ (B ∨ℋ C)) ⋖ (B ∨ℋ C)))
50		atcvat2t 5773	. . . . . . 7 ⊢ (((A ∩ (B ∨ℋ C)) ∈ Cℋ ∧ B ∈ Atoms ∧ C ∈ Atoms) → ((¬ B = C ∧ (A ∩ (B ∨ℋ C)) ⋖ (B ∨ℋ C)) → (A ∩ (B ∨ℋ C)) ∈ Atoms))
51		chinclt 5416	. . . . . . . . 9 ⊢ ((A ∈ Cℋ ∧ (B ∨ℋ C) ∈ Cℋ ) → (A ∩ (B ∨ℋ C)) ∈ Cℋ )
52	44, 51	syl 12	. . . . . . . 8 ⊢ ((B ∈ Cℋ ∧ C ∈ Cℋ ) → (A ∩ (B ∨ℋ C)) ∈ Cℋ )
53	52, 36, 37	syl2an 349	. . . . . . 7 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → (A ∩ (B ∨ℋ C)) ∈ Cℋ )
54		pm3.26 256	. . . . . . 7 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → B ∈ Atoms)
55		pm3.27 260	. . . . . . 7 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → C ∈ Atoms)
56	50, 53, 54, 55	syl3anc 629	. . . . . 6 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → ((¬ B = C ∧ (A ∩ (B ∨ℋ C)) ⋖ (B ∨ℋ C)) → (A ∩ (B ∨ℋ C)) ∈ Atoms))
57	56	exp3a 292	. . . . 5 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → (¬ B = C → ((A ∩ (B ∨ℋ C)) ⋖ (B ∨ℋ C) → (A ∩ (B ∨ℋ C)) ∈ Atoms)))
58	57	imp 277	. . . 4 ⊢ (((B ∈ Atoms ∧ C ∈ Atoms) ∧ ¬ B = C) → ((A ∩ (B ∨ℋ C)) ⋖ (B ∨ℋ C) → (A ∩ (B ∨ℋ C)) ∈ Atoms))
59	49, 58	syld 27	. . 3 ⊢ (((B ∈ Atoms ∧ C ∈ Atoms) ∧ ¬ B = C) → ((¬ C ⊆ A ∧ B ⊆ (A ∨ℋ C)) → (A ∩ (B ∨ℋ C)) ∈ Atoms))
60	59	exp4b 296	. 2 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → (¬ B = C → (¬ C ⊆ A → (B ⊆ (A ∨ℋ C) → (A ∩ (B ∨ℋ C)) ∈ Atoms))))
61	60	imp4c 284	1 ⊢ ((B ∈ Atoms ∧ C ∈ Atoms) → (((¬ B = C ∧ ¬ C ⊆ A) ∧ B ⊆ (A ∨ℋ C)) → (A ∩ (B ∨ℋ C)) ∈ Atoms))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092   ∩ cin 1486   ⊆ wss 1487   class class class wbr 2054  (class class class)co 3001   C_ℋ cch 4968   ∨_ℋ chj 4972  Atomscat 4980   ⋖ ccv 4981

This theorem is referenced by:  atcvat4 5775

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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