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Theorem cvexchlem 29803

Description: Lemma for cvexchi 29804. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
chpssat.1	⊢ 𝐴 ∈ C_ℋ
chpssat.2	⊢ 𝐵 ∈ C_ℋ

**Assertion**
Ref	Expression
cvexchlem	⊢ ((𝐴 ∩ 𝐵) ⋖_ℋ 𝐵 → 𝐴 ⋖_ℋ (𝐴 ∨_ℋ 𝐵))

Proof of Theorem cvexchlem Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		chpssat.1	. . . . 5 ⊢ 𝐴 ∈ C_ℋ
2		chpssat.2	. . . . 5 ⊢ 𝐵 ∈ C_ℋ
3	1, 2	chincli 28895	. . . 4 ⊢ (𝐴 ∩ 𝐵) ∈ C_ℋ
4		cvpss 29720	. . . 4 ⊢ (((𝐴 ∩ 𝐵) ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) → ((𝐴 ∩ 𝐵) ⋖_ℋ 𝐵 → (𝐴 ∩ 𝐵) ⊊ 𝐵))
5	3, 2, 4	mp2an 682	. . 3 ⊢ ((𝐴 ∩ 𝐵) ⋖_ℋ 𝐵 → (𝐴 ∩ 𝐵) ⊊ 𝐵)
6	3, 2	chpssati 29798	. . 3 ⊢ ((𝐴 ∩ 𝐵) ⊊ 𝐵 → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)))
7	5, 6	syl 17	. 2 ⊢ ((𝐴 ∩ 𝐵) ⋖_ℋ 𝐵 → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)))
8		ssin 4055	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ↔ 𝑥 ⊆ (𝐴 ∩ 𝐵))
9		ancom 454	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ↔ (𝑥 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴))
10	8, 9	bitr3i 269	. . . . . . . . . 10 ⊢ (𝑥 ⊆ (𝐴 ∩ 𝐵) ↔ (𝑥 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴))
11	10	baibr 532	. . . . . . . . 9 ⊢ (𝑥 ⊆ 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ (𝐴 ∩ 𝐵)))
12	11	notbid 310	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝐵 → (¬ 𝑥 ⊆ 𝐴 ↔ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)))
13	12	biimpar 471	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) → ¬ 𝑥 ⊆ 𝐴)
14		chcv1 29790	. . . . . . . . 9 ⊢ ((𝐴 ∈ C_ℋ ∧ 𝑥 ∈ HAtoms) → (¬ 𝑥 ⊆ 𝐴 ↔ 𝐴 ⋖_ℋ (𝐴 ∨_ℋ 𝑥)))
15	1, 14	mpan 680	. . . . . . . 8 ⊢ (𝑥 ∈ HAtoms → (¬ 𝑥 ⊆ 𝐴 ↔ 𝐴 ⋖_ℋ (𝐴 ∨_ℋ 𝑥)))
16	15	biimpa 470	. . . . . . 7 ⊢ ((𝑥 ∈ HAtoms ∧ ¬ 𝑥 ⊆ 𝐴) → 𝐴 ⋖_ℋ (𝐴 ∨_ℋ 𝑥))
17	13, 16	sylan2 586	. . . . . 6 ⊢ ((𝑥 ∈ HAtoms ∧ (𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵))) → 𝐴 ⋖_ℋ (𝐴 ∨_ℋ 𝑥))
18	17	adantrr 707	. . . . 5 ⊢ ((𝑥 ∈ HAtoms ∧ ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) ∧ (𝐴 ∩ 𝐵) ⋖_ℋ 𝐵)) → 𝐴 ⋖_ℋ (𝐴 ∨_ℋ 𝑥))
19		atelch 29779	. . . . . 6 ⊢ (𝑥 ∈ HAtoms → 𝑥 ∈ C_ℋ )
20	1, 2	chabs1i 28953	. . . . . . . . . 10 ⊢ (𝐴 ∨_ℋ (𝐴 ∩ 𝐵)) = 𝐴
21	20	oveq1i 6934	. . . . . . . . 9 ⊢ ((𝐴 ∨_ℋ (𝐴 ∩ 𝐵)) ∨_ℋ 𝑥) = (𝐴 ∨_ℋ 𝑥)
22		chjass 28968	. . . . . . . . . 10 ⊢ ((𝐴 ∈ C_ℋ ∧ (𝐴 ∩ 𝐵) ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → ((𝐴 ∨_ℋ (𝐴 ∩ 𝐵)) ∨_ℋ 𝑥) = (𝐴 ∨_ℋ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥)))
23	1, 3, 22	mp3an12 1524	. . . . . . . . 9 ⊢ (𝑥 ∈ C_ℋ → ((𝐴 ∨_ℋ (𝐴 ∩ 𝐵)) ∨_ℋ 𝑥) = (𝐴 ∨_ℋ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥)))
24	21, 23	syl5reqr 2829	. . . . . . . 8 ⊢ (𝑥 ∈ C_ℋ → (𝐴 ∨_ℋ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥)) = (𝐴 ∨_ℋ 𝑥))
25	24	adantr 474	. . . . . . 7 ⊢ ((𝑥 ∈ C_ℋ ∧ ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) ∧ (𝐴 ∩ 𝐵) ⋖_ℋ 𝐵)) → (𝐴 ∨_ℋ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥)) = (𝐴 ∨_ℋ 𝑥))
26		ancom 454	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) ↔ (¬ 𝑥 ⊆ (𝐴 ∩ 𝐵) ∧ 𝑥 ⊆ 𝐵))
27		chnle 28949	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∩ 𝐵) ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → (¬ 𝑥 ⊆ (𝐴 ∩ 𝐵) ↔ (𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥)))
28	3, 27	mpan 680	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ C_ℋ → (¬ 𝑥 ⊆ (𝐴 ∩ 𝐵) ↔ (𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥)))
29		inss2 4054	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
30	29	biantrur 526	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ 𝐵 ↔ ((𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐵))
31		chlub 28944	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∩ 𝐵) ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) → (((𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐵) ↔ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ⊆ 𝐵))
32	3, 2, 31	mp3an13 1525	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ C_ℋ → (((𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐵) ↔ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ⊆ 𝐵))
33	30, 32	syl5bb 275	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ⊆ 𝐵 ↔ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ⊆ 𝐵))
34	28, 33	anbi12d 624	. . . . . . . . . . 11 ⊢ (𝑥 ∈ C_ℋ → ((¬ 𝑥 ⊆ (𝐴 ∩ 𝐵) ∧ 𝑥 ⊆ 𝐵) ↔ ((𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ∧ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ⊆ 𝐵)))
35	26, 34	syl5bb 275	. . . . . . . . . 10 ⊢ (𝑥 ∈ C_ℋ → ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) ↔ ((𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ∧ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ⊆ 𝐵)))
36		chjcl 28792	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∩ 𝐵) ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ∈ C_ℋ )
37	3, 36	mpan 680	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ C_ℋ → ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ∈ C_ℋ )
38		cvnbtwn2 29722	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∩ 𝐵) ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ∧ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ∈ C_ℋ ) → ((𝐴 ∩ 𝐵) ⋖_ℋ 𝐵 → (((𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ∧ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ⊆ 𝐵) → ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) = 𝐵)))
39	3, 2, 38	mp3an12 1524	. . . . . . . . . . . 12 ⊢ (((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ∈ C_ℋ → ((𝐴 ∩ 𝐵) ⋖_ℋ 𝐵 → (((𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ∧ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ⊆ 𝐵) → ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) = 𝐵)))
40	37, 39	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ C_ℋ → ((𝐴 ∩ 𝐵) ⋖_ℋ 𝐵 → (((𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ∧ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ⊆ 𝐵) → ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) = 𝐵)))
41	40	com23 86	. . . . . . . . . 10 ⊢ (𝑥 ∈ C_ℋ → (((𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ∧ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ⊆ 𝐵) → ((𝐴 ∩ 𝐵) ⋖_ℋ 𝐵 → ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) = 𝐵)))
42	35, 41	sylbid 232	. . . . . . . . 9 ⊢ (𝑥 ∈ C_ℋ → ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) → ((𝐴 ∩ 𝐵) ⋖_ℋ 𝐵 → ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) = 𝐵)))
43	42	imp32 411	. . . . . . . 8 ⊢ ((𝑥 ∈ C_ℋ ∧ ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) ∧ (𝐴 ∩ 𝐵) ⋖_ℋ 𝐵)) → ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) = 𝐵)
44	43	oveq2d 6940	. . . . . . 7 ⊢ ((𝑥 ∈ C_ℋ ∧ ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) ∧ (𝐴 ∩ 𝐵) ⋖_ℋ 𝐵)) → (𝐴 ∨_ℋ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥)) = (𝐴 ∨_ℋ 𝐵))
45	25, 44	eqtr3d 2816	. . . . . 6 ⊢ ((𝑥 ∈ C_ℋ ∧ ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) ∧ (𝐴 ∩ 𝐵) ⋖_ℋ 𝐵)) → (𝐴 ∨_ℋ 𝑥) = (𝐴 ∨_ℋ 𝐵))
46	19, 45	sylan 575	. . . . 5 ⊢ ((𝑥 ∈ HAtoms ∧ ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) ∧ (𝐴 ∩ 𝐵) ⋖_ℋ 𝐵)) → (𝐴 ∨_ℋ 𝑥) = (𝐴 ∨_ℋ 𝐵))
47	18, 46	breqtrd 4914	. . . 4 ⊢ ((𝑥 ∈ HAtoms ∧ ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) ∧ (𝐴 ∩ 𝐵) ⋖_ℋ 𝐵)) → 𝐴 ⋖_ℋ (𝐴 ∨_ℋ 𝐵))
48	47	exp32 413	. . 3 ⊢ (𝑥 ∈ HAtoms → ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) → ((𝐴 ∩ 𝐵) ⋖_ℋ 𝐵 → 𝐴 ⋖_ℋ (𝐴 ∨_ℋ 𝐵))))
49	48	rexlimiv 3209	. 2 ⊢ (∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) → ((𝐴 ∩ 𝐵) ⋖_ℋ 𝐵 → 𝐴 ⋖_ℋ (𝐴 ∨_ℋ 𝐵)))
50	7, 49	mpcom 38	1 ⊢ ((𝐴 ∩ 𝐵) ⋖_ℋ 𝐵 → 𝐴 ⋖_ℋ (𝐴 ∨_ℋ 𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∃wrex 3091 ∩ cin 3791 ⊆ wss 3792 ⊊ wpss 3793 class class class wbr 4888 (class class class)co 6924 C_ℋ cch 28362 ∨_ℋ chj 28366 ⋖_ℋ ccv 28397 HAtomscat 28398

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cc 9594 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 ax-addf 10353 ax-mulf 10354 ax-hilex 28432 ax-hfvadd 28433 ax-hvcom 28434 ax-hvass 28435 ax-hv0cl 28436 ax-hvaddid 28437 ax-hfvmul 28438 ax-hvmulid 28439 ax-hvmulass 28440 ax-hvdistr1 28441 ax-hvdistr2 28442 ax-hvmul0 28443 ax-hfi 28512 ax-his1 28515 ax-his2 28516 ax-his3 28517 ax-his4 28518 ax-hcompl 28635

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-of 7176 df-om 7346 df-1st 7447 df-2nd 7448 df-supp 7579 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-2o 7846 df-oadd 7849 df-omul 7850 df-er 8028 df-map 8144 df-pm 8145 df-ixp 8197 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fsupp 8566 df-fi 8607 df-sup 8638 df-inf 8639 df-oi 8706 df-card 9100 df-acn 9103 df-cda 9327 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-5 11445 df-6 11446 df-7 11447 df-8 11448 df-9 11449 df-n0 11647 df-z 11733 df-dec 11850 df-uz 11997 df-q 12100 df-rp 12142 df-xneg 12261 df-xadd 12262 df-xmul 12263 df-ioo 12495 df-ico 12497 df-icc 12498 df-fz 12648 df-fzo 12789 df-fl 12916 df-seq 13124 df-exp 13183 df-hash 13440 df-cj 14250 df-re 14251 df-im 14252 df-sqrt 14386 df-abs 14387 df-clim 14631 df-rlim 14632 df-sum 14829 df-struct 16261 df-ndx 16262 df-slot 16263 df-base 16265 df-sets 16266 df-ress 16267 df-plusg 16355 df-mulr 16356 df-starv 16357 df-sca 16358 df-vsca 16359 df-ip 16360 df-tset 16361 df-ple 16362 df-ds 16364 df-unif 16365 df-hom 16366 df-cco 16367 df-rest 16473 df-topn 16474 df-0g 16492 df-gsum 16493 df-topgen 16494 df-pt 16495 df-prds 16498 df-xrs 16552 df-qtop 16557 df-imas 16558 df-xps 16560 df-mre 16636 df-mrc 16637 df-acs 16639 df-mgm 17632 df-sgrp 17674 df-mnd 17685 df-submnd 17726 df-mulg 17932 df-cntz 18137 df-cmn 18585 df-psmet 20138 df-xmet 20139 df-met 20140 df-bl 20141 df-mopn 20142 df-fbas 20143 df-fg 20144 df-cnfld 20147 df-top 21110 df-topon 21127 df-topsp 21149 df-bases 21162 df-cld 21235 df-ntr 21236 df-cls 21237 df-nei 21314 df-cn 21443 df-cnp 21444 df-lm 21445 df-haus 21531 df-tx 21778 df-hmeo 21971 df-fil 22062 df-fm 22154 df-flim 22155 df-flf 22156 df-xms 22537 df-ms 22538 df-tms 22539 df-cfil 23465 df-cau 23466 df-cmet 23467 df-grpo 27924 df-gid 27925 df-ginv 27926 df-gdiv 27927 df-ablo 27976 df-vc 27990 df-nv 28023 df-va 28026 df-ba 28027 df-sm 28028 df-0v 28029 df-vs 28030 df-nmcv 28031 df-ims 28032 df-dip 28132 df-ssp 28153 df-ph 28244 df-cbn 28295 df-hnorm 28401 df-hba 28402 df-hvsub 28404 df-hlim 28405 df-hcau 28406 df-sh 28640 df-ch 28654 df-oc 28685 df-ch0 28686 df-shs 28743 df-span 28744 df-chj 28745 df-chsup 28746 df-pjh 28830 df-cv 29714 df-at 29773

This theorem is referenced by: cvexchi 29804

W3C validator