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Theorem cvexchlem 32130

Description: Lemma for cvexchi 32131. (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 31222	. . . 4 ⊢ (𝐴 ∩ 𝐵) ∈ C_ℋ
4		cvpss 32047	. . . 4 ⊢ (((𝐴 ∩ 𝐵) ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) → ((𝐴 ∩ 𝐵) ⋖_ℋ 𝐵 → (𝐴 ∩ 𝐵) ⊊ 𝐵))
5	3, 2, 4	mp2an 689	. . 3 ⊢ ((𝐴 ∩ 𝐵) ⋖_ℋ 𝐵 → (𝐴 ∩ 𝐵) ⊊ 𝐵)
6	3, 2	chpssati 32125	. . 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 4225	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ↔ 𝑥 ⊆ (𝐴 ∩ 𝐵))
9		ancom 460	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ↔ (𝑥 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴))
10	8, 9	bitr3i 277	. . . . . . . . . 10 ⊢ (𝑥 ⊆ (𝐴 ∩ 𝐵) ↔ (𝑥 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴))
11	10	baibr 536	. . . . . . . . 9 ⊢ (𝑥 ⊆ 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ (𝐴 ∩ 𝐵)))
12	11	notbid 318	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝐵 → (¬ 𝑥 ⊆ 𝐴 ↔ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)))
13	12	biimpar 477	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) → ¬ 𝑥 ⊆ 𝐴)
14		chcv1 32117	. . . . . . . . 9 ⊢ ((𝐴 ∈ C_ℋ ∧ 𝑥 ∈ HAtoms) → (¬ 𝑥 ⊆ 𝐴 ↔ 𝐴 ⋖_ℋ (𝐴 ∨_ℋ 𝑥)))
15	1, 14	mpan 687	. . . . . . . 8 ⊢ (𝑥 ∈ HAtoms → (¬ 𝑥 ⊆ 𝐴 ↔ 𝐴 ⋖_ℋ (𝐴 ∨_ℋ 𝑥)))
16	15	biimpa 476	. . . . . . 7 ⊢ ((𝑥 ∈ HAtoms ∧ ¬ 𝑥 ⊆ 𝐴) → 𝐴 ⋖_ℋ (𝐴 ∨_ℋ 𝑥))
17	13, 16	sylan2 592	. . . . . 6 ⊢ ((𝑥 ∈ HAtoms ∧ (𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵))) → 𝐴 ⋖_ℋ (𝐴 ∨_ℋ 𝑥))
18	17	adantrr 714	. . . . 5 ⊢ ((𝑥 ∈ HAtoms ∧ ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) ∧ (𝐴 ∩ 𝐵) ⋖_ℋ 𝐵)) → 𝐴 ⋖_ℋ (𝐴 ∨_ℋ 𝑥))
19		atelch 32106	. . . . . 6 ⊢ (𝑥 ∈ HAtoms → 𝑥 ∈ C_ℋ )
20		chjass 31295	. . . . . . . . . 10 ⊢ ((𝐴 ∈ C_ℋ ∧ (𝐴 ∩ 𝐵) ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → ((𝐴 ∨_ℋ (𝐴 ∩ 𝐵)) ∨_ℋ 𝑥) = (𝐴 ∨_ℋ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥)))
21	1, 3, 20	mp3an12 1447	. . . . . . . . 9 ⊢ (𝑥 ∈ C_ℋ → ((𝐴 ∨_ℋ (𝐴 ∩ 𝐵)) ∨_ℋ 𝑥) = (𝐴 ∨_ℋ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥)))
22	1, 2	chabs1i 31280	. . . . . . . . . 10 ⊢ (𝐴 ∨_ℋ (𝐴 ∩ 𝐵)) = 𝐴
23	22	oveq1i 7415	. . . . . . . . 9 ⊢ ((𝐴 ∨_ℋ (𝐴 ∩ 𝐵)) ∨_ℋ 𝑥) = (𝐴 ∨_ℋ 𝑥)
24	21, 23	eqtr3di 2781	. . . . . . . 8 ⊢ (𝑥 ∈ C_ℋ → (𝐴 ∨_ℋ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥)) = (𝐴 ∨_ℋ 𝑥))
25	24	adantr 480	. . . . . . 7 ⊢ ((𝑥 ∈ C_ℋ ∧ ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) ∧ (𝐴 ∩ 𝐵) ⋖_ℋ 𝐵)) → (𝐴 ∨_ℋ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥)) = (𝐴 ∨_ℋ 𝑥))
26		ancom 460	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) ↔ (¬ 𝑥 ⊆ (𝐴 ∩ 𝐵) ∧ 𝑥 ⊆ 𝐵))
27		chnle 31276	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∩ 𝐵) ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → (¬ 𝑥 ⊆ (𝐴 ∩ 𝐵) ↔ (𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥)))
28	3, 27	mpan 687	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ C_ℋ → (¬ 𝑥 ⊆ (𝐴 ∩ 𝐵) ↔ (𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥)))
29		inss2 4224	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
30	29	biantrur 530	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ 𝐵 ↔ ((𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐵))
31		chlub 31271	. . . . . . . . . . . . . 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 1448	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ C_ℋ → (((𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐵) ↔ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ⊆ 𝐵))
33	30, 32	bitrid 283	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ C_ℋ → (𝑥 ⊆ 𝐵 ↔ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ⊆ 𝐵))
34	28, 33	anbi12d 630	. . . . . . . . . . 11 ⊢ (𝑥 ∈ C_ℋ → ((¬ 𝑥 ⊆ (𝐴 ∩ 𝐵) ∧ 𝑥 ⊆ 𝐵) ↔ ((𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ∧ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ⊆ 𝐵)))
35	26, 34	bitrid 283	. . . . . . . . . 10 ⊢ (𝑥 ∈ C_ℋ → ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) ↔ ((𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ∧ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ⊆ 𝐵)))
36		chjcl 31119	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∩ 𝐵) ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ∈ C_ℋ )
37	3, 36	mpan 687	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ C_ℋ → ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ∈ C_ℋ )
38		cvnbtwn2 32049	. . . . . . . . . . . . 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 1447	. . . . . . . . . . . 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 239	. . . . . . . . 9 ⊢ (𝑥 ∈ C_ℋ → ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) → ((𝐴 ∩ 𝐵) ⋖_ℋ 𝐵 → ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) = 𝐵)))
43	42	imp32 418	. . . . . . . 8 ⊢ ((𝑥 ∈ C_ℋ ∧ ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) ∧ (𝐴 ∩ 𝐵) ⋖_ℋ 𝐵)) → ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) = 𝐵)
44	43	oveq2d 7421	. . . . . . 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 2768	. . . . . 6 ⊢ ((𝑥 ∈ C_ℋ ∧ ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) ∧ (𝐴 ∩ 𝐵) ⋖_ℋ 𝐵)) → (𝐴 ∨_ℋ 𝑥) = (𝐴 ∨_ℋ 𝐵))
46	19, 45	sylan 579	. . . . 5 ⊢ ((𝑥 ∈ HAtoms ∧ ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) ∧ (𝐴 ∩ 𝐵) ⋖_ℋ 𝐵)) → (𝐴 ∨_ℋ 𝑥) = (𝐴 ∨_ℋ 𝐵))
47	18, 46	breqtrd 5167	. . . 4 ⊢ ((𝑥 ∈ HAtoms ∧ ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) ∧ (𝐴 ∩ 𝐵) ⋖_ℋ 𝐵)) → 𝐴 ⋖_ℋ (𝐴 ∨_ℋ 𝐵))
48	47	exp32 420	. . 3 ⊢ (𝑥 ∈ HAtoms → ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) → ((𝐴 ∩ 𝐵) ⋖_ℋ 𝐵 → 𝐴 ⋖_ℋ (𝐴 ∨_ℋ 𝐵))))
49	48	rexlimiv 3142	. 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 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∃wrex 3064 ∩ cin 3942 ⊆ wss 3943 ⊊ wpss 3944 class class class wbr 5141 (class class class)co 7405 C_ℋ cch 30691 ∨_ℋ chj 30695 ⋖_ℋ ccv 30726 HAtomscat 30727

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cc 10432 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192 ax-hilex 30761 ax-hfvadd 30762 ax-hvcom 30763 ax-hvass 30764 ax-hv0cl 30765 ax-hvaddid 30766 ax-hfvmul 30767 ax-hvmulid 30768 ax-hvmulass 30769 ax-hvdistr1 30770 ax-hvdistr2 30771 ax-hvmul0 30772 ax-hfi 30841 ax-his1 30844 ax-his2 30845 ax-his3 30846 ax-his4 30847 ax-hcompl 30964

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-oadd 8471 df-omul 8472 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-acn 9939 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-ioo 13334 df-ico 13336 df-icc 13337 df-fz 13491 df-fzo 13634 df-fl 13763 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-rlim 15439 df-sum 15639 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-hom 17230 df-cco 17231 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-pt 17399 df-prds 17402 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-mulg 18996 df-cntz 19233 df-cmn 19702 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-cld 22878 df-ntr 22879 df-cls 22880 df-nei 22957 df-cn 23086 df-cnp 23087 df-lm 23088 df-haus 23174 df-tx 23421 df-hmeo 23614 df-fil 23705 df-fm 23797 df-flim 23798 df-flf 23799 df-xms 24181 df-ms 24182 df-tms 24183 df-cfil 25138 df-cau 25139 df-cmet 25140 df-grpo 30255 df-gid 30256 df-ginv 30257 df-gdiv 30258 df-ablo 30307 df-vc 30321 df-nv 30354 df-va 30357 df-ba 30358 df-sm 30359 df-0v 30360 df-vs 30361 df-nmcv 30362 df-ims 30363 df-dip 30463 df-ssp 30484 df-ph 30575 df-cbn 30625 df-hnorm 30730 df-hba 30731 df-hvsub 30733 df-hlim 30734 df-hcau 30735 df-sh 30969 df-ch 30983 df-oc 31014 df-ch0 31015 df-shs 31070 df-span 31071 df-chj 31072 df-chsup 31073 df-pjh 31157 df-cv 32041 df-at 32100

This theorem is referenced by: cvexchi 32131

W3C validator