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Theorem cvexchlem 32222

Description: Lemma for cvexchi 32223. (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 31314	. . . 4 ⊢ (𝐴 ∩ 𝐵) ∈ C_ℋ
4		cvpss 32139	. . . 4 ⊢ (((𝐴 ∩ 𝐵) ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) → ((𝐴 ∩ 𝐵) ⋖_ℋ 𝐵 → (𝐴 ∩ 𝐵) ⊊ 𝐵))
5	3, 2, 4	mp2an 690	. . 3 ⊢ ((𝐴 ∩ 𝐵) ⋖_ℋ 𝐵 → (𝐴 ∩ 𝐵) ⊊ 𝐵)
6	3, 2	chpssati 32217	. . 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 459	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ↔ (𝑥 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴))
10	8, 9	bitr3i 276	. . . . . . . . . 10 ⊢ (𝑥 ⊆ (𝐴 ∩ 𝐵) ↔ (𝑥 ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐴))
11	10	baibr 535	. . . . . . . . 9 ⊢ (𝑥 ⊆ 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ (𝐴 ∩ 𝐵)))
12	11	notbid 317	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝐵 → (¬ 𝑥 ⊆ 𝐴 ↔ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)))
13	12	biimpar 476	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) → ¬ 𝑥 ⊆ 𝐴)
14		chcv1 32209	. . . . . . . . 9 ⊢ ((𝐴 ∈ C_ℋ ∧ 𝑥 ∈ HAtoms) → (¬ 𝑥 ⊆ 𝐴 ↔ 𝐴 ⋖_ℋ (𝐴 ∨_ℋ 𝑥)))
15	1, 14	mpan 688	. . . . . . . 8 ⊢ (𝑥 ∈ HAtoms → (¬ 𝑥 ⊆ 𝐴 ↔ 𝐴 ⋖_ℋ (𝐴 ∨_ℋ 𝑥)))
16	15	biimpa 475	. . . . . . 7 ⊢ ((𝑥 ∈ HAtoms ∧ ¬ 𝑥 ⊆ 𝐴) → 𝐴 ⋖_ℋ (𝐴 ∨_ℋ 𝑥))
17	13, 16	sylan2 591	. . . . . 6 ⊢ ((𝑥 ∈ HAtoms ∧ (𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵))) → 𝐴 ⋖_ℋ (𝐴 ∨_ℋ 𝑥))
18	17	adantrr 715	. . . . 5 ⊢ ((𝑥 ∈ HAtoms ∧ ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) ∧ (𝐴 ∩ 𝐵) ⋖_ℋ 𝐵)) → 𝐴 ⋖_ℋ (𝐴 ∨_ℋ 𝑥))
19		atelch 32198	. . . . . 6 ⊢ (𝑥 ∈ HAtoms → 𝑥 ∈ C_ℋ )
20		chjass 31387	. . . . . . . . . 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 31372	. . . . . . . . . 10 ⊢ (𝐴 ∨_ℋ (𝐴 ∩ 𝐵)) = 𝐴
23	22	oveq1i 7426	. . . . . . . . 9 ⊢ ((𝐴 ∨_ℋ (𝐴 ∩ 𝐵)) ∨_ℋ 𝑥) = (𝐴 ∨_ℋ 𝑥)
24	21, 23	eqtr3di 2780	. . . . . . . 8 ⊢ (𝑥 ∈ C_ℋ → (𝐴 ∨_ℋ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥)) = (𝐴 ∨_ℋ 𝑥))
25	24	adantr 479	. . . . . . 7 ⊢ ((𝑥 ∈ C_ℋ ∧ ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) ∧ (𝐴 ∩ 𝐵) ⋖_ℋ 𝐵)) → (𝐴 ∨_ℋ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥)) = (𝐴 ∨_ℋ 𝑥))
26		ancom 459	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) ↔ (¬ 𝑥 ⊆ (𝐴 ∩ 𝐵) ∧ 𝑥 ⊆ 𝐵))
27		chnle 31368	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∩ 𝐵) ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → (¬ 𝑥 ⊆ (𝐴 ∩ 𝐵) ↔ (𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥)))
28	3, 27	mpan 688	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ C_ℋ → (¬ 𝑥 ⊆ (𝐴 ∩ 𝐵) ↔ (𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥)))
29		inss2 4224	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
30	29	biantrur 529	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ 𝐵 ↔ ((𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ 𝑥 ⊆ 𝐵))
31		chlub 31363	. . . . . . . . . . . . . 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 282	. . . . . . . . . . . 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 282	. . . . . . . . . 10 ⊢ (𝑥 ∈ C_ℋ → ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) ↔ ((𝐴 ∩ 𝐵) ⊊ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ∧ ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ⊆ 𝐵)))
36		chjcl 31211	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∩ 𝐵) ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ∈ C_ℋ )
37	3, 36	mpan 688	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ C_ℋ → ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) ∈ C_ℋ )
38		cvnbtwn2 32141	. . . . . . . . . . . . 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 417	. . . . . . . 8 ⊢ ((𝑥 ∈ C_ℋ ∧ ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) ∧ (𝐴 ∩ 𝐵) ⋖_ℋ 𝐵)) → ((𝐴 ∩ 𝐵) ∨_ℋ 𝑥) = 𝐵)
44	43	oveq2d 7432	. . . . . . 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 2767	. . . . . 6 ⊢ ((𝑥 ∈ C_ℋ ∧ ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) ∧ (𝐴 ∩ 𝐵) ⋖_ℋ 𝐵)) → (𝐴 ∨_ℋ 𝑥) = (𝐴 ∨_ℋ 𝐵))
46	19, 45	sylan 578	. . . . 5 ⊢ ((𝑥 ∈ HAtoms ∧ ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) ∧ (𝐴 ∩ 𝐵) ⋖_ℋ 𝐵)) → (𝐴 ∨_ℋ 𝑥) = (𝐴 ∨_ℋ 𝐵))
47	18, 46	breqtrd 5169	. . . 4 ⊢ ((𝑥 ∈ HAtoms ∧ ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) ∧ (𝐴 ∩ 𝐵) ⋖_ℋ 𝐵)) → 𝐴 ⋖_ℋ (𝐴 ∨_ℋ 𝐵))
48	47	exp32 419	. . 3 ⊢ (𝑥 ∈ HAtoms → ((𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 ⊆ (𝐴 ∩ 𝐵)) → ((𝐴 ∩ 𝐵) ⋖_ℋ 𝐵 → 𝐴 ⋖_ℋ (𝐴 ∨_ℋ 𝐵))))
49	48	rexlimiv 3138	. 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 394 = wceq 1533 ∈ wcel 2098 ∃wrex 3060 ∩ cin 3938 ⊆ wss 3939 ⊊ wpss 3940 class class class wbr 5143 (class class class)co 7416 C_ℋ cch 30783 ∨_ℋ chj 30787 ⋖_ℋ ccv 30818 HAtomscat 30819

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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-inf2 9664 ax-cc 10458 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217 ax-mulf 11218 ax-hilex 30853 ax-hfvadd 30854 ax-hvcom 30855 ax-hvass 30856 ax-hv0cl 30857 ax-hvaddid 30858 ax-hfvmul 30859 ax-hvmulid 30860 ax-hvmulass 30861 ax-hvdistr1 30862 ax-hvdistr2 30863 ax-hvmul0 30864 ax-hfi 30933 ax-his1 30936 ax-his2 30937 ax-his3 30938 ax-his4 30939 ax-hcompl 31056

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7991 df-2nd 7992 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-oadd 8489 df-omul 8490 df-er 8723 df-map 8845 df-pm 8846 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-fi 9434 df-sup 9465 df-inf 9466 df-oi 9533 df-card 9962 df-acn 9965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-ioo 13360 df-ico 13362 df-icc 13363 df-fz 13517 df-fzo 13660 df-fl 13789 df-seq 13999 df-exp 14059 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-clim 15464 df-rlim 15465 df-sum 15665 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-hom 17256 df-cco 17257 df-rest 17403 df-topn 17404 df-0g 17422 df-gsum 17423 df-topgen 17424 df-pt 17425 df-prds 17428 df-xrs 17483 df-qtop 17488 df-imas 17489 df-xps 17491 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18740 df-mulg 19028 df-cntz 19272 df-cmn 19741 df-psmet 21275 df-xmet 21276 df-met 21277 df-bl 21278 df-mopn 21279 df-fbas 21280 df-fg 21281 df-cnfld 21284 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22867 df-cld 22941 df-ntr 22942 df-cls 22943 df-nei 23020 df-cn 23149 df-cnp 23150 df-lm 23151 df-haus 23237 df-tx 23484 df-hmeo 23677 df-fil 23768 df-fm 23860 df-flim 23861 df-flf 23862 df-xms 24244 df-ms 24245 df-tms 24246 df-cfil 25201 df-cau 25202 df-cmet 25203 df-grpo 30347 df-gid 30348 df-ginv 30349 df-gdiv 30350 df-ablo 30399 df-vc 30413 df-nv 30446 df-va 30449 df-ba 30450 df-sm 30451 df-0v 30452 df-vs 30453 df-nmcv 30454 df-ims 30455 df-dip 30555 df-ssp 30576 df-ph 30667 df-cbn 30717 df-hnorm 30822 df-hba 30823 df-hvsub 30825 df-hlim 30826 df-hcau 30827 df-sh 31061 df-ch 31075 df-oc 31106 df-ch0 31107 df-shs 31162 df-span 31163 df-chj 31164 df-chsup 31165 df-pjh 31249 df-cv 32133 df-at 32192

This theorem is referenced by: cvexchi 32223

W3C validator