Theorem lcvexchlem4 39025

Description: Lemma for lcvexch 39027. (Contributed by NM, 10-Jan-2015.)

Hypotheses

Ref	Expression
lcvexch.s	⊢ 𝑆 = (LSubSp‘𝑊)
lcvexch.p	⊢ ⊕ = (LSSum‘𝑊)
lcvexch.c	⊢ 𝐶 = ( ⋖L ‘𝑊)
lcvexch.w	⊢ (𝜑 → 𝑊 ∈ LMod)
lcvexch.t	⊢ (𝜑 → 𝑇 ∈ 𝑆)
lcvexch.u	⊢ (𝜑 → 𝑈 ∈ 𝑆)
lcvexch.f	⊢ (𝜑 → 𝑇𝐶(𝑇 ⊕ 𝑈))

Assertion

Ref	Expression
lcvexchlem4	⊢ (𝜑 → (𝑇 ∩ 𝑈)𝐶𝑈)

Proof of Theorem lcvexchlem4

Dummy variables 𝑠 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lcvexch.s	. . . 4 ⊢ 𝑆 = (LSubSp‘𝑊)
2		lcvexch.c	. . . 4 ⊢ 𝐶 = ( ⋖L ‘𝑊)
3		lcvexch.w	. . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod)
4		lcvexch.t	. . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝑆)
5		lcvexch.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑆)
6		lcvexch.p	. . . . . 6 ⊢ ⊕ = (LSSum‘𝑊)
7	1, 6	lsmcl 20996	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ⊕ 𝑈) ∈ 𝑆)
8	3, 4, 5, 7	syl3anc 1373	. . . 4 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ∈ 𝑆)
9		lcvexch.f	. . . 4 ⊢ (𝜑 → 𝑇𝐶(𝑇 ⊕ 𝑈))
10	1, 2, 3, 4, 8, 9	lcvpss 39012	. . 3 ⊢ (𝜑 → 𝑇 ⊊ (𝑇 ⊕ 𝑈))
11	1, 6, 2, 3, 4, 5	lcvexchlem1 39022	. . 3 ⊢ (𝜑 → (𝑇 ⊊ (𝑇 ⊕ 𝑈) ↔ (𝑇 ∩ 𝑈) ⊊ 𝑈))
12	10, 11	mpbid 232	. 2 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ⊊ 𝑈)
13	3	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → 𝑊 ∈ LMod)
14	1	lsssssubg 20870	. . . . . . . . 9 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊))
15	13, 14	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → 𝑆 ⊆ (SubGrp‘𝑊))
16		simp2 1137	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → 𝑠 ∈ 𝑆)
17	15, 16	sseldd 3949	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → 𝑠 ∈ (SubGrp‘𝑊))
18	4	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → 𝑇 ∈ 𝑆)
19	15, 18	sseldd 3949	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → 𝑇 ∈ (SubGrp‘𝑊))
20	6	lsmub2 19594	. . . . . . 7 ⊢ ((𝑠 ∈ (SubGrp‘𝑊) ∧ 𝑇 ∈ (SubGrp‘𝑊)) → 𝑇 ⊆ (𝑠 ⊕ 𝑇))
21	17, 19, 20	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → 𝑇 ⊆ (𝑠 ⊕ 𝑇))
22	5	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → 𝑈 ∈ 𝑆)
23	15, 22	sseldd 3949	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → 𝑈 ∈ (SubGrp‘𝑊))
24		simp3r 1203	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → 𝑠 ⊆ 𝑈)
25	6	lsmless1 19596	. . . . . . . 8 ⊢ ((𝑈 ∈ (SubGrp‘𝑊) ∧ 𝑇 ∈ (SubGrp‘𝑊) ∧ 𝑠 ⊆ 𝑈) → (𝑠 ⊕ 𝑇) ⊆ (𝑈 ⊕ 𝑇))
26	23, 19, 24, 25	syl3anc 1373	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → (𝑠 ⊕ 𝑇) ⊆ (𝑈 ⊕ 𝑇))
27		lmodabl 20821	. . . . . . . . . 10 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel)
28	3, 27	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ Abel)
29	3, 14	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊))
30	29, 4	sseldd 3949	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝑊))
31	29, 5	sseldd 3949	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊))
32	6	lsmcom 19794	. . . . . . . . 9 ⊢ ((𝑊 ∈ Abel ∧ 𝑇 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))
33	28, 30, 31, 32	syl3anc 1373	. . . . . . . 8 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))
34	33	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))
35	26, 34	sseqtrrd 3986	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → (𝑠 ⊕ 𝑇) ⊆ (𝑇 ⊕ 𝑈))
36	9	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → 𝑇𝐶(𝑇 ⊕ 𝑈))
37	1, 2, 3, 4, 8	lcvbr3 39011	. . . . . . . . . 10 ⊢ (𝜑 → (𝑇𝐶(𝑇 ⊕ 𝑈) ↔ (𝑇 ⊊ (𝑇 ⊕ 𝑈) ∧ ∀𝑟 ∈ 𝑆 ((𝑇 ⊆ 𝑟 ∧ 𝑟 ⊆ (𝑇 ⊕ 𝑈)) → (𝑟 = 𝑇 ∨ 𝑟 = (𝑇 ⊕ 𝑈))))))
38	37	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑇𝐶(𝑇 ⊕ 𝑈) ↔ (𝑇 ⊊ (𝑇 ⊕ 𝑈) ∧ ∀𝑟 ∈ 𝑆 ((𝑇 ⊆ 𝑟 ∧ 𝑟 ⊆ (𝑇 ⊕ 𝑈)) → (𝑟 = 𝑇 ∨ 𝑟 = (𝑇 ⊕ 𝑈))))))
39	3	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑊 ∈ LMod)
40		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑠 ∈ 𝑆)
41	4	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑇 ∈ 𝑆)
42	1, 6	lsmcl 20996	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ 𝑠 ∈ 𝑆 ∧ 𝑇 ∈ 𝑆) → (𝑠 ⊕ 𝑇) ∈ 𝑆)
43	39, 40, 41, 42	syl3anc 1373	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ⊕ 𝑇) ∈ 𝑆)
44		sseq2 3975	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑠 ⊕ 𝑇) → (𝑇 ⊆ 𝑟 ↔ 𝑇 ⊆ (𝑠 ⊕ 𝑇)))
45		sseq1 3974	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑠 ⊕ 𝑇) → (𝑟 ⊆ (𝑇 ⊕ 𝑈) ↔ (𝑠 ⊕ 𝑇) ⊆ (𝑇 ⊕ 𝑈)))
46	44, 45	anbi12d 632	. . . . . . . . . . . . 13 ⊢ (𝑟 = (𝑠 ⊕ 𝑇) → ((𝑇 ⊆ 𝑟 ∧ 𝑟 ⊆ (𝑇 ⊕ 𝑈)) ↔ (𝑇 ⊆ (𝑠 ⊕ 𝑇) ∧ (𝑠 ⊕ 𝑇) ⊆ (𝑇 ⊕ 𝑈))))
47		eqeq1 2734	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑠 ⊕ 𝑇) → (𝑟 = 𝑇 ↔ (𝑠 ⊕ 𝑇) = 𝑇))
48		eqeq1 2734	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑠 ⊕ 𝑇) → (𝑟 = (𝑇 ⊕ 𝑈) ↔ (𝑠 ⊕ 𝑇) = (𝑇 ⊕ 𝑈)))
49	47, 48	orbi12d 918	. . . . . . . . . . . . 13 ⊢ (𝑟 = (𝑠 ⊕ 𝑇) → ((𝑟 = 𝑇 ∨ 𝑟 = (𝑇 ⊕ 𝑈)) ↔ ((𝑠 ⊕ 𝑇) = 𝑇 ∨ (𝑠 ⊕ 𝑇) = (𝑇 ⊕ 𝑈))))
50	46, 49	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝑠 ⊕ 𝑇) → (((𝑇 ⊆ 𝑟 ∧ 𝑟 ⊆ (𝑇 ⊕ 𝑈)) → (𝑟 = 𝑇 ∨ 𝑟 = (𝑇 ⊕ 𝑈))) ↔ ((𝑇 ⊆ (𝑠 ⊕ 𝑇) ∧ (𝑠 ⊕ 𝑇) ⊆ (𝑇 ⊕ 𝑈)) → ((𝑠 ⊕ 𝑇) = 𝑇 ∨ (𝑠 ⊕ 𝑇) = (𝑇 ⊕ 𝑈)))))
51	50	rspcv 3587	. . . . . . . . . . 11 ⊢ ((𝑠 ⊕ 𝑇) ∈ 𝑆 → (∀𝑟 ∈ 𝑆 ((𝑇 ⊆ 𝑟 ∧ 𝑟 ⊆ (𝑇 ⊕ 𝑈)) → (𝑟 = 𝑇 ∨ 𝑟 = (𝑇 ⊕ 𝑈))) → ((𝑇 ⊆ (𝑠 ⊕ 𝑇) ∧ (𝑠 ⊕ 𝑇) ⊆ (𝑇 ⊕ 𝑈)) → ((𝑠 ⊕ 𝑇) = 𝑇 ∨ (𝑠 ⊕ 𝑇) = (𝑇 ⊕ 𝑈)))))
52	43, 51	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (∀𝑟 ∈ 𝑆 ((𝑇 ⊆ 𝑟 ∧ 𝑟 ⊆ (𝑇 ⊕ 𝑈)) → (𝑟 = 𝑇 ∨ 𝑟 = (𝑇 ⊕ 𝑈))) → ((𝑇 ⊆ (𝑠 ⊕ 𝑇) ∧ (𝑠 ⊕ 𝑇) ⊆ (𝑇 ⊕ 𝑈)) → ((𝑠 ⊕ 𝑇) = 𝑇 ∨ (𝑠 ⊕ 𝑇) = (𝑇 ⊕ 𝑈)))))
53	52	adantld 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((𝑇 ⊊ (𝑇 ⊕ 𝑈) ∧ ∀𝑟 ∈ 𝑆 ((𝑇 ⊆ 𝑟 ∧ 𝑟 ⊆ (𝑇 ⊕ 𝑈)) → (𝑟 = 𝑇 ∨ 𝑟 = (𝑇 ⊕ 𝑈)))) → ((𝑇 ⊆ (𝑠 ⊕ 𝑇) ∧ (𝑠 ⊕ 𝑇) ⊆ (𝑇 ⊕ 𝑈)) → ((𝑠 ⊕ 𝑇) = 𝑇 ∨ (𝑠 ⊕ 𝑇) = (𝑇 ⊕ 𝑈)))))
54	38, 53	sylbid 240	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑇𝐶(𝑇 ⊕ 𝑈) → ((𝑇 ⊆ (𝑠 ⊕ 𝑇) ∧ (𝑠 ⊕ 𝑇) ⊆ (𝑇 ⊕ 𝑈)) → ((𝑠 ⊕ 𝑇) = 𝑇 ∨ (𝑠 ⊕ 𝑇) = (𝑇 ⊕ 𝑈)))))
55	54	3adant3 1132	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → (𝑇𝐶(𝑇 ⊕ 𝑈) → ((𝑇 ⊆ (𝑠 ⊕ 𝑇) ∧ (𝑠 ⊕ 𝑇) ⊆ (𝑇 ⊕ 𝑈)) → ((𝑠 ⊕ 𝑇) = 𝑇 ∨ (𝑠 ⊕ 𝑇) = (𝑇 ⊕ 𝑈)))))
56	36, 55	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → ((𝑇 ⊆ (𝑠 ⊕ 𝑇) ∧ (𝑠 ⊕ 𝑇) ⊆ (𝑇 ⊕ 𝑈)) → ((𝑠 ⊕ 𝑇) = 𝑇 ∨ (𝑠 ⊕ 𝑇) = (𝑇 ⊕ 𝑈))))
57	21, 35, 56	mp2and 699	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → ((𝑠 ⊕ 𝑇) = 𝑇 ∨ (𝑠 ⊕ 𝑇) = (𝑇 ⊕ 𝑈)))
58		ineq1 4178	. . . . . . 7 ⊢ ((𝑠 ⊕ 𝑇) = 𝑇 → ((𝑠 ⊕ 𝑇) ∩ 𝑈) = (𝑇 ∩ 𝑈))
59		simp3l 1202	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → (𝑇 ∩ 𝑈) ⊆ 𝑠)
60	1, 6, 2, 13, 18, 22, 16, 59, 24	lcvexchlem2 39023	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → ((𝑠 ⊕ 𝑇) ∩ 𝑈) = 𝑠)
61	60	eqeq1d 2732	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → (((𝑠 ⊕ 𝑇) ∩ 𝑈) = (𝑇 ∩ 𝑈) ↔ 𝑠 = (𝑇 ∩ 𝑈)))
62	58, 61	imbitrid 244	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → ((𝑠 ⊕ 𝑇) = 𝑇 → 𝑠 = (𝑇 ∩ 𝑈)))
63		ineq1 4178	. . . . . . 7 ⊢ ((𝑠 ⊕ 𝑇) = (𝑇 ⊕ 𝑈) → ((𝑠 ⊕ 𝑇) ∩ 𝑈) = ((𝑇 ⊕ 𝑈) ∩ 𝑈))
64	6	lsmub2 19594	. . . . . . . . . 10 ⊢ ((𝑇 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → 𝑈 ⊆ (𝑇 ⊕ 𝑈))
65	19, 23, 64	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → 𝑈 ⊆ (𝑇 ⊕ 𝑈))
66		sseqin2 4188	. . . . . . . . 9 ⊢ (𝑈 ⊆ (𝑇 ⊕ 𝑈) ↔ ((𝑇 ⊕ 𝑈) ∩ 𝑈) = 𝑈)
67	65, 66	sylib 218	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → ((𝑇 ⊕ 𝑈) ∩ 𝑈) = 𝑈)
68	60, 67	eqeq12d 2746	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → (((𝑠 ⊕ 𝑇) ∩ 𝑈) = ((𝑇 ⊕ 𝑈) ∩ 𝑈) ↔ 𝑠 = 𝑈))
69	63, 68	imbitrid 244	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → ((𝑠 ⊕ 𝑇) = (𝑇 ⊕ 𝑈) → 𝑠 = 𝑈))
70	62, 69	orim12d 966	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → (((𝑠 ⊕ 𝑇) = 𝑇 ∨ (𝑠 ⊕ 𝑇) = (𝑇 ⊕ 𝑈)) → (𝑠 = (𝑇 ∩ 𝑈) ∨ 𝑠 = 𝑈)))
71	57, 70	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → (𝑠 = (𝑇 ∩ 𝑈) ∨ 𝑠 = 𝑈))
72	71	3exp 1119	. . 3 ⊢ (𝜑 → (𝑠 ∈ 𝑆 → (((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈) → (𝑠 = (𝑇 ∩ 𝑈) ∨ 𝑠 = 𝑈))))
73	72	ralrimiv 3125	. 2 ⊢ (𝜑 → ∀𝑠 ∈ 𝑆 (((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈) → (𝑠 = (𝑇 ∩ 𝑈) ∨ 𝑠 = 𝑈)))
74	1	lssincl 20877	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ∩ 𝑈) ∈ 𝑆)
75	3, 4, 5, 74	syl3anc 1373	. . 3 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ∈ 𝑆)
76	1, 2, 3, 75, 5	lcvbr3 39011	. 2 ⊢ (𝜑 → ((𝑇 ∩ 𝑈)𝐶𝑈 ↔ ((𝑇 ∩ 𝑈) ⊊ 𝑈 ∧ ∀𝑠 ∈ 𝑆 (((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈) → (𝑠 = (𝑇 ∩ 𝑈) ∨ 𝑠 = 𝑈)))))
77	12, 73, 76	mpbir2and 713	1 ⊢ (𝜑 → (𝑇 ∩ 𝑈)𝐶𝑈)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3045   ∩ cin 3915   ⊆ wss 3916   ⊊ wpss 3917   class class class wbr 5109  ‘cfv 6513  (class class class)co 7389  SubGrpcsubg 19058  LSSumclsm 19570  Abelcabl 19717  LModclmod 20772  LSubSpclss 20843   ⋖_L clcv 39006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713  ax-cnex 11130  ax-resscn 11131  ax-1cn 11132  ax-icn 11133  ax-addcl 11134  ax-addrcl 11135  ax-mulcl 11136  ax-mulrcl 11137  ax-mulcom 11138  ax-addass 11139  ax-mulass 11140  ax-distr 11141  ax-i2m1 11142  ax-1ne0 11143  ax-1rid 11144  ax-rnegex 11145  ax-rrecex 11146  ax-cnre 11147  ax-pre-lttri 11148  ax-pre-lttrn 11149  ax-pre-ltadd 11150  ax-pre-mulgt0 11151

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-pss 3936  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4913  df-iun 4959  df-iin 4960  df-br 5110  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6276  df-ord 6337  df-on 6338  df-lim 6339  df-suc 6340  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-riota 7346  df-ov 7392  df-oprab 7393  df-mpo 7394  df-om 7845  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8380  df-1o 8436  df-2o 8437  df-er 8673  df-en 8921  df-dom 8922  df-sdom 8923  df-fin 8924  df-pnf 11216  df-mnf 11217  df-xr 11218  df-ltxr 11219  df-le 11220  df-sub 11413  df-neg 11414  df-nn 12188  df-2 12250  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17186  df-ress 17207  df-plusg 17239  df-0g 17410  df-mre 17553  df-mrc 17554  df-acs 17556  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-submnd 18717  df-grp 18874  df-minusg 18875  df-sbg 18876  df-subg 19061  df-cntz 19255  df-lsm 19572  df-cmn 19718  df-abl 19719  df-mgp 20056  df-ur 20097  df-ring 20150  df-lmod 20774  df-lss 20844  df-lcv 39007

This theorem is referenced by:  lcvexch  39027  lsatcvat3  39040