Theorem lcvexchlem4 39483

Description: Lemma for lcvexch 39485. (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 21078	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ⊕ 𝑈) ∈ 𝑆)
8	3, 4, 5, 7	syl3anc 1374	. . . 4 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ∈ 𝑆)
9		lcvexch.f	. . . 4 ⊢ (𝜑 → 𝑇𝐶(𝑇 ⊕ 𝑈))
10	1, 2, 3, 4, 8, 9	lcvpss 39470	. . 3 ⊢ (𝜑 → 𝑇 ⊊ (𝑇 ⊕ 𝑈))
11	1, 6, 2, 3, 4, 5	lcvexchlem1 39480	. . 3 ⊢ (𝜑 → (𝑇 ⊊ (𝑇 ⊕ 𝑈) ↔ (𝑇 ∩ 𝑈) ⊊ 𝑈))
12	10, 11	mpbid 232	. 2 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ⊊ 𝑈)
13	3	3ad2ant1 1134	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → 𝑊 ∈ LMod)
14	1	lsssssubg 20953	. . . . . . . . 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 1138	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → 𝑠 ∈ 𝑆)
17	15, 16	sseldd 3923	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → 𝑠 ∈ (SubGrp‘𝑊))
18	4	3ad2ant1 1134	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → 𝑇 ∈ 𝑆)
19	15, 18	sseldd 3923	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → 𝑇 ∈ (SubGrp‘𝑊))
20	6	lsmub2 19633	. . . . . . 7 ⊢ ((𝑠 ∈ (SubGrp‘𝑊) ∧ 𝑇 ∈ (SubGrp‘𝑊)) → 𝑇 ⊆ (𝑠 ⊕ 𝑇))
21	17, 19, 20	syl2anc 585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → 𝑇 ⊆ (𝑠 ⊕ 𝑇))
22	5	3ad2ant1 1134	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → 𝑈 ∈ 𝑆)
23	15, 22	sseldd 3923	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → 𝑈 ∈ (SubGrp‘𝑊))
24		simp3r 1204	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → 𝑠 ⊆ 𝑈)
25	6	lsmless1 19635	. . . . . . . 8 ⊢ ((𝑈 ∈ (SubGrp‘𝑊) ∧ 𝑇 ∈ (SubGrp‘𝑊) ∧ 𝑠 ⊆ 𝑈) → (𝑠 ⊕ 𝑇) ⊆ (𝑈 ⊕ 𝑇))
26	23, 19, 24, 25	syl3anc 1374	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → (𝑠 ⊕ 𝑇) ⊆ (𝑈 ⊕ 𝑇))
27		lmodabl 20904	. . . . . . . . . 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 3923	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝑊))
31	29, 5	sseldd 3923	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊))
32	6	lsmcom 19833	. . . . . . . . 9 ⊢ ((𝑊 ∈ Abel ∧ 𝑇 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))
33	28, 30, 31, 32	syl3anc 1374	. . . . . . . 8 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))
34	33	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))
35	26, 34	sseqtrrd 3960	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → (𝑠 ⊕ 𝑇) ⊆ (𝑇 ⊕ 𝑈))
36	9	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → 𝑇𝐶(𝑇 ⊕ 𝑈))
37	1, 2, 3, 4, 8	lcvbr3 39469	. . . . . . . . . 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 21078	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ 𝑠 ∈ 𝑆 ∧ 𝑇 ∈ 𝑆) → (𝑠 ⊕ 𝑇) ∈ 𝑆)
43	39, 40, 41, 42	syl3anc 1374	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ⊕ 𝑇) ∈ 𝑆)
44		sseq2 3949	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑠 ⊕ 𝑇) → (𝑇 ⊆ 𝑟 ↔ 𝑇 ⊆ (𝑠 ⊕ 𝑇)))
45		sseq1 3948	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑠 ⊕ 𝑇) → (𝑟 ⊆ (𝑇 ⊕ 𝑈) ↔ (𝑠 ⊕ 𝑇) ⊆ (𝑇 ⊕ 𝑈)))
46	44, 45	anbi12d 633	. . . . . . . . . . . . 13 ⊢ (𝑟 = (𝑠 ⊕ 𝑇) → ((𝑇 ⊆ 𝑟 ∧ 𝑟 ⊆ (𝑇 ⊕ 𝑈)) ↔ (𝑇 ⊆ (𝑠 ⊕ 𝑇) ∧ (𝑠 ⊕ 𝑇) ⊆ (𝑇 ⊕ 𝑈))))
47		eqeq1 2741	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑠 ⊕ 𝑇) → (𝑟 = 𝑇 ↔ (𝑠 ⊕ 𝑇) = 𝑇))
48		eqeq1 2741	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑠 ⊕ 𝑇) → (𝑟 = (𝑇 ⊕ 𝑈) ↔ (𝑠 ⊕ 𝑇) = (𝑇 ⊕ 𝑈)))
49	47, 48	orbi12d 919	. . . . . . . . . . . . 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 3561	. . . . . . . . . . 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 1133	. . . . . . 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 700	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → ((𝑠 ⊕ 𝑇) = 𝑇 ∨ (𝑠 ⊕ 𝑇) = (𝑇 ⊕ 𝑈)))
58		ineq1 4154	. . . . . . 7 ⊢ ((𝑠 ⊕ 𝑇) = 𝑇 → ((𝑠 ⊕ 𝑇) ∩ 𝑈) = (𝑇 ∩ 𝑈))
59		simp3l 1203	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → (𝑇 ∩ 𝑈) ⊆ 𝑠)
60	1, 6, 2, 13, 18, 22, 16, 59, 24	lcvexchlem2 39481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → ((𝑠 ⊕ 𝑇) ∩ 𝑈) = 𝑠)
61	60	eqeq1d 2739	. . . . . . 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 4154	. . . . . . 7 ⊢ ((𝑠 ⊕ 𝑇) = (𝑇 ⊕ 𝑈) → ((𝑠 ⊕ 𝑇) ∩ 𝑈) = ((𝑇 ⊕ 𝑈) ∩ 𝑈))
64	6	lsmub2 19633	. . . . . . . . . 10 ⊢ ((𝑇 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → 𝑈 ⊆ (𝑇 ⊕ 𝑈))
65	19, 23, 64	syl2anc 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → 𝑈 ⊆ (𝑇 ⊕ 𝑈))
66		sseqin2 4164	. . . . . . . . 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 2753	. . . . . . 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 967	. . . . 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 1120	. . 3 ⊢ (𝜑 → (𝑠 ∈ 𝑆 → (((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈) → (𝑠 = (𝑇 ∩ 𝑈) ∨ 𝑠 = 𝑈))))
73	72	ralrimiv 3129	. 2 ⊢ (𝜑 → ∀𝑠 ∈ 𝑆 (((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈) → (𝑠 = (𝑇 ∩ 𝑈) ∨ 𝑠 = 𝑈)))
74	1	lssincl 20960	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ∩ 𝑈) ∈ 𝑆)
75	3, 4, 5, 74	syl3anc 1374	. . 3 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ∈ 𝑆)
76	1, 2, 3, 75, 5	lcvbr3 39469	. 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 714	1 ⊢ (𝜑 → (𝑇 ∩ 𝑈)𝐶𝑈)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ∩ cin 3889   ⊆ wss 3890   ⊊ wpss 3891   class class class wbr 5086  ‘cfv 6499  (class class class)co 7367  SubGrpcsubg 19096  LSSumclsm 19609  Abelcabl 19756  LModclmod 20855  LSubSpclss 20926   ⋖_L clcv 39464

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6266  df-ord 6327  df-on 6328  df-lim 6329  df-suc 6330  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-0g 17404  df-mre 17548  df-mrc 17549  df-acs 17551  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-submnd 18752  df-grp 18912  df-minusg 18913  df-sbg 18914  df-subg 19099  df-cntz 19292  df-lsm 19611  df-cmn 19757  df-abl 19758  df-mgp 20122  df-ur 20163  df-ring 20216  df-lmod 20857  df-lss 20927  df-lcv 39465

This theorem is referenced by:  lcvexch  39485  lsatcvat3  39498