Theorem lcvexchlem4 37499

Description: Lemma for lcvexch 37501. (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 20544	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ⊕ 𝑈) ∈ 𝑆)
8	3, 4, 5, 7	syl3anc 1371	. . . 4 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ∈ 𝑆)
9		lcvexch.f	. . . 4 ⊢ (𝜑 → 𝑇𝐶(𝑇 ⊕ 𝑈))
10	1, 2, 3, 4, 8, 9	lcvpss 37486	. . 3 ⊢ (𝜑 → 𝑇 ⊊ (𝑇 ⊕ 𝑈))
11	1, 6, 2, 3, 4, 5	lcvexchlem1 37496	. . 3 ⊢ (𝜑 → (𝑇 ⊊ (𝑇 ⊕ 𝑈) ↔ (𝑇 ∩ 𝑈) ⊊ 𝑈))
12	10, 11	mpbid 231	. 2 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ⊊ 𝑈)
13	3	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → 𝑊 ∈ LMod)
14	1	lsssssubg 20419	. . . . . . . . 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 3945	. . . . . . 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 3945	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → 𝑇 ∈ (SubGrp‘𝑊))
20	6	lsmub2 19440	. . . . . . 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 3945	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → 𝑈 ∈ (SubGrp‘𝑊))
24		simp3r 1202	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → 𝑠 ⊆ 𝑈)
25	6	lsmless1 19442	. . . . . . . 8 ⊢ ((𝑈 ∈ (SubGrp‘𝑊) ∧ 𝑇 ∈ (SubGrp‘𝑊) ∧ 𝑠 ⊆ 𝑈) → (𝑠 ⊕ 𝑇) ⊆ (𝑈 ⊕ 𝑇))
26	23, 19, 24, 25	syl3anc 1371	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → (𝑠 ⊕ 𝑇) ⊆ (𝑈 ⊕ 𝑇))
27		lmodabl 20369	. . . . . . . . . 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 3945	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝑊))
31	29, 5	sseldd 3945	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊))
32	6	lsmcom 19636	. . . . . . . . 9 ⊢ ((𝑊 ∈ Abel ∧ 𝑇 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))
33	28, 30, 31, 32	syl3anc 1371	. . . . . . . 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 3985	. . . . . 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 37485	. . . . . . . . . 10 ⊢ (𝜑 → (𝑇𝐶(𝑇 ⊕ 𝑈) ↔ (𝑇 ⊊ (𝑇 ⊕ 𝑈) ∧ ∀𝑟 ∈ 𝑆 ((𝑇 ⊆ 𝑟 ∧ 𝑟 ⊆ (𝑇 ⊕ 𝑈)) → (𝑟 = 𝑇 ∨ 𝑟 = (𝑇 ⊕ 𝑈))))))
38	37	adantr 481	. . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑊 ∈ LMod)
40		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑠 ∈ 𝑆)
41	4	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑇 ∈ 𝑆)
42	1, 6	lsmcl 20544	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ 𝑠 ∈ 𝑆 ∧ 𝑇 ∈ 𝑆) → (𝑠 ⊕ 𝑇) ∈ 𝑆)
43	39, 40, 41, 42	syl3anc 1371	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ⊕ 𝑇) ∈ 𝑆)
44		sseq2 3970	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑠 ⊕ 𝑇) → (𝑇 ⊆ 𝑟 ↔ 𝑇 ⊆ (𝑠 ⊕ 𝑇)))
45		sseq1 3969	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑠 ⊕ 𝑇) → (𝑟 ⊆ (𝑇 ⊕ 𝑈) ↔ (𝑠 ⊕ 𝑇) ⊆ (𝑇 ⊕ 𝑈)))
46	44, 45	anbi12d 631	. . . . . . . . . . . . 13 ⊢ (𝑟 = (𝑠 ⊕ 𝑇) → ((𝑇 ⊆ 𝑟 ∧ 𝑟 ⊆ (𝑇 ⊕ 𝑈)) ↔ (𝑇 ⊆ (𝑠 ⊕ 𝑇) ∧ (𝑠 ⊕ 𝑇) ⊆ (𝑇 ⊕ 𝑈))))
47		eqeq1 2740	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑠 ⊕ 𝑇) → (𝑟 = 𝑇 ↔ (𝑠 ⊕ 𝑇) = 𝑇))
48		eqeq1 2740	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑠 ⊕ 𝑇) → (𝑟 = (𝑇 ⊕ 𝑈) ↔ (𝑠 ⊕ 𝑇) = (𝑇 ⊕ 𝑈)))
49	47, 48	orbi12d 917	. . . . . . . . . . . . 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 3577	. . . . . . . . . . 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 491	. . . . . . . . 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 239	. . . . . . . 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 697	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → ((𝑠 ⊕ 𝑇) = 𝑇 ∨ (𝑠 ⊕ 𝑇) = (𝑇 ⊕ 𝑈)))
58		ineq1 4165	. . . . . . 7 ⊢ ((𝑠 ⊕ 𝑇) = 𝑇 → ((𝑠 ⊕ 𝑇) ∩ 𝑈) = (𝑇 ∩ 𝑈))
59		simp3l 1201	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → (𝑇 ∩ 𝑈) ⊆ 𝑠)
60	1, 6, 2, 13, 18, 22, 16, 59, 24	lcvexchlem2 37497	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → ((𝑠 ⊕ 𝑇) ∩ 𝑈) = 𝑠)
61	60	eqeq1d 2738	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → (((𝑠 ⊕ 𝑇) ∩ 𝑈) = (𝑇 ∩ 𝑈) ↔ 𝑠 = (𝑇 ∩ 𝑈)))
62	58, 61	imbitrid 243	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → ((𝑠 ⊕ 𝑇) = 𝑇 → 𝑠 = (𝑇 ∩ 𝑈)))
63		ineq1 4165	. . . . . . 7 ⊢ ((𝑠 ⊕ 𝑇) = (𝑇 ⊕ 𝑈) → ((𝑠 ⊕ 𝑇) ∩ 𝑈) = ((𝑇 ⊕ 𝑈) ∩ 𝑈))
64	6	lsmub2 19440	. . . . . . . . . 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 4175	. . . . . . . . 9 ⊢ (𝑈 ⊆ (𝑇 ⊕ 𝑈) ↔ ((𝑇 ⊕ 𝑈) ∩ 𝑈) = 𝑈)
67	65, 66	sylib 217	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → ((𝑇 ⊕ 𝑈) ∩ 𝑈) = 𝑈)
68	60, 67	eqeq12d 2752	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → (((𝑠 ⊕ 𝑇) ∩ 𝑈) = ((𝑇 ⊕ 𝑈) ∩ 𝑈) ↔ 𝑠 = 𝑈))
69	63, 68	imbitrid 243	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → ((𝑠 ⊕ 𝑇) = (𝑇 ⊕ 𝑈) → 𝑠 = 𝑈))
70	62, 69	orim12d 963	. . . . 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 3142	. 2 ⊢ (𝜑 → ∀𝑠 ∈ 𝑆 (((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈) → (𝑠 = (𝑇 ∩ 𝑈) ∨ 𝑠 = 𝑈)))
74	1	lssincl 20426	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ∩ 𝑈) ∈ 𝑆)
75	3, 4, 5, 74	syl3anc 1371	. . 3 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ∈ 𝑆)
76	1, 2, 3, 75, 5	lcvbr3 37485	. 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 711	1 ⊢ (𝜑 → (𝑇 ∩ 𝑈)𝐶𝑈)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3064   ∩ cin 3909   ⊆ wss 3910   ⊊ wpss 3911   class class class wbr 5105  ‘cfv 6496  (class class class)co 7357  SubGrpcsubg 18922  LSSumclsm 19416  Abelcabl 19563  LModclmod 20322  LSubSpclss 20392   ⋖_L clcv 37480

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-subg 18925  df-cntz 19097  df-lsm 19418  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-lmod 20324  df-lss 20393  df-lcv 37481

This theorem is referenced by:  lcvexch  37501  lsatcvat3  37514