Theorem lcvexchlem4 39055

Description: Lemma for lcvexch 39057. (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 21041	. . . . 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 39042	. . 3 ⊢ (𝜑 → 𝑇 ⊊ (𝑇 ⊕ 𝑈))
11	1, 6, 2, 3, 4, 5	lcvexchlem1 39052	. . 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 20915	. . . . . . . . 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 3959	. . . . . . 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 3959	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → 𝑇 ∈ (SubGrp‘𝑊))
20	6	lsmub2 19639	. . . . . . 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 3959	. . . . . . . 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 19641	. . . . . . . 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 20866	. . . . . . . . . 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 3959	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝑊))
31	29, 5	sseldd 3959	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊))
32	6	lsmcom 19839	. . . . . . . . 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 3996	. . . . . 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 39041	. . . . . . . . . 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 21041	. . . . . . . . . . . 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 3985	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑠 ⊕ 𝑇) → (𝑇 ⊆ 𝑟 ↔ 𝑇 ⊆ (𝑠 ⊕ 𝑇)))
45		sseq1 3984	. . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑠 ⊕ 𝑇) → (𝑟 = 𝑇 ↔ (𝑠 ⊕ 𝑇) = 𝑇))
48		eqeq1 2739	. . . . . . . . . . . . . 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 3597	. . . . . . . . . . 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 4188	. . . . . . 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 39053	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ ((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈)) → ((𝑠 ⊕ 𝑇) ∩ 𝑈) = 𝑠)
61	60	eqeq1d 2737	. . . . . . 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 4188	. . . . . . 7 ⊢ ((𝑠 ⊕ 𝑇) = (𝑇 ⊕ 𝑈) → ((𝑠 ⊕ 𝑇) ∩ 𝑈) = ((𝑇 ⊕ 𝑈) ∩ 𝑈))
64	6	lsmub2 19639	. . . . . . . . . 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 4198	. . . . . . . . 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 2751	. . . . . . 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 3131	. 2 ⊢ (𝜑 → ∀𝑠 ∈ 𝑆 (((𝑇 ∩ 𝑈) ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑈) → (𝑠 = (𝑇 ∩ 𝑈) ∨ 𝑠 = 𝑈)))
74	1	lssincl 20922	. . . 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 39041	. 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 2108  ∀wral 3051   ∩ cin 3925   ⊆ wss 3926   ⊊ wpss 3927   class class class wbr 5119  ‘cfv 6531  (class class class)co 7405  SubGrpcsubg 19103  LSSumclsm 19615  Abelcabl 19762  LModclmod 20817  LSubSpclss 20888   ⋖_L clcv 39036

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-iin 4970  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17252  df-plusg 17284  df-0g 17455  df-mre 17598  df-mrc 17599  df-acs 17601  df-mgm 18618  df-sgrp 18697  df-mnd 18713  df-submnd 18762  df-grp 18919  df-minusg 18920  df-sbg 18921  df-subg 19106  df-cntz 19300  df-lsm 19617  df-cmn 19763  df-abl 19764  df-mgp 20101  df-ur 20142  df-ring 20195  df-lmod 20819  df-lss 20889  df-lcv 39037

This theorem is referenced by:  lcvexch  39057  lsatcvat3  39070