Theorem lcvexchlem5 36979

Description: Lemma for lcvexch 36980. (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.g	⊢ (𝜑 → (𝑇 ∩ 𝑈)𝐶𝑈)

Assertion

Ref	Expression
lcvexchlem5	⊢ (𝜑 → 𝑇𝐶(𝑇 ⊕ 𝑈))

Proof of Theorem lcvexchlem5

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	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝑆)
5		lcvexch.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑆)
6	1	lssincl 20142	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ∩ 𝑈) ∈ 𝑆)
7	3, 4, 5, 6	syl3anc 1369	. . . 4 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ∈ 𝑆)
8		lcvexch.g	. . . 4 ⊢ (𝜑 → (𝑇 ∩ 𝑈)𝐶𝑈)
9	1, 2, 3, 7, 5, 8	lcvpss 36965	. . 3 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ⊊ 𝑈)
10		lcvexch.p	. . . 4 ⊢ ⊕ = (LSSum‘𝑊)
11	1, 10, 2, 3, 4, 5	lcvexchlem1 36975	. . 3 ⊢ (𝜑 → (𝑇 ⊊ (𝑇 ⊕ 𝑈) ↔ (𝑇 ∩ 𝑈) ⊊ 𝑈))
12	9, 11	mpbird 256	. 2 ⊢ (𝜑 → 𝑇 ⊊ (𝑇 ⊕ 𝑈))
13		simp3l 1199	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → 𝑇 ⊆ 𝑠)
14	13	ssrind 4166	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → (𝑇 ∩ 𝑈) ⊆ (𝑠 ∩ 𝑈))
15		inss2 4160	. . . . . . 7 ⊢ (𝑠 ∩ 𝑈) ⊆ 𝑈
16	14, 15	jctir 520	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → ((𝑇 ∩ 𝑈) ⊆ (𝑠 ∩ 𝑈) ∧ (𝑠 ∩ 𝑈) ⊆ 𝑈))
17	8	3ad2ant1 1131	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → (𝑇 ∩ 𝑈)𝐶𝑈)
18	1, 2, 3, 7, 5	lcvbr3 36964	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑇 ∩ 𝑈)𝐶𝑈 ↔ ((𝑇 ∩ 𝑈) ⊊ 𝑈 ∧ ∀𝑟 ∈ 𝑆 (((𝑇 ∩ 𝑈) ⊆ 𝑟 ∧ 𝑟 ⊆ 𝑈) → (𝑟 = (𝑇 ∩ 𝑈) ∨ 𝑟 = 𝑈)))))
19	18	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((𝑇 ∩ 𝑈)𝐶𝑈 ↔ ((𝑇 ∩ 𝑈) ⊊ 𝑈 ∧ ∀𝑟 ∈ 𝑆 (((𝑇 ∩ 𝑈) ⊆ 𝑟 ∧ 𝑟 ⊆ 𝑈) → (𝑟 = (𝑇 ∩ 𝑈) ∨ 𝑟 = 𝑈)))))
20	3	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑊 ∈ LMod)
21		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑠 ∈ 𝑆)
22	5	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑈 ∈ 𝑆)
23	1	lssincl 20142	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ 𝑠 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑠 ∩ 𝑈) ∈ 𝑆)
24	20, 21, 22, 23	syl3anc 1369	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∩ 𝑈) ∈ 𝑆)
25		sseq2 3943	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑠 ∩ 𝑈) → ((𝑇 ∩ 𝑈) ⊆ 𝑟 ↔ (𝑇 ∩ 𝑈) ⊆ (𝑠 ∩ 𝑈)))
26		sseq1 3942	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑠 ∩ 𝑈) → (𝑟 ⊆ 𝑈 ↔ (𝑠 ∩ 𝑈) ⊆ 𝑈))
27	25, 26	anbi12d 630	. . . . . . . . . . . . 13 ⊢ (𝑟 = (𝑠 ∩ 𝑈) → (((𝑇 ∩ 𝑈) ⊆ 𝑟 ∧ 𝑟 ⊆ 𝑈) ↔ ((𝑇 ∩ 𝑈) ⊆ (𝑠 ∩ 𝑈) ∧ (𝑠 ∩ 𝑈) ⊆ 𝑈)))
28		eqeq1 2742	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑠 ∩ 𝑈) → (𝑟 = (𝑇 ∩ 𝑈) ↔ (𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈)))
29		eqeq1 2742	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑠 ∩ 𝑈) → (𝑟 = 𝑈 ↔ (𝑠 ∩ 𝑈) = 𝑈))
30	28, 29	orbi12d 915	. . . . . . . . . . . . 13 ⊢ (𝑟 = (𝑠 ∩ 𝑈) → ((𝑟 = (𝑇 ∩ 𝑈) ∨ 𝑟 = 𝑈) ↔ ((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) ∨ (𝑠 ∩ 𝑈) = 𝑈)))
31	27, 30	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝑠 ∩ 𝑈) → ((((𝑇 ∩ 𝑈) ⊆ 𝑟 ∧ 𝑟 ⊆ 𝑈) → (𝑟 = (𝑇 ∩ 𝑈) ∨ 𝑟 = 𝑈)) ↔ (((𝑇 ∩ 𝑈) ⊆ (𝑠 ∩ 𝑈) ∧ (𝑠 ∩ 𝑈) ⊆ 𝑈) → ((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) ∨ (𝑠 ∩ 𝑈) = 𝑈))))
32	31	rspcv 3547	. . . . . . . . . . 11 ⊢ ((𝑠 ∩ 𝑈) ∈ 𝑆 → (∀𝑟 ∈ 𝑆 (((𝑇 ∩ 𝑈) ⊆ 𝑟 ∧ 𝑟 ⊆ 𝑈) → (𝑟 = (𝑇 ∩ 𝑈) ∨ 𝑟 = 𝑈)) → (((𝑇 ∩ 𝑈) ⊆ (𝑠 ∩ 𝑈) ∧ (𝑠 ∩ 𝑈) ⊆ 𝑈) → ((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) ∨ (𝑠 ∩ 𝑈) = 𝑈))))
33	24, 32	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (∀𝑟 ∈ 𝑆 (((𝑇 ∩ 𝑈) ⊆ 𝑟 ∧ 𝑟 ⊆ 𝑈) → (𝑟 = (𝑇 ∩ 𝑈) ∨ 𝑟 = 𝑈)) → (((𝑇 ∩ 𝑈) ⊆ (𝑠 ∩ 𝑈) ∧ (𝑠 ∩ 𝑈) ⊆ 𝑈) → ((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) ∨ (𝑠 ∩ 𝑈) = 𝑈))))
34	33	adantld 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (((𝑇 ∩ 𝑈) ⊊ 𝑈 ∧ ∀𝑟 ∈ 𝑆 (((𝑇 ∩ 𝑈) ⊆ 𝑟 ∧ 𝑟 ⊆ 𝑈) → (𝑟 = (𝑇 ∩ 𝑈) ∨ 𝑟 = 𝑈))) → (((𝑇 ∩ 𝑈) ⊆ (𝑠 ∩ 𝑈) ∧ (𝑠 ∩ 𝑈) ⊆ 𝑈) → ((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) ∨ (𝑠 ∩ 𝑈) = 𝑈))))
35	19, 34	sylbid 239	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((𝑇 ∩ 𝑈)𝐶𝑈 → (((𝑇 ∩ 𝑈) ⊆ (𝑠 ∩ 𝑈) ∧ (𝑠 ∩ 𝑈) ⊆ 𝑈) → ((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) ∨ (𝑠 ∩ 𝑈) = 𝑈))))
36	35	3adant3 1130	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → ((𝑇 ∩ 𝑈)𝐶𝑈 → (((𝑇 ∩ 𝑈) ⊆ (𝑠 ∩ 𝑈) ∧ (𝑠 ∩ 𝑈) ⊆ 𝑈) → ((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) ∨ (𝑠 ∩ 𝑈) = 𝑈))))
37	17, 36	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → (((𝑇 ∩ 𝑈) ⊆ (𝑠 ∩ 𝑈) ∧ (𝑠 ∩ 𝑈) ⊆ 𝑈) → ((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) ∨ (𝑠 ∩ 𝑈) = 𝑈)))
38	16, 37	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → ((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) ∨ (𝑠 ∩ 𝑈) = 𝑈))
39		oveq1 7262	. . . . . . 7 ⊢ ((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) → ((𝑠 ∩ 𝑈) ⊕ 𝑇) = ((𝑇 ∩ 𝑈) ⊕ 𝑇))
40	3	3ad2ant1 1131	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → 𝑊 ∈ LMod)
41	4	3ad2ant1 1131	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → 𝑇 ∈ 𝑆)
42	5	3ad2ant1 1131	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → 𝑈 ∈ 𝑆)
43		simp2 1135	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → 𝑠 ∈ 𝑆)
44		simp3r 1200	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → 𝑠 ⊆ (𝑇 ⊕ 𝑈))
45	1, 10, 2, 40, 41, 42, 43, 13, 44	lcvexchlem3 36977	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → ((𝑠 ∩ 𝑈) ⊕ 𝑇) = 𝑠)
46	1	lsssssubg 20135	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊))
47	3, 46	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊))
48	47, 7	sseldd 3918	. . . . . . . . . 10 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ∈ (SubGrp‘𝑊))
49	47, 4	sseldd 3918	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝑊))
50		inss1 4159	. . . . . . . . . . 11 ⊢ (𝑇 ∩ 𝑈) ⊆ 𝑇
51	50	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ⊆ 𝑇)
52	10	lsmss1 19186	. . . . . . . . . 10 ⊢ (((𝑇 ∩ 𝑈) ∈ (SubGrp‘𝑊) ∧ 𝑇 ∈ (SubGrp‘𝑊) ∧ (𝑇 ∩ 𝑈) ⊆ 𝑇) → ((𝑇 ∩ 𝑈) ⊕ 𝑇) = 𝑇)
53	48, 49, 51, 52	syl3anc 1369	. . . . . . . . 9 ⊢ (𝜑 → ((𝑇 ∩ 𝑈) ⊕ 𝑇) = 𝑇)
54	53	3ad2ant1 1131	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → ((𝑇 ∩ 𝑈) ⊕ 𝑇) = 𝑇)
55	45, 54	eqeq12d 2754	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → (((𝑠 ∩ 𝑈) ⊕ 𝑇) = ((𝑇 ∩ 𝑈) ⊕ 𝑇) ↔ 𝑠 = 𝑇))
56	39, 55	syl5ib 243	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → ((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) → 𝑠 = 𝑇))
57		oveq1 7262	. . . . . . 7 ⊢ ((𝑠 ∩ 𝑈) = 𝑈 → ((𝑠 ∩ 𝑈) ⊕ 𝑇) = (𝑈 ⊕ 𝑇))
58		lmodabl 20085	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel)
59	3, 58	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ Abel)
60	47, 5	sseldd 3918	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊))
61	10	lsmcom 19374	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Abel ∧ 𝑈 ∈ (SubGrp‘𝑊) ∧ 𝑇 ∈ (SubGrp‘𝑊)) → (𝑈 ⊕ 𝑇) = (𝑇 ⊕ 𝑈))
62	59, 60, 49, 61	syl3anc 1369	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 ⊕ 𝑇) = (𝑇 ⊕ 𝑈))
63	62	3ad2ant1 1131	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → (𝑈 ⊕ 𝑇) = (𝑇 ⊕ 𝑈))
64	45, 63	eqeq12d 2754	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → (((𝑠 ∩ 𝑈) ⊕ 𝑇) = (𝑈 ⊕ 𝑇) ↔ 𝑠 = (𝑇 ⊕ 𝑈)))
65	57, 64	syl5ib 243	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → ((𝑠 ∩ 𝑈) = 𝑈 → 𝑠 = (𝑇 ⊕ 𝑈)))
66	56, 65	orim12d 961	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → (((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) ∨ (𝑠 ∩ 𝑈) = 𝑈) → (𝑠 = 𝑇 ∨ 𝑠 = (𝑇 ⊕ 𝑈))))
67	38, 66	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → (𝑠 = 𝑇 ∨ 𝑠 = (𝑇 ⊕ 𝑈)))
68	67	3exp 1117	. . 3 ⊢ (𝜑 → (𝑠 ∈ 𝑆 → ((𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈)) → (𝑠 = 𝑇 ∨ 𝑠 = (𝑇 ⊕ 𝑈)))))
69	68	ralrimiv 3106	. 2 ⊢ (𝜑 → ∀𝑠 ∈ 𝑆 ((𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈)) → (𝑠 = 𝑇 ∨ 𝑠 = (𝑇 ⊕ 𝑈))))
70	1, 10	lsmcl 20260	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ⊕ 𝑈) ∈ 𝑆)
71	3, 4, 5, 70	syl3anc 1369	. . 3 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ∈ 𝑆)
72	1, 2, 3, 4, 71	lcvbr3 36964	. 2 ⊢ (𝜑 → (𝑇𝐶(𝑇 ⊕ 𝑈) ↔ (𝑇 ⊊ (𝑇 ⊕ 𝑈) ∧ ∀𝑠 ∈ 𝑆 ((𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈)) → (𝑠 = 𝑇 ∨ 𝑠 = (𝑇 ⊕ 𝑈))))))
73	12, 69, 72	mpbir2and 709	1 ⊢ (𝜑 → 𝑇𝐶(𝑇 ⊕ 𝑈))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063   ∩ cin 3882   ⊆ wss 3883   ⊊ wpss 3884   class class class wbr 5070  ‘cfv 6418  (class class class)co 7255  SubGrpcsubg 18664  LSSumclsm 19154  Abelcabl 19302  LModclmod 20038  LSubSpclss 20108   ⋖_L clcv 36959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-tpos 8013  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-0g 17069  df-mre 17212  df-mrc 17213  df-acs 17215  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-submnd 18346  df-grp 18495  df-minusg 18496  df-sbg 18497  df-subg 18667  df-cntz 18838  df-oppg 18865  df-lsm 19156  df-cmn 19303  df-abl 19304  df-mgp 19636  df-ur 19653  df-ring 19700  df-lmod 20040  df-lss 20109  df-lcv 36960

This theorem is referenced by:  lcvexch  36980