Theorem lcvexchlem5 40138

Description: Lemma for lcvexch 40139. (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 21240	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ∩ 𝑈) ∈ 𝑆)
7	3, 4, 5, 6	syl3anc 1398	. . . 4 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ∈ 𝑆)
8		lcvexch.g	. . . 4 ⊢ (𝜑 → (𝑇 ∩ 𝑈)𝐶𝑈)
9	1, 2, 3, 7, 5, 8	lcvpss 40124	. . 3 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ⊊ 𝑈)
10		lcvexch.p	. . . 4 ⊢ ⊕ = (LSSum‘𝑊)
11	1, 10, 2, 3, 4, 5	lcvexchlem1 40134	. . 3 ⊢ (𝜑 → (𝑇 ⊊ (𝑇 ⊕ 𝑈) ↔ (𝑇 ∩ 𝑈) ⊊ 𝑈))
12	9, 11	mpbird 260	. 2 ⊢ (𝜑 → 𝑇 ⊊ (𝑇 ⊕ 𝑈))
13		simp3l 1220	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → 𝑇 ⊆ 𝑠)
14	13	ssrind 4140	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → (𝑇 ∩ 𝑈) ⊆ (𝑠 ∩ 𝑈))
15		inss2 4134	. . . . . . 7 ⊢ (𝑠 ∩ 𝑈) ⊆ 𝑈
16	14, 15	jctir 530	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → ((𝑇 ∩ 𝑈) ⊆ (𝑠 ∩ 𝑈) ∧ (𝑠 ∩ 𝑈) ⊆ 𝑈))
17	8	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → (𝑇 ∩ 𝑈)𝐶𝑈)
18	1, 2, 3, 7, 5	lcvbr3 40123	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑇 ∩ 𝑈)𝐶𝑈 ↔ ((𝑇 ∩ 𝑈) ⊊ 𝑈 ∧ ∀𝑟 ∈ 𝑆 (((𝑇 ∩ 𝑈) ⊆ 𝑟 ∧ 𝑟 ⊆ 𝑈) → (𝑟 = (𝑇 ∩ 𝑈) ∨ 𝑟 = 𝑈)))))
19	18	adantr 486	. . . . . . . . 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 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑊 ∈ LMod)
21		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑠 ∈ 𝑆)
22	5	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑈 ∈ 𝑆)
23	1	lssincl 21240	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ 𝑠 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑠 ∩ 𝑈) ∈ 𝑆)
24	20, 21, 22, 23	syl3anc 1398	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∩ 𝑈) ∈ 𝑆)
25		sseq2 3934	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑠 ∩ 𝑈) → ((𝑇 ∩ 𝑈) ⊆ 𝑟 ↔ (𝑇 ∩ 𝑈) ⊆ (𝑠 ∩ 𝑈)))
26		sseq1 3933	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑠 ∩ 𝑈) → (𝑟 ⊆ 𝑈 ↔ (𝑠 ∩ 𝑈) ⊆ 𝑈))
27	25, 26	anbi12d 644	. . . . . . . . . . . . 13 ⊢ (𝑟 = (𝑠 ∩ 𝑈) → (((𝑇 ∩ 𝑈) ⊆ 𝑟 ∧ 𝑟 ⊆ 𝑈) ↔ ((𝑇 ∩ 𝑈) ⊆ (𝑠 ∩ 𝑈) ∧ (𝑠 ∩ 𝑈) ⊆ 𝑈)))
28		eqeq1 2764	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑠 ∩ 𝑈) → (𝑟 = (𝑇 ∩ 𝑈) ↔ (𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈)))
29		eqeq1 2764	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑠 ∩ 𝑈) → (𝑟 = 𝑈 ↔ (𝑠 ∩ 𝑈) = 𝑈))
30	28, 29	orbi12d 932	. . . . . . . . . . . . 13 ⊢ (𝑟 = (𝑠 ∩ 𝑈) → ((𝑟 = (𝑇 ∩ 𝑈) ∨ 𝑟 = 𝑈) ↔ ((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) ∨ (𝑠 ∩ 𝑈) = 𝑈)))
31	27, 30	imbi12d 347	. . . . . . . . . . . 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 3572	. . . . . . . . . . 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 18	. . . . . . . . . 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 496	. . . . . . . . 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 243	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((𝑇 ∩ 𝑈)𝐶𝑈 → (((𝑇 ∩ 𝑈) ⊆ (𝑠 ∩ 𝑈) ∧ (𝑠 ∩ 𝑈) ⊆ 𝑈) → ((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) ∨ (𝑠 ∩ 𝑈) = 𝑈))))
36	35	3adant3 1150	. . . . . . 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 16	. . . . . 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 16	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → ((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) ∨ (𝑠 ∩ 𝑈) = 𝑈))
39		oveq1 7427	. . . . . . 7 ⊢ ((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) → ((𝑠 ∩ 𝑈) ⊕ 𝑇) = ((𝑇 ∩ 𝑈) ⊕ 𝑇))
40	3	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → 𝑊 ∈ LMod)
41	4	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → 𝑇 ∈ 𝑆)
42	5	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → 𝑈 ∈ 𝑆)
43		simp2 1155	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → 𝑠 ∈ 𝑆)
44		simp3r 1221	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → 𝑠 ⊆ (𝑇 ⊕ 𝑈))
45	1, 10, 2, 40, 41, 42, 43, 13, 44	lcvexchlem3 40136	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → ((𝑠 ∩ 𝑈) ⊕ 𝑇) = 𝑠)
46	1	lsssssubg 21233	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊))
47	3, 46	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊))
48	47, 7	sseldd 3909	. . . . . . . . . 10 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ∈ (SubGrp‘𝑊))
49	47, 4	sseldd 3909	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝑊))
50		inss1 4133	. . . . . . . . . . 11 ⊢ (𝑇 ∩ 𝑈) ⊆ 𝑇
51	50	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ⊆ 𝑇)
52	10	lsmss1 19879	. . . . . . . . . 10 ⊢ (((𝑇 ∩ 𝑈) ∈ (SubGrp‘𝑊) ∧ 𝑇 ∈ (SubGrp‘𝑊) ∧ (𝑇 ∩ 𝑈) ⊆ 𝑇) → ((𝑇 ∩ 𝑈) ⊕ 𝑇) = 𝑇)
53	48, 49, 51, 52	syl3anc 1398	. . . . . . . . 9 ⊢ (𝜑 → ((𝑇 ∩ 𝑈) ⊕ 𝑇) = 𝑇)
54	53	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → ((𝑇 ∩ 𝑈) ⊕ 𝑇) = 𝑇)
55	45, 54	eqeq12d 2776	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → (((𝑠 ∩ 𝑈) ⊕ 𝑇) = ((𝑇 ∩ 𝑈) ⊕ 𝑇) ↔ 𝑠 = 𝑇))
56	39, 55	imbitrid 247	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → ((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) → 𝑠 = 𝑇))
57		oveq1 7427	. . . . . . 7 ⊢ ((𝑠 ∩ 𝑈) = 𝑈 → ((𝑠 ∩ 𝑈) ⊕ 𝑇) = (𝑈 ⊕ 𝑇))
58		lmodabl 21184	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel)
59	3, 58	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ Abel)
60	47, 5	sseldd 3909	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊))
61	10	lsmcom 20072	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Abel ∧ 𝑈 ∈ (SubGrp‘𝑊) ∧ 𝑇 ∈ (SubGrp‘𝑊)) → (𝑈 ⊕ 𝑇) = (𝑇 ⊕ 𝑈))
62	59, 60, 49, 61	syl3anc 1398	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 ⊕ 𝑇) = (𝑇 ⊕ 𝑈))
63	62	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → (𝑈 ⊕ 𝑇) = (𝑇 ⊕ 𝑈))
64	45, 63	eqeq12d 2776	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → (((𝑠 ∩ 𝑈) ⊕ 𝑇) = (𝑈 ⊕ 𝑇) ↔ 𝑠 = (𝑇 ⊕ 𝑈)))
65	57, 64	imbitrid 247	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → ((𝑠 ∩ 𝑈) = 𝑈 → 𝑠 = (𝑇 ⊕ 𝑈)))
66	56, 65	orim12d 979	. . . . 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 16	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → (𝑠 = 𝑇 ∨ 𝑠 = (𝑇 ⊕ 𝑈)))
68	67	3exp 1137	. . 3 ⊢ (𝜑 → (𝑠 ∈ 𝑆 → ((𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈)) → (𝑠 = 𝑇 ∨ 𝑠 = (𝑇 ⊕ 𝑈)))))
69	68	ralrimiv 3153	. 2 ⊢ (𝜑 → ∀𝑠 ∈ 𝑆 ((𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈)) → (𝑠 = 𝑇 ∨ 𝑠 = (𝑇 ⊕ 𝑈))))
70	1, 10	lsmcl 21358	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ⊕ 𝑈) ∈ 𝑆)
71	3, 4, 5, 70	syl3anc 1398	. . 3 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ∈ 𝑆)
72	1, 2, 3, 4, 71	lcvbr3 40123	. 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 726	1 ⊢ (𝜑 → 𝑇𝐶(𝑇 ⊕ 𝑈))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895   ⊊ wpss 4013   ∩ cin 4089   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420  SubGrpcsubg 19330  LSSumclsm 19848  Abelcabl 19995  LModclmod 21135  LSubSpclss 21206   ⋖_L clcv 40118

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-tpos 8243  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-0g 17612  df-mre 17756  df-mrc 17757  df-acs 17759  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-submnd 18979  df-grp 19147  df-minusg 19148  df-sbg 19149  df-subg 19333  df-cntz 19531  df-oppg 19560  df-lsm 19850  df-cmn 19996  df-abl 19997  df-mgp 20361  df-ur 20408  df-ring 20461  df-lmod 21137  df-lss 21207  df-lcv 40119

This theorem is used by:  lcvexch  40139