Theorem lcvexchlem5 34325

Description: Lemma for lcvexch 34326. (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 18965	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ∩ 𝑈) ∈ 𝑆)
7	3, 4, 5, 6	syl3anc 1326	. . . 4 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ∈ 𝑆)
8		lcvexch.g	. . . 4 ⊢ (𝜑 → (𝑇 ∩ 𝑈)𝐶𝑈)
9	1, 2, 3, 7, 5, 8	lcvpss 34311	. . 3 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ⊊ 𝑈)
10		lcvexch.p	. . . 4 ⊢ ⊕ = (LSSum‘𝑊)
11	1, 10, 2, 3, 4, 5	lcvexchlem1 34321	. . 3 ⊢ (𝜑 → (𝑇 ⊊ (𝑇 ⊕ 𝑈) ↔ (𝑇 ∩ 𝑈) ⊊ 𝑈))
12	9, 11	mpbird 247	. 2 ⊢ (𝜑 → 𝑇 ⊊ (𝑇 ⊕ 𝑈))
13		simp3l 1089	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → 𝑇 ⊆ 𝑠)
14		ssrin 3838	. . . . . . . 8 ⊢ (𝑇 ⊆ 𝑠 → (𝑇 ∩ 𝑈) ⊆ (𝑠 ∩ 𝑈))
15	13, 14	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → (𝑇 ∩ 𝑈) ⊆ (𝑠 ∩ 𝑈))
16		inss2 3834	. . . . . . 7 ⊢ (𝑠 ∩ 𝑈) ⊆ 𝑈
17	15, 16	jctir 561	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → ((𝑇 ∩ 𝑈) ⊆ (𝑠 ∩ 𝑈) ∧ (𝑠 ∩ 𝑈) ⊆ 𝑈))
18	8	3ad2ant1 1082	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → (𝑇 ∩ 𝑈)𝐶𝑈)
19	1, 2, 3, 7, 5	lcvbr3 34310	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑇 ∩ 𝑈)𝐶𝑈 ↔ ((𝑇 ∩ 𝑈) ⊊ 𝑈 ∧ ∀𝑟 ∈ 𝑆 (((𝑇 ∩ 𝑈) ⊆ 𝑟 ∧ 𝑟 ⊆ 𝑈) → (𝑟 = (𝑇 ∩ 𝑈) ∨ 𝑟 = 𝑈)))))
20	19	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((𝑇 ∩ 𝑈)𝐶𝑈 ↔ ((𝑇 ∩ 𝑈) ⊊ 𝑈 ∧ ∀𝑟 ∈ 𝑆 (((𝑇 ∩ 𝑈) ⊆ 𝑟 ∧ 𝑟 ⊆ 𝑈) → (𝑟 = (𝑇 ∩ 𝑈) ∨ 𝑟 = 𝑈)))))
21	3	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑊 ∈ LMod)
22		simpr 477	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑠 ∈ 𝑆)
23	5	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑈 ∈ 𝑆)
24	1	lssincl 18965	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ 𝑠 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑠 ∩ 𝑈) ∈ 𝑆)
25	21, 22, 23, 24	syl3anc 1326	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∩ 𝑈) ∈ 𝑆)
26		sseq2 3627	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑠 ∩ 𝑈) → ((𝑇 ∩ 𝑈) ⊆ 𝑟 ↔ (𝑇 ∩ 𝑈) ⊆ (𝑠 ∩ 𝑈)))
27		sseq1 3626	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑠 ∩ 𝑈) → (𝑟 ⊆ 𝑈 ↔ (𝑠 ∩ 𝑈) ⊆ 𝑈))
28	26, 27	anbi12d 747	. . . . . . . . . . . . 13 ⊢ (𝑟 = (𝑠 ∩ 𝑈) → (((𝑇 ∩ 𝑈) ⊆ 𝑟 ∧ 𝑟 ⊆ 𝑈) ↔ ((𝑇 ∩ 𝑈) ⊆ (𝑠 ∩ 𝑈) ∧ (𝑠 ∩ 𝑈) ⊆ 𝑈)))
29		eqeq1 2626	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑠 ∩ 𝑈) → (𝑟 = (𝑇 ∩ 𝑈) ↔ (𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈)))
30		eqeq1 2626	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑠 ∩ 𝑈) → (𝑟 = 𝑈 ↔ (𝑠 ∩ 𝑈) = 𝑈))
31	29, 30	orbi12d 746	. . . . . . . . . . . . 13 ⊢ (𝑟 = (𝑠 ∩ 𝑈) → ((𝑟 = (𝑇 ∩ 𝑈) ∨ 𝑟 = 𝑈) ↔ ((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) ∨ (𝑠 ∩ 𝑈) = 𝑈)))
32	28, 31	imbi12d 334	. . . . . . . . . . . 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))))
33	32	rspcv 3305	. . . . . . . . . . 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))))
34	25, 33	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))))
35	34	adantld 483	. . . . . . . . 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))))
36	20, 35	sylbid 230	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((𝑇 ∩ 𝑈)𝐶𝑈 → (((𝑇 ∩ 𝑈) ⊆ (𝑠 ∩ 𝑈) ∧ (𝑠 ∩ 𝑈) ⊆ 𝑈) → ((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) ∨ (𝑠 ∩ 𝑈) = 𝑈))))
37	36	3adant3 1081	. . . . . . 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))))
38	18, 37	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)))
39	17, 38	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → ((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) ∨ (𝑠 ∩ 𝑈) = 𝑈))
40		oveq1 6657	. . . . . . 7 ⊢ ((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) → ((𝑠 ∩ 𝑈) ⊕ 𝑇) = ((𝑇 ∩ 𝑈) ⊕ 𝑇))
41	3	3ad2ant1 1082	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → 𝑊 ∈ LMod)
42	4	3ad2ant1 1082	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → 𝑇 ∈ 𝑆)
43	5	3ad2ant1 1082	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → 𝑈 ∈ 𝑆)
44		simp2 1062	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → 𝑠 ∈ 𝑆)
45		simp3r 1090	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → 𝑠 ⊆ (𝑇 ⊕ 𝑈))
46	1, 10, 2, 41, 42, 43, 44, 13, 45	lcvexchlem3 34323	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → ((𝑠 ∩ 𝑈) ⊕ 𝑇) = 𝑠)
47	1	lsssssubg 18958	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊))
48	3, 47	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊))
49	48, 7	sseldd 3604	. . . . . . . . . 10 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ∈ (SubGrp‘𝑊))
50	48, 4	sseldd 3604	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝑊))
51		inss1 3833	. . . . . . . . . . 11 ⊢ (𝑇 ∩ 𝑈) ⊆ 𝑇
52	51	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝑇 ∩ 𝑈) ⊆ 𝑇)
53	10	lsmss1 18079	. . . . . . . . . 10 ⊢ (((𝑇 ∩ 𝑈) ∈ (SubGrp‘𝑊) ∧ 𝑇 ∈ (SubGrp‘𝑊) ∧ (𝑇 ∩ 𝑈) ⊆ 𝑇) → ((𝑇 ∩ 𝑈) ⊕ 𝑇) = 𝑇)
54	49, 50, 52, 53	syl3anc 1326	. . . . . . . . 9 ⊢ (𝜑 → ((𝑇 ∩ 𝑈) ⊕ 𝑇) = 𝑇)
55	54	3ad2ant1 1082	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → ((𝑇 ∩ 𝑈) ⊕ 𝑇) = 𝑇)
56	46, 55	eqeq12d 2637	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → (((𝑠 ∩ 𝑈) ⊕ 𝑇) = ((𝑇 ∩ 𝑈) ⊕ 𝑇) ↔ 𝑠 = 𝑇))
57	40, 56	syl5ib 234	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → ((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) → 𝑠 = 𝑇))
58		oveq1 6657	. . . . . . 7 ⊢ ((𝑠 ∩ 𝑈) = 𝑈 → ((𝑠 ∩ 𝑈) ⊕ 𝑇) = (𝑈 ⊕ 𝑇))
59		lmodabl 18910	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel)
60	3, 59	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ Abel)
61	48, 5	sseldd 3604	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊))
62	10	lsmcom 18261	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Abel ∧ 𝑈 ∈ (SubGrp‘𝑊) ∧ 𝑇 ∈ (SubGrp‘𝑊)) → (𝑈 ⊕ 𝑇) = (𝑇 ⊕ 𝑈))
63	60, 61, 50, 62	syl3anc 1326	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 ⊕ 𝑇) = (𝑇 ⊕ 𝑈))
64	63	3ad2ant1 1082	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → (𝑈 ⊕ 𝑇) = (𝑇 ⊕ 𝑈))
65	46, 64	eqeq12d 2637	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → (((𝑠 ∩ 𝑈) ⊕ 𝑇) = (𝑈 ⊕ 𝑇) ↔ 𝑠 = (𝑇 ⊕ 𝑈)))
66	58, 65	syl5ib 234	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → ((𝑠 ∩ 𝑈) = 𝑈 → 𝑠 = (𝑇 ⊕ 𝑈)))
67	57, 66	orim12d 883	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → (((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) ∨ (𝑠 ∩ 𝑈) = 𝑈) → (𝑠 = 𝑇 ∨ 𝑠 = (𝑇 ⊕ 𝑈))))
68	39, 67	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → (𝑠 = 𝑇 ∨ 𝑠 = (𝑇 ⊕ 𝑈)))
69	68	3exp 1264	. . 3 ⊢ (𝜑 → (𝑠 ∈ 𝑆 → ((𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈)) → (𝑠 = 𝑇 ∨ 𝑠 = (𝑇 ⊕ 𝑈)))))
70	69	ralrimiv 2965	. 2 ⊢ (𝜑 → ∀𝑠 ∈ 𝑆 ((𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈)) → (𝑠 = 𝑇 ∨ 𝑠 = (𝑇 ⊕ 𝑈))))
71	1, 10	lsmcl 19083	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ⊕ 𝑈) ∈ 𝑆)
72	3, 4, 5, 71	syl3anc 1326	. . 3 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ∈ 𝑆)
73	1, 2, 3, 4, 72	lcvbr3 34310	. 2 ⊢ (𝜑 → (𝑇𝐶(𝑇 ⊕ 𝑈) ↔ (𝑇 ⊊ (𝑇 ⊕ 𝑈) ∧ ∀𝑠 ∈ 𝑆 ((𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈)) → (𝑠 = 𝑇 ∨ 𝑠 = (𝑇 ⊕ 𝑈))))))
74	12, 70, 73	mpbir2and 957	1 ⊢ (𝜑 → 𝑇𝐶(𝑇 ⊕ 𝑈))

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ∩ cin 3573   ⊆ wss 3574   ⊊ wpss 3575   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  SubGrpcsubg 17588  LSSumclsm 18049  Abelcabl 18194  LModclmod 18863  LSubSpclss 18932   ⋖_L clcv 34305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-cntz 17750  df-oppg 17776  df-lsm 18051  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-lmod 18865  df-lss 18933  df-lcv 34306

This theorem is referenced by:  lcvexch  34326