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Theorem lsmcv 19915

Description: Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 29431 analog.) TODO: ugly proof; can it be shortened? (Contributed by NM, 2-Oct-2014.)

**Hypotheses**
Ref	Expression
lsmcv.v	⊢ 𝑉 = (Base‘𝑊)
lsmcv.s	⊢ 𝑆 = (LSubSp‘𝑊)
lsmcv.n	⊢ 𝑁 = (LSpan‘𝑊)
lsmcv.p	⊢ ⊕ = (LSSum‘𝑊)
lsmcv.w	⊢ (𝜑 → 𝑊 ∈ LVec)
lsmcv.t	⊢ (𝜑 → 𝑇 ∈ 𝑆)
lsmcv.u	⊢ (𝜑 → 𝑈 ∈ 𝑆)
lsmcv.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)

**Assertion**
Ref	Expression
lsmcv	⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → 𝑈 = (𝑇 ⊕ (𝑁‘{𝑋})))

Proof of Theorem lsmcv Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3 1134	. 2 ⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋})))
2		simp2 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → 𝑇 ⊊ 𝑈)
3		pssss 4074	. . . 4 ⊢ (𝑇 ⊊ 𝑈 → 𝑇 ⊆ 𝑈)
4	2, 3	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → 𝑇 ⊆ 𝑈)
5		pssnel 4422	. . . . 5 ⊢ (𝑇 ⊊ 𝑈 → ∃𝑥(𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇))
6	2, 5	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → ∃𝑥(𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇))
7		simpl3 1189	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) → 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋})))
8		simprl 769	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) → 𝑥 ∈ 𝑈)
9	7, 8	sseldd 3970	. . . . . 6 ⊢ (((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) → 𝑥 ∈ (𝑇 ⊕ (𝑁‘{𝑋})))
10		lsmcv.w	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 ∈ LVec)
11		lveclmod 19880	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
12	10, 11	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 ∈ LMod)
13		lsmcv.s	. . . . . . . . . . . 12 ⊢ 𝑆 = (LSubSp‘𝑊)
14	13	lsssssubg 19732	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊))
15	12, 14	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊))
16		lsmcv.t	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 ∈ 𝑆)
17	15, 16	sseldd 3970	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝑊))
18		lsmcv.x	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
19		lsmcv.v	. . . . . . . . . . . 12 ⊢ 𝑉 = (Base‘𝑊)
20		lsmcv.n	. . . . . . . . . . . 12 ⊢ 𝑁 = (LSpan‘𝑊)
21	19, 13, 20	lspsncl 19751	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ 𝑆)
22	12, 18, 21	syl2anc 586	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ 𝑆)
23	15, 22	sseldd 3970	. . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊))
24		eqid 2823	. . . . . . . . . 10 ⊢ (+_g‘𝑊) = (+_g‘𝑊)
25		lsmcv.p	. . . . . . . . . 10 ⊢ ⊕ = (LSSum‘𝑊)
26	24, 25	lsmelval 18776	. . . . . . . . 9 ⊢ ((𝑇 ∈ (SubGrp‘𝑊) ∧ (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊)) → (𝑥 ∈ (𝑇 ⊕ (𝑁‘{𝑋})) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ (𝑁‘{𝑋})𝑥 = (𝑦(+_g‘𝑊)𝑧)))
27	17, 23, 26	syl2anc 586	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑇 ⊕ (𝑁‘{𝑋})) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ (𝑁‘{𝑋})𝑥 = (𝑦(+_g‘𝑊)𝑧)))
28	27	3ad2ant1 1129	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → (𝑥 ∈ (𝑇 ⊕ (𝑁‘{𝑋})) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ (𝑁‘{𝑋})𝑥 = (𝑦(+_g‘𝑊)𝑧)))
29	28	adantr 483	. . . . . 6 ⊢ (((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) → (𝑥 ∈ (𝑇 ⊕ (𝑁‘{𝑋})) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ (𝑁‘{𝑋})𝑥 = (𝑦(+_g‘𝑊)𝑧)))
30	9, 29	mpbid 234	. . . . 5 ⊢ (((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) → ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ (𝑁‘{𝑋})𝑥 = (𝑦(+_g‘𝑊)𝑧))
31		simp1rr 1235	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) → ¬ 𝑥 ∈ 𝑇)
32		simp2l 1195	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) → 𝑦 ∈ 𝑇)
33		oveq2 7166	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (0_g‘𝑊) → (𝑦(+_g‘𝑊)𝑧) = (𝑦(+_g‘𝑊)(0_g‘𝑊)))
34	33	eqeq2d 2834	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (0_g‘𝑊) → (𝑥 = (𝑦(+_g‘𝑊)𝑧) ↔ 𝑥 = (𝑦(+_g‘𝑊)(0_g‘𝑊))))
35	34	biimpac 481	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = (𝑦(+_g‘𝑊)𝑧) ∧ 𝑧 = (0_g‘𝑊)) → 𝑥 = (𝑦(+_g‘𝑊)(0_g‘𝑊)))
36	12	3ad2ant1 1129	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → 𝑊 ∈ LMod)
37	36	ad2antrr 724	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋}))) → 𝑊 ∈ LMod)
38	16	3ad2ant1 1129	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → 𝑇 ∈ 𝑆)
39	38	ad2antrr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋}))) → 𝑇 ∈ 𝑆)
40		simprl 769	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋}))) → 𝑦 ∈ 𝑇)
41	19, 13	lssel 19711	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑇 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇) → 𝑦 ∈ 𝑉)
42	39, 40, 41	syl2anc 586	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋}))) → 𝑦 ∈ 𝑉)
43		eqid 2823	. . . . . . . . . . . . . . . . . . 19 ⊢ (0_g‘𝑊) = (0_g‘𝑊)
44	19, 24, 43	lmod0vrid 19667	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ LMod ∧ 𝑦 ∈ 𝑉) → (𝑦(+_g‘𝑊)(0_g‘𝑊)) = 𝑦)
45	37, 42, 44	syl2anc 586	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋}))) → (𝑦(+_g‘𝑊)(0_g‘𝑊)) = 𝑦)
46	45	eqeq2d 2834	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋}))) → (𝑥 = (𝑦(+_g‘𝑊)(0_g‘𝑊)) ↔ 𝑥 = 𝑦))
47	46	biimpd 231	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋}))) → (𝑥 = (𝑦(+_g‘𝑊)(0_g‘𝑊)) → 𝑥 = 𝑦))
48	47	ex 415	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) → ((𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) → (𝑥 = (𝑦(+_g‘𝑊)(0_g‘𝑊)) → 𝑥 = 𝑦)))
49	35, 48	syl7 74	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) → ((𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) → ((𝑥 = (𝑦(+_g‘𝑊)𝑧) ∧ 𝑧 = (0_g‘𝑊)) → 𝑥 = 𝑦)))
50	49	exp4a 434	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) → ((𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) → (𝑥 = (𝑦(+_g‘𝑊)𝑧) → (𝑧 = (0_g‘𝑊) → 𝑥 = 𝑦))))
51	50	3imp 1107	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) → (𝑧 = (0_g‘𝑊) → 𝑥 = 𝑦))
52		eleq1 2902	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑇 ↔ 𝑦 ∈ 𝑇))
53	52	biimparc 482	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝑇 ∧ 𝑥 = 𝑦) → 𝑥 ∈ 𝑇)
54	32, 51, 53	syl6an 682	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) → (𝑧 = (0_g‘𝑊) → 𝑥 ∈ 𝑇))
55	54	necon3bd 3032	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) → (¬ 𝑥 ∈ 𝑇 → 𝑧 ≠ (0_g‘𝑊)))
56	31, 55	mpd 15	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) → 𝑧 ≠ (0_g‘𝑊))
57	10	3ad2ant1 1129	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → 𝑊 ∈ LVec)
58	57	adantr 483	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) → 𝑊 ∈ LVec)
59	58	3ad2ant1 1129	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) → 𝑊 ∈ LVec)
60		lmodabl 19683	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel)
61	11, 60	syl 17	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ Abel)
62	59, 61	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) → 𝑊 ∈ Abel)
63		simp1l1 1262	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) → 𝜑)
64	63, 16	syl 17	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) → 𝑇 ∈ 𝑆)
65	64, 32, 41	syl2anc 586	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) → 𝑦 ∈ 𝑉)
66	59, 11	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) → 𝑊 ∈ LMod)
67	63, 18	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) → 𝑋 ∈ 𝑉)
68	66, 67, 21	syl2anc 586	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) → (𝑁‘{𝑋}) ∈ 𝑆)
69		simp2r 1196	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) → 𝑧 ∈ (𝑁‘{𝑋}))
70	19, 13	lssel 19711	. . . . . . . . . . 11 ⊢ (((𝑁‘{𝑋}) ∈ 𝑆 ∧ 𝑧 ∈ (𝑁‘{𝑋})) → 𝑧 ∈ 𝑉)
71	68, 69, 70	syl2anc 586	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) → 𝑧 ∈ 𝑉)
72		eqid 2823	. . . . . . . . . . 11 ⊢ (-_g‘𝑊) = (-_g‘𝑊)
73	19, 24, 72	ablpncan2 18938	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Abel ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝑦(+_g‘𝑊)𝑧)(-_g‘𝑊)𝑦) = 𝑧)
74	62, 65, 71, 73	syl3anc 1367	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) → ((𝑦(+_g‘𝑊)𝑧)(-_g‘𝑊)𝑦) = 𝑧)
75		lsmcv.u	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ 𝑆)
76	63, 75	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) → 𝑈 ∈ 𝑆)
77		simp3 1134	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) → 𝑥 = (𝑦(+_g‘𝑊)𝑧))
78		simp1rl 1234	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) → 𝑥 ∈ 𝑈)
79	77, 78	eqeltrrd 2916	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) → (𝑦(+_g‘𝑊)𝑧) ∈ 𝑈)
80		simp1l2 1263	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) → 𝑇 ⊊ 𝑈)
81	3	sselda 3969	. . . . . . . . . . 11 ⊢ ((𝑇 ⊊ 𝑈 ∧ 𝑦 ∈ 𝑇) → 𝑦 ∈ 𝑈)
82	80, 32, 81	syl2anc 586	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) → 𝑦 ∈ 𝑈)
83	72, 13	lssvsubcl 19717	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ ((𝑦(+_g‘𝑊)𝑧) ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → ((𝑦(+_g‘𝑊)𝑧)(-_g‘𝑊)𝑦) ∈ 𝑈)
84	66, 76, 79, 82, 83	syl22anc 836	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) → ((𝑦(+_g‘𝑊)𝑧)(-_g‘𝑊)𝑦) ∈ 𝑈)
85	74, 84	eqeltrrd 2916	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) → 𝑧 ∈ 𝑈)
86	59	3ad2ant1 1129	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) ∧ 𝑧 ≠ (0_g‘𝑊) ∧ 𝑧 ∈ 𝑈) → 𝑊 ∈ LVec)
87	63	3ad2ant1 1129	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) ∧ 𝑧 ≠ (0_g‘𝑊) ∧ 𝑧 ∈ 𝑈) → 𝜑)
88	87, 18	syl 17	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) ∧ 𝑧 ≠ (0_g‘𝑊) ∧ 𝑧 ∈ 𝑈) → 𝑋 ∈ 𝑉)
89		simp12r 1283	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) ∧ 𝑧 ≠ (0_g‘𝑊) ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ (𝑁‘{𝑋}))
90		simp2 1133	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) ∧ 𝑧 ≠ (0_g‘𝑊) ∧ 𝑧 ∈ 𝑈) → 𝑧 ≠ (0_g‘𝑊))
91	19, 43, 20, 86, 88, 89, 90	lspsneleq 19889	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) ∧ 𝑧 ≠ (0_g‘𝑊) ∧ 𝑧 ∈ 𝑈) → (𝑁‘{𝑧}) = (𝑁‘{𝑋}))
92	86, 11	syl 17	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) ∧ 𝑧 ≠ (0_g‘𝑊) ∧ 𝑧 ∈ 𝑈) → 𝑊 ∈ LMod)
93	87, 75	syl 17	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) ∧ 𝑧 ≠ (0_g‘𝑊) ∧ 𝑧 ∈ 𝑈) → 𝑈 ∈ 𝑆)
94		simp3 1134	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) ∧ 𝑧 ≠ (0_g‘𝑊) ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ 𝑈)
95	13, 20, 92, 93, 94	lspsnel5a 19770	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) ∧ 𝑧 ≠ (0_g‘𝑊) ∧ 𝑧 ∈ 𝑈) → (𝑁‘{𝑧}) ⊆ 𝑈)
96	91, 95	eqsstrrd 4008	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) ∧ 𝑧 ≠ (0_g‘𝑊) ∧ 𝑧 ∈ 𝑈) → (𝑁‘{𝑋}) ⊆ 𝑈)
97	56, 85, 96	mpd3an23 1459	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+_g‘𝑊)𝑧)) → (𝑁‘{𝑋}) ⊆ 𝑈)
98	97	3exp 1115	. . . . . 6 ⊢ (((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) → ((𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) → (𝑥 = (𝑦(+_g‘𝑊)𝑧) → (𝑁‘{𝑋}) ⊆ 𝑈)))
99	98	rexlimdvv 3295	. . . . 5 ⊢ (((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) → (∃𝑦 ∈ 𝑇 ∃𝑧 ∈ (𝑁‘{𝑋})𝑥 = (𝑦(+_g‘𝑊)𝑧) → (𝑁‘{𝑋}) ⊆ 𝑈))
100	30, 99	mpd 15	. . . 4 ⊢ (((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) → (𝑁‘{𝑋}) ⊆ 𝑈)
101	6, 100	exlimddv 1936	. . 3 ⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → (𝑁‘{𝑋}) ⊆ 𝑈)
102	15, 75	sseldd 3970	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊))
103	25	lsmlub 18792	. . . . 5 ⊢ ((𝑇 ∈ (SubGrp‘𝑊) ∧ (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → ((𝑇 ⊆ 𝑈 ∧ (𝑁‘{𝑋}) ⊆ 𝑈) ↔ (𝑇 ⊕ (𝑁‘{𝑋})) ⊆ 𝑈))
104	17, 23, 102, 103	syl3anc 1367	. . . 4 ⊢ (𝜑 → ((𝑇 ⊆ 𝑈 ∧ (𝑁‘{𝑋}) ⊆ 𝑈) ↔ (𝑇 ⊕ (𝑁‘{𝑋})) ⊆ 𝑈))
105	104	3ad2ant1 1129	. . 3 ⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → ((𝑇 ⊆ 𝑈 ∧ (𝑁‘{𝑋}) ⊆ 𝑈) ↔ (𝑇 ⊕ (𝑁‘{𝑋})) ⊆ 𝑈))
106	4, 101, 105	mpbi2and 710	. 2 ⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → (𝑇 ⊕ (𝑁‘{𝑋})) ⊆ 𝑈)
107	1, 106	eqssd 3986	1 ⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → 𝑈 = (𝑇 ⊕ (𝑁‘{𝑋})))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3018 ∃wrex 3141 ⊆ wss 3938 ⊊ wpss 3939 {csn 4569 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 +_gcplusg 16567 0_gc0g 16715 -_gcsg 18107 SubGrpcsubg 18275 LSSumclsm 18761 Abelcabl 18909 LModclmod 19636 LSubSpclss 19705 LSpanclspn 19745 LVecclvec 19876

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-subg 18278 df-lsm 18763 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-drng 19506 df-lmod 19638 df-lss 19706 df-lsp 19746 df-lvec 19877

This theorem is referenced by: lshpnelb 36122 lshpcmp 36126 lsmsatcv 36148 lsmcv2 36167 dochshpncl 38522

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