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Theorem lsmcv 21108

Description: Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 31739 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 1139	. 2 ⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋})))
2		simp2 1138	. . . 4 ⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → 𝑇 ⊊ 𝑈)
3		pssss 4052	. . . 4 ⊢ (𝑇 ⊊ 𝑈 → 𝑇 ⊆ 𝑈)
4	2, 3	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → 𝑇 ⊆ 𝑈)
5		pssnel 4425	. . . . 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 1195	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) → 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋})))
8		simprl 771	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) → 𝑥 ∈ 𝑈)
9	7, 8	sseldd 3936	. . . . . 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 21070	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
12	10, 11	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 ∈ LMod)
13		lsmcv.s	. . . . . . . . . . . 12 ⊢ 𝑆 = (LSubSp‘𝑊)
14	13	lsssssubg 20921	. . . . . . . . . . 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 3936	. . . . . . . . 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 20940	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ 𝑆)
22	12, 18, 21	syl2anc 585	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ 𝑆)
23	15, 22	sseldd 3936	. . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊))
24		eqid 2737	. . . . . . . . . 10 ⊢ (+_g‘𝑊) = (+_g‘𝑊)
25		lsmcv.p	. . . . . . . . . 10 ⊢ ⊕ = (LSSum‘𝑊)
26	24, 25	lsmelval 19590	. . . . . . . . 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 585	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑇 ⊕ (𝑁‘{𝑋})) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ (𝑁‘{𝑋})𝑥 = (𝑦(+_g‘𝑊)𝑧)))
28	27	3ad2ant1 1134	. . . . . . 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 480	. . . . . 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 232	. . . . 5 ⊢ (((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) → ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ (𝑁‘{𝑋})𝑥 = (𝑦(+_g‘𝑊)𝑧))
31		simp1rr 1241	. . . . . . . . 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 1201	. . . . . . . . . . 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 7376	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (0_g‘𝑊) → (𝑦(+_g‘𝑊)𝑧) = (𝑦(+_g‘𝑊)(0_g‘𝑊)))
34	33	eqeq2d 2748	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (0_g‘𝑊) → (𝑥 = (𝑦(+_g‘𝑊)𝑧) ↔ 𝑥 = (𝑦(+_g‘𝑊)(0_g‘𝑊))))
35	34	biimpac 478	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = (𝑦(+_g‘𝑊)𝑧) ∧ 𝑧 = (0_g‘𝑊)) → 𝑥 = (𝑦(+_g‘𝑊)(0_g‘𝑊)))
36	12	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → 𝑊 ∈ LMod)
37	36	ad2antrr 727	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋}))) → 𝑊 ∈ LMod)
38	16	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → 𝑇 ∈ 𝑆)
39	38	ad2antrr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋}))) → 𝑇 ∈ 𝑆)
40		simprl 771	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋}))) → 𝑦 ∈ 𝑇)
41	19, 13	lssel 20900	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑇 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇) → 𝑦 ∈ 𝑉)
42	39, 40, 41	syl2anc 585	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋}))) → 𝑦 ∈ 𝑉)
43		eqid 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ (0_g‘𝑊) = (0_g‘𝑊)
44	19, 24, 43	lmod0vrid 20856	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ LMod ∧ 𝑦 ∈ 𝑉) → (𝑦(+_g‘𝑊)(0_g‘𝑊)) = 𝑦)
45	37, 42, 44	syl2anc 585	. . . . . . . . . . . . . . . . 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 2748	. . . . . . . . . . . . . . . 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 229	. . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . 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 431	. . . . . . . . . . . 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 1111	. . . . . . . . . . 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 2825	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑇 ↔ 𝑦 ∈ 𝑇))
53	52	biimparc 479	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝑇 ∧ 𝑥 = 𝑦) → 𝑥 ∈ 𝑇)
54	32, 51, 53	syl6an 685	. . . . . . . . . 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 2947	. . . . . . . . 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 1134	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → 𝑊 ∈ LVec)
58	57	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) → 𝑊 ∈ LVec)
59	58	3ad2ant1 1134	. . . . . . . . . . 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 20872	. . . . . . . . . . . 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 1268	. . . . . . . . . . . 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 585	. . . . . . . . . 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 585	. . . . . . . . . . 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 1202	. . . . . . . . . . 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 20900	. . . . . . . . . . 11 ⊢ (((𝑁‘{𝑋}) ∈ 𝑆 ∧ 𝑧 ∈ (𝑁‘{𝑋})) → 𝑧 ∈ 𝑉)
71	68, 69, 70	syl2anc 585	. . . . . . . . . 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 2737	. . . . . . . . . . 11 ⊢ (-_g‘𝑊) = (-_g‘𝑊)
73	19, 24, 72	ablpncan2 19756	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Abel ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝑦(+_g‘𝑊)𝑧)(-_g‘𝑊)𝑦) = 𝑧)
74	62, 65, 71, 73	syl3anc 1374	. . . . . . . . 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 1139	. . . . . . . . . . 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 1240	. . . . . . . . . . 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 2838	. . . . . . . . . 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 1269	. . . . . . . . . . 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 3935	. . . . . . . . . . 11 ⊢ ((𝑇 ⊊ 𝑈 ∧ 𝑦 ∈ 𝑇) → 𝑦 ∈ 𝑈)
82	80, 32, 81	syl2anc 585	. . . . . . . . . 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 20907	. . . . . . . . . 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 839	. . . . . . . . 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 2838	. . . . . . . 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 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) → 𝑊 ∈ LVec)
87	63	3ad2ant1 1134	. . . . . . . . . . 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 1289	. . . . . . . . . 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 1138	. . . . . . . . . 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 21082	. . . . . . . . 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 1139	. . . . . . . . . 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	ellspsn5 20959	. . . . . . . . 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 3971	. . . . . . . 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 1466	. . . . . . 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 1120	. . . . . 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 3194	. . . . 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 1937	. . 3 ⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → (𝑁‘{𝑋}) ⊆ 𝑈)
102	15, 75	sseldd 3936	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊))
103	25	lsmlub 19605	. . . . 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 1374	. . . 4 ⊢ (𝜑 → ((𝑇 ⊆ 𝑈 ∧ (𝑁‘{𝑋}) ⊆ 𝑈) ↔ (𝑇 ⊕ (𝑁‘{𝑋})) ⊆ 𝑈))
105	104	3ad2ant1 1134	. . 3 ⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → ((𝑇 ⊆ 𝑈 ∧ (𝑁‘{𝑋}) ⊆ 𝑈) ↔ (𝑇 ⊕ (𝑁‘{𝑋})) ⊆ 𝑈))
106	4, 101, 105	mpbi2and 713	. 2 ⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → (𝑇 ⊕ (𝑁‘{𝑋})) ⊆ 𝑈)
107	1, 106	eqssd 3953	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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∃wex 1781 ∈ wcel 2114 ≠ wne 2933 ∃wrex 3062 ⊆ wss 3903 ⊊ wpss 3904 {csn 4582 ‘cfv 6500 (class class class)co 7368 Basecbs 17148 +_gcplusg 17189 0_gc0g 17371 -_gcsg 18877 SubGrpcsubg 19062 LSSumclsm 19575 Abelcabl 19722 LModclmod 20823 LSubSpclss 20894 LSpanclspn 20934 LVecclvec 21066

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-tpos 8178 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-0g 17373 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-grp 18878 df-minusg 18879 df-sbg 18880 df-subg 19065 df-lsm 19577 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-ring 20182 df-oppr 20285 df-dvdsr 20305 df-unit 20306 df-invr 20336 df-drng 20676 df-lmod 20825 df-lss 20895 df-lsp 20935 df-lvec 21067

This theorem is referenced by: lshpnelb 39354 lshpcmp 39358 lsmsatcv 39380 lsmcv2 39399 dochshpncl 41754

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