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Theorem lspfixed 19634

Description: Show membership in the span of the sum of two vectors, one of which (𝑌) is fixed in advance. (Contributed by NM, 27-May-2015.) (Revised by AV, 12-Jul-2022.)

**Hypotheses**
Ref	Expression
lspfixed.v	⊢ 𝑉 = (Base‘𝑊)
lspfixed.p	⊢ + = (+_g‘𝑊)
lspfixed.o	⊢ 0 = (0_g‘𝑊)
lspfixed.n	⊢ 𝑁 = (LSpan‘𝑊)
lspfixed.w	⊢ (𝜑 → 𝑊 ∈ LVec)
lspfixed.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
lspfixed.z	⊢ (𝜑 → 𝑍 ∈ 𝑉)
lspfixed.e	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌}))
lspfixed.f	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑍}))
lspfixed.g	⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))

**Assertion**
Ref	Expression
lspfixed	⊢ (𝜑 → ∃𝑧 ∈ ((𝑁‘{𝑍}) ∖ { 0 })𝑋 ∈ (𝑁‘{(𝑌 + 𝑧)}))

Distinct variable groups: 𝑧,𝑁 𝑧, 0 𝑧, + 𝑧,𝑊 𝑧,𝑋 𝑧,𝑌 𝑧,𝑍 Allowed substitution hints: 𝜑(𝑧) 𝑉(𝑧)

Proof of Theorem lspfixed Dummy variables 𝑘 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lspfixed.g	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
2		lspfixed.v	. . . 4 ⊢ 𝑉 = (Base‘𝑊)
3		lspfixed.p	. . . 4 ⊢ + = (+_g‘𝑊)
4		eqid 2771	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2771	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6		eqid 2771	. . . 4 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑊)
7		lspfixed.n	. . . 4 ⊢ 𝑁 = (LSpan‘𝑊)
8		lspfixed.w	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ LVec)
9		lveclmod 19612	. . . . 5 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
10	8, 9	syl 17	. . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod)
11		lspfixed.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
12		lspfixed.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
13	2, 3, 4, 5, 6, 7, 10, 11, 12	lspprel 19600	. . 3 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑌, 𝑍}) ↔ ∃𝑘 ∈ (Base‘(Scalar‘𝑊))∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))))
14	1, 13	mpbid 224	. 2 ⊢ (𝜑 → ∃𝑘 ∈ (Base‘(Scalar‘𝑊))∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍)))
15	10	3ad2ant1 1114	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → 𝑊 ∈ LMod)
16		eqid 2771	. . . . . . . . . 10 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
17	2, 16, 7	lspsncl 19483	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝑍 ∈ 𝑉) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
18	10, 12, 17	syl2anc 576	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
19	18	3ad2ant1 1114	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
20	8	3ad2ant1 1114	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → 𝑊 ∈ LVec)
21	4	lvecdrng 19611	. . . . . . . . 9 ⊢ (𝑊 ∈ LVec → (Scalar‘𝑊) ∈ DivRing)
22	20, 21	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → (Scalar‘𝑊) ∈ DivRing)
23		simp2l 1180	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → 𝑘 ∈ (Base‘(Scalar‘𝑊)))
24		lspfixed.f	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑍}))
25	24	3ad2ant1 1114	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → ¬ 𝑋 ∈ (𝑁‘{𝑍}))
26		simpl3 1174	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0_g‘(Scalar‘𝑊))) → 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍)))
27		simpr 477	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0_g‘(Scalar‘𝑊))) → 𝑘 = (0_g‘(Scalar‘𝑊)))
28	27	oveq1d 6989	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0_g‘(Scalar‘𝑊))) → (𝑘( ·_𝑠 ‘𝑊)𝑌) = ((0_g‘(Scalar‘𝑊))( ·_𝑠 ‘𝑊)𝑌))
29		simpl1 1172	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0_g‘(Scalar‘𝑊))) → 𝜑)
30	29, 10	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0_g‘(Scalar‘𝑊))) → 𝑊 ∈ LMod)
31	29, 11	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0_g‘(Scalar‘𝑊))) → 𝑌 ∈ 𝑉)
32		eqid 2771	. . . . . . . . . . . . . . . . 17 ⊢ (0_g‘(Scalar‘𝑊)) = (0_g‘(Scalar‘𝑊))
33		lspfixed.o	. . . . . . . . . . . . . . . . 17 ⊢ 0 = (0_g‘𝑊)
34	2, 4, 6, 32, 33	lmod0vs 19401	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → ((0_g‘(Scalar‘𝑊))( ·_𝑠 ‘𝑊)𝑌) = 0 )
35	30, 31, 34	syl2anc 576	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0_g‘(Scalar‘𝑊))) → ((0_g‘(Scalar‘𝑊))( ·_𝑠 ‘𝑊)𝑌) = 0 )
36	28, 35	eqtrd 2807	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0_g‘(Scalar‘𝑊))) → (𝑘( ·_𝑠 ‘𝑊)𝑌) = 0 )
37	36	oveq1d 6989	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0_g‘(Scalar‘𝑊))) → ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍)) = ( 0 + (𝑙( ·_𝑠 ‘𝑊)𝑍)))
38		simp2r 1181	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → 𝑙 ∈ (Base‘(Scalar‘𝑊)))
39	12	3ad2ant1 1114	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → 𝑍 ∈ 𝑉)
40	2, 4, 6, 5	lmodvscl 19385	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑍 ∈ 𝑉) → (𝑙( ·_𝑠 ‘𝑊)𝑍) ∈ 𝑉)
41	15, 38, 39, 40	syl3anc 1352	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → (𝑙( ·_𝑠 ‘𝑊)𝑍) ∈ 𝑉)
42	41	adantr 473	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0_g‘(Scalar‘𝑊))) → (𝑙( ·_𝑠 ‘𝑊)𝑍) ∈ 𝑉)
43	2, 3, 33	lmod0vlid 19398	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LMod ∧ (𝑙( ·_𝑠 ‘𝑊)𝑍) ∈ 𝑉) → ( 0 + (𝑙( ·_𝑠 ‘𝑊)𝑍)) = (𝑙( ·_𝑠 ‘𝑊)𝑍))
44	30, 42, 43	syl2anc 576	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0_g‘(Scalar‘𝑊))) → ( 0 + (𝑙( ·_𝑠 ‘𝑊)𝑍)) = (𝑙( ·_𝑠 ‘𝑊)𝑍))
45	26, 37, 44	3eqtrd 2811	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0_g‘(Scalar‘𝑊))) → 𝑋 = (𝑙( ·_𝑠 ‘𝑊)𝑍))
46	29, 18	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0_g‘(Scalar‘𝑊))) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
47		simpl2r 1208	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0_g‘(Scalar‘𝑊))) → 𝑙 ∈ (Base‘(Scalar‘𝑊)))
48	2, 7	lspsnid 19499	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ LMod ∧ 𝑍 ∈ 𝑉) → 𝑍 ∈ (𝑁‘{𝑍}))
49	10, 12, 48	syl2anc 576	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 ∈ (𝑁‘{𝑍}))
50	29, 49	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0_g‘(Scalar‘𝑊))) → 𝑍 ∈ (𝑁‘{𝑍}))
51	4, 6, 5, 16	lssvscl 19461	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LMod ∧ (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊)) ∧ (𝑙 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑍 ∈ (𝑁‘{𝑍}))) → (𝑙( ·_𝑠 ‘𝑊)𝑍) ∈ (𝑁‘{𝑍}))
52	30, 46, 47, 50, 51	syl22anc 827	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0_g‘(Scalar‘𝑊))) → (𝑙( ·_𝑠 ‘𝑊)𝑍) ∈ (𝑁‘{𝑍}))
53	45, 52	eqeltrd 2859	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) ∧ 𝑘 = (0_g‘(Scalar‘𝑊))) → 𝑋 ∈ (𝑁‘{𝑍}))
54	53	ex 405	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → (𝑘 = (0_g‘(Scalar‘𝑊)) → 𝑋 ∈ (𝑁‘{𝑍})))
55	54	necon3bd 2974	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → (¬ 𝑋 ∈ (𝑁‘{𝑍}) → 𝑘 ≠ (0_g‘(Scalar‘𝑊))))
56	25, 55	mpd 15	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → 𝑘 ≠ (0_g‘(Scalar‘𝑊)))
57		eqid 2771	. . . . . . . . 9 ⊢ (inv_r‘(Scalar‘𝑊)) = (inv_r‘(Scalar‘𝑊))
58	5, 32, 57	drnginvrcl 19254	. . . . . . . 8 ⊢ (((Scalar‘𝑊) ∈ DivRing ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ≠ (0_g‘(Scalar‘𝑊))) → ((inv_r‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)))
59	22, 23, 56, 58	syl3anc 1352	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → ((inv_r‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)))
60	49	3ad2ant1 1114	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → 𝑍 ∈ (𝑁‘{𝑍}))
61	15, 19, 38, 60, 51	syl22anc 827	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → (𝑙( ·_𝑠 ‘𝑊)𝑍) ∈ (𝑁‘{𝑍}))
62	4, 6, 5, 16	lssvscl 19461	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊)) ∧ (((inv_r‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑙( ·_𝑠 ‘𝑊)𝑍) ∈ (𝑁‘{𝑍}))) → (((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)(𝑙( ·_𝑠 ‘𝑊)𝑍)) ∈ (𝑁‘{𝑍}))
63	15, 19, 59, 61, 62	syl22anc 827	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → (((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)(𝑙( ·_𝑠 ‘𝑊)𝑍)) ∈ (𝑁‘{𝑍}))
64	5, 32, 57	drnginvrn0 19255	. . . . . . . 8 ⊢ (((Scalar‘𝑊) ∈ DivRing ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ≠ (0_g‘(Scalar‘𝑊))) → ((inv_r‘(Scalar‘𝑊))‘𝑘) ≠ (0_g‘(Scalar‘𝑊)))
65	22, 23, 56, 64	syl3anc 1352	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → ((inv_r‘(Scalar‘𝑊))‘𝑘) ≠ (0_g‘(Scalar‘𝑊)))
66		lspfixed.e	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌}))
67	66	3ad2ant1 1114	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → ¬ 𝑋 ∈ (𝑁‘{𝑌}))
68		simpl3 1174	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) ∧ 𝑙 = (0_g‘(Scalar‘𝑊))) → 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍)))
69		oveq1 6981	. . . . . . . . . . . . . . 15 ⊢ (𝑙 = (0_g‘(Scalar‘𝑊)) → (𝑙( ·_𝑠 ‘𝑊)𝑍) = ((0_g‘(Scalar‘𝑊))( ·_𝑠 ‘𝑊)𝑍))
70	2, 4, 6, 32, 33	lmod0vs 19401	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑍 ∈ 𝑉) → ((0_g‘(Scalar‘𝑊))( ·_𝑠 ‘𝑊)𝑍) = 0 )
71	15, 39, 70	syl2anc 576	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → ((0_g‘(Scalar‘𝑊))( ·_𝑠 ‘𝑊)𝑍) = 0 )
72	69, 71	sylan9eqr 2829	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) ∧ 𝑙 = (0_g‘(Scalar‘𝑊))) → (𝑙( ·_𝑠 ‘𝑊)𝑍) = 0 )
73	72	oveq2d 6990	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) ∧ 𝑙 = (0_g‘(Scalar‘𝑊))) → ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍)) = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + 0 ))
74	11	3ad2ant1 1114	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → 𝑌 ∈ 𝑉)
75	2, 4, 6, 5	lmodvscl 19385	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌 ∈ 𝑉) → (𝑘( ·_𝑠 ‘𝑊)𝑌) ∈ 𝑉)
76	15, 23, 74, 75	syl3anc 1352	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → (𝑘( ·_𝑠 ‘𝑊)𝑌) ∈ 𝑉)
77	2, 3, 33	lmod0vrid 19399	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ LMod ∧ (𝑘( ·_𝑠 ‘𝑊)𝑌) ∈ 𝑉) → ((𝑘( ·_𝑠 ‘𝑊)𝑌) + 0 ) = (𝑘( ·_𝑠 ‘𝑊)𝑌))
78	15, 76, 77	syl2anc 576	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → ((𝑘( ·_𝑠 ‘𝑊)𝑌) + 0 ) = (𝑘( ·_𝑠 ‘𝑊)𝑌))
79	78	adantr 473	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) ∧ 𝑙 = (0_g‘(Scalar‘𝑊))) → ((𝑘( ·_𝑠 ‘𝑊)𝑌) + 0 ) = (𝑘( ·_𝑠 ‘𝑊)𝑌))
80	68, 73, 79	3eqtrd 2811	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) ∧ 𝑙 = (0_g‘(Scalar‘𝑊))) → 𝑋 = (𝑘( ·_𝑠 ‘𝑊)𝑌))
81	2, 16, 7	lspsncl 19483	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
82	10, 11, 81	syl2anc 576	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
83	82	3ad2ant1 1114	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
84	2, 7	lspsnid 19499	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → 𝑌 ∈ (𝑁‘{𝑌}))
85	10, 11, 84	syl2anc 576	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑌}))
86	85	3ad2ant1 1114	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → 𝑌 ∈ (𝑁‘{𝑌}))
87	4, 6, 5, 16	lssvscl 19461	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LMod ∧ (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌 ∈ (𝑁‘{𝑌}))) → (𝑘( ·_𝑠 ‘𝑊)𝑌) ∈ (𝑁‘{𝑌}))
88	15, 83, 23, 86, 87	syl22anc 827	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → (𝑘( ·_𝑠 ‘𝑊)𝑌) ∈ (𝑁‘{𝑌}))
89	88	adantr 473	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) ∧ 𝑙 = (0_g‘(Scalar‘𝑊))) → (𝑘( ·_𝑠 ‘𝑊)𝑌) ∈ (𝑁‘{𝑌}))
90	80, 89	eqeltrd 2859	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) ∧ 𝑙 = (0_g‘(Scalar‘𝑊))) → 𝑋 ∈ (𝑁‘{𝑌}))
91	90	ex 405	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → (𝑙 = (0_g‘(Scalar‘𝑊)) → 𝑋 ∈ (𝑁‘{𝑌})))
92	91	necon3bd 2974	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → (¬ 𝑋 ∈ (𝑁‘{𝑌}) → 𝑙 ≠ (0_g‘(Scalar‘𝑊))))
93	67, 92	mpd 15	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → 𝑙 ≠ (0_g‘(Scalar‘𝑊)))
94		simpl1 1172	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) ∧ 𝑍 = 0 ) → 𝜑)
95	94, 1	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) ∧ 𝑍 = 0 ) → 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
96		preq2 4540	. . . . . . . . . . . . . 14 ⊢ (𝑍 = 0 → {𝑌, 𝑍} = {𝑌, 0 })
97	96	fveq2d 6500	. . . . . . . . . . . . 13 ⊢ (𝑍 = 0 → (𝑁‘{𝑌, 𝑍}) = (𝑁‘{𝑌, 0 }))
98	2, 33, 7, 15, 74	lsppr0 19598	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → (𝑁‘{𝑌, 0 }) = (𝑁‘{𝑌}))
99	97, 98	sylan9eqr 2829	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) ∧ 𝑍 = 0 ) → (𝑁‘{𝑌, 𝑍}) = (𝑁‘{𝑌}))
100	95, 99	eleqtrd 2861	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) ∧ 𝑍 = 0 ) → 𝑋 ∈ (𝑁‘{𝑌}))
101	100	ex 405	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → (𝑍 = 0 → 𝑋 ∈ (𝑁‘{𝑌})))
102	101	necon3bd 2974	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → (¬ 𝑋 ∈ (𝑁‘{𝑌}) → 𝑍 ≠ 0 ))
103	67, 102	mpd 15	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → 𝑍 ≠ 0 )
104	2, 6, 4, 5, 32, 33, 20, 38, 39	lvecvsn0 19615	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → ((𝑙( ·_𝑠 ‘𝑊)𝑍) ≠ 0 ↔ (𝑙 ≠ (0_g‘(Scalar‘𝑊)) ∧ 𝑍 ≠ 0 )))
105	93, 103, 104	mpbir2and 701	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → (𝑙( ·_𝑠 ‘𝑊)𝑍) ≠ 0 )
106	2, 6, 4, 5, 32, 33, 20, 59, 41	lvecvsn0 19615	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → ((((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)(𝑙( ·_𝑠 ‘𝑊)𝑍)) ≠ 0 ↔ (((inv_r‘(Scalar‘𝑊))‘𝑘) ≠ (0_g‘(Scalar‘𝑊)) ∧ (𝑙( ·_𝑠 ‘𝑊)𝑍) ≠ 0 )))
107	65, 105, 106	mpbir2and 701	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → (((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)(𝑙( ·_𝑠 ‘𝑊)𝑍)) ≠ 0 )
108		eldifsn 4589	. . . . . 6 ⊢ ((((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)(𝑙( ·_𝑠 ‘𝑊)𝑍)) ∈ ((𝑁‘{𝑍}) ∖ { 0 }) ↔ ((((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)(𝑙( ·_𝑠 ‘𝑊)𝑍)) ∈ (𝑁‘{𝑍}) ∧ (((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)(𝑙( ·_𝑠 ‘𝑊)𝑍)) ≠ 0 ))
109	63, 107, 108	sylanbrc 575	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → (((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)(𝑙( ·_𝑠 ‘𝑊)𝑍)) ∈ ((𝑁‘{𝑍}) ∖ { 0 }))
110		simp3 1119	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍)))
111	2, 3	lmodvacl 19382	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ (𝑘( ·_𝑠 ‘𝑊)𝑌) ∈ 𝑉 ∧ (𝑙( ·_𝑠 ‘𝑊)𝑍) ∈ 𝑉) → ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍)) ∈ 𝑉)
112	15, 76, 41, 111	syl3anc 1352	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍)) ∈ 𝑉)
113	2, 7	lspsnid 19499	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍)) ∈ 𝑉) → ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍)) ∈ (𝑁‘{((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))}))
114	15, 112, 113	syl2anc 576	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍)) ∈ (𝑁‘{((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))}))
115	110, 114	eqeltrd 2859	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → 𝑋 ∈ (𝑁‘{((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))}))
116	2, 4, 6, 5, 32, 7	lspsnvs 19620	. . . . . . . 8 ⊢ ((𝑊 ∈ LVec ∧ (((inv_r‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)) ∧ ((inv_r‘(Scalar‘𝑊))‘𝑘) ≠ (0_g‘(Scalar‘𝑊))) ∧ ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍)) ∈ 𝑉) → (𝑁‘{(((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍)))}) = (𝑁‘{((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))}))
117	20, 59, 65, 112, 116	syl121anc 1356	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → (𝑁‘{(((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍)))}) = (𝑁‘{((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))}))
118	2, 3, 4, 6, 5	lmodvsdi 19391	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ LMod ∧ (((inv_r‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)) ∧ (𝑘( ·_𝑠 ‘𝑊)𝑌) ∈ 𝑉 ∧ (𝑙( ·_𝑠 ‘𝑊)𝑍) ∈ 𝑉)) → (((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) = ((((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)(𝑘( ·_𝑠 ‘𝑊)𝑌)) + (((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)(𝑙( ·_𝑠 ‘𝑊)𝑍))))
119	15, 59, 76, 41, 118	syl13anc 1353	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → (((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) = ((((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)(𝑘( ·_𝑠 ‘𝑊)𝑌)) + (((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)(𝑙( ·_𝑠 ‘𝑊)𝑍))))
120		eqid 2771	. . . . . . . . . . . . . . 15 ⊢ (._r‘(Scalar‘𝑊)) = (._r‘(Scalar‘𝑊))
121		eqid 2771	. . . . . . . . . . . . . . 15 ⊢ (1_r‘(Scalar‘𝑊)) = (1_r‘(Scalar‘𝑊))
122	5, 32, 120, 121, 57	drnginvrl 19256	. . . . . . . . . . . . . 14 ⊢ (((Scalar‘𝑊) ∈ DivRing ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ≠ (0_g‘(Scalar‘𝑊))) → (((inv_r‘(Scalar‘𝑊))‘𝑘)(._r‘(Scalar‘𝑊))𝑘) = (1_r‘(Scalar‘𝑊)))
123	22, 23, 56, 122	syl3anc 1352	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → (((inv_r‘(Scalar‘𝑊))‘𝑘)(._r‘(Scalar‘𝑊))𝑘) = (1_r‘(Scalar‘𝑊)))
124	123	oveq1d 6989	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → ((((inv_r‘(Scalar‘𝑊))‘𝑘)(._r‘(Scalar‘𝑊))𝑘)( ·_𝑠 ‘𝑊)𝑌) = ((1_r‘(Scalar‘𝑊))( ·_𝑠 ‘𝑊)𝑌))
125	2, 4, 6, 5, 120	lmodvsass 19393	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ (((inv_r‘(Scalar‘𝑊))‘𝑘) ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌 ∈ 𝑉)) → ((((inv_r‘(Scalar‘𝑊))‘𝑘)(._r‘(Scalar‘𝑊))𝑘)( ·_𝑠 ‘𝑊)𝑌) = (((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)(𝑘( ·_𝑠 ‘𝑊)𝑌)))
126	15, 59, 23, 74, 125	syl13anc 1353	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → ((((inv_r‘(Scalar‘𝑊))‘𝑘)(._r‘(Scalar‘𝑊))𝑘)( ·_𝑠 ‘𝑊)𝑌) = (((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)(𝑘( ·_𝑠 ‘𝑊)𝑌)))
127	2, 4, 6, 121	lmodvs1 19396	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → ((1_r‘(Scalar‘𝑊))( ·_𝑠 ‘𝑊)𝑌) = 𝑌)
128	15, 74, 127	syl2anc 576	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → ((1_r‘(Scalar‘𝑊))( ·_𝑠 ‘𝑊)𝑌) = 𝑌)
129	124, 126, 128	3eqtr3d 2815	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → (((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)(𝑘( ·_𝑠 ‘𝑊)𝑌)) = 𝑌)
130	129	oveq1d 6989	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → ((((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)(𝑘( ·_𝑠 ‘𝑊)𝑌)) + (((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)(𝑙( ·_𝑠 ‘𝑊)𝑍))) = (𝑌 + (((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)(𝑙( ·_𝑠 ‘𝑊)𝑍))))
131	119, 130	eqtrd 2807	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → (((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) = (𝑌 + (((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)(𝑙( ·_𝑠 ‘𝑊)𝑍))))
132	131	sneqd 4447	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → {(((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍)))} = {(𝑌 + (((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)(𝑙( ·_𝑠 ‘𝑊)𝑍)))})
133	132	fveq2d 6500	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → (𝑁‘{(((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍)))}) = (𝑁‘{(𝑌 + (((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)(𝑙( ·_𝑠 ‘𝑊)𝑍)))}))
134	117, 133	eqtr3d 2809	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → (𝑁‘{((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))}) = (𝑁‘{(𝑌 + (((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)(𝑙( ·_𝑠 ‘𝑊)𝑍)))}))
135	115, 134	eleqtrd 2861	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → 𝑋 ∈ (𝑁‘{(𝑌 + (((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)(𝑙( ·_𝑠 ‘𝑊)𝑍)))}))
136		oveq2 6982	. . . . . . . . 9 ⊢ (𝑧 = (((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)(𝑙( ·_𝑠 ‘𝑊)𝑍)) → (𝑌 + 𝑧) = (𝑌 + (((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)(𝑙( ·_𝑠 ‘𝑊)𝑍))))
137	136	sneqd 4447	. . . . . . . 8 ⊢ (𝑧 = (((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)(𝑙( ·_𝑠 ‘𝑊)𝑍)) → {(𝑌 + 𝑧)} = {(𝑌 + (((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)(𝑙( ·_𝑠 ‘𝑊)𝑍)))})
138	137	fveq2d 6500	. . . . . . 7 ⊢ (𝑧 = (((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)(𝑙( ·_𝑠 ‘𝑊)𝑍)) → (𝑁‘{(𝑌 + 𝑧)}) = (𝑁‘{(𝑌 + (((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)(𝑙( ·_𝑠 ‘𝑊)𝑍)))}))
139	138	eleq2d 2844	. . . . . 6 ⊢ (𝑧 = (((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)(𝑙( ·_𝑠 ‘𝑊)𝑍)) → (𝑋 ∈ (𝑁‘{(𝑌 + 𝑧)}) ↔ 𝑋 ∈ (𝑁‘{(𝑌 + (((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)(𝑙( ·_𝑠 ‘𝑊)𝑍)))})))
140	139	rspcev 3528	. . . . 5 ⊢ (((((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)(𝑙( ·_𝑠 ‘𝑊)𝑍)) ∈ ((𝑁‘{𝑍}) ∖ { 0 }) ∧ 𝑋 ∈ (𝑁‘{(𝑌 + (((inv_r‘(Scalar‘𝑊))‘𝑘)( ·_𝑠 ‘𝑊)(𝑙( ·_𝑠 ‘𝑊)𝑍)))})) → ∃𝑧 ∈ ((𝑁‘{𝑍}) ∖ { 0 })𝑋 ∈ (𝑁‘{(𝑌 + 𝑧)}))
141	109, 135, 140	syl2anc 576	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → ∃𝑧 ∈ ((𝑁‘{𝑍}) ∖ { 0 })𝑋 ∈ (𝑁‘{(𝑌 + 𝑧)}))
142	141	3exp 1100	. . 3 ⊢ (𝜑 → ((𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) → (𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍)) → ∃𝑧 ∈ ((𝑁‘{𝑍}) ∖ { 0 })𝑋 ∈ (𝑁‘{(𝑌 + 𝑧)}))))
143	142	rexlimdvv 3231	. 2 ⊢ (𝜑 → (∃𝑘 ∈ (Base‘(Scalar‘𝑊))∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍)) → ∃𝑧 ∈ ((𝑁‘{𝑍}) ∖ { 0 })𝑋 ∈ (𝑁‘{(𝑌 + 𝑧)})))
144	14, 143	mpd 15	1 ⊢ (𝜑 → ∃𝑧 ∈ ((𝑁‘{𝑍}) ∖ { 0 })𝑋 ∈ (𝑁‘{(𝑌 + 𝑧)}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 ≠ wne 2960 ∃wrex 3082 ∖ cdif 3819 {csn 4435 {cpr 4437 ‘cfv 6185 (class class class)co 6974 Basecbs 16337 +_gcplusg 16419 ._rcmulr 16420 Scalarcsca 16422 ·_𝑠 cvsca 16423 0_gc0g 16567 1_rcur 18986 inv_rcinvr 19156 DivRingcdr 19237 LModclmod 19368 LSubSpclss 19437 LSpanclspn 19477 LVecclvec 19608

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410

This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-tpos 7693 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-2 11501 df-3 11502 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-ress 16345 df-plusg 16432 df-mulr 16433 df-0g 16569 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-submnd 17816 df-grp 17906 df-minusg 17907 df-sbg 17908 df-subg 18072 df-cntz 18230 df-lsm 18534 df-cmn 18680 df-abl 18681 df-mgp 18975 df-ur 18987 df-ring 19034 df-oppr 19108 df-dvdsr 19126 df-unit 19127 df-invr 19157 df-drng 19239 df-lmod 19370 df-lss 19438 df-lsp 19478 df-lvec 19609

This theorem is referenced by: lsatfixedN 35627

W3C validator