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Theorem lspfixed 21045

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 2730	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2730	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6		eqid 2730	. . . 4 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑊)
7		lspfixed.n	. . . 4 ⊢ 𝑁 = (LSpan‘𝑊)
8		lspfixed.w	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ LVec)
9		lveclmod 21020	. . . . 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 21008	. . 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 232	. 2 ⊢ (𝜑 → ∃𝑘 ∈ (Base‘(Scalar‘𝑊))∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍)))
15	10	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → 𝑊 ∈ LMod)
16		eqid 2730	. . . . . . . . . 10 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
17	2, 16, 7	lspsncl 20890	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝑍 ∈ 𝑉) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
18	10, 12, 17	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑊))
19	18	3ad2ant1 1133	. . . . . . 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 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → 𝑊 ∈ LVec)
21	4	lvecdrng 21019	. . . . . . . . 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 1200	. . . . . . . 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 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → ¬ 𝑋 ∈ (𝑁‘{𝑍}))
26		simpl3 1194	. . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . 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 7405	. . . . . . . . . . . . . . 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 1192	. . . . . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . . . . 17 ⊢ (0_g‘(Scalar‘𝑊)) = (0_g‘(Scalar‘𝑊))
33		lspfixed.o	. . . . . . . . . . . . . . . . 17 ⊢ 0 = (0_g‘𝑊)
34	2, 4, 6, 32, 33	lmod0vs 20808	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → ((0_g‘(Scalar‘𝑊))( ·_𝑠 ‘𝑊)𝑌) = 0 )
35	30, 31, 34	syl2anc 584	. . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . 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 7405	. . . . . . . . . . . . 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 1201	. . . . . . . . . . . . . . . 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 1133	. . . . . . . . . . . . . . . 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 20791	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑍 ∈ 𝑉) → (𝑙( ·_𝑠 ‘𝑊)𝑍) ∈ 𝑉)
41	15, 38, 39, 40	syl3anc 1373	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 20805	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LMod ∧ (𝑙( ·_𝑠 ‘𝑊)𝑍) ∈ 𝑉) → ( 0 + (𝑙( ·_𝑠 ‘𝑊)𝑍)) = (𝑙( ·_𝑠 ‘𝑊)𝑍))
44	30, 42, 43	syl2anc 584	. . . . . . . . . . . . 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 2769	. . . . . . . . . . . 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 1228	. . . . . . . . . . . . 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 20906	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ LMod ∧ 𝑍 ∈ 𝑉) → 𝑍 ∈ (𝑁‘{𝑍}))
49	10, 12, 48	syl2anc 584	. . . . . . . . . . . . . 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 20868	. . . . . . . . . . . . 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 838	. . . . . . . . . . . 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 2829	. . . . . . . . . . 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 412	. . . . . . . . . 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 2940	. . . . . . . . 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 2730	. . . . . . . . 9 ⊢ (inv_r‘(Scalar‘𝑊)) = (inv_r‘(Scalar‘𝑊))
58	5, 32, 57	drnginvrcl 20669	. . . . . . . 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 1373	. . . . . . 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 1133	. . . . . . . 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 838	. . . . . . 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 20868	. . . . . . 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 838	. . . . . 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 20670	. . . . . . . 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 1373	. . . . . . 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 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑙 ∈ (Base‘(Scalar‘𝑊))) ∧ 𝑋 = ((𝑘( ·_𝑠 ‘𝑊)𝑌) + (𝑙( ·_𝑠 ‘𝑊)𝑍))) → ¬ 𝑋 ∈ (𝑁‘{𝑌}))
68		simpl3 1194	. . . . . . . . . . . . 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 7397	. . . . . . . . . . . . . . 15 ⊢ (𝑙 = (0_g‘(Scalar‘𝑊)) → (𝑙( ·_𝑠 ‘𝑊)𝑍) = ((0_g‘(Scalar‘𝑊))( ·_𝑠 ‘𝑊)𝑍))
70	2, 4, 6, 32, 33	lmod0vs 20808	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑍 ∈ 𝑉) → ((0_g‘(Scalar‘𝑊))( ·_𝑠 ‘𝑊)𝑍) = 0 )
71	15, 39, 70	syl2anc 584	. . . . . . . . . . . . . . 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 2787	. . . . . . . . . . . . . 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 7406	. . . . . . . . . . . . 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 1133	. . . . . . . . . . . . . . . 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 20791	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑌 ∈ 𝑉) → (𝑘( ·_𝑠 ‘𝑊)𝑌) ∈ 𝑉)
76	15, 23, 74, 75	syl3anc 1373	. . . . . . . . . . . . . . 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 20806	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ LMod ∧ (𝑘( ·_𝑠 ‘𝑊)𝑌) ∈ 𝑉) → ((𝑘( ·_𝑠 ‘𝑊)𝑌) + 0 ) = (𝑘( ·_𝑠 ‘𝑊)𝑌))
78	15, 76, 77	syl2anc 584	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 2769	. . . . . . . . . . . 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 20890	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
82	10, 11, 81	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊))
83	82	3ad2ant1 1133	. . . . . . . . . . . . . 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 20906	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → 𝑌 ∈ (𝑁‘{𝑌}))
85	10, 11, 84	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑌}))
86	85	3ad2ant1 1133	. . . . . . . . . . . . . 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 20868	. . . . . . . . . . . . . 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 838	. . . . . . . . . . . . 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 480	. . . . . . . . . . . 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 2829	. . . . . . . . . . 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 412	. . . . . . . . . 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 2940	. . . . . . . . 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 1192	. . . . . . . . . . . . 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 4701	. . . . . . . . . . . . . 14 ⊢ (𝑍 = 0 → {𝑌, 𝑍} = {𝑌, 0 })
97	96	fveq2d 6865	. . . . . . . . . . . . 13 ⊢ (𝑍 = 0 → (𝑁‘{𝑌, 𝑍}) = (𝑁‘{𝑌, 0 }))
98	2, 33, 7, 15, 74	lsppr0 21006	. . . . . . . . . . . . 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 2787	. . . . . . . . . . . 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 2831	. . . . . . . . . . 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 412	. . . . . . . . . 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 2940	. . . . . . . . 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 21026	. . . . . . . 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 713	. . . . . . 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 21026	. . . . . . 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 713	. . . . . 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 4753	. . . . . 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 583	. . . . 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 1138	. . . . . . 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 20788	. . . . . . . . 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 1373	. . . . . . . 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 20906	. . . . . . . 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 584	. . . . . . 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 2829	. . . . . 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 21031	. . . . . . . 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 1377	. . . . . . 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 20798	. . . . . . . . . . 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 1374	. . . . . . . . . 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 2730	. . . . . . . . . . . . . . 15 ⊢ (._r‘(Scalar‘𝑊)) = (._r‘(Scalar‘𝑊))
121		eqid 2730	. . . . . . . . . . . . . . 15 ⊢ (1_r‘(Scalar‘𝑊)) = (1_r‘(Scalar‘𝑊))
122	5, 32, 120, 121, 57	drnginvrl 20672	. . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . 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 7405	. . . . . . . . . . . 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 20800	. . . . . . . . . . . . 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 1374	. . . . . . . . . . . 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 20803	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → ((1_r‘(Scalar‘𝑊))( ·_𝑠 ‘𝑊)𝑌) = 𝑌)
128	15, 74, 127	syl2anc 584	. . . . . . . . . . . 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 2773	. . . . . . . . . . 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 7405	. . . . . . . . . 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 2765	. . . . . . . . 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 4604	. . . . . . . 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 6865	. . . . . . 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 2767	. . . . . 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 2831	. . . . 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 7398	. . . . . . . . 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 4604	. . . . . . . 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 6865	. . . . . . 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 2815	. . . . . 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 3591	. . . . 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 584	. . . 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 1119	. . 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 3194	. 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 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 ∃wrex 3054 ∖ cdif 3914 {csn 4592 {cpr 4594 ‘cfv 6514 (class class class)co 7390 Basecbs 17186 +_gcplusg 17227 ._rcmulr 17228 Scalarcsca 17230 ·_𝑠 cvsca 17231 0_gc0g 17409 1_rcur 20097 inv_rcinvr 20303 DivRingcdr 20645 LModclmod 20773 LSubSpclss 20844 LSpanclspn 20884 LVecclvec 21016

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-tpos 8208 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-3 12257 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-0g 17411 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-grp 18875 df-minusg 18876 df-sbg 18877 df-subg 19062 df-cntz 19256 df-lsm 19573 df-cmn 19719 df-abl 19720 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-oppr 20253 df-dvdsr 20273 df-unit 20274 df-invr 20304 df-drng 20647 df-lmod 20775 df-lss 20845 df-lsp 20885 df-lvec 21017

This theorem is referenced by: lsatfixedN 39009

W3C validator