linc1 - Mathbox for Alexander van der Vekens

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Theorem linc1 47195

Description: A vector is a linear combination of a set containing this vector. (Contributed by AV, 18-Apr-2019.) (Proof shortened by AV, 28-Jul-2019.)

**Hypotheses**
Ref	Expression
linc1.b	⊢ 𝐵 = (Base‘𝑀)
linc1.s	⊢ 𝑆 = (Scalar‘𝑀)
linc1.0	⊢ 0 = (0_g‘𝑆)
linc1.1	⊢ 1 = (1_r‘𝑆)
linc1.f	⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 ))

**Assertion**
Ref	Expression
linc1	⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹( linC ‘𝑀)𝑉) = 𝑋)

Distinct variable groups: 𝑥,𝐵 𝑥,𝑀 𝑥,𝑉 𝑥,𝑋 𝑥, 0 𝑥, 1 Allowed substitution hints: 𝑆(𝑥) 𝐹(𝑥)

Proof of Theorem linc1 Dummy variables 𝑣 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1134	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑀 ∈ LMod)
2		linc1.s	. . . . . . . . . 10 ⊢ 𝑆 = (Scalar‘𝑀)
3	2	lmodring 20624	. . . . . . . . 9 ⊢ (𝑀 ∈ LMod → 𝑆 ∈ Ring)
4	2	eqcomi 2739	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑀) = 𝑆
5	4	fveq2i 6895	. . . . . . . . . . 11 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘𝑆)
6		linc1.1	. . . . . . . . . . 11 ⊢ 1 = (1_r‘𝑆)
7	5, 6	ringidcl 20156	. . . . . . . . . 10 ⊢ (𝑆 ∈ Ring → 1 ∈ (Base‘(Scalar‘𝑀)))
8		linc1.0	. . . . . . . . . . 11 ⊢ 0 = (0_g‘𝑆)
9	5, 8	ring0cl 20157	. . . . . . . . . 10 ⊢ (𝑆 ∈ Ring → 0 ∈ (Base‘(Scalar‘𝑀)))
10	7, 9	jca 510	. . . . . . . . 9 ⊢ (𝑆 ∈ Ring → ( 1 ∈ (Base‘(Scalar‘𝑀)) ∧ 0 ∈ (Base‘(Scalar‘𝑀))))
11	3, 10	syl 17	. . . . . . . 8 ⊢ (𝑀 ∈ LMod → ( 1 ∈ (Base‘(Scalar‘𝑀)) ∧ 0 ∈ (Base‘(Scalar‘𝑀))))
12	11	3ad2ant1 1131	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ( 1 ∈ (Base‘(Scalar‘𝑀)) ∧ 0 ∈ (Base‘(Scalar‘𝑀))))
13	12	adantr 479	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) → ( 1 ∈ (Base‘(Scalar‘𝑀)) ∧ 0 ∈ (Base‘(Scalar‘𝑀))))
14		ifcl 4574	. . . . . 6 ⊢ (( 1 ∈ (Base‘(Scalar‘𝑀)) ∧ 0 ∈ (Base‘(Scalar‘𝑀))) → if(𝑥 = 𝑋, 1 , 0 ) ∈ (Base‘(Scalar‘𝑀)))
15	13, 14	syl 17	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) → if(𝑥 = 𝑋, 1 , 0 ) ∈ (Base‘(Scalar‘𝑀)))
16		linc1.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 ))
17	15, 16	fmptd 7116	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝐹:𝑉⟶(Base‘(Scalar‘𝑀)))
18		fvex 6905	. . . . 5 ⊢ (Base‘(Scalar‘𝑀)) ∈ V
19		simp2 1135	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑉 ∈ 𝒫 𝐵)
20		elmapg 8837	. . . . 5 ⊢ (((Base‘(Scalar‘𝑀)) ∈ V ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ↔ 𝐹:𝑉⟶(Base‘(Scalar‘𝑀))))
21	18, 19, 20	sylancr 585	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ↔ 𝐹:𝑉⟶(Base‘(Scalar‘𝑀))))
22	17, 21	mpbird 256	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉))
23		linc1.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑀)
24	23	pweqi 4619	. . . . . 6 ⊢ 𝒫 𝐵 = 𝒫 (Base‘𝑀)
25	24	eleq2i 2823	. . . . 5 ⊢ (𝑉 ∈ 𝒫 𝐵 ↔ 𝑉 ∈ 𝒫 (Base‘𝑀))
26	25	biimpi 215	. . . 4 ⊢ (𝑉 ∈ 𝒫 𝐵 → 𝑉 ∈ 𝒫 (Base‘𝑀))
27	26	3ad2ant2 1132	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑉 ∈ 𝒫 (Base‘𝑀))
28		lincval 47179	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝐹( linC ‘𝑀)𝑉) = (𝑀 Σ_g (𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·_𝑠 ‘𝑀)𝑦))))
29	1, 22, 27, 28	syl3anc 1369	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹( linC ‘𝑀)𝑉) = (𝑀 Σ_g (𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·_𝑠 ‘𝑀)𝑦))))
30		eqid 2730	. . 3 ⊢ (0_g‘𝑀) = (0_g‘𝑀)
31		lmodgrp 20623	. . . . 5 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp)
32	31	grpmndd 18870	. . . 4 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Mnd)
33	32	3ad2ant1 1131	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑀 ∈ Mnd)
34		simp3 1136	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉)
35	1	adantr 479	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → 𝑀 ∈ LMod)
36		eqeq1 2734	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑋 ↔ 𝑦 = 𝑋))
37	36	ifbid 4552	. . . . . . 7 ⊢ (𝑥 = 𝑦 → if(𝑥 = 𝑋, 1 , 0 ) = if(𝑦 = 𝑋, 1 , 0 ))
38		simpr 483	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝑉)
39		eqid 2730	. . . . . . . . . . 11 ⊢ (Base‘𝑆) = (Base‘𝑆)
40	2, 39, 6	lmod1cl 20645	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → 1 ∈ (Base‘𝑆))
41	40	3ad2ant1 1131	. . . . . . . . 9 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 1 ∈ (Base‘𝑆))
42	41	adantr 479	. . . . . . . 8 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → 1 ∈ (Base‘𝑆))
43	2, 39, 8	lmod0cl 20644	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → 0 ∈ (Base‘𝑆))
44	43	3ad2ant1 1131	. . . . . . . . 9 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 0 ∈ (Base‘𝑆))
45	44	adantr 479	. . . . . . . 8 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → 0 ∈ (Base‘𝑆))
46	42, 45	ifcld 4575	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → if(𝑦 = 𝑋, 1 , 0 ) ∈ (Base‘𝑆))
47	16, 37, 38, 46	fvmptd3 7022	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → (𝐹‘𝑦) = if(𝑦 = 𝑋, 1 , 0 ))
48	47, 46	eqeltrd 2831	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → (𝐹‘𝑦) ∈ (Base‘𝑆))
49		elelpwi 4613	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑦 ∈ 𝐵)
50	49	expcom 412	. . . . . . 7 ⊢ (𝑉 ∈ 𝒫 𝐵 → (𝑦 ∈ 𝑉 → 𝑦 ∈ 𝐵))
51	50	3ad2ant2 1132	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑦 ∈ 𝑉 → 𝑦 ∈ 𝐵))
52	51	imp 405	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝐵)
53		eqid 2730	. . . . . 6 ⊢ ( ·_𝑠 ‘𝑀) = ( ·_𝑠 ‘𝑀)
54	23, 2, 53, 39	lmodvscl 20634	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ (𝐹‘𝑦) ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵) → ((𝐹‘𝑦)( ·_𝑠 ‘𝑀)𝑦) ∈ 𝐵)
55	35, 48, 52, 54	syl3anc 1369	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → ((𝐹‘𝑦)( ·_𝑠 ‘𝑀)𝑦) ∈ 𝐵)
56		eqid 2730	. . . 4 ⊢ (𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·_𝑠 ‘𝑀)𝑦)) = (𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·_𝑠 ‘𝑀)𝑦))
57	55, 56	fmptd 7116	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·_𝑠 ‘𝑀)𝑦)):𝑉⟶𝐵)
58		fveq2 6892	. . . . . . 7 ⊢ (𝑦 = 𝑣 → (𝐹‘𝑦) = (𝐹‘𝑣))
59		id 22	. . . . . . 7 ⊢ (𝑦 = 𝑣 → 𝑦 = 𝑣)
60	58, 59	oveq12d 7431	. . . . . 6 ⊢ (𝑦 = 𝑣 → ((𝐹‘𝑦)( ·_𝑠 ‘𝑀)𝑦) = ((𝐹‘𝑣)( ·_𝑠 ‘𝑀)𝑣))
61	60	cbvmptv 5262	. . . . 5 ⊢ (𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·_𝑠 ‘𝑀)𝑦)) = (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·_𝑠 ‘𝑀)𝑣))
62		fvexd 6907	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (0_g‘𝑀) ∈ V)
63		ovexd 7448	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → ((𝐹‘𝑣)( ·_𝑠 ‘𝑀)𝑣) ∈ V)
64	61, 19, 62, 63	mptsuppd 8176	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ((𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·_𝑠 ‘𝑀)𝑦)) supp (0_g‘𝑀)) = {𝑣 ∈ 𝑉 ∣ ((𝐹‘𝑣)( ·_𝑠 ‘𝑀)𝑣) ≠ (0_g‘𝑀)})
65		2a1 28	. . . . . . 7 ⊢ (𝑣 = 𝑋 → (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → (((𝐹‘𝑣)( ·_𝑠 ‘𝑀)𝑣) ≠ (0_g‘𝑀) → 𝑣 = 𝑋)))
66		simprr 769	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → 𝑣 ∈ 𝑉)
67	6	fvexi 6906	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ V
68	8	fvexi 6906	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
69	67, 68	ifex 4579	. . . . . . . . . . . . . 14 ⊢ if(𝑣 = 𝑋, 1 , 0 ) ∈ V
70		eqeq1 2734	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑣 → (𝑥 = 𝑋 ↔ 𝑣 = 𝑋))
71	70	ifbid 4552	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑣 → if(𝑥 = 𝑋, 1 , 0 ) = if(𝑣 = 𝑋, 1 , 0 ))
72	71, 16	fvmptg 6997	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ∈ 𝑉 ∧ if(𝑣 = 𝑋, 1 , 0 ) ∈ V) → (𝐹‘𝑣) = if(𝑣 = 𝑋, 1 , 0 ))
73	66, 69, 72	sylancl 584	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → (𝐹‘𝑣) = if(𝑣 = 𝑋, 1 , 0 ))
74		iffalse 4538	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑣 = 𝑋 → if(𝑣 = 𝑋, 1 , 0 ) = 0 )
75	74	adantr 479	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → if(𝑣 = 𝑋, 1 , 0 ) = 0 )
76	73, 75	eqtrd 2770	. . . . . . . . . . . 12 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → (𝐹‘𝑣) = 0 )
77	76	oveq1d 7428	. . . . . . . . . . 11 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → ((𝐹‘𝑣)( ·_𝑠 ‘𝑀)𝑣) = ( 0 ( ·_𝑠 ‘𝑀)𝑣))
78	1	adantr 479	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → 𝑀 ∈ LMod)
79	78	adantl 480	. . . . . . . . . . . 12 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → 𝑀 ∈ LMod)
80		elelpwi 4613	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑣 ∈ 𝐵)
81	80	expcom 412	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ 𝒫 𝐵 → (𝑣 ∈ 𝑉 → 𝑣 ∈ 𝐵))
82	81	3ad2ant2 1132	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑣 ∈ 𝑉 → 𝑣 ∈ 𝐵))
83	82	imp 405	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝐵)
84	83	adantl 480	. . . . . . . . . . . 12 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → 𝑣 ∈ 𝐵)
85	23, 2, 53, 8, 30	lmod0vs 20651	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ LMod ∧ 𝑣 ∈ 𝐵) → ( 0 ( ·_𝑠 ‘𝑀)𝑣) = (0_g‘𝑀))
86	79, 84, 85	syl2anc 582	. . . . . . . . . . 11 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → ( 0 ( ·_𝑠 ‘𝑀)𝑣) = (0_g‘𝑀))
87	77, 86	eqtrd 2770	. . . . . . . . . 10 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → ((𝐹‘𝑣)( ·_𝑠 ‘𝑀)𝑣) = (0_g‘𝑀))
88	87	neeq1d 2998	. . . . . . . . 9 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → (((𝐹‘𝑣)( ·_𝑠 ‘𝑀)𝑣) ≠ (0_g‘𝑀) ↔ (0_g‘𝑀) ≠ (0_g‘𝑀)))
89		eqneqall 2949	. . . . . . . . . 10 ⊢ ((0_g‘𝑀) = (0_g‘𝑀) → ((0_g‘𝑀) ≠ (0_g‘𝑀) → 𝑣 = 𝑋))
90	30, 89	ax-mp 5	. . . . . . . . 9 ⊢ ((0_g‘𝑀) ≠ (0_g‘𝑀) → 𝑣 = 𝑋)
91	88, 90	syl6bi 252	. . . . . . . 8 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → (((𝐹‘𝑣)( ·_𝑠 ‘𝑀)𝑣) ≠ (0_g‘𝑀) → 𝑣 = 𝑋))
92	91	ex 411	. . . . . . 7 ⊢ (¬ 𝑣 = 𝑋 → (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → (((𝐹‘𝑣)( ·_𝑠 ‘𝑀)𝑣) ≠ (0_g‘𝑀) → 𝑣 = 𝑋)))
93	65, 92	pm2.61i 182	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → (((𝐹‘𝑣)( ·_𝑠 ‘𝑀)𝑣) ≠ (0_g‘𝑀) → 𝑣 = 𝑋))
94	93	ralrimiva 3144	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ∀𝑣 ∈ 𝑉 (((𝐹‘𝑣)( ·_𝑠 ‘𝑀)𝑣) ≠ (0_g‘𝑀) → 𝑣 = 𝑋))
95		rabsssn 4671	. . . . 5 ⊢ ({𝑣 ∈ 𝑉 ∣ ((𝐹‘𝑣)( ·_𝑠 ‘𝑀)𝑣) ≠ (0_g‘𝑀)} ⊆ {𝑋} ↔ ∀𝑣 ∈ 𝑉 (((𝐹‘𝑣)( ·_𝑠 ‘𝑀)𝑣) ≠ (0_g‘𝑀) → 𝑣 = 𝑋))
96	94, 95	sylibr 233	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → {𝑣 ∈ 𝑉 ∣ ((𝐹‘𝑣)( ·_𝑠 ‘𝑀)𝑣) ≠ (0_g‘𝑀)} ⊆ {𝑋})
97	64, 96	eqsstrd 4021	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ((𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·_𝑠 ‘𝑀)𝑦)) supp (0_g‘𝑀)) ⊆ {𝑋})
98	23, 30, 33, 19, 34, 57, 97	gsumpt 19873	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑀 Σ_g (𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·_𝑠 ‘𝑀)𝑦))) = ((𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·_𝑠 ‘𝑀)𝑦))‘𝑋))
99		ovex 7446	. . . 4 ⊢ ((𝐹‘𝑋)( ·_𝑠 ‘𝑀)𝑋) ∈ V
100		fveq2 6892	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝐹‘𝑦) = (𝐹‘𝑋))
101		id 22	. . . . . 6 ⊢ (𝑦 = 𝑋 → 𝑦 = 𝑋)
102	100, 101	oveq12d 7431	. . . . 5 ⊢ (𝑦 = 𝑋 → ((𝐹‘𝑦)( ·_𝑠 ‘𝑀)𝑦) = ((𝐹‘𝑋)( ·_𝑠 ‘𝑀)𝑋))
103	102, 56	fvmptg 6997	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ ((𝐹‘𝑋)( ·_𝑠 ‘𝑀)𝑋) ∈ V) → ((𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·_𝑠 ‘𝑀)𝑦))‘𝑋) = ((𝐹‘𝑋)( ·_𝑠 ‘𝑀)𝑋))
104	34, 99, 103	sylancl 584	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ((𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·_𝑠 ‘𝑀)𝑦))‘𝑋) = ((𝐹‘𝑋)( ·_𝑠 ‘𝑀)𝑋))
105		iftrue 4535	. . . . . 6 ⊢ (𝑥 = 𝑋 → if(𝑥 = 𝑋, 1 , 0 ) = 1 )
106	105, 16	fvmptg 6997	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 1 ∈ V) → (𝐹‘𝑋) = 1 )
107	34, 67, 106	sylancl 584	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹‘𝑋) = 1 )
108	107	oveq1d 7428	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ((𝐹‘𝑋)( ·_𝑠 ‘𝑀)𝑋) = ( 1 ( ·_𝑠 ‘𝑀)𝑋))
109		elelpwi 4613	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑋 ∈ 𝐵)
110	109	ancoms 457	. . . . 5 ⊢ ((𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝐵)
111	110	3adant1 1128	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝐵)
112	23, 2, 53, 6	lmodvs1 20646	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → ( 1 ( ·_𝑠 ‘𝑀)𝑋) = 𝑋)
113	1, 111, 112	syl2anc 582	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ( 1 ( ·_𝑠 ‘𝑀)𝑋) = 𝑋)
114	104, 108, 113	3eqtrd 2774	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ((𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·_𝑠 ‘𝑀)𝑦))‘𝑋) = 𝑋)
115	29, 98, 114	3eqtrd 2774	1 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹( linC ‘𝑀)𝑉) = 𝑋)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ∀wral 3059 {crab 3430 Vcvv 3472 ⊆ wss 3949 ifcif 4529 𝒫 cpw 4603 {csn 4629 ↦ cmpt 5232 ⟶wf 6540 ‘cfv 6544 (class class class)co 7413 supp csupp 8150 ↑_m cmap 8824 Basecbs 17150 Scalarcsca 17206 ·_𝑠 cvsca 17207 0_gc0g 17391 Σ_g cgsu 17392 Mndcmnd 18661 1_rcur 20077 Ringcrg 20129 LModclmod 20616 linC clinc 47174

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8151 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9366 df-oi 9509 df-card 9938 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11452 df-neg 11453 df-nn 12219 df-2 12281 df-n0 12479 df-z 12565 df-uz 12829 df-fz 13491 df-fzo 13634 df-seq 13973 df-hash 14297 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-0g 17393 df-gsum 17394 df-mre 17536 df-mrc 17537 df-acs 17539 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18708 df-grp 18860 df-mulg 18989 df-cntz 19224 df-cmn 19693 df-mgp 20031 df-ur 20078 df-ring 20131 df-lmod 20618 df-linc 47176

This theorem is referenced by: lcoss 47206

W3C validator