Theorem linc1 43853

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 = (0g‘𝑆)
linc1.1	⊢ 1 = (1r‘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 1116	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑀 ∈ LMod)
2		linc1.s	. . . . . . . . . 10 ⊢ 𝑆 = (Scalar‘𝑀)
3	2	lmodring 19364	. . . . . . . . 9 ⊢ (𝑀 ∈ LMod → 𝑆 ∈ Ring)
4	2	eqcomi 2787	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑀) = 𝑆
5	4	fveq2i 6502	. . . . . . . . . . 11 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘𝑆)
6		linc1.1	. . . . . . . . . . 11 ⊢ 1 = (1r‘𝑆)
7	5, 6	ringidcl 19041	. . . . . . . . . 10 ⊢ (𝑆 ∈ Ring → 1 ∈ (Base‘(Scalar‘𝑀)))
8		linc1.0	. . . . . . . . . . 11 ⊢ 0 = (0g‘𝑆)
9	5, 8	ring0cl 19042	. . . . . . . . . 10 ⊢ (𝑆 ∈ Ring → 0 ∈ (Base‘(Scalar‘𝑀)))
10	7, 9	jca 504	. . . . . . . . 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 1113	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ( 1 ∈ (Base‘(Scalar‘𝑀)) ∧ 0 ∈ (Base‘(Scalar‘𝑀))))
13	12	adantr 473	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) → ( 1 ∈ (Base‘(Scalar‘𝑀)) ∧ 0 ∈ (Base‘(Scalar‘𝑀))))
14		ifcl 4394	. . . . . 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 6701	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝐹:𝑉⟶(Base‘(Scalar‘𝑀)))
18		fvex 6512	. . . . 5 ⊢ (Base‘(Scalar‘𝑀)) ∈ V
19		simp2 1117	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑉 ∈ 𝒫 𝐵)
20		elmapg 8219	. . . . 5 ⊢ (((Base‘(Scalar‘𝑀)) ∈ V ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ↔ 𝐹:𝑉⟶(Base‘(Scalar‘𝑀))))
21	18, 19, 20	sylancr 578	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ↔ 𝐹:𝑉⟶(Base‘(Scalar‘𝑀))))
22	17, 21	mpbird 249	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
23		linc1.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑀)
24	23	pweqi 4426	. . . . . 6 ⊢ 𝒫 𝐵 = 𝒫 (Base‘𝑀)
25	24	eleq2i 2857	. . . . 5 ⊢ (𝑉 ∈ 𝒫 𝐵 ↔ 𝑉 ∈ 𝒫 (Base‘𝑀))
26	25	biimpi 208	. . . 4 ⊢ (𝑉 ∈ 𝒫 𝐵 → 𝑉 ∈ 𝒫 (Base‘𝑀))
27	26	3ad2ant2 1114	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑉 ∈ 𝒫 (Base‘𝑀))
28		lincval 43837	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝐹( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦))))
29	1, 22, 27, 28	syl3anc 1351	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦))))
30		eqid 2778	. . 3 ⊢ (0g‘𝑀) = (0g‘𝑀)
31		lmodgrp 19363	. . . . 5 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp)
32		grpmnd 17898	. . . . 5 ⊢ (𝑀 ∈ Grp → 𝑀 ∈ Mnd)
33	31, 32	syl 17	. . . 4 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Mnd)
34	33	3ad2ant1 1113	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑀 ∈ Mnd)
35		simp3 1118	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉)
36	1	adantr 473	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → 𝑀 ∈ LMod)
37		eqeq1 2782	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑋 ↔ 𝑦 = 𝑋))
38	37	ifbid 4372	. . . . . . 7 ⊢ (𝑥 = 𝑦 → if(𝑥 = 𝑋, 1 , 0 ) = if(𝑦 = 𝑋, 1 , 0 ))
39		simpr 477	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝑉)
40		eqid 2778	. . . . . . . . . . 11 ⊢ (Base‘𝑆) = (Base‘𝑆)
41	2, 40, 6	lmod1cl 19383	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → 1 ∈ (Base‘𝑆))
42	41	3ad2ant1 1113	. . . . . . . . 9 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 1 ∈ (Base‘𝑆))
43	42	adantr 473	. . . . . . . 8 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → 1 ∈ (Base‘𝑆))
44	2, 40, 8	lmod0cl 19382	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → 0 ∈ (Base‘𝑆))
45	44	3ad2ant1 1113	. . . . . . . . 9 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 0 ∈ (Base‘𝑆))
46	45	adantr 473	. . . . . . . 8 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → 0 ∈ (Base‘𝑆))
47	43, 46	ifcld 4395	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → if(𝑦 = 𝑋, 1 , 0 ) ∈ (Base‘𝑆))
48	16, 38, 39, 47	fvmptd3 6617	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → (𝐹‘𝑦) = if(𝑦 = 𝑋, 1 , 0 ))
49	48, 47	eqeltrd 2866	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → (𝐹‘𝑦) ∈ (Base‘𝑆))
50		elelpwi 4435	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑦 ∈ 𝐵)
51	50	expcom 406	. . . . . . 7 ⊢ (𝑉 ∈ 𝒫 𝐵 → (𝑦 ∈ 𝑉 → 𝑦 ∈ 𝐵))
52	51	3ad2ant2 1114	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑦 ∈ 𝑉 → 𝑦 ∈ 𝐵))
53	52	imp 398	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝐵)
54		eqid 2778	. . . . . 6 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
55	23, 2, 54, 40	lmodvscl 19373	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ (𝐹‘𝑦) ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵) → ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦) ∈ 𝐵)
56	36, 49, 53, 55	syl3anc 1351	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦) ∈ 𝐵)
57		eqid 2778	. . . 4 ⊢ (𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦)) = (𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦))
58	56, 57	fmptd 6701	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦)):𝑉⟶𝐵)
59		fveq2 6499	. . . . . . 7 ⊢ (𝑦 = 𝑣 → (𝐹‘𝑦) = (𝐹‘𝑣))
60		id 22	. . . . . . 7 ⊢ (𝑦 = 𝑣 → 𝑦 = 𝑣)
61	59, 60	oveq12d 6994	. . . . . 6 ⊢ (𝑦 = 𝑣 → ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦) = ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣))
62	61	cbvmptv 5028	. . . . 5 ⊢ (𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦)) = (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣))
63		fvexd 6514	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (0g‘𝑀) ∈ V)
64		ovexd 7010	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) ∈ V)
65	62, 19, 63, 64	mptsuppd 7656	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ((𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦)) supp (0g‘𝑀)) = {𝑣 ∈ 𝑉 ∣ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) ≠ (0g‘𝑀)})
66		2a1 28	. . . . . . 7 ⊢ (𝑣 = 𝑋 → (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → (((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) ≠ (0g‘𝑀) → 𝑣 = 𝑋)))
67		simprr 760	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → 𝑣 ∈ 𝑉)
68	6	fvexi 6513	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ V
69	8	fvexi 6513	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
70	68, 69	ifex 4398	. . . . . . . . . . . . . 14 ⊢ if(𝑣 = 𝑋, 1 , 0 ) ∈ V
71		eqeq1 2782	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑣 → (𝑥 = 𝑋 ↔ 𝑣 = 𝑋))
72	71	ifbid 4372	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑣 → if(𝑥 = 𝑋, 1 , 0 ) = if(𝑣 = 𝑋, 1 , 0 ))
73	72, 16	fvmptg 6593	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ∈ 𝑉 ∧ if(𝑣 = 𝑋, 1 , 0 ) ∈ V) → (𝐹‘𝑣) = if(𝑣 = 𝑋, 1 , 0 ))
74	67, 70, 73	sylancl 577	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → (𝐹‘𝑣) = if(𝑣 = 𝑋, 1 , 0 ))
75		iffalse 4359	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑣 = 𝑋 → if(𝑣 = 𝑋, 1 , 0 ) = 0 )
76	75	adantr 473	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → if(𝑣 = 𝑋, 1 , 0 ) = 0 )
77	74, 76	eqtrd 2814	. . . . . . . . . . . 12 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → (𝐹‘𝑣) = 0 )
78	77	oveq1d 6991	. . . . . . . . . . 11 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) = ( 0 ( ·𝑠 ‘𝑀)𝑣))
79	1	adantr 473	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → 𝑀 ∈ LMod)
80	79	adantl 474	. . . . . . . . . . . 12 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → 𝑀 ∈ LMod)
81		elelpwi 4435	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑣 ∈ 𝐵)
82	81	expcom 406	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ 𝒫 𝐵 → (𝑣 ∈ 𝑉 → 𝑣 ∈ 𝐵))
83	82	3ad2ant2 1114	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑣 ∈ 𝑉 → 𝑣 ∈ 𝐵))
84	83	imp 398	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝐵)
85	84	adantl 474	. . . . . . . . . . . 12 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → 𝑣 ∈ 𝐵)
86	23, 2, 54, 8, 30	lmod0vs 19389	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ LMod ∧ 𝑣 ∈ 𝐵) → ( 0 ( ·𝑠 ‘𝑀)𝑣) = (0g‘𝑀))
87	80, 85, 86	syl2anc 576	. . . . . . . . . . 11 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → ( 0 ( ·𝑠 ‘𝑀)𝑣) = (0g‘𝑀))
88	78, 87	eqtrd 2814	. . . . . . . . . 10 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) = (0g‘𝑀))
89	88	neeq1d 3026	. . . . . . . . 9 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → (((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) ≠ (0g‘𝑀) ↔ (0g‘𝑀) ≠ (0g‘𝑀)))
90		eqneqall 2978	. . . . . . . . . 10 ⊢ ((0g‘𝑀) = (0g‘𝑀) → ((0g‘𝑀) ≠ (0g‘𝑀) → 𝑣 = 𝑋))
91	30, 90	ax-mp 5	. . . . . . . . 9 ⊢ ((0g‘𝑀) ≠ (0g‘𝑀) → 𝑣 = 𝑋)
92	89, 91	syl6bi 245	. . . . . . . 8 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → (((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) ≠ (0g‘𝑀) → 𝑣 = 𝑋))
93	92	ex 405	. . . . . . 7 ⊢ (¬ 𝑣 = 𝑋 → (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → (((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) ≠ (0g‘𝑀) → 𝑣 = 𝑋)))
94	66, 93	pm2.61i 177	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → (((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) ≠ (0g‘𝑀) → 𝑣 = 𝑋))
95	94	ralrimiva 3132	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ∀𝑣 ∈ 𝑉 (((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) ≠ (0g‘𝑀) → 𝑣 = 𝑋))
96		rabsssn 4479	. . . . 5 ⊢ ({𝑣 ∈ 𝑉 ∣ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) ≠ (0g‘𝑀)} ⊆ {𝑋} ↔ ∀𝑣 ∈ 𝑉 (((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) ≠ (0g‘𝑀) → 𝑣 = 𝑋))
97	95, 96	sylibr 226	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → {𝑣 ∈ 𝑉 ∣ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) ≠ (0g‘𝑀)} ⊆ {𝑋})
98	65, 97	eqsstrd 3895	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ((𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦)) supp (0g‘𝑀)) ⊆ {𝑋})
99	23, 30, 34, 19, 35, 58, 98	gsumpt 18835	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑀 Σg (𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦))) = ((𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦))‘𝑋))
100		ovex 7008	. . . 4 ⊢ ((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋) ∈ V
101		fveq2 6499	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝐹‘𝑦) = (𝐹‘𝑋))
102		id 22	. . . . . 6 ⊢ (𝑦 = 𝑋 → 𝑦 = 𝑋)
103	101, 102	oveq12d 6994	. . . . 5 ⊢ (𝑦 = 𝑋 → ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦) = ((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋))
104	103, 57	fvmptg 6593	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ ((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋) ∈ V) → ((𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦))‘𝑋) = ((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋))
105	35, 100, 104	sylancl 577	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ((𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦))‘𝑋) = ((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋))
106		iftrue 4356	. . . . . 6 ⊢ (𝑥 = 𝑋 → if(𝑥 = 𝑋, 1 , 0 ) = 1 )
107	106, 16	fvmptg 6593	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 1 ∈ V) → (𝐹‘𝑋) = 1 )
108	35, 68, 107	sylancl 577	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹‘𝑋) = 1 )
109	108	oveq1d 6991	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋) = ( 1 ( ·𝑠 ‘𝑀)𝑋))
110		elelpwi 4435	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑋 ∈ 𝐵)
111	110	ancoms 451	. . . . 5 ⊢ ((𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝐵)
112	111	3adant1 1110	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝐵)
113	23, 2, 54, 6	lmodvs1 19384	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → ( 1 ( ·𝑠 ‘𝑀)𝑋) = 𝑋)
114	1, 112, 113	syl2anc 576	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ( 1 ( ·𝑠 ‘𝑀)𝑋) = 𝑋)
115	105, 109, 114	3eqtrd 2818	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ((𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦))‘𝑋) = 𝑋)
116	29, 99, 115	3eqtrd 2818	1 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹( linC ‘𝑀)𝑉) = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387   ∧ w3a 1068   = wceq 1507   ∈ wcel 2050   ≠ wne 2967  ∀wral 3088  {crab 3092  Vcvv 3415   ⊆ wss 3829  ifcif 4350  𝒫 cpw 4422  {csn 4441   ↦ cmpt 5008  ⟶wf 6184  ‘cfv 6188  (class class class)co 6976   supp csupp 7633   ↑_𝑚 cmap 8206  Basecbs 16339  Scalarcsca 16424   ·_𝑠 cvsca 16425  0_gc0g 16569   Σ_g cgsu 16570  Mndcmnd 17762  Grpcgrp 17891  1_rcur 18974  Ringcrg 19020  LModclmod 19356   linC clinc 43832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-cnex 10391  ax-resscn 10392  ax-1cn 10393  ax-icn 10394  ax-addcl 10395  ax-addrcl 10396  ax-mulcl 10397  ax-mulrcl 10398  ax-mulcom 10399  ax-addass 10400  ax-mulass 10401  ax-distr 10402  ax-i2m1 10403  ax-1ne0 10404  ax-1rid 10405  ax-rnegex 10406  ax-rrecex 10407  ax-cnre 10408  ax-pre-lttri 10409  ax-pre-lttrn 10410  ax-pre-ltadd 10411  ax-pre-mulgt0 10412

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-iin 4795  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-se 5367  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-isom 6197  df-riota 6937  df-ov 6979  df-oprab 6980  df-mpo 6981  df-om 7397  df-1st 7501  df-2nd 7502  df-supp 7634  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-1o 7905  df-oadd 7909  df-er 8089  df-map 8208  df-en 8307  df-dom 8308  df-sdom 8309  df-fin 8310  df-fsupp 8629  df-oi 8769  df-card 9162  df-pnf 10476  df-mnf 10477  df-xr 10478  df-ltxr 10479  df-le 10480  df-sub 10672  df-neg 10673  df-nn 11440  df-2 11503  df-n0 11708  df-z 11794  df-uz 12059  df-fz 12709  df-fzo 12850  df-seq 13185  df-hash 13506  df-ndx 16342  df-slot 16343  df-base 16345  df-sets 16346  df-ress 16347  df-plusg 16434  df-0g 16571  df-gsum 16572  df-mre 16715  df-mrc 16716  df-acs 16718  df-mgm 17710  df-sgrp 17752  df-mnd 17763  df-submnd 17804  df-grp 17894  df-mulg 18012  df-cntz 18218  df-cmn 18668  df-mgp 18963  df-ur 18975  df-ring 19022  df-lmod 19358  df-linc 43834

This theorem is referenced by:  lcoss  43864