Theorem linc1 45766

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 1135	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑀 ∈ LMod)
2		linc1.s	. . . . . . . . . 10 ⊢ 𝑆 = (Scalar‘𝑀)
3	2	lmodring 20131	. . . . . . . . 9 ⊢ (𝑀 ∈ LMod → 𝑆 ∈ Ring)
4	2	eqcomi 2747	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑀) = 𝑆
5	4	fveq2i 6777	. . . . . . . . . . 11 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘𝑆)
6		linc1.1	. . . . . . . . . . 11 ⊢ 1 = (1r‘𝑆)
7	5, 6	ringidcl 19807	. . . . . . . . . 10 ⊢ (𝑆 ∈ Ring → 1 ∈ (Base‘(Scalar‘𝑀)))
8		linc1.0	. . . . . . . . . . 11 ⊢ 0 = (0g‘𝑆)
9	5, 8	ring0cl 19808	. . . . . . . . . 10 ⊢ (𝑆 ∈ Ring → 0 ∈ (Base‘(Scalar‘𝑀)))
10	7, 9	jca 512	. . . . . . . . 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 1132	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ( 1 ∈ (Base‘(Scalar‘𝑀)) ∧ 0 ∈ (Base‘(Scalar‘𝑀))))
13	12	adantr 481	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) → ( 1 ∈ (Base‘(Scalar‘𝑀)) ∧ 0 ∈ (Base‘(Scalar‘𝑀))))
14		ifcl 4504	. . . . . 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 6988	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝐹:𝑉⟶(Base‘(Scalar‘𝑀)))
18		fvex 6787	. . . . 5 ⊢ (Base‘(Scalar‘𝑀)) ∈ V
19		simp2 1136	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑉 ∈ 𝒫 𝐵)
20		elmapg 8628	. . . . 5 ⊢ (((Base‘(Scalar‘𝑀)) ∈ V ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ↔ 𝐹:𝑉⟶(Base‘(Scalar‘𝑀))))
21	18, 19, 20	sylancr 587	. . . 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 4551	. . . . . 6 ⊢ 𝒫 𝐵 = 𝒫 (Base‘𝑀)
25	24	eleq2i 2830	. . . . 5 ⊢ (𝑉 ∈ 𝒫 𝐵 ↔ 𝑉 ∈ 𝒫 (Base‘𝑀))
26	25	biimpi 215	. . . 4 ⊢ (𝑉 ∈ 𝒫 𝐵 → 𝑉 ∈ 𝒫 (Base‘𝑀))
27	26	3ad2ant2 1133	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑉 ∈ 𝒫 (Base‘𝑀))
28		lincval 45750	. . 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 1370	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦))))
30		eqid 2738	. . 3 ⊢ (0g‘𝑀) = (0g‘𝑀)
31		lmodgrp 20130	. . . . 5 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp)
32	31	grpmndd 18589	. . . 4 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Mnd)
33	32	3ad2ant1 1132	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑀 ∈ Mnd)
34		simp3 1137	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉)
35	1	adantr 481	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → 𝑀 ∈ LMod)
36		eqeq1 2742	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑋 ↔ 𝑦 = 𝑋))
37	36	ifbid 4482	. . . . . . 7 ⊢ (𝑥 = 𝑦 → if(𝑥 = 𝑋, 1 , 0 ) = if(𝑦 = 𝑋, 1 , 0 ))
38		simpr 485	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝑉)
39		eqid 2738	. . . . . . . . . . 11 ⊢ (Base‘𝑆) = (Base‘𝑆)
40	2, 39, 6	lmod1cl 20150	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → 1 ∈ (Base‘𝑆))
41	40	3ad2ant1 1132	. . . . . . . . 9 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 1 ∈ (Base‘𝑆))
42	41	adantr 481	. . . . . . . 8 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → 1 ∈ (Base‘𝑆))
43	2, 39, 8	lmod0cl 20149	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → 0 ∈ (Base‘𝑆))
44	43	3ad2ant1 1132	. . . . . . . . 9 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 0 ∈ (Base‘𝑆))
45	44	adantr 481	. . . . . . . 8 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → 0 ∈ (Base‘𝑆))
46	42, 45	ifcld 4505	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → if(𝑦 = 𝑋, 1 , 0 ) ∈ (Base‘𝑆))
47	16, 37, 38, 46	fvmptd3 6898	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → (𝐹‘𝑦) = if(𝑦 = 𝑋, 1 , 0 ))
48	47, 46	eqeltrd 2839	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → (𝐹‘𝑦) ∈ (Base‘𝑆))
49		elelpwi 4545	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑦 ∈ 𝐵)
50	49	expcom 414	. . . . . . 7 ⊢ (𝑉 ∈ 𝒫 𝐵 → (𝑦 ∈ 𝑉 → 𝑦 ∈ 𝐵))
51	50	3ad2ant2 1133	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑦 ∈ 𝑉 → 𝑦 ∈ 𝐵))
52	51	imp 407	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝐵)
53		eqid 2738	. . . . . 6 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
54	23, 2, 53, 39	lmodvscl 20140	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ (𝐹‘𝑦) ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵) → ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦) ∈ 𝐵)
55	35, 48, 52, 54	syl3anc 1370	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦) ∈ 𝐵)
56		eqid 2738	. . . 4 ⊢ (𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦)) = (𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦))
57	55, 56	fmptd 6988	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦)):𝑉⟶𝐵)
58		fveq2 6774	. . . . . . 7 ⊢ (𝑦 = 𝑣 → (𝐹‘𝑦) = (𝐹‘𝑣))
59		id 22	. . . . . . 7 ⊢ (𝑦 = 𝑣 → 𝑦 = 𝑣)
60	58, 59	oveq12d 7293	. . . . . 6 ⊢ (𝑦 = 𝑣 → ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦) = ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣))
61	60	cbvmptv 5187	. . . . 5 ⊢ (𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦)) = (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣))
62		fvexd 6789	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (0g‘𝑀) ∈ V)
63		ovexd 7310	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) ∈ V)
64	61, 19, 62, 63	mptsuppd 8003	. . . 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)})
65		2a1 28	. . . . . . 7 ⊢ (𝑣 = 𝑋 → (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → (((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) ≠ (0g‘𝑀) → 𝑣 = 𝑋)))
66		simprr 770	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → 𝑣 ∈ 𝑉)
67	6	fvexi 6788	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ V
68	8	fvexi 6788	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
69	67, 68	ifex 4509	. . . . . . . . . . . . . 14 ⊢ if(𝑣 = 𝑋, 1 , 0 ) ∈ V
70		eqeq1 2742	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑣 → (𝑥 = 𝑋 ↔ 𝑣 = 𝑋))
71	70	ifbid 4482	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑣 → if(𝑥 = 𝑋, 1 , 0 ) = if(𝑣 = 𝑋, 1 , 0 ))
72	71, 16	fvmptg 6873	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ∈ 𝑉 ∧ if(𝑣 = 𝑋, 1 , 0 ) ∈ V) → (𝐹‘𝑣) = if(𝑣 = 𝑋, 1 , 0 ))
73	66, 69, 72	sylancl 586	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → (𝐹‘𝑣) = if(𝑣 = 𝑋, 1 , 0 ))
74		iffalse 4468	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑣 = 𝑋 → if(𝑣 = 𝑋, 1 , 0 ) = 0 )
75	74	adantr 481	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → if(𝑣 = 𝑋, 1 , 0 ) = 0 )
76	73, 75	eqtrd 2778	. . . . . . . . . . . 12 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → (𝐹‘𝑣) = 0 )
77	76	oveq1d 7290	. . . . . . . . . . 11 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) = ( 0 ( ·𝑠 ‘𝑀)𝑣))
78	1	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → 𝑀 ∈ LMod)
79	78	adantl 482	. . . . . . . . . . . 12 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → 𝑀 ∈ LMod)
80		elelpwi 4545	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑣 ∈ 𝐵)
81	80	expcom 414	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ 𝒫 𝐵 → (𝑣 ∈ 𝑉 → 𝑣 ∈ 𝐵))
82	81	3ad2ant2 1133	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑣 ∈ 𝑉 → 𝑣 ∈ 𝐵))
83	82	imp 407	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝐵)
84	83	adantl 482	. . . . . . . . . . . 12 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → 𝑣 ∈ 𝐵)
85	23, 2, 53, 8, 30	lmod0vs 20156	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ LMod ∧ 𝑣 ∈ 𝐵) → ( 0 ( ·𝑠 ‘𝑀)𝑣) = (0g‘𝑀))
86	79, 84, 85	syl2anc 584	. . . . . . . . . . 11 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → ( 0 ( ·𝑠 ‘𝑀)𝑣) = (0g‘𝑀))
87	77, 86	eqtrd 2778	. . . . . . . . . 10 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) = (0g‘𝑀))
88	87	neeq1d 3003	. . . . . . . . 9 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → (((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) ≠ (0g‘𝑀) ↔ (0g‘𝑀) ≠ (0g‘𝑀)))
89		eqneqall 2954	. . . . . . . . . 10 ⊢ ((0g‘𝑀) = (0g‘𝑀) → ((0g‘𝑀) ≠ (0g‘𝑀) → 𝑣 = 𝑋))
90	30, 89	ax-mp 5	. . . . . . . . 9 ⊢ ((0g‘𝑀) ≠ (0g‘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) ≠ (0g‘𝑀) → 𝑣 = 𝑋))
92	91	ex 413	. . . . . . 7 ⊢ (¬ 𝑣 = 𝑋 → (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → (((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) ≠ (0g‘𝑀) → 𝑣 = 𝑋)))
93	65, 92	pm2.61i 182	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → (((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) ≠ (0g‘𝑀) → 𝑣 = 𝑋))
94	93	ralrimiva 3103	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ∀𝑣 ∈ 𝑉 (((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) ≠ (0g‘𝑀) → 𝑣 = 𝑋))
95		rabsssn 4603	. . . . 5 ⊢ ({𝑣 ∈ 𝑉 ∣ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) ≠ (0g‘𝑀)} ⊆ {𝑋} ↔ ∀𝑣 ∈ 𝑉 (((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) ≠ (0g‘𝑀) → 𝑣 = 𝑋))
96	94, 95	sylibr 233	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → {𝑣 ∈ 𝑉 ∣ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) ≠ (0g‘𝑀)} ⊆ {𝑋})
97	64, 96	eqsstrd 3959	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ((𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦)) supp (0g‘𝑀)) ⊆ {𝑋})
98	23, 30, 33, 19, 34, 57, 97	gsumpt 19563	. 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))
99		ovex 7308	. . . 4 ⊢ ((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋) ∈ V
100		fveq2 6774	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝐹‘𝑦) = (𝐹‘𝑋))
101		id 22	. . . . . 6 ⊢ (𝑦 = 𝑋 → 𝑦 = 𝑋)
102	100, 101	oveq12d 7293	. . . . 5 ⊢ (𝑦 = 𝑋 → ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦) = ((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋))
103	102, 56	fvmptg 6873	. . . 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 586	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ((𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦))‘𝑋) = ((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋))
105		iftrue 4465	. . . . . 6 ⊢ (𝑥 = 𝑋 → if(𝑥 = 𝑋, 1 , 0 ) = 1 )
106	105, 16	fvmptg 6873	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 1 ∈ V) → (𝐹‘𝑋) = 1 )
107	34, 67, 106	sylancl 586	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹‘𝑋) = 1 )
108	107	oveq1d 7290	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋) = ( 1 ( ·𝑠 ‘𝑀)𝑋))
109		elelpwi 4545	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑋 ∈ 𝐵)
110	109	ancoms 459	. . . . 5 ⊢ ((𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝐵)
111	110	3adant1 1129	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝐵)
112	23, 2, 53, 6	lmodvs1 20151	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → ( 1 ( ·𝑠 ‘𝑀)𝑋) = 𝑋)
113	1, 111, 112	syl2anc 584	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ( 1 ( ·𝑠 ‘𝑀)𝑋) = 𝑋)
114	104, 108, 113	3eqtrd 2782	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ((𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦))‘𝑋) = 𝑋)
115	29, 98, 114	3eqtrd 2782	1 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹( linC ‘𝑀)𝑉) = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  {crab 3068  Vcvv 3432   ⊆ wss 3887  ifcif 4459  𝒫 cpw 4533  {csn 4561   ↦ cmpt 5157  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   supp csupp 7977   ↑_m cmap 8615  Basecbs 16912  Scalarcsca 16965   ·_𝑠 cvsca 16966  0_gc0g 17150   Σ_g cgsu 17151  Mndcmnd 18385  1_rcur 19737  Ringcrg 19783  LModclmod 20123   linC clinc 45745

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fsupp 9129  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-fzo 13383  df-seq 13722  df-hash 14045  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-0g 17152  df-gsum 17153  df-mre 17295  df-mrc 17296  df-acs 17298  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-submnd 18431  df-grp 18580  df-mulg 18701  df-cntz 18923  df-cmn 19388  df-mgp 19721  df-ur 19738  df-ring 19785  df-lmod 20125  df-linc 45747

This theorem is referenced by:  lcoss  45777