Theorem linc1 49536

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 1154	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑀 ∈ LMod)
2		linc1.s	. . . . . . . . . 10 ⊢ 𝑆 = (Scalar‘𝑀)
3	2	lmodring 21143	. . . . . . . . 9 ⊢ (𝑀 ∈ LMod → 𝑆 ∈ Ring)
4	2	eqcomi 2770	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑀) = 𝑆
5	4	fveq2i 6888	. . . . . . . . . . 11 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘𝑆)
6		linc1.1	. . . . . . . . . . 11 ⊢ 1 = (1r‘𝑆)
7	5, 6	ringidcl 20494	. . . . . . . . . 10 ⊢ (𝑆 ∈ Ring → 1 ∈ (Base‘(Scalar‘𝑀)))
8		linc1.0	. . . . . . . . . . 11 ⊢ 0 = (0g‘𝑆)
9	5, 8	ring0cl 20496	. . . . . . . . . 10 ⊢ (𝑆 ∈ Ring → 0 ∈ (Base‘(Scalar‘𝑀)))
10	7, 9	jca 521	. . . . . . . . 9 ⊢ (𝑆 ∈ Ring → ( 1 ∈ (Base‘(Scalar‘𝑀)) ∧ 0 ∈ (Base‘(Scalar‘𝑀))))
11	3, 10	syl 18	. . . . . . . 8 ⊢ (𝑀 ∈ LMod → ( 1 ∈ (Base‘(Scalar‘𝑀)) ∧ 0 ∈ (Base‘(Scalar‘𝑀))))
12	11	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ( 1 ∈ (Base‘(Scalar‘𝑀)) ∧ 0 ∈ (Base‘(Scalar‘𝑀))))
13	12	adantr 486	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) → ( 1 ∈ (Base‘(Scalar‘𝑀)) ∧ 0 ∈ (Base‘(Scalar‘𝑀))))
14		ifcl 4528	. . . . . 6 ⊢ (( 1 ∈ (Base‘(Scalar‘𝑀)) ∧ 0 ∈ (Base‘(Scalar‘𝑀))) → if(𝑥 = 𝑋, 1 , 0 ) ∈ (Base‘(Scalar‘𝑀)))
15	13, 14	syl 18	. . . . 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 7114	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝐹:𝑉⟶(Base‘(Scalar‘𝑀)))
18		fvex 6898	. . . . 5 ⊢ (Base‘(Scalar‘𝑀)) ∈ V
19		simp2 1155	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑉 ∈ 𝒫 𝐵)
20		elmapg 8859	. . . . 5 ⊢ (((Base‘(Scalar‘𝑀)) ∈ V ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ↔ 𝐹:𝑉⟶(Base‘(Scalar‘𝑀))))
21	18, 19, 20	sylancr 599	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ↔ 𝐹:𝑉⟶(Base‘(Scalar‘𝑀))))
22	17, 21	mpbird 260	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
23		linc1.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑀)
24	23	pweqi 4573	. . . . . 6 ⊢ 𝒫 𝐵 = 𝒫 (Base‘𝑀)
25	24	eleq2i 2853	. . . . 5 ⊢ (𝑉 ∈ 𝒫 𝐵 ↔ 𝑉 ∈ 𝒫 (Base‘𝑀))
26	25	biimpi 219	. . . 4 ⊢ (𝑉 ∈ 𝒫 𝐵 → 𝑉 ∈ 𝒫 (Base‘𝑀))
27	26	3ad2ant2 1152	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑉 ∈ 𝒫 (Base‘𝑀))
28		lincval 49520	. . 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 1398	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦))))
30		eqid 2761	. . 3 ⊢ (0g‘𝑀) = (0g‘𝑀)
31		lmodgrp 21142	. . . . 5 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Grp)
32	31	grpmndd 19157	. . . 4 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ Mnd)
33	32	3ad2ant1 1151	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑀 ∈ Mnd)
34		simp3 1156	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉)
35	1	adantr 486	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → 𝑀 ∈ LMod)
36		eqeq1 2765	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑋 ↔ 𝑦 = 𝑋))
37	36	ifbid 4506	. . . . . . 7 ⊢ (𝑥 = 𝑦 → if(𝑥 = 𝑋, 1 , 0 ) = if(𝑦 = 𝑋, 1 , 0 ))
38		simpr 490	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝑉)
39		eqid 2761	. . . . . . . . . . 11 ⊢ (Base‘𝑆) = (Base‘𝑆)
40	2, 39, 6	lmod1cl 21164	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → 1 ∈ (Base‘𝑆))
41	40	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 1 ∈ (Base‘𝑆))
42	41	adantr 486	. . . . . . . 8 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → 1 ∈ (Base‘𝑆))
43	2, 39, 8	lmod0cl 21163	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → 0 ∈ (Base‘𝑆))
44	43	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 0 ∈ (Base‘𝑆))
45	44	adantr 486	. . . . . . . 8 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → 0 ∈ (Base‘𝑆))
46	42, 45	ifcld 4529	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → if(𝑦 = 𝑋, 1 , 0 ) ∈ (Base‘𝑆))
47	16, 37, 38, 46	fvmptd3 7017	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → (𝐹‘𝑦) = if(𝑦 = 𝑋, 1 , 0 ))
48	47, 46	eqeltrd 2861	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → (𝐹‘𝑦) ∈ (Base‘𝑆))
49		elelpwi 4567	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑦 ∈ 𝐵)
50	49	expcom 419	. . . . . . 7 ⊢ (𝑉 ∈ 𝒫 𝐵 → (𝑦 ∈ 𝑉 → 𝑦 ∈ 𝐵))
51	50	3ad2ant2 1152	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑦 ∈ 𝑉 → 𝑦 ∈ 𝐵))
52	51	imp 412	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝐵)
53		eqid 2761	. . . . . 6 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
54	23, 2, 53, 39	lmodvscl 21153	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ (𝐹‘𝑦) ∈ (Base‘𝑆) ∧ 𝑦 ∈ 𝐵) → ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦) ∈ 𝐵)
55	35, 48, 52, 54	syl3anc 1398	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦) ∈ 𝐵)
56		eqid 2761	. . . 4 ⊢ (𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦)) = (𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦))
57	55, 56	fmptd 7114	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦)):𝑉⟶𝐵)
58		fveq2 6885	. . . . . . 7 ⊢ (𝑦 = 𝑣 → (𝐹‘𝑦) = (𝐹‘𝑣))
59		id 23	. . . . . . 7 ⊢ (𝑦 = 𝑣 → 𝑦 = 𝑣)
60	58, 59	oveq12d 7438	. . . . . 6 ⊢ (𝑦 = 𝑣 → ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦) = ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣))
61	60	cbvmptv 5209	. . . . 5 ⊢ (𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦)) = (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣))
62		fvexd 6900	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (0g‘𝑀) ∈ V)
63		ovexd 7455	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) ∈ V)
64	61, 19, 62, 63	mptsuppd 8204	. . . 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 29	. . . . . . 7 ⊢ (𝑣 = 𝑋 → (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → (((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) ≠ (0g‘𝑀) → 𝑣 = 𝑋)))
66		simprr 785	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → 𝑣 ∈ 𝑉)
67	6	fvexi 6899	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ V
68	8	fvexi 6899	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
69	67, 68	ifex 4533	. . . . . . . . . . . . . 14 ⊢ if(𝑣 = 𝑋, 1 , 0 ) ∈ V
70		eqeq1 2765	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑣 → (𝑥 = 𝑋 ↔ 𝑣 = 𝑋))
71	70	ifbid 4506	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑣 → if(𝑥 = 𝑋, 1 , 0 ) = if(𝑣 = 𝑋, 1 , 0 ))
72	71, 16	fvmptg 6991	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ∈ 𝑉 ∧ if(𝑣 = 𝑋, 1 , 0 ) ∈ V) → (𝐹‘𝑣) = if(𝑣 = 𝑋, 1 , 0 ))
73	66, 69, 72	sylancl 598	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → (𝐹‘𝑣) = if(𝑣 = 𝑋, 1 , 0 ))
74		iffalse 4491	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑣 = 𝑋 → if(𝑣 = 𝑋, 1 , 0 ) = 0 )
75	74	adantr 486	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → if(𝑣 = 𝑋, 1 , 0 ) = 0 )
76	73, 75	eqtrd 2796	. . . . . . . . . . . 12 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → (𝐹‘𝑣) = 0 )
77	76	oveq1d 7435	. . . . . . . . . . 11 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) = ( 0 ( ·𝑠 ‘𝑀)𝑣))
78	1	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → 𝑀 ∈ LMod)
79	78	adantl 487	. . . . . . . . . . . 12 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → 𝑀 ∈ LMod)
80		elelpwi 4567	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑣 ∈ 𝐵)
81	80	expcom 419	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ 𝒫 𝐵 → (𝑣 ∈ 𝑉 → 𝑣 ∈ 𝐵))
82	81	3ad2ant2 1152	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑣 ∈ 𝑉 → 𝑣 ∈ 𝐵))
83	82	imp 412	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝐵)
84	83	adantl 487	. . . . . . . . . . . 12 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → 𝑣 ∈ 𝐵)
85	23, 2, 53, 8, 30	lmod0vs 21170	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ LMod ∧ 𝑣 ∈ 𝐵) → ( 0 ( ·𝑠 ‘𝑀)𝑣) = (0g‘𝑀))
86	79, 84, 85	syl2anc 596	. . . . . . . . . . 11 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → ( 0 ( ·𝑠 ‘𝑀)𝑣) = (0g‘𝑀))
87	77, 86	eqtrd 2796	. . . . . . . . . 10 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) = (0g‘𝑀))
88	87	neeq1d 3015	. . . . . . . . 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 2967	. . . . . . . . . 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	biimtrdi 256	. . . . . . . 8 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → (((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) ≠ (0g‘𝑀) → 𝑣 = 𝑋))
92	91	ex 418	. . . . . . 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 184	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → (((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) ≠ (0g‘𝑀) → 𝑣 = 𝑋))
94	93	ralrimiva 3155	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ∀𝑣 ∈ 𝑉 (((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) ≠ (0g‘𝑀) → 𝑣 = 𝑋))
95		rabsssn 4629	. . . . 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 237	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → {𝑣 ∈ 𝑉 ∣ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) ≠ (0g‘𝑀)} ⊆ {𝑋})
97	64, 96	eqsstrd 3965	. . 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 20176	. 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 7453	. . . 4 ⊢ ((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋) ∈ V
100		fveq2 6885	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝐹‘𝑦) = (𝐹‘𝑋))
101		id 23	. . . . . 6 ⊢ (𝑦 = 𝑋 → 𝑦 = 𝑋)
102	100, 101	oveq12d 7438	. . . . 5 ⊢ (𝑦 = 𝑋 → ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦) = ((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋))
103	102, 56	fvmptg 6991	. . . 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 598	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ((𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦))‘𝑋) = ((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋))
105		iftrue 4488	. . . . . 6 ⊢ (𝑥 = 𝑋 → if(𝑥 = 𝑋, 1 , 0 ) = 1 )
106	105, 16	fvmptg 6991	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 1 ∈ V) → (𝐹‘𝑋) = 1 )
107	34, 67, 106	sylancl 598	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹‘𝑋) = 1 )
108	107	oveq1d 7435	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ((𝐹‘𝑋)( ·𝑠 ‘𝑀)𝑋) = ( 1 ( ·𝑠 ‘𝑀)𝑋))
109		elelpwi 4567	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑋 ∈ 𝐵)
110	109	ancoms 464	. . . . 5 ⊢ ((𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝐵)
111	110	3adant1 1148	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝐵)
112	23, 2, 53, 6	lmodvs1 21165	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → ( 1 ( ·𝑠 ‘𝑀)𝑋) = 𝑋)
113	1, 111, 112	syl2anc 596	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ( 1 ( ·𝑠 ‘𝑀)𝑋) = 𝑋)
114	104, 108, 113	3eqtrd 2800	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ((𝑦 ∈ 𝑉 ↦ ((𝐹‘𝑦)( ·𝑠 ‘𝑀)𝑦))‘𝑋) = 𝑋)
115	29, 98, 114	3eqtrd 2800	1 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹( linC ‘𝑀)𝑉) = 𝑋)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  {crab 3413  Vcvv 3451   ⊆ wss 3899  ifcif 4482  𝒫 cpw 4557  {csn 4584   ↦ cmpt 5186  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   supp csupp 8177   ↑_m cmap 8847  Basecbs 17387  Scalarcsca 17431   ·_𝑠 cvsca 17432  0_gc0g 17610   Σ_g cgsu 17611  Mndcmnd 18923  1_rcur 20407  Ringcrg 20459  LModclmod 21135   linC clinc 49515

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-supp 8178  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-er 8717  df-map 8849  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-fsupp 9354  df-oi 9504  df-card 10020  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-n0 12607  df-z 12694  df-uz 12966  df-fz 13640  df-fzo 13789  df-seq 14145  df-hash 14475  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-0g 17612  df-gsum 17613  df-mre 17756  df-mrc 17757  df-acs 17759  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-submnd 18979  df-grp 19147  df-mulg 19278  df-cntz 19531  df-cmn 19996  df-mgp 20361  df-ur 20408  df-ring 20461  df-lmod 21137  df-linc 49517

This theorem is used by:  lcoss  49547