Theorem lincdifsn 49047

Description: A vector is a linear combination of a set containing this vector. (Contributed by AV, 21-Apr-2019.) (Revised by AV, 28-Jul-2019.)

Hypotheses

Ref	Expression
lincdifsn.b	⊢ 𝐵 = (Base‘𝑀)
lincdifsn.r	⊢ 𝑅 = (Scalar‘𝑀)
lincdifsn.s	⊢ 𝑆 = (Base‘𝑅)
lincdifsn.t	⊢ · = ( ·𝑠 ‘𝑀)
lincdifsn.p	⊢ + = (+g‘𝑀)
lincdifsn.0	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
lincdifsn	⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → (𝐹( linC ‘𝑀)𝑉) = ((𝐺( linC ‘𝑀)(𝑉 ∖ {𝑋})) + ((𝐹‘𝑋) · 𝑋)))

Proof of Theorem lincdifsn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp11 1218	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → 𝑀 ∈ LMod)
2		lincdifsn.s	. . . . . . . . 9 ⊢ 𝑆 = (Base‘𝑅)
3		lincdifsn.r	. . . . . . . . . 10 ⊢ 𝑅 = (Scalar‘𝑀)
4	3	fveq2i 6871	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘(Scalar‘𝑀))
5	2, 4	eqtri 2786	. . . . . . . 8 ⊢ 𝑆 = (Base‘(Scalar‘𝑀))
6	5	oveq1i 7407	. . . . . . 7 ⊢ (𝑆 ↑m 𝑉) = ((Base‘(Scalar‘𝑀)) ↑m 𝑉)
7	6	eleq2i 2855	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 ↑m 𝑉) ↔ 𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
8	7	biimpi 218	. . . . 5 ⊢ (𝐹 ∈ (𝑆 ↑m 𝑉) → 𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
9	8	adantr 484	. . . 4 ⊢ ((𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) → 𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
10	9	3ad2ant2 1148	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → 𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
11		lincdifsn.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑀)
12	11	pweqi 4572	. . . . . . 7 ⊢ 𝒫 𝐵 = 𝒫 (Base‘𝑀)
13	12	eleq2i 2855	. . . . . 6 ⊢ (𝑉 ∈ 𝒫 𝐵 ↔ 𝑉 ∈ 𝒫 (Base‘𝑀))
14	13	biimpi 218	. . . . 5 ⊢ (𝑉 ∈ 𝒫 𝐵 → 𝑉 ∈ 𝒫 (Base‘𝑀))
15	14	3ad2ant2 1148	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑉 ∈ 𝒫 (Base‘𝑀))
16	15	3ad2ant1 1147	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → 𝑉 ∈ 𝒫 (Base‘𝑀))
17		lincval 49032	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝐹( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥 ∈ 𝑉 ↦ ((𝐹‘𝑥)( ·𝑠 ‘𝑀)𝑥))))
18	1, 10, 16, 17	syl3anc 1391	. 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → (𝐹( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥 ∈ 𝑉 ↦ ((𝐹‘𝑥)( ·𝑠 ‘𝑀)𝑥))))
19		lincdifsn.p	. . . 4 ⊢ + = (+g‘𝑀)
20		lmodcmn 20978	. . . . . 6 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ CMnd)
21	20	3ad2ant1 1147	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑀 ∈ CMnd)
22	21	3ad2ant1 1147	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → 𝑀 ∈ CMnd)
23		simp12 1219	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → 𝑉 ∈ 𝒫 𝐵)
24	14	anim2i 626	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)))
25	24	3adant3 1146	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)))
26	25	3ad2ant1 1147	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)))
27		simp2l 1214	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → 𝐹 ∈ (𝑆 ↑m 𝑉))
28		lincdifsn.0	. . . . . . . . 9 ⊢ 0 = (0g‘𝑅)
29	28	breq2i 5109	. . . . . . . 8 ⊢ (𝐹 finSupp 0 ↔ 𝐹 finSupp (0g‘𝑅))
30	29	biimpi 218	. . . . . . 7 ⊢ (𝐹 finSupp 0 → 𝐹 finSupp (0g‘𝑅))
31	30	adantl 485	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) → 𝐹 finSupp (0g‘𝑅))
32	31	3ad2ant2 1148	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → 𝐹 finSupp (0g‘𝑅))
33	3, 2	scmfsupp 48998	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp (0g‘𝑅)) → (𝑥 ∈ 𝑉 ↦ ((𝐹‘𝑥)( ·𝑠 ‘𝑀)𝑥)) finSupp (0g‘𝑀))
34	26, 27, 32, 33	syl3anc 1391	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → (𝑥 ∈ 𝑉 ↦ ((𝐹‘𝑥)( ·𝑠 ‘𝑀)𝑥)) finSupp (0g‘𝑀))
35		simpl1 1206	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 )) → 𝑀 ∈ LMod)
36	35	adantr 484	. . . . . 6 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 )) ∧ 𝑥 ∈ 𝑉) → 𝑀 ∈ LMod)
37		elmapi 8831	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑆 ↑m 𝑉) → 𝐹:𝑉⟶𝑆)
38		ffvelcdm 7063	. . . . . . . . . . . 12 ⊢ ((𝐹:𝑉⟶𝑆 ∧ 𝑥 ∈ 𝑉) → (𝐹‘𝑥) ∈ 𝑆)
39	38	ex 416	. . . . . . . . . . 11 ⊢ (𝐹:𝑉⟶𝑆 → (𝑥 ∈ 𝑉 → (𝐹‘𝑥) ∈ 𝑆))
40	39	a1d 25	. . . . . . . . . 10 ⊢ (𝐹:𝑉⟶𝑆 → ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑥 ∈ 𝑉 → (𝐹‘𝑥) ∈ 𝑆)))
41	37, 40	syl 17	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 ↑m 𝑉) → ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑥 ∈ 𝑉 → (𝐹‘𝑥) ∈ 𝑆)))
42	41	adantr 484	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) → ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑥 ∈ 𝑉 → (𝐹‘𝑥) ∈ 𝑆)))
43	42	impcom 411	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 )) → (𝑥 ∈ 𝑉 → (𝐹‘𝑥) ∈ 𝑆))
44	43	imp 410	. . . . . 6 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 )) ∧ 𝑥 ∈ 𝑉) → (𝐹‘𝑥) ∈ 𝑆)
45		elelpwi 4566	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑥 ∈ 𝐵)
46	45	expcom 417	. . . . . . . . 9 ⊢ (𝑉 ∈ 𝒫 𝐵 → (𝑥 ∈ 𝑉 → 𝑥 ∈ 𝐵))
47	46	3ad2ant2 1148	. . . . . . . 8 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑥 ∈ 𝑉 → 𝑥 ∈ 𝐵))
48	47	adantr 484	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 )) → (𝑥 ∈ 𝑉 → 𝑥 ∈ 𝐵))
49	48	imp 410	. . . . . 6 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 )) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ 𝐵)
50		eqid 2763	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
51	11, 3, 50, 2	lmodvscl 20946	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ (𝐹‘𝑥) ∈ 𝑆 ∧ 𝑥 ∈ 𝐵) → ((𝐹‘𝑥)( ·𝑠 ‘𝑀)𝑥) ∈ 𝐵)
52	36, 44, 49, 51	syl3anc 1391	. . . . 5 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 )) ∧ 𝑥 ∈ 𝑉) → ((𝐹‘𝑥)( ·𝑠 ‘𝑀)𝑥) ∈ 𝐵)
53	52	3adantl3 1183	. . . 4 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) ∧ 𝑥 ∈ 𝑉) → ((𝐹‘𝑥)( ·𝑠 ‘𝑀)𝑥) ∈ 𝐵)
54		simp13 1220	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → 𝑋 ∈ 𝑉)
55		ffvelcdm 7063	. . . . . . . . . . 11 ⊢ ((𝐹:𝑉⟶𝑆 ∧ 𝑋 ∈ 𝑉) → (𝐹‘𝑋) ∈ 𝑆)
56	55	expcom 417	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑉 → (𝐹:𝑉⟶𝑆 → (𝐹‘𝑋) ∈ 𝑆))
57	56	3ad2ant3 1149	. . . . . . . . 9 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹:𝑉⟶𝑆 → (𝐹‘𝑋) ∈ 𝑆))
58	37, 57	syl5com 31	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑆 ↑m 𝑉) → ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹‘𝑋) ∈ 𝑆))
59	58	adantr 484	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) → ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹‘𝑋) ∈ 𝑆))
60	59	impcom 411	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 )) → (𝐹‘𝑋) ∈ 𝑆)
61		elelpwi 4566	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑋 ∈ 𝐵)
62	61	ancoms 462	. . . . . . . 8 ⊢ ((𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝐵)
63	62	3adant1 1144	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝐵)
64	63	adantr 484	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 )) → 𝑋 ∈ 𝐵)
65		lincdifsn.t	. . . . . . 7 ⊢ · = ( ·𝑠 ‘𝑀)
66	11, 3, 65, 2	lmodvscl 20946	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ (𝐹‘𝑋) ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → ((𝐹‘𝑋) · 𝑋) ∈ 𝐵)
67	35, 60, 64, 66	syl3anc 1391	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 )) → ((𝐹‘𝑋) · 𝑋) ∈ 𝐵)
68	67	3adant3 1146	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → ((𝐹‘𝑋) · 𝑋) ∈ 𝐵)
69	65	eqcomi 2772	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑀) = ·
70	69	a1i 11	. . . . . 6 ⊢ (𝑥 = 𝑋 → ( ·𝑠 ‘𝑀) = · )
71		fveq2 6868	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
72		id 22	. . . . . 6 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
73	70, 71, 72	oveq123d 7418	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥)( ·𝑠 ‘𝑀)𝑥) = ((𝐹‘𝑋) · 𝑋))
74	73	adantl 485	. . . 4 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) ∧ 𝑥 = 𝑋) → ((𝐹‘𝑥)( ·𝑠 ‘𝑀)𝑥) = ((𝐹‘𝑋) · 𝑋))
75	11, 19, 22, 23, 34, 53, 54, 68, 74	gsumdifsnd 20002	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → (𝑀 Σg (𝑥 ∈ 𝑉 ↦ ((𝐹‘𝑥)( ·𝑠 ‘𝑀)𝑥))) = ((𝑀 Σg (𝑥 ∈ (𝑉 ∖ {𝑋}) ↦ ((𝐹‘𝑥)( ·𝑠 ‘𝑀)𝑥))) + ((𝐹‘𝑋) · 𝑋)))
76		fveq1 6867	. . . . . . . . . 10 ⊢ (𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋})) → (𝐺‘𝑥) = ((𝐹 ↾ (𝑉 ∖ {𝑋}))‘𝑥))
77	76	3ad2ant3 1149	. . . . . . . . 9 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → (𝐺‘𝑥) = ((𝐹 ↾ (𝑉 ∖ {𝑋}))‘𝑥))
78		fvres 6887	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑉 ∖ {𝑋}) → ((𝐹 ↾ (𝑉 ∖ {𝑋}))‘𝑥) = (𝐹‘𝑥))
79	77, 78	sylan9eq 2818	. . . . . . . 8 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) ∧ 𝑥 ∈ (𝑉 ∖ {𝑋})) → (𝐺‘𝑥) = (𝐹‘𝑥))
80	79	oveq1d 7412	. . . . . . 7 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) ∧ 𝑥 ∈ (𝑉 ∖ {𝑋})) → ((𝐺‘𝑥)( ·𝑠 ‘𝑀)𝑥) = ((𝐹‘𝑥)( ·𝑠 ‘𝑀)𝑥))
81	80	mpteq2dva 5194	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → (𝑥 ∈ (𝑉 ∖ {𝑋}) ↦ ((𝐺‘𝑥)( ·𝑠 ‘𝑀)𝑥)) = (𝑥 ∈ (𝑉 ∖ {𝑋}) ↦ ((𝐹‘𝑥)( ·𝑠 ‘𝑀)𝑥)))
82	81	eqcomd 2769	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → (𝑥 ∈ (𝑉 ∖ {𝑋}) ↦ ((𝐹‘𝑥)( ·𝑠 ‘𝑀)𝑥)) = (𝑥 ∈ (𝑉 ∖ {𝑋}) ↦ ((𝐺‘𝑥)( ·𝑠 ‘𝑀)𝑥)))
83	82	oveq2d 7413	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → (𝑀 Σg (𝑥 ∈ (𝑉 ∖ {𝑋}) ↦ ((𝐹‘𝑥)( ·𝑠 ‘𝑀)𝑥))) = (𝑀 Σg (𝑥 ∈ (𝑉 ∖ {𝑋}) ↦ ((𝐺‘𝑥)( ·𝑠 ‘𝑀)𝑥))))
84	83	oveq1d 7412	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → ((𝑀 Σg (𝑥 ∈ (𝑉 ∖ {𝑋}) ↦ ((𝐹‘𝑥)( ·𝑠 ‘𝑀)𝑥))) + ((𝐹‘𝑋) · 𝑋)) = ((𝑀 Σg (𝑥 ∈ (𝑉 ∖ {𝑋}) ↦ ((𝐺‘𝑥)( ·𝑠 ‘𝑀)𝑥))) + ((𝐹‘𝑋) · 𝑋)))
85	75, 84	eqtrd 2798	. 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → (𝑀 Σg (𝑥 ∈ 𝑉 ↦ ((𝐹‘𝑥)( ·𝑠 ‘𝑀)𝑥))) = ((𝑀 Σg (𝑥 ∈ (𝑉 ∖ {𝑋}) ↦ ((𝐺‘𝑥)( ·𝑠 ‘𝑀)𝑥))) + ((𝐹‘𝑋) · 𝑋)))
86		eqid 2763	. . . . . . . . . . . 12 ⊢ 𝑉 = 𝑉
87	86, 5	feq23i 6686	. . . . . . . . . . 11 ⊢ (𝐹:𝑉⟶𝑆 ↔ 𝐹:𝑉⟶(Base‘(Scalar‘𝑀)))
88	37, 87	sylib 220	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑆 ↑m 𝑉) → 𝐹:𝑉⟶(Base‘(Scalar‘𝑀)))
89	88	adantr 484	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) → 𝐹:𝑉⟶(Base‘(Scalar‘𝑀)))
90	89	3ad2ant2 1148	. . . . . . . 8 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → 𝐹:𝑉⟶(Base‘(Scalar‘𝑀)))
91		difssd 4091	. . . . . . . 8 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → (𝑉 ∖ {𝑋}) ⊆ 𝑉)
92	90, 91	fssresd 6732	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → (𝐹 ↾ (𝑉 ∖ {𝑋})):(𝑉 ∖ {𝑋})⟶(Base‘(Scalar‘𝑀)))
93		feq1 6670	. . . . . . . 8 ⊢ (𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋})) → (𝐺:(𝑉 ∖ {𝑋})⟶(Base‘(Scalar‘𝑀)) ↔ (𝐹 ↾ (𝑉 ∖ {𝑋})):(𝑉 ∖ {𝑋})⟶(Base‘(Scalar‘𝑀))))
94	93	3ad2ant3 1149	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → (𝐺:(𝑉 ∖ {𝑋})⟶(Base‘(Scalar‘𝑀)) ↔ (𝐹 ↾ (𝑉 ∖ {𝑋})):(𝑉 ∖ {𝑋})⟶(Base‘(Scalar‘𝑀))))
95	92, 94	mpbird 259	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → 𝐺:(𝑉 ∖ {𝑋})⟶(Base‘(Scalar‘𝑀)))
96		fvex 6881	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑀)) ∈ V
97		difexg 5286	. . . . . . . . 9 ⊢ (𝑉 ∈ 𝒫 𝐵 → (𝑉 ∖ {𝑋}) ∈ V)
98	97	3ad2ant2 1148	. . . . . . . 8 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑉 ∖ {𝑋}) ∈ V)
99	98	3ad2ant1 1147	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → (𝑉 ∖ {𝑋}) ∈ V)
100		elmapg 8821	. . . . . . 7 ⊢ (((Base‘(Scalar‘𝑀)) ∈ V ∧ (𝑉 ∖ {𝑋}) ∈ V) → (𝐺 ∈ ((Base‘(Scalar‘𝑀)) ↑m (𝑉 ∖ {𝑋})) ↔ 𝐺:(𝑉 ∖ {𝑋})⟶(Base‘(Scalar‘𝑀))))
101	96, 99, 100	sylancr 596	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → (𝐺 ∈ ((Base‘(Scalar‘𝑀)) ↑m (𝑉 ∖ {𝑋})) ↔ 𝐺:(𝑉 ∖ {𝑋})⟶(Base‘(Scalar‘𝑀))))
102	95, 101	mpbird 259	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → 𝐺 ∈ ((Base‘(Scalar‘𝑀)) ↑m (𝑉 ∖ {𝑋})))
103		elpwi 4563	. . . . . . . . . 10 ⊢ (𝑉 ∈ 𝒫 𝐵 → 𝑉 ⊆ 𝐵)
104	11	sseq2i 3966	. . . . . . . . . . . 12 ⊢ (𝑉 ⊆ 𝐵 ↔ 𝑉 ⊆ (Base‘𝑀))
105	104	biimpi 218	. . . . . . . . . . 11 ⊢ (𝑉 ⊆ 𝐵 → 𝑉 ⊆ (Base‘𝑀))
106	105	ssdifssd 4101	. . . . . . . . . 10 ⊢ (𝑉 ⊆ 𝐵 → (𝑉 ∖ {𝑋}) ⊆ (Base‘𝑀))
107	103, 106	syl 17	. . . . . . . . 9 ⊢ (𝑉 ∈ 𝒫 𝐵 → (𝑉 ∖ {𝑋}) ⊆ (Base‘𝑀))
108	107	adantl 485	. . . . . . . 8 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑉 ∖ {𝑋}) ⊆ (Base‘𝑀))
109	97	adantl 485	. . . . . . . . 9 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑉 ∖ {𝑋}) ∈ V)
110		elpwg 4559	. . . . . . . . 9 ⊢ ((𝑉 ∖ {𝑋}) ∈ V → ((𝑉 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀) ↔ (𝑉 ∖ {𝑋}) ⊆ (Base‘𝑀)))
111	109, 110	syl 17	. . . . . . . 8 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → ((𝑉 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀) ↔ (𝑉 ∖ {𝑋}) ⊆ (Base‘𝑀)))
112	108, 111	mpbird 259	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑉 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
113	112	3adant3 1146	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑉 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
114	113	3ad2ant1 1147	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → (𝑉 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
115		lincval 49032	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝐺 ∈ ((Base‘(Scalar‘𝑀)) ↑m (𝑉 ∖ {𝑋})) ∧ (𝑉 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)) → (𝐺( linC ‘𝑀)(𝑉 ∖ {𝑋})) = (𝑀 Σg (𝑥 ∈ (𝑉 ∖ {𝑋}) ↦ ((𝐺‘𝑥)( ·𝑠 ‘𝑀)𝑥))))
116	1, 102, 114, 115	syl3anc 1391	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → (𝐺( linC ‘𝑀)(𝑉 ∖ {𝑋})) = (𝑀 Σg (𝑥 ∈ (𝑉 ∖ {𝑋}) ↦ ((𝐺‘𝑥)( ·𝑠 ‘𝑀)𝑥))))
117	116	eqcomd 2769	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → (𝑀 Σg (𝑥 ∈ (𝑉 ∖ {𝑋}) ↦ ((𝐺‘𝑥)( ·𝑠 ‘𝑀)𝑥))) = (𝐺( linC ‘𝑀)(𝑉 ∖ {𝑋})))
118	117	oveq1d 7412	. 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → ((𝑀 Σg (𝑥 ∈ (𝑉 ∖ {𝑋}) ↦ ((𝐺‘𝑥)( ·𝑠 ‘𝑀)𝑥))) + ((𝐹‘𝑋) · 𝑋)) = ((𝐺( linC ‘𝑀)(𝑉 ∖ {𝑋})) + ((𝐹‘𝑋) · 𝑋)))
119	18, 85, 118	3eqtrd 2802	1 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → (𝐹( linC ‘𝑀)𝑉) = ((𝐺( linC ‘𝑀)(𝑉 ∖ {𝑋})) + ((𝐹‘𝑋) · 𝑋)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1561   ∈ wcel 2143  Vcvv 3455   ∖ cdif 3902   ⊆ wss 3905  𝒫 cpw 4556  {csn 4583   class class class wbr 5101   ↦ cmpt 5182   ↾ cres 5650  ⟶wf 6518  ‘cfv 6522  (class class class)co 7397   ↑_m cmap 8809   finSupp cfsupp 9308  Basecbs 17246  +_gcplusg 17287  Scalarcsca 17290   ·_𝑠 cvsca 17291  0_gc0g 17469   Σ_g cgsu 17470  CMndccmn 19821  LModclmod 20928   linC clinc 49027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719  ax-cnex 11130  ax-resscn 11131  ax-1cn 11132  ax-icn 11133  ax-addcl 11134  ax-addrcl 11135  ax-mulcl 11136  ax-mulrcl 11137  ax-mulcom 11138  ax-addass 11139  ax-mulass 11140  ax-distr 11141  ax-i2m1 11142  ax-1ne0 11143  ax-1rid 11144  ax-rnegex 11145  ax-rrecex 11146  ax-cnre 11147  ax-pre-lttri 11148  ax-pre-lttrn 11149  ax-pre-ltadd 11150  ax-pre-mulgt0 11151

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-se 5602  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6289  df-ord 6350  df-on 6351  df-lim 6352  df-suc 6353  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-isom 6531  df-riota 7354  df-ov 7400  df-oprab 7401  df-mpo 7402  df-of 7661  df-om 7848  df-1st 7971  df-2nd 7972  df-supp 8142  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8382  df-1o 8438  df-2o 8439  df-er 8679  df-map 8811  df-en 8929  df-dom 8930  df-sdom 8931  df-fin 8932  df-fsupp 9309  df-oi 9459  df-card 9898  df-pnf 11219  df-mnf 11220  df-xr 11221  df-ltxr 11222  df-le 11223  df-sub 11417  df-neg 11418  df-nn 12212  df-2 12281  df-n0 12483  df-z 12570  df-uz 12841  df-fz 13514  df-fzo 13661  df-seq 14016  df-hash 14345  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17247  df-ress 17268  df-plusg 17300  df-0g 17471  df-gsum 17472  df-mre 17615  df-mrc 17616  df-acs 17618  df-mgm 18675  df-sgrp 18754  df-mnd 18770  df-submnd 18819  df-grp 18979  df-minusg 18980  df-mulg 19111  df-cntz 19358  df-cmn 19823  df-abl 19824  df-mgp 20188  df-ur 20233  df-ring 20286  df-lmod 20930  df-linc 49029

This theorem is referenced by:  lincext3  49079  lindslinindimp2lem4  49084  lincresunit3  49104