Theorem lincdifsn 48419

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 1204	. . 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 6825	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘(Scalar‘𝑀))
5	2, 4	eqtri 2752	. . . . . . . 8 ⊢ 𝑆 = (Base‘(Scalar‘𝑀))
6	5	oveq1i 7359	. . . . . . 7 ⊢ (𝑆 ↑m 𝑉) = ((Base‘(Scalar‘𝑀)) ↑m 𝑉)
7	6	eleq2i 2820	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 ↑m 𝑉) ↔ 𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
8	7	biimpi 216	. . . . 5 ⊢ (𝐹 ∈ (𝑆 ↑m 𝑉) → 𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
9	8	adantr 480	. . . 4 ⊢ ((𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) → 𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
10	9	3ad2ant2 1134	. . 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 4567	. . . . . . 7 ⊢ 𝒫 𝐵 = 𝒫 (Base‘𝑀)
13	12	eleq2i 2820	. . . . . 6 ⊢ (𝑉 ∈ 𝒫 𝐵 ↔ 𝑉 ∈ 𝒫 (Base‘𝑀))
14	13	biimpi 216	. . . . 5 ⊢ (𝑉 ∈ 𝒫 𝐵 → 𝑉 ∈ 𝒫 (Base‘𝑀))
15	14	3ad2ant2 1134	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑉 ∈ 𝒫 (Base‘𝑀))
16	15	3ad2ant1 1133	. . 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 48404	. . 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 1373	. 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 20813	. . . . . 6 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ CMnd)
21	20	3ad2ant1 1133	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑀 ∈ CMnd)
22	21	3ad2ant1 1133	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → 𝑀 ∈ CMnd)
23		simp12 1205	. . . 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 617	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)))
25	24	3adant3 1132	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)))
26	25	3ad2ant1 1133	. . . . 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 1200	. . . . 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 5100	. . . . . . . 8 ⊢ (𝐹 finSupp 0 ↔ 𝐹 finSupp (0g‘𝑅))
30	29	biimpi 216	. . . . . . 7 ⊢ (𝐹 finSupp 0 → 𝐹 finSupp (0g‘𝑅))
31	30	adantl 481	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) → 𝐹 finSupp (0g‘𝑅))
32	31	3ad2ant2 1134	. . . . 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 48369	. . . . 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 1373	. . . 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 1192	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 )) → 𝑀 ∈ LMod)
36	35	adantr 480	. . . . . 6 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 )) ∧ 𝑥 ∈ 𝑉) → 𝑀 ∈ LMod)
37		elmapi 8776	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑆 ↑m 𝑉) → 𝐹:𝑉⟶𝑆)
38		ffvelcdm 7015	. . . . . . . . . . . 12 ⊢ ((𝐹:𝑉⟶𝑆 ∧ 𝑥 ∈ 𝑉) → (𝐹‘𝑥) ∈ 𝑆)
39	38	ex 412	. . . . . . . . . . 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 480	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) → ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑥 ∈ 𝑉 → (𝐹‘𝑥) ∈ 𝑆)))
43	42	impcom 407	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 )) → (𝑥 ∈ 𝑉 → (𝐹‘𝑥) ∈ 𝑆))
44	43	imp 406	. . . . . 6 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 )) ∧ 𝑥 ∈ 𝑉) → (𝐹‘𝑥) ∈ 𝑆)
45		elelpwi 4561	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑥 ∈ 𝐵)
46	45	expcom 413	. . . . . . . . 9 ⊢ (𝑉 ∈ 𝒫 𝐵 → (𝑥 ∈ 𝑉 → 𝑥 ∈ 𝐵))
47	46	3ad2ant2 1134	. . . . . . . 8 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑥 ∈ 𝑉 → 𝑥 ∈ 𝐵))
48	47	adantr 480	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 )) → (𝑥 ∈ 𝑉 → 𝑥 ∈ 𝐵))
49	48	imp 406	. . . . . 6 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 )) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ 𝐵)
50		eqid 2729	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
51	11, 3, 50, 2	lmodvscl 20781	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ (𝐹‘𝑥) ∈ 𝑆 ∧ 𝑥 ∈ 𝐵) → ((𝐹‘𝑥)( ·𝑠 ‘𝑀)𝑥) ∈ 𝐵)
52	36, 44, 49, 51	syl3anc 1373	. . . . 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 1169	. . . 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 1206	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → 𝑋 ∈ 𝑉)
55		ffvelcdm 7015	. . . . . . . . . . 11 ⊢ ((𝐹:𝑉⟶𝑆 ∧ 𝑋 ∈ 𝑉) → (𝐹‘𝑋) ∈ 𝑆)
56	55	expcom 413	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑉 → (𝐹:𝑉⟶𝑆 → (𝐹‘𝑋) ∈ 𝑆))
57	56	3ad2ant3 1135	. . . . . . . . 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 480	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) → ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹‘𝑋) ∈ 𝑆))
60	59	impcom 407	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 )) → (𝐹‘𝑋) ∈ 𝑆)
61		elelpwi 4561	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑋 ∈ 𝐵)
62	61	ancoms 458	. . . . . . . 8 ⊢ ((𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝐵)
63	62	3adant1 1130	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝐵)
64	63	adantr 480	. . . . . 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 20781	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ (𝐹‘𝑋) ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → ((𝐹‘𝑋) · 𝑋) ∈ 𝐵)
67	35, 60, 64, 66	syl3anc 1373	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 )) → ((𝐹‘𝑋) · 𝑋) ∈ 𝐵)
68	67	3adant3 1132	. . . 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 2738	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑀) = ·
70	69	a1i 11	. . . . . 6 ⊢ (𝑥 = 𝑋 → ( ·𝑠 ‘𝑀) = · )
71		fveq2 6822	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
72		id 22	. . . . . 6 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
73	70, 71, 72	oveq123d 7370	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥)( ·𝑠 ‘𝑀)𝑥) = ((𝐹‘𝑋) · 𝑋))
74	73	adantl 481	. . . 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 19840	. . 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 6821	. . . . . . . . . 10 ⊢ (𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋})) → (𝐺‘𝑥) = ((𝐹 ↾ (𝑉 ∖ {𝑋}))‘𝑥))
77	76	3ad2ant3 1135	. . . . . . . . 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 6841	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑉 ∖ {𝑋}) → ((𝐹 ↾ (𝑉 ∖ {𝑋}))‘𝑥) = (𝐹‘𝑥))
79	77, 78	sylan9eq 2784	. . . . . . . 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 7364	. . . . . . 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 5185	. . . . . 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 2735	. . . . 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 7365	. . . 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 7364	. . 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 2764	. 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 2729	. . . . . . . . . . . 12 ⊢ 𝑉 = 𝑉
87	86, 5	feq23i 6646	. . . . . . . . . . 11 ⊢ (𝐹:𝑉⟶𝑆 ↔ 𝐹:𝑉⟶(Base‘(Scalar‘𝑀)))
88	37, 87	sylib 218	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑆 ↑m 𝑉) → 𝐹:𝑉⟶(Base‘(Scalar‘𝑀)))
89	88	adantr 480	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) → 𝐹:𝑉⟶(Base‘(Scalar‘𝑀)))
90	89	3ad2ant2 1134	. . . . . . . 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 4088	. . . . . . . 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 6691	. . . . . . 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 6630	. . . . . . . 8 ⊢ (𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋})) → (𝐺:(𝑉 ∖ {𝑋})⟶(Base‘(Scalar‘𝑀)) ↔ (𝐹 ↾ (𝑉 ∖ {𝑋})):(𝑉 ∖ {𝑋})⟶(Base‘(Scalar‘𝑀))))
94	93	3ad2ant3 1135	. . . . . . 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 257	. . . . . 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 6835	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑀)) ∈ V
97		difexg 5268	. . . . . . . . 9 ⊢ (𝑉 ∈ 𝒫 𝐵 → (𝑉 ∖ {𝑋}) ∈ V)
98	97	3ad2ant2 1134	. . . . . . . 8 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑉 ∖ {𝑋}) ∈ V)
99	98	3ad2ant1 1133	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → (𝑉 ∖ {𝑋}) ∈ V)
100		elmapg 8766	. . . . . . 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 587	. . . . . 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 257	. . . . 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 4558	. . . . . . . . . 10 ⊢ (𝑉 ∈ 𝒫 𝐵 → 𝑉 ⊆ 𝐵)
104	11	sseq2i 3965	. . . . . . . . . . . 12 ⊢ (𝑉 ⊆ 𝐵 ↔ 𝑉 ⊆ (Base‘𝑀))
105	104	biimpi 216	. . . . . . . . . . 11 ⊢ (𝑉 ⊆ 𝐵 → 𝑉 ⊆ (Base‘𝑀))
106	105	ssdifssd 4098	. . . . . . . . . 10 ⊢ (𝑉 ⊆ 𝐵 → (𝑉 ∖ {𝑋}) ⊆ (Base‘𝑀))
107	103, 106	syl 17	. . . . . . . . 9 ⊢ (𝑉 ∈ 𝒫 𝐵 → (𝑉 ∖ {𝑋}) ⊆ (Base‘𝑀))
108	107	adantl 481	. . . . . . . 8 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑉 ∖ {𝑋}) ⊆ (Base‘𝑀))
109	97	adantl 481	. . . . . . . . 9 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑉 ∖ {𝑋}) ∈ V)
110		elpwg 4554	. . . . . . . . 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 257	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑉 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
113	112	3adant3 1132	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑉 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀))
114	113	3ad2ant1 1133	. . . . 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 48404	. . . . 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 1373	. . . 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 2735	. . 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 7364	. 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 2768	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  Vcvv 3436   ∖ cdif 3900   ⊆ wss 3903  𝒫 cpw 4551  {csn 4577   class class class wbr 5092   ↦ cmpt 5173   ↾ cres 5621  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349   ↑_m cmap 8753   finSupp cfsupp 9251  Basecbs 17120  +_gcplusg 17161  Scalarcsca 17164   ·_𝑠 cvsca 17165  0_gc0g 17343   Σ_g cgsu 17344  CMndccmn 19659  LModclmod 20763   linC clinc 48399

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-of 7613  df-om 7800  df-1st 7924  df-2nd 7925  df-supp 8094  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-er 8625  df-map 8755  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-fsupp 9252  df-oi 9402  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-n0 12385  df-z 12472  df-uz 12736  df-fz 13411  df-fzo 13558  df-seq 13909  df-hash 14238  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-0g 17345  df-gsum 17346  df-mre 17488  df-mrc 17489  df-acs 17491  df-mgm 18514  df-sgrp 18593  df-mnd 18609  df-submnd 18658  df-grp 18815  df-minusg 18816  df-mulg 18947  df-cntz 19196  df-cmn 19661  df-abl 19662  df-mgp 20026  df-ur 20067  df-ring 20120  df-lmod 20765  df-linc 48401

This theorem is referenced by:  lincext3  48451  lindslinindimp2lem4  48456  lincresunit3  48476