Step | Hyp | Ref
| Expression |
1 | | simp11 1202 |
. . 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 6779 |
. . . . . . . . 9
⊢
(Base‘𝑅) =
(Base‘(Scalar‘𝑀)) |
5 | 2, 4 | eqtri 2766 |
. . . . . . . 8
⊢ 𝑆 =
(Base‘(Scalar‘𝑀)) |
6 | 5 | oveq1i 7287 |
. . . . . . 7
⊢ (𝑆 ↑m 𝑉) =
((Base‘(Scalar‘𝑀)) ↑m 𝑉) |
7 | 6 | eleq2i 2830 |
. . . . . 6
⊢ (𝐹 ∈ (𝑆 ↑m 𝑉) ↔ 𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉)) |
8 | 7 | biimpi 215 |
. . . . 5
⊢ (𝐹 ∈ (𝑆 ↑m 𝑉) → 𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉)) |
9 | 8 | adantr 481 |
. . . 4
⊢ ((𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) → 𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉)) |
10 | 9 | 3ad2ant2 1133 |
. . 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 4553 |
. . . . . . 7
⊢ 𝒫
𝐵 = 𝒫
(Base‘𝑀) |
13 | 12 | eleq2i 2830 |
. . . . . 6
⊢ (𝑉 ∈ 𝒫 𝐵 ↔ 𝑉 ∈ 𝒫 (Base‘𝑀)) |
14 | 13 | biimpi 215 |
. . . . 5
⊢ (𝑉 ∈ 𝒫 𝐵 → 𝑉 ∈ 𝒫 (Base‘𝑀)) |
15 | 14 | 3ad2ant2 1133 |
. . . 4
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑉 ∈ 𝒫 (Base‘𝑀)) |
16 | 15 | 3ad2ant1 1132 |
. . 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 45717 |
. . 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 1370 |
. 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 20169 |
. . . . . 6
⊢ (𝑀 ∈ LMod → 𝑀 ∈ CMnd) |
21 | 20 | 3ad2ant1 1132 |
. . . . 5
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑀 ∈ CMnd) |
22 | 21 | 3ad2ant1 1132 |
. . . 4
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → 𝑀 ∈ CMnd) |
23 | | simp12 1203 |
. . . 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 1131 |
. . . . . 6
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) |
26 | 25 | 3ad2ant1 1132 |
. . . . 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 1198 |
. . . . 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 5084 |
. . . . . . . 8
⊢ (𝐹 finSupp 0 ↔ 𝐹 finSupp (0g‘𝑅)) |
30 | 29 | biimpi 215 |
. . . . . . 7
⊢ (𝐹 finSupp 0 → 𝐹 finSupp (0g‘𝑅)) |
31 | 30 | adantl 482 |
. . . . . 6
⊢ ((𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) → 𝐹 finSupp (0g‘𝑅)) |
32 | 31 | 3ad2ant2 1133 |
. . . . 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 45681 |
. . . . 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 1370 |
. . . 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 1190 |
. . . . . . 7
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 )) → 𝑀 ∈ LMod) |
36 | 35 | adantr 481 |
. . . . . 6
⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 )) ∧ 𝑥 ∈ 𝑉) → 𝑀 ∈ LMod) |
37 | | elmapi 8635 |
. . . . . . . . . 10
⊢ (𝐹 ∈ (𝑆 ↑m 𝑉) → 𝐹:𝑉⟶𝑆) |
38 | | ffvelrn 6961 |
. . . . . . . . . . . 12
⊢ ((𝐹:𝑉⟶𝑆 ∧ 𝑥 ∈ 𝑉) → (𝐹‘𝑥) ∈ 𝑆) |
39 | 38 | ex 413 |
. . . . . . . . . . 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 481 |
. . . . . . . 8
⊢ ((𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) → ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑥 ∈ 𝑉 → (𝐹‘𝑥) ∈ 𝑆))) |
43 | 42 | impcom 408 |
. . . . . . 7
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 )) → (𝑥 ∈ 𝑉 → (𝐹‘𝑥) ∈ 𝑆)) |
44 | 43 | imp 407 |
. . . . . 6
⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 )) ∧ 𝑥 ∈ 𝑉) → (𝐹‘𝑥) ∈ 𝑆) |
45 | | elelpwi 4547 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑥 ∈ 𝐵) |
46 | 45 | expcom 414 |
. . . . . . . . 9
⊢ (𝑉 ∈ 𝒫 𝐵 → (𝑥 ∈ 𝑉 → 𝑥 ∈ 𝐵)) |
47 | 46 | 3ad2ant2 1133 |
. . . . . . . 8
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑥 ∈ 𝑉 → 𝑥 ∈ 𝐵)) |
48 | 47 | adantr 481 |
. . . . . . 7
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 )) → (𝑥 ∈ 𝑉 → 𝑥 ∈ 𝐵)) |
49 | 48 | imp 407 |
. . . . . 6
⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 )) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ 𝐵) |
50 | | eqid 2738 |
. . . . . . 7
⊢ (
·𝑠 ‘𝑀) = ( ·𝑠
‘𝑀) |
51 | 11, 3, 50, 2 | lmodvscl 20138 |
. . . . . 6
⊢ ((𝑀 ∈ LMod ∧ (𝐹‘𝑥) ∈ 𝑆 ∧ 𝑥 ∈ 𝐵) → ((𝐹‘𝑥)( ·𝑠
‘𝑀)𝑥) ∈ 𝐵) |
52 | 36, 44, 49, 51 | syl3anc 1370 |
. . . . 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 1167 |
. . . 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 1204 |
. . . 4
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → 𝑋 ∈ 𝑉) |
55 | | ffvelrn 6961 |
. . . . . . . . . . 11
⊢ ((𝐹:𝑉⟶𝑆 ∧ 𝑋 ∈ 𝑉) → (𝐹‘𝑋) ∈ 𝑆) |
56 | 55 | expcom 414 |
. . . . . . . . . 10
⊢ (𝑋 ∈ 𝑉 → (𝐹:𝑉⟶𝑆 → (𝐹‘𝑋) ∈ 𝑆)) |
57 | 56 | 3ad2ant3 1134 |
. . . . . . . . 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 481 |
. . . . . . 7
⊢ ((𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) → ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹‘𝑋) ∈ 𝑆)) |
60 | 59 | impcom 408 |
. . . . . 6
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 )) → (𝐹‘𝑋) ∈ 𝑆) |
61 | | elelpwi 4547 |
. . . . . . . . 9
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 𝐵) → 𝑋 ∈ 𝐵) |
62 | 61 | ancoms 459 |
. . . . . . . 8
⊢ ((𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝐵) |
63 | 62 | 3adant1 1129 |
. . . . . . 7
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝐵) |
64 | 63 | adantr 481 |
. . . . . 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 20138 |
. . . . . 6
⊢ ((𝑀 ∈ LMod ∧ (𝐹‘𝑋) ∈ 𝑆 ∧ 𝑋 ∈ 𝐵) → ((𝐹‘𝑋) · 𝑋) ∈ 𝐵) |
67 | 35, 60, 64, 66 | syl3anc 1370 |
. . . . 5
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 )) → ((𝐹‘𝑋) · 𝑋) ∈ 𝐵) |
68 | 67 | 3adant3 1131 |
. . . 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 2747 |
. . . . . . 7
⊢ (
·𝑠 ‘𝑀) = · |
70 | 69 | a1i 11 |
. . . . . 6
⊢ (𝑥 = 𝑋 → (
·𝑠 ‘𝑀) = · ) |
71 | | fveq2 6776 |
. . . . . 6
⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋)) |
72 | | id 22 |
. . . . . 6
⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋) |
73 | 70, 71, 72 | oveq123d 7298 |
. . . . 5
⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥)( ·𝑠
‘𝑀)𝑥) = ((𝐹‘𝑋) · 𝑋)) |
74 | 73 | adantl 482 |
. . . 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 19560 |
. . 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 6775 |
. . . . . . . . . 10
⊢ (𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋})) → (𝐺‘𝑥) = ((𝐹 ↾ (𝑉 ∖ {𝑋}))‘𝑥)) |
77 | 76 | 3ad2ant3 1134 |
. . . . . . . . 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 6795 |
. . . . . . . . 9
⊢ (𝑥 ∈ (𝑉 ∖ {𝑋}) → ((𝐹 ↾ (𝑉 ∖ {𝑋}))‘𝑥) = (𝐹‘𝑥)) |
79 | 77, 78 | sylan9eq 2798 |
. . . . . . . 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 7292 |
. . . . . . 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 5176 |
. . . . . 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 2744 |
. . . . 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 7293 |
. . . 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 7292 |
. . 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 2778 |
. 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 2738 |
. . . . . . . . . . . 12
⊢ 𝑉 = 𝑉 |
87 | 86, 5 | feq23i 6596 |
. . . . . . . . . . 11
⊢ (𝐹:𝑉⟶𝑆 ↔ 𝐹:𝑉⟶(Base‘(Scalar‘𝑀))) |
88 | 37, 87 | sylib 217 |
. . . . . . . . . 10
⊢ (𝐹 ∈ (𝑆 ↑m 𝑉) → 𝐹:𝑉⟶(Base‘(Scalar‘𝑀))) |
89 | 88 | adantr 481 |
. . . . . . . . 9
⊢ ((𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) → 𝐹:𝑉⟶(Base‘(Scalar‘𝑀))) |
90 | 89 | 3ad2ant2 1133 |
. . . . . . . 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 4068 |
. . . . . . . 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 6643 |
. . . . . . 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 6583 |
. . . . . . . 8
⊢ (𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋})) → (𝐺:(𝑉 ∖ {𝑋})⟶(Base‘(Scalar‘𝑀)) ↔ (𝐹 ↾ (𝑉 ∖ {𝑋})):(𝑉 ∖ {𝑋})⟶(Base‘(Scalar‘𝑀)))) |
94 | 93 | 3ad2ant3 1134 |
. . . . . . 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 256 |
. . . . . 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 6789 |
. . . . . . 7
⊢
(Base‘(Scalar‘𝑀)) ∈ V |
97 | | difexg 5253 |
. . . . . . . . 9
⊢ (𝑉 ∈ 𝒫 𝐵 → (𝑉 ∖ {𝑋}) ∈ V) |
98 | 97 | 3ad2ant2 1133 |
. . . . . . . 8
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑉 ∖ {𝑋}) ∈ V) |
99 | 98 | 3ad2ant1 1132 |
. . . . . . 7
⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ (𝐹 ∈ (𝑆 ↑m 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → (𝑉 ∖ {𝑋}) ∈ V) |
100 | | elmapg 8626 |
. . . . . . 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 256 |
. . . . 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 4544 |
. . . . . . . . . 10
⊢ (𝑉 ∈ 𝒫 𝐵 → 𝑉 ⊆ 𝐵) |
104 | 11 | sseq2i 3951 |
. . . . . . . . . . . 12
⊢ (𝑉 ⊆ 𝐵 ↔ 𝑉 ⊆ (Base‘𝑀)) |
105 | 104 | biimpi 215 |
. . . . . . . . . . 11
⊢ (𝑉 ⊆ 𝐵 → 𝑉 ⊆ (Base‘𝑀)) |
106 | 105 | ssdifssd 4078 |
. . . . . . . . . 10
⊢ (𝑉 ⊆ 𝐵 → (𝑉 ∖ {𝑋}) ⊆ (Base‘𝑀)) |
107 | 103, 106 | syl 17 |
. . . . . . . . 9
⊢ (𝑉 ∈ 𝒫 𝐵 → (𝑉 ∖ {𝑋}) ⊆ (Base‘𝑀)) |
108 | 107 | adantl 482 |
. . . . . . . 8
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑉 ∖ {𝑋}) ⊆ (Base‘𝑀)) |
109 | 97 | adantl 482 |
. . . . . . . . 9
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑉 ∖ {𝑋}) ∈ V) |
110 | | elpwg 4538 |
. . . . . . . . 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 256 |
. . . . . . 7
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑉 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)) |
113 | 112 | 3adant3 1131 |
. . . . . 6
⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝑉 ∖ {𝑋}) ∈ 𝒫 (Base‘𝑀)) |
114 | 113 | 3ad2ant1 1132 |
. . . . 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 45717 |
. . . . 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 1370 |
. . . 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 2744 |
. . 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 7292 |
. 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 2782 |
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))) |