Step | Hyp | Ref
| Expression |
1 | | matgsum.j |
. . . 4
⊢ (𝜑 → 𝐽 ∈ 𝑊) |
2 | 1 | mptexd 7040 |
. . 3
⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈)) ∈ V) |
3 | | matgsum.a |
. . . . 5
⊢ 𝐴 = (𝑁 Mat 𝑅) |
4 | 3 | ovexi 7247 |
. . . 4
⊢ 𝐴 ∈ V |
5 | 4 | a1i 11 |
. . 3
⊢ (𝜑 → 𝐴 ∈ V) |
6 | | ovexd 7248 |
. . 3
⊢ (𝜑 → (𝑅 freeLMod (𝑁 × 𝑁)) ∈ V) |
7 | | matgsum.i |
. . . . 5
⊢ (𝜑 → 𝑁 ∈ Fin) |
8 | | matgsum.r |
. . . . 5
⊢ (𝜑 → 𝑅 ∈ Ring) |
9 | | eqid 2737 |
. . . . . 6
⊢ (𝑅 freeLMod (𝑁 × 𝑁)) = (𝑅 freeLMod (𝑁 × 𝑁)) |
10 | 3, 9 | matbas 21310 |
. . . . 5
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) →
(Base‘(𝑅 freeLMod
(𝑁 × 𝑁))) = (Base‘𝐴)) |
11 | 7, 8, 10 | syl2anc 587 |
. . . 4
⊢ (𝜑 → (Base‘(𝑅 freeLMod (𝑁 × 𝑁))) = (Base‘𝐴)) |
12 | 11 | eqcomd 2743 |
. . 3
⊢ (𝜑 → (Base‘𝐴) = (Base‘(𝑅 freeLMod (𝑁 × 𝑁)))) |
13 | 3, 9 | matplusg 21311 |
. . . . 5
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) →
(+g‘(𝑅
freeLMod (𝑁 × 𝑁))) = (+g‘𝐴)) |
14 | 7, 8, 13 | syl2anc 587 |
. . . 4
⊢ (𝜑 →
(+g‘(𝑅
freeLMod (𝑁 × 𝑁))) = (+g‘𝐴)) |
15 | 14 | eqcomd 2743 |
. . 3
⊢ (𝜑 → (+g‘𝐴) = (+g‘(𝑅 freeLMod (𝑁 × 𝑁)))) |
16 | 2, 5, 6, 12, 15 | gsumpropd 18150 |
. 2
⊢ (𝜑 → (𝐴 Σg (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈))) = ((𝑅 freeLMod (𝑁 × 𝑁)) Σg (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈)))) |
17 | | mpompts 7835 |
. . . . . 6
⊢ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈) = (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st
‘𝑧) / 𝑖⦌⦋(2nd
‘𝑧) / 𝑗⦌𝑈) |
18 | 17 | a1i 11 |
. . . . 5
⊢ (𝜑 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈) = (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st
‘𝑧) / 𝑖⦌⦋(2nd
‘𝑧) / 𝑗⦌𝑈)) |
19 | 18 | mpteq2dv 5151 |
. . . 4
⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈)) = (𝑦 ∈ 𝐽 ↦ (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st
‘𝑧) / 𝑖⦌⦋(2nd
‘𝑧) / 𝑗⦌𝑈))) |
20 | 19 | oveq2d 7229 |
. . 3
⊢ (𝜑 → ((𝑅 freeLMod (𝑁 × 𝑁)) Σg (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈))) = ((𝑅 freeLMod (𝑁 × 𝑁)) Σg (𝑦 ∈ 𝐽 ↦ (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st
‘𝑧) / 𝑖⦌⦋(2nd
‘𝑧) / 𝑗⦌𝑈)))) |
21 | | eqid 2737 |
. . . 4
⊢
(Base‘(𝑅
freeLMod (𝑁 × 𝑁))) = (Base‘(𝑅 freeLMod (𝑁 × 𝑁))) |
22 | | eqid 2737 |
. . . 4
⊢
(0g‘(𝑅 freeLMod (𝑁 × 𝑁))) = (0g‘(𝑅 freeLMod (𝑁 × 𝑁))) |
23 | | xpfi 8942 |
. . . . 5
⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑁 × 𝑁) ∈ Fin) |
24 | 7, 7, 23 | syl2anc 587 |
. . . 4
⊢ (𝜑 → (𝑁 × 𝑁) ∈ Fin) |
25 | | matgsum.f |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈) ∈ 𝐵) |
26 | | matgsum.b |
. . . . . 6
⊢ 𝐵 = (Base‘𝐴) |
27 | 25, 26 | eleqtrdi 2848 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈) ∈ (Base‘𝐴)) |
28 | 17 | eqcomi 2746 |
. . . . . 6
⊢ (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st
‘𝑧) / 𝑖⦌⦋(2nd
‘𝑧) / 𝑗⦌𝑈) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈) |
29 | 28 | a1i 11 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st
‘𝑧) / 𝑖⦌⦋(2nd
‘𝑧) / 𝑗⦌𝑈) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈)) |
30 | 7, 8 | jca 515 |
. . . . . . 7
⊢ (𝜑 → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring)) |
31 | 30 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring)) |
32 | 31, 10 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → (Base‘(𝑅 freeLMod (𝑁 × 𝑁))) = (Base‘𝐴)) |
33 | 27, 29, 32 | 3eltr4d 2853 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st
‘𝑧) / 𝑖⦌⦋(2nd
‘𝑧) / 𝑗⦌𝑈) ∈ (Base‘(𝑅 freeLMod (𝑁 × 𝑁)))) |
34 | | matgsum.w |
. . . . . 6
⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈)) finSupp 0 ) |
35 | 28 | mpteq2i 5147 |
. . . . . 6
⊢ (𝑦 ∈ 𝐽 ↦ (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st
‘𝑧) / 𝑖⦌⦋(2nd
‘𝑧) / 𝑗⦌𝑈)) = (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈)) |
36 | | matgsum.z |
. . . . . . 7
⊢ 0 =
(0g‘𝐴) |
37 | 36 | eqcomi 2746 |
. . . . . 6
⊢
(0g‘𝐴) = 0 |
38 | 34, 35, 37 | 3brtr4g 5087 |
. . . . 5
⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st
‘𝑧) / 𝑖⦌⦋(2nd
‘𝑧) / 𝑗⦌𝑈)) finSupp (0g‘𝐴)) |
39 | 3, 9 | mat0 21314 |
. . . . . 6
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) →
(0g‘(𝑅
freeLMod (𝑁 × 𝑁))) = (0g‘𝐴)) |
40 | 7, 8, 39 | syl2anc 587 |
. . . . 5
⊢ (𝜑 →
(0g‘(𝑅
freeLMod (𝑁 × 𝑁))) = (0g‘𝐴)) |
41 | 38, 40 | breqtrrd 5081 |
. . . 4
⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st
‘𝑧) / 𝑖⦌⦋(2nd
‘𝑧) / 𝑗⦌𝑈)) finSupp (0g‘(𝑅 freeLMod (𝑁 × 𝑁)))) |
42 | 9, 21, 22, 24, 1, 8, 33, 41 | frlmgsum 20734 |
. . 3
⊢ (𝜑 → ((𝑅 freeLMod (𝑁 × 𝑁)) Σg (𝑦 ∈ 𝐽 ↦ (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st
‘𝑧) / 𝑖⦌⦋(2nd
‘𝑧) / 𝑗⦌𝑈))) = (𝑧 ∈ (𝑁 × 𝑁) ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ ⦋(1st
‘𝑧) / 𝑖⦌⦋(2nd
‘𝑧) / 𝑗⦌𝑈)))) |
43 | 20, 42 | eqtrd 2777 |
. 2
⊢ (𝜑 → ((𝑅 freeLMod (𝑁 × 𝑁)) Σg (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈))) = (𝑧 ∈ (𝑁 × 𝑁) ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ ⦋(1st
‘𝑧) / 𝑖⦌⦋(2nd
‘𝑧) / 𝑗⦌𝑈)))) |
44 | | fvex 6730 |
. . . . . . . 8
⊢
(2nd ‘𝑧) ∈ V |
45 | | csbov2g 7259 |
. . . . . . . 8
⊢
((2nd ‘𝑧) ∈ V →
⦋(2nd ‘𝑧) / 𝑗⦌(𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈)) = (𝑅 Σg
⦋(2nd ‘𝑧) / 𝑗⦌(𝑦 ∈ 𝐽 ↦ 𝑈))) |
46 | 44, 45 | ax-mp 5 |
. . . . . . 7
⊢
⦋(2nd ‘𝑧) / 𝑗⦌(𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈)) = (𝑅 Σg
⦋(2nd ‘𝑧) / 𝑗⦌(𝑦 ∈ 𝐽 ↦ 𝑈)) |
47 | 46 | csbeq2i 3819 |
. . . . . 6
⊢
⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd
‘𝑧) / 𝑗⦌(𝑅 Σg
(𝑦 ∈ 𝐽 ↦ 𝑈)) = ⦋(1st
‘𝑧) / 𝑖⦌(𝑅 Σg
⦋(2nd ‘𝑧) / 𝑗⦌(𝑦 ∈ 𝐽 ↦ 𝑈)) |
48 | | fvex 6730 |
. . . . . . 7
⊢
(1st ‘𝑧) ∈ V |
49 | | csbov2g 7259 |
. . . . . . 7
⊢
((1st ‘𝑧) ∈ V →
⦋(1st ‘𝑧) / 𝑖⦌(𝑅 Σg
⦋(2nd ‘𝑧) / 𝑗⦌(𝑦 ∈ 𝐽 ↦ 𝑈)) = (𝑅 Σg
⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd
‘𝑧) / 𝑗⦌(𝑦 ∈ 𝐽 ↦ 𝑈))) |
50 | 48, 49 | ax-mp 5 |
. . . . . 6
⊢
⦋(1st ‘𝑧) / 𝑖⦌(𝑅 Σg
⦋(2nd ‘𝑧) / 𝑗⦌(𝑦 ∈ 𝐽 ↦ 𝑈)) = (𝑅 Σg
⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd
‘𝑧) / 𝑗⦌(𝑦 ∈ 𝐽 ↦ 𝑈)) |
51 | | csbmpt2 5439 |
. . . . . . . . . 10
⊢
((2nd ‘𝑧) ∈ V →
⦋(2nd ‘𝑧) / 𝑗⦌(𝑦 ∈ 𝐽 ↦ 𝑈) = (𝑦 ∈ 𝐽 ↦ ⦋(2nd
‘𝑧) / 𝑗⦌𝑈)) |
52 | 44, 51 | ax-mp 5 |
. . . . . . . . 9
⊢
⦋(2nd ‘𝑧) / 𝑗⦌(𝑦 ∈ 𝐽 ↦ 𝑈) = (𝑦 ∈ 𝐽 ↦ ⦋(2nd
‘𝑧) / 𝑗⦌𝑈) |
53 | 52 | csbeq2i 3819 |
. . . . . . . 8
⊢
⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd
‘𝑧) / 𝑗⦌(𝑦 ∈ 𝐽 ↦ 𝑈) = ⦋(1st
‘𝑧) / 𝑖⦌(𝑦 ∈ 𝐽 ↦ ⦋(2nd
‘𝑧) / 𝑗⦌𝑈) |
54 | | csbmpt2 5439 |
. . . . . . . . 9
⊢
((1st ‘𝑧) ∈ V →
⦋(1st ‘𝑧) / 𝑖⦌(𝑦 ∈ 𝐽 ↦ ⦋(2nd
‘𝑧) / 𝑗⦌𝑈) = (𝑦 ∈ 𝐽 ↦ ⦋(1st
‘𝑧) / 𝑖⦌⦋(2nd
‘𝑧) / 𝑗⦌𝑈)) |
55 | 48, 54 | ax-mp 5 |
. . . . . . . 8
⊢
⦋(1st ‘𝑧) / 𝑖⦌(𝑦 ∈ 𝐽 ↦ ⦋(2nd
‘𝑧) / 𝑗⦌𝑈) = (𝑦 ∈ 𝐽 ↦ ⦋(1st
‘𝑧) / 𝑖⦌⦋(2nd
‘𝑧) / 𝑗⦌𝑈) |
56 | 53, 55 | eqtri 2765 |
. . . . . . 7
⊢
⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd
‘𝑧) / 𝑗⦌(𝑦 ∈ 𝐽 ↦ 𝑈) = (𝑦 ∈ 𝐽 ↦ ⦋(1st
‘𝑧) / 𝑖⦌⦋(2nd
‘𝑧) / 𝑗⦌𝑈) |
57 | 56 | oveq2i 7224 |
. . . . . 6
⊢ (𝑅 Σg
⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd
‘𝑧) / 𝑗⦌(𝑦 ∈ 𝐽 ↦ 𝑈)) = (𝑅 Σg (𝑦 ∈ 𝐽 ↦ ⦋(1st
‘𝑧) / 𝑖⦌⦋(2nd
‘𝑧) / 𝑗⦌𝑈)) |
58 | 47, 50, 57 | 3eqtrri 2770 |
. . . . 5
⊢ (𝑅 Σg
(𝑦 ∈ 𝐽 ↦ ⦋(1st
‘𝑧) / 𝑖⦌⦋(2nd
‘𝑧) / 𝑗⦌𝑈)) = ⦋(1st
‘𝑧) / 𝑖⦌⦋(2nd
‘𝑧) / 𝑗⦌(𝑅 Σg
(𝑦 ∈ 𝐽 ↦ 𝑈)) |
59 | 58 | mpteq2i 5147 |
. . . 4
⊢ (𝑧 ∈ (𝑁 × 𝑁) ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ ⦋(1st
‘𝑧) / 𝑖⦌⦋(2nd
‘𝑧) / 𝑗⦌𝑈))) = (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st
‘𝑧) / 𝑖⦌⦋(2nd
‘𝑧) / 𝑗⦌(𝑅 Σg
(𝑦 ∈ 𝐽 ↦ 𝑈))) |
60 | | mpompts 7835 |
. . . 4
⊢ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈))) = (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st
‘𝑧) / 𝑖⦌⦋(2nd
‘𝑧) / 𝑗⦌(𝑅 Σg
(𝑦 ∈ 𝐽 ↦ 𝑈))) |
61 | 59, 60 | eqtr4i 2768 |
. . 3
⊢ (𝑧 ∈ (𝑁 × 𝑁) ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ ⦋(1st
‘𝑧) / 𝑖⦌⦋(2nd
‘𝑧) / 𝑗⦌𝑈))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈))) |
62 | 61 | a1i 11 |
. 2
⊢ (𝜑 → (𝑧 ∈ (𝑁 × 𝑁) ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ ⦋(1st
‘𝑧) / 𝑖⦌⦋(2nd
‘𝑧) / 𝑗⦌𝑈))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈)))) |
63 | 16, 43, 62 | 3eqtrd 2781 |
1
⊢ (𝜑 → (𝐴 Σg (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈)))) |