Proof of Theorem mulmarep1gsum1
Step | Hyp | Ref
| Expression |
1 | | simp1 1061 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐽 ≠ 𝐾)) → 𝑅 ∈ Ring) |
2 | 1 | adantr 481 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐽 ≠ 𝐾)) ∧ 𝑙 ∈ 𝑁) → 𝑅 ∈ Ring) |
3 | | simp2 1062 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐽 ≠ 𝐾)) → (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) |
4 | 3 | adantr 481 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐽 ≠ 𝐾)) ∧ 𝑙 ∈ 𝑁) → (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) |
5 | | simp1 1061 |
. . . . . . 7
⊢ ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐽 ≠ 𝐾) → 𝐼 ∈ 𝑁) |
6 | 5 | 3ad2ant3 1084 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐽 ≠ 𝐾)) → 𝐼 ∈ 𝑁) |
7 | 6 | adantr 481 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐽 ≠ 𝐾)) ∧ 𝑙 ∈ 𝑁) → 𝐼 ∈ 𝑁) |
8 | | simp2 1062 |
. . . . . . 7
⊢ ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐽 ≠ 𝐾) → 𝐽 ∈ 𝑁) |
9 | 8 | 3ad2ant3 1084 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐽 ≠ 𝐾)) → 𝐽 ∈ 𝑁) |
10 | 9 | adantr 481 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐽 ≠ 𝐾)) ∧ 𝑙 ∈ 𝑁) → 𝐽 ∈ 𝑁) |
11 | | simpr 477 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐽 ≠ 𝐾)) ∧ 𝑙 ∈ 𝑁) → 𝑙 ∈ 𝑁) |
12 | | marepvcl.a |
. . . . . 6
⊢ 𝐴 = (𝑁 Mat 𝑅) |
13 | | marepvcl.b |
. . . . . 6
⊢ 𝐵 = (Base‘𝐴) |
14 | | marepvcl.v |
. . . . . 6
⊢ 𝑉 = ((Base‘𝑅) ↑𝑚
𝑁) |
15 | | ma1repvcl.1 |
. . . . . 6
⊢ 1 =
(1r‘𝐴) |
16 | | mulmarep1el.0 |
. . . . . 6
⊢ 0 =
(0g‘𝑅) |
17 | | mulmarep1el.e |
. . . . . 6
⊢ 𝐸 = (( 1 (𝑁 matRepV 𝑅)𝐶)‘𝐾) |
18 | 12, 13, 14, 15, 16, 17 | mulmarep1el 20378 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁)) → ((𝐼𝑋𝑙)(.r‘𝑅)(𝑙𝐸𝐽)) = if(𝐽 = 𝐾, ((𝐼𝑋𝑙)(.r‘𝑅)(𝐶‘𝑙)), if(𝐽 = 𝑙, (𝐼𝑋𝑙), 0 ))) |
19 | 2, 4, 7, 10, 11, 18 | syl113anc 1338 |
. . . 4
⊢ (((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐽 ≠ 𝐾)) ∧ 𝑙 ∈ 𝑁) → ((𝐼𝑋𝑙)(.r‘𝑅)(𝑙𝐸𝐽)) = if(𝐽 = 𝐾, ((𝐼𝑋𝑙)(.r‘𝑅)(𝐶‘𝑙)), if(𝐽 = 𝑙, (𝐼𝑋𝑙), 0 ))) |
20 | 19 | mpteq2dva 4744 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐽 ≠ 𝐾)) → (𝑙 ∈ 𝑁 ↦ ((𝐼𝑋𝑙)(.r‘𝑅)(𝑙𝐸𝐽))) = (𝑙 ∈ 𝑁 ↦ if(𝐽 = 𝐾, ((𝐼𝑋𝑙)(.r‘𝑅)(𝐶‘𝑙)), if(𝐽 = 𝑙, (𝐼𝑋𝑙), 0 )))) |
21 | 20 | oveq2d 6666 |
. 2
⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐽 ≠ 𝐾)) → (𝑅 Σg (𝑙 ∈ 𝑁 ↦ ((𝐼𝑋𝑙)(.r‘𝑅)(𝑙𝐸𝐽)))) = (𝑅 Σg (𝑙 ∈ 𝑁 ↦ if(𝐽 = 𝐾, ((𝐼𝑋𝑙)(.r‘𝑅)(𝐶‘𝑙)), if(𝐽 = 𝑙, (𝐼𝑋𝑙), 0 ))))) |
22 | | df-ne 2795 |
. . . . . . . 8
⊢ (𝐽 ≠ 𝐾 ↔ ¬ 𝐽 = 𝐾) |
23 | 22 | biimpi 206 |
. . . . . . 7
⊢ (𝐽 ≠ 𝐾 → ¬ 𝐽 = 𝐾) |
24 | 23 | 3ad2ant3 1084 |
. . . . . 6
⊢ ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐽 ≠ 𝐾) → ¬ 𝐽 = 𝐾) |
25 | 24 | 3ad2ant3 1084 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐽 ≠ 𝐾)) → ¬ 𝐽 = 𝐾) |
26 | 25 | iffalsed 4097 |
. . . 4
⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐽 ≠ 𝐾)) → if(𝐽 = 𝐾, ((𝐼𝑋𝑙)(.r‘𝑅)(𝐶‘𝑙)), if(𝐽 = 𝑙, (𝐼𝑋𝑙), 0 )) = if(𝐽 = 𝑙, (𝐼𝑋𝑙), 0 )) |
27 | 26 | mpteq2dv 4745 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐽 ≠ 𝐾)) → (𝑙 ∈ 𝑁 ↦ if(𝐽 = 𝐾, ((𝐼𝑋𝑙)(.r‘𝑅)(𝐶‘𝑙)), if(𝐽 = 𝑙, (𝐼𝑋𝑙), 0 ))) = (𝑙 ∈ 𝑁 ↦ if(𝐽 = 𝑙, (𝐼𝑋𝑙), 0 ))) |
28 | 27 | oveq2d 6666 |
. 2
⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐽 ≠ 𝐾)) → (𝑅 Σg (𝑙 ∈ 𝑁 ↦ if(𝐽 = 𝐾, ((𝐼𝑋𝑙)(.r‘𝑅)(𝐶‘𝑙)), if(𝐽 = 𝑙, (𝐼𝑋𝑙), 0 )))) = (𝑅 Σg (𝑙 ∈ 𝑁 ↦ if(𝐽 = 𝑙, (𝐼𝑋𝑙), 0 )))) |
29 | | ringmnd 18556 |
. . . 4
⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd) |
30 | 29 | 3ad2ant1 1082 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐽 ≠ 𝐾)) → 𝑅 ∈ Mnd) |
31 | 12, 13 | matrcl 20218 |
. . . . . 6
⊢ (𝑋 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
32 | 31 | simpld 475 |
. . . . 5
⊢ (𝑋 ∈ 𝐵 → 𝑁 ∈ Fin) |
33 | 32 | 3ad2ant1 1082 |
. . . 4
⊢ ((𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) → 𝑁 ∈ Fin) |
34 | 33 | 3ad2ant2 1083 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐽 ≠ 𝐾)) → 𝑁 ∈ Fin) |
35 | | eqcom 2629 |
. . . . 5
⊢ (𝐽 = 𝑙 ↔ 𝑙 = 𝐽) |
36 | | ifbi 4107 |
. . . . . 6
⊢ ((𝐽 = 𝑙 ↔ 𝑙 = 𝐽) → if(𝐽 = 𝑙, (𝐼𝑋𝑙), 0 ) = if(𝑙 = 𝐽, (𝐼𝑋𝑙), 0 )) |
37 | | oveq2 6658 |
. . . . . . . 8
⊢ (𝑙 = 𝐽 → (𝐼𝑋𝑙) = (𝐼𝑋𝐽)) |
38 | 37 | adantl 482 |
. . . . . . 7
⊢ (((𝐽 = 𝑙 ↔ 𝑙 = 𝐽) ∧ 𝑙 = 𝐽) → (𝐼𝑋𝑙) = (𝐼𝑋𝐽)) |
39 | 38 | ifeq1da 4116 |
. . . . . 6
⊢ ((𝐽 = 𝑙 ↔ 𝑙 = 𝐽) → if(𝑙 = 𝐽, (𝐼𝑋𝑙), 0 ) = if(𝑙 = 𝐽, (𝐼𝑋𝐽), 0 )) |
40 | 36, 39 | eqtrd 2656 |
. . . . 5
⊢ ((𝐽 = 𝑙 ↔ 𝑙 = 𝐽) → if(𝐽 = 𝑙, (𝐼𝑋𝑙), 0 ) = if(𝑙 = 𝐽, (𝐼𝑋𝐽), 0 )) |
41 | 35, 40 | ax-mp 5 |
. . . 4
⊢ if(𝐽 = 𝑙, (𝐼𝑋𝑙), 0 ) = if(𝑙 = 𝐽, (𝐼𝑋𝐽), 0 ) |
42 | 41 | mpteq2i 4741 |
. . 3
⊢ (𝑙 ∈ 𝑁 ↦ if(𝐽 = 𝑙, (𝐼𝑋𝑙), 0 )) = (𝑙 ∈ 𝑁 ↦ if(𝑙 = 𝐽, (𝐼𝑋𝐽), 0 )) |
43 | 13 | eleq2i 2693 |
. . . . . . 7
⊢ (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ (Base‘𝐴)) |
44 | 43 | biimpi 206 |
. . . . . 6
⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ (Base‘𝐴)) |
45 | 44 | 3ad2ant1 1082 |
. . . . 5
⊢ ((𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) → 𝑋 ∈ (Base‘𝐴)) |
46 | 45 | 3ad2ant2 1083 |
. . . 4
⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐽 ≠ 𝐾)) → 𝑋 ∈ (Base‘𝐴)) |
47 | | eqid 2622 |
. . . . 5
⊢
(Base‘𝑅) =
(Base‘𝑅) |
48 | 12, 47 | matecl 20231 |
. . . 4
⊢ ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝑋 ∈ (Base‘𝐴)) → (𝐼𝑋𝐽) ∈ (Base‘𝑅)) |
49 | 6, 9, 46, 48 | syl3anc 1326 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐽 ≠ 𝐾)) → (𝐼𝑋𝐽) ∈ (Base‘𝑅)) |
50 | 16, 30, 34, 9, 42, 49 | gsummptif1n0 18365 |
. 2
⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐽 ≠ 𝐾)) → (𝑅 Σg (𝑙 ∈ 𝑁 ↦ if(𝐽 = 𝑙, (𝐼𝑋𝑙), 0 ))) = (𝐼𝑋𝐽)) |
51 | 21, 28, 50 | 3eqtrd 2660 |
1
⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐽 ≠ 𝐾)) → (𝑅 Σg (𝑙 ∈ 𝑁 ↦ ((𝐼𝑋𝑙)(.r‘𝑅)(𝑙𝐸𝐽)))) = (𝐼𝑋𝐽)) |