Proof of Theorem mulmarep1gsum2
Step | Hyp | Ref
| Expression |
1 | | simp1 1135 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍)) → 𝑅 ∈ Ring) |
2 | 1 | adantr 481 |
. . . . . . . . . 10
⊢ (((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍)) ∧ 𝑙 ∈ 𝑁) → 𝑅 ∈ Ring) |
3 | | simpl2 1191 |
. . . . . . . . . 10
⊢ (((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍)) ∧ 𝑙 ∈ 𝑁) → (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) |
4 | | simp1 1135 |
. . . . . . . . . . . . 13
⊢ ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍) → 𝐼 ∈ 𝑁) |
5 | 4 | 3ad2ant3 1134 |
. . . . . . . . . . . 12
⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍)) → 𝐼 ∈ 𝑁) |
6 | 5 | adantr 481 |
. . . . . . . . . . 11
⊢ (((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍)) ∧ 𝑙 ∈ 𝑁) → 𝐼 ∈ 𝑁) |
7 | | simpl32 1254 |
. . . . . . . . . . 11
⊢ (((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍)) ∧ 𝑙 ∈ 𝑁) → 𝐽 ∈ 𝑁) |
8 | | simpr 485 |
. . . . . . . . . . 11
⊢ (((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍)) ∧ 𝑙 ∈ 𝑁) → 𝑙 ∈ 𝑁) |
9 | 6, 7, 8 | 3jca 1127 |
. . . . . . . . . 10
⊢ (((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍)) ∧ 𝑙 ∈ 𝑁) → (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁)) |
10 | 2, 3, 9 | 3jca 1127 |
. . . . . . . . 9
⊢ (((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍)) ∧ 𝑙 ∈ 𝑁) → (𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁))) |
11 | 10 | adantll 711 |
. . . . . . . 8
⊢ (((𝐽 = 𝐾 ∧ (𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍))) ∧ 𝑙 ∈ 𝑁) → (𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁))) |
12 | | marepvcl.a |
. . . . . . . . 9
⊢ 𝐴 = (𝑁 Mat 𝑅) |
13 | | marepvcl.b |
. . . . . . . . 9
⊢ 𝐵 = (Base‘𝐴) |
14 | | marepvcl.v |
. . . . . . . . 9
⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁) |
15 | | ma1repvcl.1 |
. . . . . . . . 9
⊢ 1 =
(1r‘𝐴) |
16 | | mulmarep1el.0 |
. . . . . . . . 9
⊢ 0 =
(0g‘𝑅) |
17 | | mulmarep1el.e |
. . . . . . . . 9
⊢ 𝐸 = (( 1 (𝑁 matRepV 𝑅)𝐶)‘𝐾) |
18 | 12, 13, 14, 15, 16, 17 | mulmarep1el 21721 |
. . . . . . . 8
⊢ ((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 | 11, 18 | syl 17 |
. . . . . . 7
⊢ (((𝐽 = 𝐾 ∧ (𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍))) ∧ 𝑙 ∈ 𝑁) → ((𝐼𝑋𝑙)(.r‘𝑅)(𝑙𝐸𝐽)) = if(𝐽 = 𝐾, ((𝐼𝑋𝑙)(.r‘𝑅)(𝐶‘𝑙)), if(𝐽 = 𝑙, (𝐼𝑋𝑙), 0 ))) |
20 | | iftrue 4465 |
. . . . . . . . 9
⊢ (𝐽 = 𝐾 → if(𝐽 = 𝐾, ((𝐼𝑋𝑙)(.r‘𝑅)(𝐶‘𝑙)), if(𝐽 = 𝑙, (𝐼𝑋𝑙), 0 )) = ((𝐼𝑋𝑙)(.r‘𝑅)(𝐶‘𝑙))) |
21 | 20 | adantr 481 |
. . . . . . . 8
⊢ ((𝐽 = 𝐾 ∧ (𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍))) → if(𝐽 = 𝐾, ((𝐼𝑋𝑙)(.r‘𝑅)(𝐶‘𝑙)), if(𝐽 = 𝑙, (𝐼𝑋𝑙), 0 )) = ((𝐼𝑋𝑙)(.r‘𝑅)(𝐶‘𝑙))) |
22 | 21 | adantr 481 |
. . . . . . 7
⊢ (((𝐽 = 𝐾 ∧ (𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍))) ∧ 𝑙 ∈ 𝑁) → if(𝐽 = 𝐾, ((𝐼𝑋𝑙)(.r‘𝑅)(𝐶‘𝑙)), if(𝐽 = 𝑙, (𝐼𝑋𝑙), 0 )) = ((𝐼𝑋𝑙)(.r‘𝑅)(𝐶‘𝑙))) |
23 | 19, 22 | eqtrd 2778 |
. . . . . 6
⊢ (((𝐽 = 𝐾 ∧ (𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍))) ∧ 𝑙 ∈ 𝑁) → ((𝐼𝑋𝑙)(.r‘𝑅)(𝑙𝐸𝐽)) = ((𝐼𝑋𝑙)(.r‘𝑅)(𝐶‘𝑙))) |
24 | 23 | mpteq2dva 5174 |
. . . . 5
⊢ ((𝐽 = 𝐾 ∧ (𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍))) → (𝑙 ∈ 𝑁 ↦ ((𝐼𝑋𝑙)(.r‘𝑅)(𝑙𝐸𝐽))) = (𝑙 ∈ 𝑁 ↦ ((𝐼𝑋𝑙)(.r‘𝑅)(𝐶‘𝑙)))) |
25 | 24 | oveq2d 7291 |
. . . 4
⊢ ((𝐽 = 𝐾 ∧ (𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍))) → (𝑅 Σg (𝑙 ∈ 𝑁 ↦ ((𝐼𝑋𝑙)(.r‘𝑅)(𝑙𝐸𝐽)))) = (𝑅 Σg (𝑙 ∈ 𝑁 ↦ ((𝐼𝑋𝑙)(.r‘𝑅)(𝐶‘𝑙))))) |
26 | | fveq1 6773 |
. . . . . . . . 9
⊢ ((𝑋 × 𝐶) = 𝑍 → ((𝑋 × 𝐶)‘𝐼) = (𝑍‘𝐼)) |
27 | 26 | eqcomd 2744 |
. . . . . . . 8
⊢ ((𝑋 × 𝐶) = 𝑍 → (𝑍‘𝐼) = ((𝑋 × 𝐶)‘𝐼)) |
28 | 27 | 3ad2ant3 1134 |
. . . . . . 7
⊢ ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍) → (𝑍‘𝐼) = ((𝑋 × 𝐶)‘𝐼)) |
29 | 28 | 3ad2ant3 1134 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍)) → (𝑍‘𝐼) = ((𝑋 × 𝐶)‘𝐼)) |
30 | 29 | adantl 482 |
. . . . 5
⊢ ((𝐽 = 𝐾 ∧ (𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍))) → (𝑍‘𝐼) = ((𝑋 × 𝐶)‘𝐼)) |
31 | | mulmarep1gsum2.x |
. . . . . 6
⊢ × =
(𝑅 maVecMul 〈𝑁, 𝑁〉) |
32 | | eqid 2738 |
. . . . . 6
⊢
(Base‘𝑅) =
(Base‘𝑅) |
33 | | eqid 2738 |
. . . . . 6
⊢
(.r‘𝑅) = (.r‘𝑅) |
34 | 1 | adantl 482 |
. . . . . 6
⊢ ((𝐽 = 𝐾 ∧ (𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍))) → 𝑅 ∈ Ring) |
35 | 12, 13 | matrcl 21559 |
. . . . . . . . . 10
⊢ (𝑋 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
36 | 35 | simpld 495 |
. . . . . . . . 9
⊢ (𝑋 ∈ 𝐵 → 𝑁 ∈ Fin) |
37 | 36 | 3ad2ant1 1132 |
. . . . . . . 8
⊢ ((𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) → 𝑁 ∈ Fin) |
38 | 37 | 3ad2ant2 1133 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍)) → 𝑁 ∈ Fin) |
39 | 38 | adantl 482 |
. . . . . 6
⊢ ((𝐽 = 𝐾 ∧ (𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍))) → 𝑁 ∈ Fin) |
40 | 13 | eleq2i 2830 |
. . . . . . . . . 10
⊢ (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ (Base‘𝐴)) |
41 | 40 | biimpi 215 |
. . . . . . . . 9
⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ (Base‘𝐴)) |
42 | 41 | 3ad2ant1 1132 |
. . . . . . . 8
⊢ ((𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) → 𝑋 ∈ (Base‘𝐴)) |
43 | 42 | 3ad2ant2 1133 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍)) → 𝑋 ∈ (Base‘𝐴)) |
44 | 43 | adantl 482 |
. . . . . 6
⊢ ((𝐽 = 𝐾 ∧ (𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍))) → 𝑋 ∈ (Base‘𝐴)) |
45 | 14 | eleq2i 2830 |
. . . . . . . . . 10
⊢ (𝐶 ∈ 𝑉 ↔ 𝐶 ∈ ((Base‘𝑅) ↑m 𝑁)) |
46 | 45 | biimpi 215 |
. . . . . . . . 9
⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ ((Base‘𝑅) ↑m 𝑁)) |
47 | 46 | 3ad2ant2 1133 |
. . . . . . . 8
⊢ ((𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) → 𝐶 ∈ ((Base‘𝑅) ↑m 𝑁)) |
48 | 47 | 3ad2ant2 1133 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍)) → 𝐶 ∈ ((Base‘𝑅) ↑m 𝑁)) |
49 | 48 | adantl 482 |
. . . . . 6
⊢ ((𝐽 = 𝐾 ∧ (𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍))) → 𝐶 ∈ ((Base‘𝑅) ↑m 𝑁)) |
50 | 5 | adantl 482 |
. . . . . 6
⊢ ((𝐽 = 𝐾 ∧ (𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍))) → 𝐼 ∈ 𝑁) |
51 | 12, 31, 32, 33, 34, 39, 44, 49, 50 | mavmulfv 21695 |
. . . . 5
⊢ ((𝐽 = 𝐾 ∧ (𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍))) → ((𝑋 × 𝐶)‘𝐼) = (𝑅 Σg (𝑙 ∈ 𝑁 ↦ ((𝐼𝑋𝑙)(.r‘𝑅)(𝐶‘𝑙))))) |
52 | 30, 51 | eqtrd 2778 |
. . . 4
⊢ ((𝐽 = 𝐾 ∧ (𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍))) → (𝑍‘𝐼) = (𝑅 Σg (𝑙 ∈ 𝑁 ↦ ((𝐼𝑋𝑙)(.r‘𝑅)(𝐶‘𝑙))))) |
53 | | iftrue 4465 |
. . . . . 6
⊢ (𝐽 = 𝐾 → if(𝐽 = 𝐾, (𝑍‘𝐼), (𝐼𝑋𝐽)) = (𝑍‘𝐼)) |
54 | 53 | eqcomd 2744 |
. . . . 5
⊢ (𝐽 = 𝐾 → (𝑍‘𝐼) = if(𝐽 = 𝐾, (𝑍‘𝐼), (𝐼𝑋𝐽))) |
55 | 54 | adantr 481 |
. . . 4
⊢ ((𝐽 = 𝐾 ∧ (𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍))) → (𝑍‘𝐼) = if(𝐽 = 𝐾, (𝑍‘𝐼), (𝐼𝑋𝐽))) |
56 | 25, 52, 55 | 3eqtr2d 2784 |
. . 3
⊢ ((𝐽 = 𝐾 ∧ (𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍))) → (𝑅 Σg (𝑙 ∈ 𝑁 ↦ ((𝐼𝑋𝑙)(.r‘𝑅)(𝑙𝐸𝐽)))) = if(𝐽 = 𝐾, (𝑍‘𝐼), (𝐼𝑋𝐽))) |
57 | 56 | ex 413 |
. 2
⊢ (𝐽 = 𝐾 → ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍)) → (𝑅 Σg (𝑙 ∈ 𝑁 ↦ ((𝐼𝑋𝑙)(.r‘𝑅)(𝑙𝐸𝐽)))) = if(𝐽 = 𝐾, (𝑍‘𝐼), (𝐼𝑋𝐽)))) |
58 | 1 | adantr 481 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍)) ∧ 𝐽 ≠ 𝐾) → 𝑅 ∈ Ring) |
59 | | simpl2 1191 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍)) ∧ 𝐽 ≠ 𝐾) → (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) |
60 | 5 | adantr 481 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍)) ∧ 𝐽 ≠ 𝐾) → 𝐼 ∈ 𝑁) |
61 | | simpl32 1254 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍)) ∧ 𝐽 ≠ 𝐾) → 𝐽 ∈ 𝑁) |
62 | | simpr 485 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍)) ∧ 𝐽 ≠ 𝐾) → 𝐽 ≠ 𝐾) |
63 | 12, 13, 14, 15, 16, 17 | mulmarep1gsum1 21722 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐽 ≠ 𝐾)) → (𝑅 Σg (𝑙 ∈ 𝑁 ↦ ((𝐼𝑋𝑙)(.r‘𝑅)(𝑙𝐸𝐽)))) = (𝐼𝑋𝐽)) |
64 | 58, 59, 60, 61, 62, 63 | syl113anc 1381 |
. . . 4
⊢ (((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍)) ∧ 𝐽 ≠ 𝐾) → (𝑅 Σg (𝑙 ∈ 𝑁 ↦ ((𝐼𝑋𝑙)(.r‘𝑅)(𝑙𝐸𝐽)))) = (𝐼𝑋𝐽)) |
65 | | df-ne 2944 |
. . . . . 6
⊢ (𝐽 ≠ 𝐾 ↔ ¬ 𝐽 = 𝐾) |
66 | | iffalse 4468 |
. . . . . . 7
⊢ (¬
𝐽 = 𝐾 → if(𝐽 = 𝐾, (𝑍‘𝐼), (𝐼𝑋𝐽)) = (𝐼𝑋𝐽)) |
67 | 66 | eqcomd 2744 |
. . . . . 6
⊢ (¬
𝐽 = 𝐾 → (𝐼𝑋𝐽) = if(𝐽 = 𝐾, (𝑍‘𝐼), (𝐼𝑋𝐽))) |
68 | 65, 67 | sylbi 216 |
. . . . 5
⊢ (𝐽 ≠ 𝐾 → (𝐼𝑋𝐽) = if(𝐽 = 𝐾, (𝑍‘𝐼), (𝐼𝑋𝐽))) |
69 | 68 | adantl 482 |
. . . 4
⊢ (((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍)) ∧ 𝐽 ≠ 𝐾) → (𝐼𝑋𝐽) = if(𝐽 = 𝐾, (𝑍‘𝐼), (𝐼𝑋𝐽))) |
70 | 64, 69 | eqtrd 2778 |
. . 3
⊢ (((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍)) ∧ 𝐽 ≠ 𝐾) → (𝑅 Σg (𝑙 ∈ 𝑁 ↦ ((𝐼𝑋𝑙)(.r‘𝑅)(𝑙𝐸𝐽)))) = if(𝐽 = 𝐾, (𝑍‘𝐼), (𝐼𝑋𝐽))) |
71 | 70 | expcom 414 |
. 2
⊢ (𝐽 ≠ 𝐾 → ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍)) → (𝑅 Σg (𝑙 ∈ 𝑁 ↦ ((𝐼𝑋𝑙)(.r‘𝑅)(𝑙𝐸𝐽)))) = if(𝐽 = 𝐾, (𝑍‘𝐼), (𝐼𝑋𝐽)))) |
72 | 57, 71 | pm2.61ine 3028 |
1
⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍)) → (𝑅 Σg (𝑙 ∈ 𝑁 ↦ ((𝐼𝑋𝑙)(.r‘𝑅)(𝑙𝐸𝐽)))) = if(𝐽 = 𝐾, (𝑍‘𝐼), (𝐼𝑋𝐽))) |