| Step | Hyp | Ref
| Expression |
| 1 | | 1marepvsma1.x |
. . . . . 6
⊢ 𝑋 = (( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼) |
| 2 | 1 | oveqi 7444 |
. . . . 5
⊢ (𝑖𝑋𝑗) = (𝑖(( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼)𝑗) |
| 3 | 2 | a1i 11 |
. . . 4
⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ (𝑁 ∖ {𝐼}) ∧ 𝑗 ∈ (𝑁 ∖ {𝐼})) → (𝑖𝑋𝑗) = (𝑖(( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼)𝑗)) |
| 4 | | eqid 2737 |
. . . . . . . . 9
⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅) |
| 5 | | eqid 2737 |
. . . . . . . . 9
⊢
(Base‘(𝑁 Mat
𝑅)) = (Base‘(𝑁 Mat 𝑅)) |
| 6 | | 1marepvsma1.1 |
. . . . . . . . 9
⊢ 1 =
(1r‘(𝑁 Mat
𝑅)) |
| 7 | 4, 5, 6 | mat1bas 22455 |
. . . . . . . 8
⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 1 ∈
(Base‘(𝑁 Mat 𝑅))) |
| 8 | 7 | adantr 480 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → 1 ∈ (Base‘(𝑁 Mat 𝑅))) |
| 9 | | simprr 773 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → 𝑍 ∈ 𝑉) |
| 10 | | simprl 771 |
. . . . . . 7
⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → 𝐼 ∈ 𝑁) |
| 11 | 8, 9, 10 | 3jca 1129 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → ( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍 ∈ 𝑉 ∧ 𝐼 ∈ 𝑁)) |
| 12 | 11 | 3ad2ant1 1134 |
. . . . 5
⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ (𝑁 ∖ {𝐼}) ∧ 𝑗 ∈ (𝑁 ∖ {𝐼})) → ( 1 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝑍 ∈ 𝑉 ∧ 𝐼 ∈ 𝑁)) |
| 13 | | eldifi 4131 |
. . . . . . 7
⊢ (𝑖 ∈ (𝑁 ∖ {𝐼}) → 𝑖 ∈ 𝑁) |
| 14 | | eldifi 4131 |
. . . . . . 7
⊢ (𝑗 ∈ (𝑁 ∖ {𝐼}) → 𝑗 ∈ 𝑁) |
| 15 | 13, 14 | anim12i 613 |
. . . . . 6
⊢ ((𝑖 ∈ (𝑁 ∖ {𝐼}) ∧ 𝑗 ∈ (𝑁 ∖ {𝐼})) → (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) |
| 16 | 15 | 3adant1 1131 |
. . . . 5
⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ (𝑁 ∖ {𝐼}) ∧ 𝑗 ∈ (𝑁 ∖ {𝐼})) → (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) |
| 17 | | eqid 2737 |
. . . . . 6
⊢ (𝑁 matRepV 𝑅) = (𝑁 matRepV 𝑅) |
| 18 | | 1marepvsma1.v |
. . . . . 6
⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁) |
| 19 | 4, 5, 17, 18 | marepveval 22574 |
. . . . 5
⊢ ((( 1 ∈
(Base‘(𝑁 Mat 𝑅)) ∧ 𝑍 ∈ 𝑉 ∧ 𝐼 ∈ 𝑁) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼)𝑗) = if(𝑗 = 𝐼, (𝑍‘𝑖), (𝑖 1 𝑗))) |
| 20 | 12, 16, 19 | syl2anc 584 |
. . . 4
⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ (𝑁 ∖ {𝐼}) ∧ 𝑗 ∈ (𝑁 ∖ {𝐼})) → (𝑖(( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼)𝑗) = if(𝑗 = 𝐼, (𝑍‘𝑖), (𝑖 1 𝑗))) |
| 21 | | eldifsni 4790 |
. . . . . . . 8
⊢ (𝑗 ∈ (𝑁 ∖ {𝐼}) → 𝑗 ≠ 𝐼) |
| 22 | 21 | neneqd 2945 |
. . . . . . 7
⊢ (𝑗 ∈ (𝑁 ∖ {𝐼}) → ¬ 𝑗 = 𝐼) |
| 23 | 22 | 3ad2ant3 1136 |
. . . . . 6
⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ (𝑁 ∖ {𝐼}) ∧ 𝑗 ∈ (𝑁 ∖ {𝐼})) → ¬ 𝑗 = 𝐼) |
| 24 | 23 | iffalsed 4536 |
. . . . 5
⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ (𝑁 ∖ {𝐼}) ∧ 𝑗 ∈ (𝑁 ∖ {𝐼})) → if(𝑗 = 𝐼, (𝑍‘𝑖), (𝑖 1 𝑗)) = (𝑖 1 𝑗)) |
| 25 | | eqid 2737 |
. . . . . 6
⊢
(1r‘𝑅) = (1r‘𝑅) |
| 26 | | eqid 2737 |
. . . . . 6
⊢
(0g‘𝑅) = (0g‘𝑅) |
| 27 | | simp1lr 1238 |
. . . . . 6
⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ (𝑁 ∖ {𝐼}) ∧ 𝑗 ∈ (𝑁 ∖ {𝐼})) → 𝑁 ∈ Fin) |
| 28 | | simp1ll 1237 |
. . . . . 6
⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ (𝑁 ∖ {𝐼}) ∧ 𝑗 ∈ (𝑁 ∖ {𝐼})) → 𝑅 ∈ Ring) |
| 29 | 13 | 3ad2ant2 1135 |
. . . . . 6
⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ (𝑁 ∖ {𝐼}) ∧ 𝑗 ∈ (𝑁 ∖ {𝐼})) → 𝑖 ∈ 𝑁) |
| 30 | 14 | 3ad2ant3 1136 |
. . . . . 6
⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ (𝑁 ∖ {𝐼}) ∧ 𝑗 ∈ (𝑁 ∖ {𝐼})) → 𝑗 ∈ 𝑁) |
| 31 | 4, 25, 26, 27, 28, 29, 30, 6 | mat1ov 22454 |
. . . . 5
⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ (𝑁 ∖ {𝐼}) ∧ 𝑗 ∈ (𝑁 ∖ {𝐼})) → (𝑖 1 𝑗) = if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅))) |
| 32 | 24, 31 | eqtrd 2777 |
. . . 4
⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ (𝑁 ∖ {𝐼}) ∧ 𝑗 ∈ (𝑁 ∖ {𝐼})) → if(𝑗 = 𝐼, (𝑍‘𝑖), (𝑖 1 𝑗)) = if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅))) |
| 33 | 3, 20, 32 | 3eqtrd 2781 |
. . 3
⊢ ((((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) ∧ 𝑖 ∈ (𝑁 ∖ {𝐼}) ∧ 𝑗 ∈ (𝑁 ∖ {𝐼})) → (𝑖𝑋𝑗) = if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅))) |
| 34 | 33 | mpoeq3dva 7510 |
. 2
⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → (𝑖 ∈ (𝑁 ∖ {𝐼}), 𝑗 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑖𝑋𝑗)) = (𝑖 ∈ (𝑁 ∖ {𝐼}), 𝑗 ∈ (𝑁 ∖ {𝐼}) ↦ if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅)))) |
| 35 | 4, 5, 18, 6 | ma1repvcl 22576 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝑍 ∈ 𝑉 ∧ 𝐼 ∈ 𝑁)) → (( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼) ∈ (Base‘(𝑁 Mat 𝑅))) |
| 36 | 35 | ancom2s 650 |
. . . 4
⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → (( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼) ∈ (Base‘(𝑁 Mat 𝑅))) |
| 37 | 1, 36 | eqeltrid 2845 |
. . 3
⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → 𝑋 ∈ (Base‘(𝑁 Mat 𝑅))) |
| 38 | | eqid 2737 |
. . . 4
⊢ (𝑁 subMat 𝑅) = (𝑁 subMat 𝑅) |
| 39 | 4, 38, 5 | submaval 22587 |
. . 3
⊢ ((𝑋 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝐼 ∈ 𝑁 ∧ 𝐼 ∈ 𝑁) → (𝐼((𝑁 subMat 𝑅)‘𝑋)𝐼) = (𝑖 ∈ (𝑁 ∖ {𝐼}), 𝑗 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑖𝑋𝑗))) |
| 40 | 37, 10, 10, 39 | syl3anc 1373 |
. 2
⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → (𝐼((𝑁 subMat 𝑅)‘𝑋)𝐼) = (𝑖 ∈ (𝑁 ∖ {𝐼}), 𝑗 ∈ (𝑁 ∖ {𝐼}) ↦ (𝑖𝑋𝑗))) |
| 41 | | diffi 9215 |
. . . . . 6
⊢ (𝑁 ∈ Fin → (𝑁 ∖ {𝐼}) ∈ Fin) |
| 42 | 41 | anim2i 617 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → (𝑅 ∈ Ring ∧ (𝑁 ∖ {𝐼}) ∈ Fin)) |
| 43 | 42 | ancomd 461 |
. . . 4
⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → ((𝑁 ∖ {𝐼}) ∈ Fin ∧ 𝑅 ∈ Ring)) |
| 44 | 43 | adantr 480 |
. . 3
⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → ((𝑁 ∖ {𝐼}) ∈ Fin ∧ 𝑅 ∈ Ring)) |
| 45 | | eqid 2737 |
. . . 4
⊢ ((𝑁 ∖ {𝐼}) Mat 𝑅) = ((𝑁 ∖ {𝐼}) Mat 𝑅) |
| 46 | 45, 25, 26 | mat1 22453 |
. . 3
⊢ (((𝑁 ∖ {𝐼}) ∈ Fin ∧ 𝑅 ∈ Ring) →
(1r‘((𝑁
∖ {𝐼}) Mat 𝑅)) = (𝑖 ∈ (𝑁 ∖ {𝐼}), 𝑗 ∈ (𝑁 ∖ {𝐼}) ↦ if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅)))) |
| 47 | 44, 46 | syl 17 |
. 2
⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → (1r‘((𝑁 ∖ {𝐼}) Mat 𝑅)) = (𝑖 ∈ (𝑁 ∖ {𝐼}), 𝑗 ∈ (𝑁 ∖ {𝐼}) ↦ if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅)))) |
| 48 | 34, 40, 47 | 3eqtr4d 2787 |
1
⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → (𝐼((𝑁 subMat 𝑅)‘𝑋)𝐼) = (1r‘((𝑁 ∖ {𝐼}) Mat 𝑅))) |