| Step | Hyp | Ref
| Expression |
| 1 | | snfi 9083 |
. . . . . 6
⊢ {𝐴} ∈ Fin |
| 2 | | eleq1 2829 |
. . . . . 6
⊢ (𝑁 = {𝐴} → (𝑁 ∈ Fin ↔ {𝐴} ∈ Fin)) |
| 3 | 1, 2 | mpbiri 258 |
. . . . 5
⊢ (𝑁 = {𝐴} → 𝑁 ∈ Fin) |
| 4 | 3 | adantr 480 |
. . . 4
⊢ ((𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) → 𝑁 ∈ Fin) |
| 5 | 4 | 3ad2ant2 1135 |
. . 3
⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → 𝑁 ∈ Fin) |
| 6 | | simp1 1137 |
. . 3
⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ 𝑉) |
| 7 | | simp3 1139 |
. . 3
⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵) |
| 8 | | d1mat2pmat.t |
. . . 4
⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) |
| 9 | | eqid 2737 |
. . . 4
⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅) |
| 10 | | d1mat2pmat.b |
. . . 4
⊢ 𝐵 = (Base‘(𝑁 Mat 𝑅)) |
| 11 | | d1mat2pmat.p |
. . . 4
⊢ 𝑃 = (Poly1‘𝑅) |
| 12 | | d1mat2pmat.s |
. . . 4
⊢ 𝑆 = (algSc‘𝑃) |
| 13 | 8, 9, 10, 11, 12 | mat2pmatval 22730 |
. . 3
⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑆‘(𝑖𝑀𝑗)))) |
| 14 | 5, 6, 7, 13 | syl3anc 1373 |
. 2
⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑆‘(𝑖𝑀𝑗)))) |
| 15 | | id 22 |
. . . . . . 7
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉) |
| 16 | | fvexd 6921 |
. . . . . . 7
⊢ (𝐴 ∈ 𝑉 → (𝑆‘(𝐴𝑀𝐴)) ∈ V) |
| 17 | 15, 15, 16 | 3jca 1129 |
. . . . . 6
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ (𝑆‘(𝐴𝑀𝐴)) ∈ V)) |
| 18 | 17 | adantl 481 |
. . . . 5
⊢ ((𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) → (𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ (𝑆‘(𝐴𝑀𝐴)) ∈ V)) |
| 19 | 18 | 3ad2ant2 1135 |
. . . 4
⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → (𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ (𝑆‘(𝐴𝑀𝐴)) ∈ V)) |
| 20 | | eqid 2737 |
. . . . 5
⊢ (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) |
| 21 | | fvoveq1 7454 |
. . . . 5
⊢ (𝑖 = 𝐴 → (𝑆‘(𝑖𝑀𝑗)) = (𝑆‘(𝐴𝑀𝑗))) |
| 22 | | oveq2 7439 |
. . . . . 6
⊢ (𝑗 = 𝐴 → (𝐴𝑀𝑗) = (𝐴𝑀𝐴)) |
| 23 | 22 | fveq2d 6910 |
. . . . 5
⊢ (𝑗 = 𝐴 → (𝑆‘(𝐴𝑀𝑗)) = (𝑆‘(𝐴𝑀𝐴))) |
| 24 | 20, 21, 23 | mposn 8128 |
. . . 4
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ (𝑆‘(𝐴𝑀𝐴)) ∈ V) → (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = {〈〈𝐴, 𝐴〉, (𝑆‘(𝐴𝑀𝐴))〉}) |
| 25 | 19, 24 | syl 17 |
. . 3
⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = {〈〈𝐴, 𝐴〉, (𝑆‘(𝐴𝑀𝐴))〉}) |
| 26 | | mpoeq12 7506 |
. . . . . . 7
⊢ ((𝑁 = {𝐴} ∧ 𝑁 = {𝐴}) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗)))) |
| 27 | 26 | eqeq1d 2739 |
. . . . . 6
⊢ ((𝑁 = {𝐴} ∧ 𝑁 = {𝐴}) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = {〈〈𝐴, 𝐴〉, (𝑆‘(𝐴𝑀𝐴))〉} ↔ (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = {〈〈𝐴, 𝐴〉, (𝑆‘(𝐴𝑀𝐴))〉})) |
| 28 | 27 | anidms 566 |
. . . . 5
⊢ (𝑁 = {𝐴} → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = {〈〈𝐴, 𝐴〉, (𝑆‘(𝐴𝑀𝐴))〉} ↔ (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = {〈〈𝐴, 𝐴〉, (𝑆‘(𝐴𝑀𝐴))〉})) |
| 29 | 28 | adantr 480 |
. . . 4
⊢ ((𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = {〈〈𝐴, 𝐴〉, (𝑆‘(𝐴𝑀𝐴))〉} ↔ (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = {〈〈𝐴, 𝐴〉, (𝑆‘(𝐴𝑀𝐴))〉})) |
| 30 | 29 | 3ad2ant2 1135 |
. . 3
⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = {〈〈𝐴, 𝐴〉, (𝑆‘(𝐴𝑀𝐴))〉} ↔ (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = {〈〈𝐴, 𝐴〉, (𝑆‘(𝐴𝑀𝐴))〉})) |
| 31 | 25, 30 | mpbird 257 |
. 2
⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = {〈〈𝐴, 𝐴〉, (𝑆‘(𝐴𝑀𝐴))〉}) |
| 32 | 14, 31 | eqtrd 2777 |
1
⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) = {〈〈𝐴, 𝐴〉, (𝑆‘(𝐴𝑀𝐴))〉}) |