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| Mirrors > Home > MPE Home > Th. List > mat2pmatval | Structured version Visualization version GIF version | ||
| Description: The result of a matrix transformation. (Contributed by AV, 31-Jul-2019.) |
| Ref | Expression |
|---|---|
| mat2pmatfval.t | ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) |
| mat2pmatfval.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
| mat2pmatfval.b | ⊢ 𝐵 = (Base‘𝐴) |
| mat2pmatfval.p | ⊢ 𝑃 = (Poly1‘𝑅) |
| mat2pmatfval.s | ⊢ 𝑆 = (algSc‘𝑃) |
| Ref | Expression |
|---|---|
| mat2pmatval | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑀𝑦)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mat2pmatfval.t | . . . 4 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) | |
| 2 | mat2pmatfval.a | . . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
| 3 | mat2pmatfval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐴) | |
| 4 | mat2pmatfval.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
| 5 | mat2pmatfval.s | . . . 4 ⊢ 𝑆 = (algSc‘𝑃) | |
| 6 | 1, 2, 3, 4, 5 | mat2pmatfval 22729 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑇 = (𝑚 ∈ 𝐵 ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑚𝑦))))) |
| 7 | 6 | 3adant3 1133 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝑇 = (𝑚 ∈ 𝐵 ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑚𝑦))))) |
| 8 | oveq 7437 | . . . . 5 ⊢ (𝑚 = 𝑀 → (𝑥𝑚𝑦) = (𝑥𝑀𝑦)) | |
| 9 | 8 | fveq2d 6910 | . . . 4 ⊢ (𝑚 = 𝑀 → (𝑆‘(𝑥𝑚𝑦)) = (𝑆‘(𝑥𝑀𝑦))) |
| 10 | 9 | mpoeq3dv 7512 | . . 3 ⊢ (𝑚 = 𝑀 → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑚𝑦))) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑀𝑦)))) |
| 11 | 10 | adantl 481 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) ∧ 𝑚 = 𝑀) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑚𝑦))) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑀𝑦)))) |
| 12 | simp3 1139 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵) | |
| 13 | simp1 1137 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝑁 ∈ Fin) | |
| 14 | mpoexga 8102 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑀𝑦))) ∈ V) | |
| 15 | 13, 13, 14 | syl2anc 584 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑀𝑦))) ∈ V) |
| 16 | 7, 11, 12, 15 | fvmptd 7023 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑀𝑦)))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 Fincfn 8985 Basecbs 17247 algSccascl 21872 Poly1cpl1 22178 Mat cmat 22411 matToPolyMat cmat2pmat 22710 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-mat2pmat 22713 |
| This theorem is referenced by: mat2pmatvalel 22731 mat2pmatbas 22732 mat2pmatghm 22736 mat2pmatmul 22737 d0mat2pmat 22744 d1mat2pmat 22745 m2cpminvid2 22761 pmatcollpwlem 22786 pmatcollpwscmatlem2 22796 |
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