![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 22068 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑇 = (𝑚 ∈ 𝐵 ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑚𝑦))))) |
7 | 6 | 3adant3 1132 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝑇 = (𝑚 ∈ 𝐵 ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑚𝑦))))) |
8 | oveq 7360 | . . . . 5 ⊢ (𝑚 = 𝑀 → (𝑥𝑚𝑦) = (𝑥𝑀𝑦)) | |
9 | 8 | fveq2d 6844 | . . . 4 ⊢ (𝑚 = 𝑀 → (𝑆‘(𝑥𝑚𝑦)) = (𝑆‘(𝑥𝑀𝑦))) |
10 | 9 | mpoeq3dv 7433 | . . 3 ⊢ (𝑚 = 𝑀 → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑚𝑦))) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑀𝑦)))) |
11 | 10 | adantl 482 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) ∧ 𝑚 = 𝑀) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑚𝑦))) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑀𝑦)))) |
12 | simp3 1138 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵) | |
13 | simp1 1136 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝑁 ∈ Fin) | |
14 | mpoexga 8007 | . . 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 6953 | 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 1541 ∈ wcel 2106 Vcvv 3444 ↦ cmpt 5187 ‘cfv 6494 (class class class)co 7354 ∈ cmpo 7356 Fincfn 8880 Basecbs 17080 algSccascl 21254 Poly1cpl1 21544 Mat cmat 21750 matToPolyMat cmat2pmat 22049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-ov 7357 df-oprab 7358 df-mpo 7359 df-1st 7918 df-2nd 7919 df-mat2pmat 22052 |
This theorem is referenced by: mat2pmatvalel 22070 mat2pmatbas 22071 mat2pmatghm 22075 mat2pmatmul 22076 d0mat2pmat 22083 d1mat2pmat 22084 m2cpminvid2 22100 pmatcollpwlem 22125 pmatcollpwscmatlem2 22135 |
Copyright terms: Public domain | W3C validator |