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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 22072 | . . 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 7363 | . . . . 5 ⊢ (𝑚 = 𝑀 → (𝑥𝑚𝑦) = (𝑥𝑀𝑦)) | |
9 | 8 | fveq2d 6846 | . . . 4 ⊢ (𝑚 = 𝑀 → (𝑆‘(𝑥𝑚𝑦)) = (𝑆‘(𝑥𝑀𝑦))) |
10 | 9 | mpoeq3dv 7436 | . . 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 8010 | . . 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 6955 | 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 3445 ↦ cmpt 5188 ‘cfv 6496 (class class class)co 7357 ∈ cmpo 7359 Fincfn 8883 Basecbs 17083 algSccascl 21258 Poly1cpl1 21548 Mat cmat 21754 matToPolyMat cmat2pmat 22053 |
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 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 |
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 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7360 df-oprab 7361 df-mpo 7362 df-1st 7921 df-2nd 7922 df-mat2pmat 22056 |
This theorem is referenced by: mat2pmatvalel 22074 mat2pmatbas 22075 mat2pmatghm 22079 mat2pmatmul 22080 d0mat2pmat 22087 d1mat2pmat 22088 m2cpminvid2 22104 pmatcollpwlem 22129 pmatcollpwscmatlem2 22139 |
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