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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 22745 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑇 = (𝑚 ∈ 𝐵 ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑚𝑦))))) |
7 | 6 | 3adant3 1131 | . 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 6911 | . . . 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 1137 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵) | |
13 | simp1 1135 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝑁 ∈ Fin) | |
14 | mpoexga 8101 | . . 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 1086 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 Fincfn 8984 Basecbs 17245 algSccascl 21890 Poly1cpl1 22194 Mat cmat 22427 matToPolyMat cmat2pmat 22726 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-mat2pmat 22729 |
This theorem is referenced by: mat2pmatvalel 22747 mat2pmatbas 22748 mat2pmatghm 22752 mat2pmatmul 22753 d0mat2pmat 22760 d1mat2pmat 22761 m2cpminvid2 22777 pmatcollpwlem 22802 pmatcollpwscmatlem2 22812 |
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