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Mirrors > Home > MPE Home > Th. List > cpm2mf | Structured version Visualization version GIF version |
Description: The inverse matrix transformation is a function from the constant polynomial matrices to the matrices over the base ring of the polynomials. (Contributed by AV, 24-Nov-2019.) (Revised by AV, 15-Dec-2019.) |
Ref | Expression |
---|---|
cpm2mf.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
cpm2mf.k | ⊢ 𝐾 = (Base‘𝐴) |
cpm2mf.s | ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) |
cpm2mf.i | ⊢ 𝐼 = (𝑁 cPolyMatToMat 𝑅) |
Ref | Expression |
---|---|
cpm2mf | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐼:𝑆⟶𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cpm2mf.i | . . 3 ⊢ 𝐼 = (𝑁 cPolyMatToMat 𝑅) | |
2 | cpm2mf.s | . . 3 ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) | |
3 | 1, 2 | cpm2mfval 21351 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐼 = (𝑚 ∈ 𝑆 ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))) |
4 | cpm2mf.a | . . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
5 | eqid 2821 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | cpm2mf.k | . . 3 ⊢ 𝐾 = (Base‘𝐴) | |
7 | simpll 765 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝑆) → 𝑁 ∈ Fin) | |
8 | simplr 767 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝑆) → 𝑅 ∈ Ring) | |
9 | eqid 2821 | . . . . 5 ⊢ (𝑁 Mat (Poly1‘𝑅)) = (𝑁 Mat (Poly1‘𝑅)) | |
10 | eqid 2821 | . . . . 5 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅)) | |
11 | eqid 2821 | . . . . 5 ⊢ (Base‘(𝑁 Mat (Poly1‘𝑅))) = (Base‘(𝑁 Mat (Poly1‘𝑅))) | |
12 | simp2 1133 | . . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑥 ∈ 𝑁) | |
13 | simp3 1134 | . . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑦 ∈ 𝑁) | |
14 | eqid 2821 | . . . . . . . 8 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅) | |
15 | 2, 14, 9, 11 | cpmatpmat 21312 | . . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑚 ∈ 𝑆) → 𝑚 ∈ (Base‘(𝑁 Mat (Poly1‘𝑅)))) |
16 | 15 | 3expa 1114 | . . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝑆) → 𝑚 ∈ (Base‘(𝑁 Mat (Poly1‘𝑅)))) |
17 | 16 | 3ad2ant1 1129 | . . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑚 ∈ (Base‘(𝑁 Mat (Poly1‘𝑅)))) |
18 | 9, 10, 11, 12, 13, 17 | matecld 21029 | . . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑥𝑚𝑦) ∈ (Base‘(Poly1‘𝑅))) |
19 | 0nn0 11906 | . . . 4 ⊢ 0 ∈ ℕ0 | |
20 | eqid 2821 | . . . . 5 ⊢ (coe1‘(𝑥𝑚𝑦)) = (coe1‘(𝑥𝑚𝑦)) | |
21 | 20, 10, 14, 5 | coe1fvalcl 20374 | . . . 4 ⊢ (((𝑥𝑚𝑦) ∈ (Base‘(Poly1‘𝑅)) ∧ 0 ∈ ℕ0) → ((coe1‘(𝑥𝑚𝑦))‘0) ∈ (Base‘𝑅)) |
22 | 18, 19, 21 | sylancl 588 | . . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → ((coe1‘(𝑥𝑚𝑦))‘0) ∈ (Base‘𝑅)) |
23 | 4, 5, 6, 7, 8, 22 | matbas2d 21026 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝑆) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)) ∈ 𝐾) |
24 | 3, 23 | fmpt3d 6874 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐼:𝑆⟶𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ∈ cmpo 7152 Fincfn 8503 0cc0 10531 ℕ0cn0 11891 Basecbs 16477 Ringcrg 19291 Poly1cpl1 20339 coe1cco1 20340 Mat cmat 21010 ConstPolyMat ccpmat 21305 cPolyMatToMat ccpmat2mat 21307 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-ot 4569 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-hom 16583 df-cco 16584 df-0g 16709 df-prds 16715 df-pws 16717 df-sra 19938 df-rgmod 19939 df-psr 20130 df-opsr 20134 df-psr1 20342 df-ply1 20344 df-coe1 20345 df-dsmm 20870 df-frlm 20885 df-mat 21011 df-cpmat 21308 df-cpmat2mat 21310 |
This theorem is referenced by: m2cpminv 21362 cpmadumatpolylem1 21483 cpmadumatpolylem2 21484 chcoeffeqlem 21487 cayhamlem4 21490 |
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