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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 21045 | . 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 2797 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | cpm2mf.k | . . 3 ⊢ 𝐾 = (Base‘𝐴) | |
7 | simpll 763 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝑆) → 𝑁 ∈ Fin) | |
8 | simplr 765 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝑆) → 𝑅 ∈ Ring) | |
9 | eqid 2797 | . . . . 5 ⊢ (𝑁 Mat (Poly1‘𝑅)) = (𝑁 Mat (Poly1‘𝑅)) | |
10 | eqid 2797 | . . . . 5 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅)) | |
11 | eqid 2797 | . . . . 5 ⊢ (Base‘(𝑁 Mat (Poly1‘𝑅))) = (Base‘(𝑁 Mat (Poly1‘𝑅))) | |
12 | simp2 1130 | . . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑥 ∈ 𝑁) | |
13 | simp3 1131 | . . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑦 ∈ 𝑁) | |
14 | eqid 2797 | . . . . . . . 8 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅) | |
15 | 2, 14, 9, 11 | cpmatpmat 21006 | . . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑚 ∈ 𝑆) → 𝑚 ∈ (Base‘(𝑁 Mat (Poly1‘𝑅)))) |
16 | 15 | 3expa 1111 | . . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝑆) → 𝑚 ∈ (Base‘(𝑁 Mat (Poly1‘𝑅)))) |
17 | 16 | 3ad2ant1 1126 | . . . . 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 20723 | . . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑥𝑚𝑦) ∈ (Base‘(Poly1‘𝑅))) |
19 | 0nn0 11766 | . . . 4 ⊢ 0 ∈ ℕ0 | |
20 | eqid 2797 | . . . . 5 ⊢ (coe1‘(𝑥𝑚𝑦)) = (coe1‘(𝑥𝑚𝑦)) | |
21 | 20, 10, 14, 5 | coe1fvalcl 20067 | . . . 4 ⊢ (((𝑥𝑚𝑦) ∈ (Base‘(Poly1‘𝑅)) ∧ 0 ∈ ℕ0) → ((coe1‘(𝑥𝑚𝑦))‘0) ∈ (Base‘𝑅)) |
22 | 18, 19, 21 | sylancl 586 | . . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → ((coe1‘(𝑥𝑚𝑦))‘0) ∈ (Base‘𝑅)) |
23 | 4, 5, 6, 7, 8, 22 | matbas2d 20720 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝑆) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)) ∈ 𝐾) |
24 | 3, 23 | fmpt3d 6750 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐼:𝑆⟶𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1080 = wceq 1525 ∈ wcel 2083 ⟶wf 6228 ‘cfv 6232 (class class class)co 7023 ∈ cmpo 7025 Fincfn 8364 0cc0 10390 ℕ0cn0 11751 Basecbs 16316 Ringcrg 18991 Poly1cpl1 20032 coe1cco1 20033 Mat cmat 20704 ConstPolyMat ccpmat 20999 cPolyMatToMat ccpmat2mat 21001 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-ot 4487 df-uni 4752 df-int 4789 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-of 7274 df-om 7444 df-1st 7552 df-2nd 7553 df-supp 7689 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-oadd 7964 df-er 8146 df-map 8265 df-ixp 8318 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-fsupp 8687 df-sup 8759 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-nn 11493 df-2 11554 df-3 11555 df-4 11556 df-5 11557 df-6 11558 df-7 11559 df-8 11560 df-9 11561 df-n0 11752 df-z 11836 df-dec 11953 df-uz 12098 df-fz 12747 df-struct 16318 df-ndx 16319 df-slot 16320 df-base 16322 df-sets 16323 df-ress 16324 df-plusg 16411 df-mulr 16412 df-sca 16414 df-vsca 16415 df-ip 16416 df-tset 16417 df-ple 16418 df-ds 16420 df-hom 16422 df-cco 16423 df-0g 16548 df-prds 16554 df-pws 16556 df-sra 19638 df-rgmod 19639 df-psr 19828 df-opsr 19832 df-psr1 20035 df-ply1 20037 df-coe1 20038 df-dsmm 20562 df-frlm 20577 df-mat 20705 df-cpmat 21002 df-cpmat2mat 21004 |
This theorem is referenced by: m2cpminv 21056 cpmadumatpolylem1 21177 cpmadumatpolylem2 21178 chcoeffeqlem 21181 cayhamlem4 21184 |
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