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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 22602 | . 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 2726 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
6 | cpm2mf.k | . . 3 ⊢ 𝐾 = (Base‘𝐴) | |
7 | simpll 764 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝑆) → 𝑁 ∈ Fin) | |
8 | simplr 766 | . . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝑆) → 𝑅 ∈ Ring) | |
9 | eqid 2726 | . . . . 5 ⊢ (𝑁 Mat (Poly1‘𝑅)) = (𝑁 Mat (Poly1‘𝑅)) | |
10 | eqid 2726 | . . . . 5 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅)) | |
11 | eqid 2726 | . . . . 5 ⊢ (Base‘(𝑁 Mat (Poly1‘𝑅))) = (Base‘(𝑁 Mat (Poly1‘𝑅))) | |
12 | simp2 1134 | . . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑥 ∈ 𝑁) | |
13 | simp3 1135 | . . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝑦 ∈ 𝑁) | |
14 | eqid 2726 | . . . . . . . 8 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅) | |
15 | 2, 14, 9, 11 | cpmatpmat 22563 | . . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑚 ∈ 𝑆) → 𝑚 ∈ (Base‘(𝑁 Mat (Poly1‘𝑅)))) |
16 | 15 | 3expa 1115 | . . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝑆) → 𝑚 ∈ (Base‘(𝑁 Mat (Poly1‘𝑅)))) |
17 | 16 | 3ad2ant1 1130 | . . . . 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 22279 | . . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → (𝑥𝑚𝑦) ∈ (Base‘(Poly1‘𝑅))) |
19 | 0nn0 12488 | . . . 4 ⊢ 0 ∈ ℕ0 | |
20 | eqid 2726 | . . . . 5 ⊢ (coe1‘(𝑥𝑚𝑦)) = (coe1‘(𝑥𝑚𝑦)) | |
21 | 20, 10, 14, 5 | coe1fvalcl 22082 | . . . 4 ⊢ (((𝑥𝑚𝑦) ∈ (Base‘(Poly1‘𝑅)) ∧ 0 ∈ ℕ0) → ((coe1‘(𝑥𝑚𝑦))‘0) ∈ (Base‘𝑅)) |
22 | 18, 19, 21 | sylancl 585 | . . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝑆) ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → ((coe1‘(𝑥𝑚𝑦))‘0) ∈ (Base‘𝑅)) |
23 | 4, 5, 6, 7, 8, 22 | matbas2d 22276 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑚 ∈ 𝑆) → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)) ∈ 𝐾) |
24 | 3, 23 | fmpt3d 7110 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐼:𝑆⟶𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ⟶wf 6532 ‘cfv 6536 (class class class)co 7404 ∈ cmpo 7406 Fincfn 8938 0cc0 11109 ℕ0cn0 12473 Basecbs 17151 Ringcrg 20136 Poly1cpl1 22047 coe1cco1 22048 Mat cmat 22258 ConstPolyMat ccpmat 22556 cPolyMatToMat ccpmat2mat 22558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-ot 4632 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-sup 9436 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-fz 13488 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-hom 17228 df-cco 17229 df-0g 17394 df-prds 17400 df-pws 17402 df-sra 21019 df-rgmod 21020 df-dsmm 21623 df-frlm 21638 df-psr 21799 df-opsr 21803 df-psr1 22050 df-ply1 22052 df-coe1 22053 df-mat 22259 df-cpmat 22559 df-cpmat2mat 22561 |
This theorem is referenced by: m2cpminv 22613 cpmadumatpolylem1 22734 cpmadumatpolylem2 22735 chcoeffeqlem 22738 cayhamlem4 22741 |
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