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Mirrors > Home > MPE Home > Th. List > m2cpmrngiso | Structured version Visualization version GIF version |
Description: The transformation of matrices into constant polynomial matrices is a ring isomorphism. (Contributed by AV, 19-Nov-2019.) |
Ref | Expression |
---|---|
m2cpmfo.s | ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) |
m2cpmfo.t | ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) |
m2cpmfo.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
m2cpmfo.k | ⊢ 𝐾 = (Base‘𝐴) |
m2cpmrngiso.p | ⊢ 𝑃 = (Poly1‘𝑅) |
m2cpmrngiso.c | ⊢ 𝐶 = (𝑁 Mat 𝑃) |
m2cpmrngiso.u | ⊢ 𝑈 = (𝐶 ↾s 𝑆) |
Ref | Expression |
---|---|
m2cpmrngiso | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingIso 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | m2cpmfo.s | . . 3 ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) | |
2 | m2cpmfo.t | . . 3 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) | |
3 | m2cpmfo.a | . . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
4 | m2cpmfo.k | . . 3 ⊢ 𝐾 = (Base‘𝐴) | |
5 | m2cpmrngiso.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
6 | m2cpmrngiso.c | . . 3 ⊢ 𝐶 = (𝑁 Mat 𝑃) | |
7 | m2cpmrngiso.u | . . 3 ⊢ 𝑈 = (𝐶 ↾s 𝑆) | |
8 | 1, 2, 3, 4, 5, 6, 7 | m2cpmrhm 22603 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingHom 𝑈)) |
9 | crngring 20150 | . . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
10 | 1, 2, 3, 4 | m2cpmf1o 22614 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐾–1-1-onto→𝑆) |
11 | eqid 2726 | . . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
12 | 1, 5, 6, 11 | cpmatpmat 22567 | . . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑚 ∈ 𝑆) → 𝑚 ∈ (Base‘𝐶)) |
13 | 12 | 3expia 1118 | . . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑚 ∈ 𝑆 → 𝑚 ∈ (Base‘𝐶))) |
14 | 13 | ssrdv 3983 | . . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ⊆ (Base‘𝐶)) |
15 | 7, 11 | ressbas2 17191 | . . . . . . 7 ⊢ (𝑆 ⊆ (Base‘𝐶) → 𝑆 = (Base‘𝑈)) |
16 | 14, 15 | syl 17 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 = (Base‘𝑈)) |
17 | 16 | eqcomd 2732 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘𝑈) = 𝑆) |
18 | 17 | f1oeq3d 6824 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇:𝐾–1-1-onto→(Base‘𝑈) ↔ 𝑇:𝐾–1-1-onto→𝑆)) |
19 | 10, 18 | mpbird 257 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐾–1-1-onto→(Base‘𝑈)) |
20 | 9, 19 | sylan2 592 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇:𝐾–1-1-onto→(Base‘𝑈)) |
21 | eqid 2726 | . . 3 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
22 | 4, 21 | isrim 20394 | . 2 ⊢ (𝑇 ∈ (𝐴 RingIso 𝑈) ↔ (𝑇 ∈ (𝐴 RingHom 𝑈) ∧ 𝑇:𝐾–1-1-onto→(Base‘𝑈))) |
23 | 8, 20, 22 | sylanbrc 582 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingIso 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ⊆ wss 3943 –1-1-onto→wf1o 6536 ‘cfv 6537 (class class class)co 7405 Fincfn 8941 Basecbs 17153 ↾s cress 17182 Ringcrg 20138 CRingccrg 20139 RingHom crh 20371 RingIso crs 20372 Poly1cpl1 22051 Mat cmat 22262 ConstPolyMat ccpmat 22560 matToPolyMat cmat2pmat 22561 |
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 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
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-rmo 3370 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-int 4944 df-iun 4992 df-iin 4993 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-se 5625 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 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-ofr 7668 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13491 df-fzo 13634 df-seq 13973 df-hash 14296 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-hom 17230 df-cco 17231 df-0g 17396 df-gsum 17397 df-prds 17402 df-pws 17404 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-submnd 18714 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18996 df-subg 19050 df-ghm 19139 df-cntz 19233 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-srg 20092 df-ring 20140 df-cring 20141 df-rhm 20374 df-rim 20375 df-subrng 20446 df-subrg 20471 df-lmod 20708 df-lss 20779 df-sra 21021 df-rgmod 21022 df-dsmm 21627 df-frlm 21642 df-assa 21748 df-ascl 21750 df-psr 21803 df-mvr 21804 df-mpl 21805 df-opsr 21807 df-psr1 22054 df-vr1 22055 df-ply1 22056 df-coe1 22057 df-mamu 22241 df-mat 22263 df-cpmat 22563 df-mat2pmat 22564 df-cpmat2mat 22565 |
This theorem is referenced by: matcpmric 22616 |
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