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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 22641 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingHom 𝑈)) |
9 | crngring 20178 | . . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
10 | 1, 2, 3, 4 | m2cpmf1o 22652 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐾–1-1-onto→𝑆) |
11 | eqid 2728 | . . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
12 | 1, 5, 6, 11 | cpmatpmat 22605 | . . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑚 ∈ 𝑆) → 𝑚 ∈ (Base‘𝐶)) |
13 | 12 | 3expia 1119 | . . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑚 ∈ 𝑆 → 𝑚 ∈ (Base‘𝐶))) |
14 | 13 | ssrdv 3984 | . . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ⊆ (Base‘𝐶)) |
15 | 7, 11 | ressbas2 17211 | . . . . . . 7 ⊢ (𝑆 ⊆ (Base‘𝐶) → 𝑆 = (Base‘𝑈)) |
16 | 14, 15 | syl 17 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 = (Base‘𝑈)) |
17 | 16 | eqcomd 2734 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘𝑈) = 𝑆) |
18 | 17 | f1oeq3d 6830 | . . . 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 2728 | . . 3 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
22 | 4, 21 | isrim 20424 | . 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 1534 ∈ wcel 2099 ⊆ wss 3945 –1-1-onto→wf1o 6541 ‘cfv 6542 (class class class)co 7414 Fincfn 8957 Basecbs 17173 ↾s cress 17202 Ringcrg 20166 CRingccrg 20167 RingHom crh 20401 RingIso crs 20402 Poly1cpl1 22089 Mat cmat 22300 ConstPolyMat ccpmat 22598 matToPolyMat cmat2pmat 22599 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-ofr 7680 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9380 df-sup 9459 df-oi 9527 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-fz 13511 df-fzo 13654 df-seq 13993 df-hash 14316 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-hom 17250 df-cco 17251 df-0g 17416 df-gsum 17417 df-prds 17422 df-pws 17424 df-mre 17559 df-mrc 17560 df-acs 17562 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-mhm 18733 df-submnd 18734 df-grp 18886 df-minusg 18887 df-sbg 18888 df-mulg 19017 df-subg 19071 df-ghm 19161 df-cntz 19261 df-cmn 19730 df-abl 19731 df-mgp 20068 df-rng 20086 df-ur 20115 df-srg 20120 df-ring 20168 df-cring 20169 df-rhm 20404 df-rim 20405 df-subrng 20476 df-subrg 20501 df-lmod 20738 df-lss 20809 df-sra 21051 df-rgmod 21052 df-dsmm 21659 df-frlm 21674 df-assa 21780 df-ascl 21782 df-psr 21835 df-mvr 21836 df-mpl 21837 df-opsr 21839 df-psr1 22092 df-vr1 22093 df-ply1 22094 df-coe1 22095 df-mamu 22279 df-mat 22301 df-cpmat 22601 df-mat2pmat 22602 df-cpmat2mat 22603 |
This theorem is referenced by: matcpmric 22654 |
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