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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 22664 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingHom 𝑈)) |
9 | crngring 20187 | . . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
10 | 1, 2, 3, 4 | m2cpmf1o 22675 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐾–1-1-onto→𝑆) |
11 | eqid 2725 | . . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
12 | 1, 5, 6, 11 | cpmatpmat 22628 | . . . . . . . . 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 3978 | . . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ⊆ (Base‘𝐶)) |
15 | 7, 11 | ressbas2 17215 | . . . . . . 7 ⊢ (𝑆 ⊆ (Base‘𝐶) → 𝑆 = (Base‘𝑈)) |
16 | 14, 15 | syl 17 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 = (Base‘𝑈)) |
17 | 16 | eqcomd 2731 | . . . . 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 256 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐾–1-1-onto→(Base‘𝑈)) |
20 | 9, 19 | sylan2 591 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇:𝐾–1-1-onto→(Base‘𝑈)) |
21 | eqid 2725 | . . 3 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
22 | 4, 21 | isrim 20433 | . 2 ⊢ (𝑇 ∈ (𝐴 RingIso 𝑈) ↔ (𝑇 ∈ (𝐴 RingHom 𝑈) ∧ 𝑇:𝐾–1-1-onto→(Base‘𝑈))) |
23 | 8, 20, 22 | sylanbrc 581 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingIso 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ⊆ wss 3940 –1-1-onto→wf1o 6541 ‘cfv 6542 (class class class)co 7415 Fincfn 8960 Basecbs 17177 ↾s cress 17206 Ringcrg 20175 CRingccrg 20176 RingHom crh 20410 RingIso crs 20411 Poly1cpl1 22102 Mat cmat 22323 ConstPolyMat ccpmat 22621 matToPolyMat cmat2pmat 22622 |
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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 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 5144 df-opab 5206 df-mpt 5227 df-tr 5261 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7681 df-ofr 7682 df-om 7868 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-map 8843 df-pm 8844 df-ixp 8913 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-fsupp 9384 df-sup 9463 df-oi 9531 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-uz 12851 df-fz 13515 df-fzo 13658 df-seq 13997 df-hash 14320 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-hom 17254 df-cco 17255 df-0g 17420 df-gsum 17421 df-prds 17426 df-pws 17428 df-mre 17563 df-mrc 17564 df-acs 17566 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18737 df-submnd 18738 df-grp 18895 df-minusg 18896 df-sbg 18897 df-mulg 19026 df-subg 19080 df-ghm 19170 df-cntz 19270 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-ur 20124 df-srg 20129 df-ring 20177 df-cring 20178 df-rhm 20413 df-rim 20414 df-subrng 20485 df-subrg 20510 df-lmod 20747 df-lss 20818 df-sra 21060 df-rgmod 21061 df-dsmm 21668 df-frlm 21683 df-assa 21789 df-ascl 21791 df-psr 21844 df-mvr 21845 df-mpl 21846 df-opsr 21848 df-psr1 22105 df-vr1 22106 df-ply1 22107 df-coe1 22108 df-mamu 22307 df-mat 22324 df-cpmat 22624 df-mat2pmat 22625 df-cpmat2mat 22626 |
This theorem is referenced by: matcpmric 22677 |
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