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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 22111 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingHom 𝑈)) |
9 | crngring 19981 | . . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
10 | 1, 2, 3, 4 | m2cpmf1o 22122 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐾–1-1-onto→𝑆) |
11 | eqid 2733 | . . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
12 | 1, 5, 6, 11 | cpmatpmat 22075 | . . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑚 ∈ 𝑆) → 𝑚 ∈ (Base‘𝐶)) |
13 | 12 | 3expia 1122 | . . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑚 ∈ 𝑆 → 𝑚 ∈ (Base‘𝐶))) |
14 | 13 | ssrdv 3951 | . . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ⊆ (Base‘𝐶)) |
15 | 7, 11 | ressbas2 17125 | . . . . . . 7 ⊢ (𝑆 ⊆ (Base‘𝐶) → 𝑆 = (Base‘𝑈)) |
16 | 14, 15 | syl 17 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 = (Base‘𝑈)) |
17 | 16 | eqcomd 2739 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘𝑈) = 𝑆) |
18 | 17 | f1oeq3d 6782 | . . . 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 594 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇:𝐾–1-1-onto→(Base‘𝑈)) |
21 | eqid 2733 | . . 3 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
22 | 4, 21 | isrim 20172 | . 2 ⊢ (𝑇 ∈ (𝐴 RingIso 𝑈) ↔ (𝑇 ∈ (𝐴 RingHom 𝑈) ∧ 𝑇:𝐾–1-1-onto→(Base‘𝑈))) |
23 | 8, 20, 22 | sylanbrc 584 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingIso 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⊆ wss 3911 –1-1-onto→wf1o 6496 ‘cfv 6497 (class class class)co 7358 Fincfn 8886 Basecbs 17088 ↾s cress 17117 Ringcrg 19969 CRingccrg 19970 RingHom crh 20150 RingIso crs 20151 Poly1cpl1 21564 Mat cmat 21770 ConstPolyMat ccpmat 22068 matToPolyMat cmat2pmat 22069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-ot 4596 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-ofr 7619 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-pm 8771 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9309 df-sup 9383 df-oi 9451 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-fz 13431 df-fzo 13574 df-seq 13913 df-hash 14237 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-sca 17154 df-vsca 17155 df-ip 17156 df-tset 17157 df-ple 17158 df-ds 17160 df-hom 17162 df-cco 17163 df-0g 17328 df-gsum 17329 df-prds 17334 df-pws 17336 df-mre 17471 df-mrc 17472 df-acs 17474 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-mhm 18606 df-submnd 18607 df-grp 18756 df-minusg 18757 df-sbg 18758 df-mulg 18878 df-subg 18930 df-ghm 19011 df-cntz 19102 df-cmn 19569 df-abl 19570 df-mgp 19902 df-ur 19919 df-srg 19923 df-ring 19971 df-cring 19972 df-rnghom 20153 df-rngiso 20154 df-subrg 20234 df-lmod 20338 df-lss 20408 df-sra 20649 df-rgmod 20650 df-dsmm 21154 df-frlm 21169 df-assa 21275 df-ascl 21277 df-psr 21327 df-mvr 21328 df-mpl 21329 df-opsr 21331 df-psr1 21567 df-vr1 21568 df-ply1 21569 df-coe1 21570 df-mamu 21749 df-mat 21771 df-cpmat 22071 df-mat2pmat 22072 df-cpmat2mat 22073 |
This theorem is referenced by: matcpmric 22124 |
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