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Mirrors > Home > MPE Home > Th. List > m2cpmrhm | Structured version Visualization version GIF version |
Description: The transformation of matrices into constant polynomial matrices is a ring homomorphism. (Contributed by AV, 18-Nov-2019.) |
Ref | Expression |
---|---|
m2cpm.s | ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) |
m2cpm.t | ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) |
m2cpm.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
m2cpm.b | ⊢ 𝐵 = (Base‘𝐴) |
m2cpmghm.p | ⊢ 𝑃 = (Poly1‘𝑅) |
m2cpmghm.c | ⊢ 𝐶 = (𝑁 Mat 𝑃) |
m2cpmghm.u | ⊢ 𝑈 = (𝐶 ↾s 𝑆) |
Ref | Expression |
---|---|
m2cpmrhm | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingHom 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crngring 19431 | . . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
2 | m2cpm.a | . . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
3 | 2 | matring 21197 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
4 | 1, 3 | sylan2 596 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring) |
5 | m2cpm.s | . . . . 5 ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) | |
6 | m2cpmghm.p | . . . . 5 ⊢ 𝑃 = (Poly1‘𝑅) | |
7 | m2cpmghm.c | . . . . 5 ⊢ 𝐶 = (𝑁 Mat 𝑃) | |
8 | 5, 6, 7 | cpmatsrgpmat 21475 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubRing‘𝐶)) |
9 | 1, 8 | sylan2 596 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑆 ∈ (SubRing‘𝐶)) |
10 | m2cpmghm.u | . . . 4 ⊢ 𝑈 = (𝐶 ↾s 𝑆) | |
11 | 10 | subrgring 19660 | . . 3 ⊢ (𝑆 ∈ (SubRing‘𝐶) → 𝑈 ∈ Ring) |
12 | 9, 11 | syl 17 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑈 ∈ Ring) |
13 | m2cpm.t | . . . . 5 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) | |
14 | m2cpm.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐴) | |
15 | 5, 13, 2, 14, 6, 7, 10 | m2cpmghm 21498 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐴 GrpHom 𝑈)) |
16 | 1, 15 | sylan2 596 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 GrpHom 𝑈)) |
17 | 5, 13, 2, 14, 6, 7, 10 | m2cpmmhm 21499 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑈))) |
18 | 16, 17 | jca 515 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑇 ∈ (𝐴 GrpHom 𝑈) ∧ 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑈)))) |
19 | eqid 2739 | . . 3 ⊢ (mulGrp‘𝐴) = (mulGrp‘𝐴) | |
20 | eqid 2739 | . . 3 ⊢ (mulGrp‘𝑈) = (mulGrp‘𝑈) | |
21 | 19, 20 | isrhm 19598 | . 2 ⊢ (𝑇 ∈ (𝐴 RingHom 𝑈) ↔ ((𝐴 ∈ Ring ∧ 𝑈 ∈ Ring) ∧ (𝑇 ∈ (𝐴 GrpHom 𝑈) ∧ 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑈))))) |
22 | 4, 12, 18, 21 | syl21anbrc 1345 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingHom 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ‘cfv 6340 (class class class)co 7173 Fincfn 8558 Basecbs 16589 ↾s cress 16590 MndHom cmhm 18073 GrpHom cghm 18476 mulGrpcmgp 19361 Ringcrg 19419 CRingccrg 19420 RingHom crh 19589 SubRingcsubrg 19653 Poly1cpl1 20955 Mat cmat 21161 ConstPolyMat ccpmat 21457 matToPolyMat cmat2pmat 21458 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-ot 4526 df-uni 4798 df-int 4838 df-iun 4884 df-iin 4885 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-se 5485 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-isom 6349 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-of 7428 df-ofr 7429 df-om 7603 df-1st 7717 df-2nd 7718 df-supp 7860 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-1o 8134 df-er 8323 df-map 8442 df-pm 8443 df-ixp 8511 df-en 8559 df-dom 8560 df-sdom 8561 df-fin 8562 df-fsupp 8910 df-sup 8982 df-oi 9050 df-card 9444 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-nn 11720 df-2 11782 df-3 11783 df-4 11784 df-5 11785 df-6 11786 df-7 11787 df-8 11788 df-9 11789 df-n0 11980 df-z 12066 df-dec 12183 df-uz 12328 df-fz 12985 df-fzo 13128 df-seq 13464 df-hash 13786 df-struct 16591 df-ndx 16592 df-slot 16593 df-base 16595 df-sets 16596 df-ress 16597 df-plusg 16684 df-mulr 16685 df-sca 16687 df-vsca 16688 df-ip 16689 df-tset 16690 df-ple 16691 df-ds 16693 df-hom 16695 df-cco 16696 df-0g 16821 df-gsum 16822 df-prds 16827 df-pws 16829 df-mre 16963 df-mrc 16964 df-acs 16966 df-mgm 17971 df-sgrp 18020 df-mnd 18031 df-mhm 18075 df-submnd 18076 df-grp 18225 df-minusg 18226 df-sbg 18227 df-mulg 18346 df-subg 18397 df-ghm 18477 df-cntz 18568 df-cmn 19029 df-abl 19030 df-mgp 19362 df-ur 19374 df-srg 19378 df-ring 19421 df-cring 19422 df-rnghom 19592 df-subrg 19655 df-lmod 19758 df-lss 19826 df-sra 20066 df-rgmod 20067 df-dsmm 20551 df-frlm 20566 df-assa 20672 df-ascl 20674 df-psr 20725 df-mvr 20726 df-mpl 20727 df-opsr 20729 df-psr1 20958 df-vr1 20959 df-ply1 20960 df-coe1 20961 df-mamu 21140 df-mat 21162 df-cpmat 21460 df-mat2pmat 21461 |
This theorem is referenced by: m2cpmrngiso 21512 |
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