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Mirrors > Home > MPE Home > Th. List > pm2mprhm | Structured version Visualization version GIF version |
Description: The transformation of polynomial matrices into polynomials over matrices is a ring homomorphism. (Contributed by AV, 22-Oct-2019.) |
Ref | Expression |
---|---|
pm2mpmhm.p | ⊢ 𝑃 = (Poly1‘𝑅) |
pm2mpmhm.c | ⊢ 𝐶 = (𝑁 Mat 𝑃) |
pm2mpmhm.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
pm2mpmhm.q | ⊢ 𝑄 = (Poly1‘𝐴) |
pm2mpmhm.t | ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅) |
Ref | Expression |
---|---|
pm2mprhm | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐶 RingHom 𝑄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2mpmhm.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | pm2mpmhm.c | . . 3 ⊢ 𝐶 = (𝑁 Mat 𝑃) | |
3 | 1, 2 | pmatring 21293 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring) |
4 | pm2mpmhm.a | . . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
5 | 4 | matring 21044 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
6 | pm2mpmhm.q | . . . 4 ⊢ 𝑄 = (Poly1‘𝐴) | |
7 | 6 | ply1ring 20408 | . . 3 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ Ring) |
8 | 5, 7 | syl 17 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ Ring) |
9 | eqid 2819 | . . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
10 | eqid 2819 | . . . 4 ⊢ ( ·𝑠 ‘𝑄) = ( ·𝑠 ‘𝑄) | |
11 | eqid 2819 | . . . 4 ⊢ (.g‘(mulGrp‘𝑄)) = (.g‘(mulGrp‘𝑄)) | |
12 | eqid 2819 | . . . 4 ⊢ (var1‘𝐴) = (var1‘𝐴) | |
13 | eqid 2819 | . . . 4 ⊢ (Base‘𝑄) = (Base‘𝑄) | |
14 | pm2mpmhm.t | . . . 4 ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅) | |
15 | 1, 2, 9, 10, 11, 12, 4, 6, 13, 14 | pm2mpghm 21416 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐶 GrpHom 𝑄)) |
16 | 1, 2, 4, 6, 14 | pm2mpmhm 21420 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ ((mulGrp‘𝐶) MndHom (mulGrp‘𝑄))) |
17 | 15, 16 | jca 514 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇 ∈ (𝐶 GrpHom 𝑄) ∧ 𝑇 ∈ ((mulGrp‘𝐶) MndHom (mulGrp‘𝑄)))) |
18 | eqid 2819 | . . 3 ⊢ (mulGrp‘𝐶) = (mulGrp‘𝐶) | |
19 | eqid 2819 | . . 3 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄) | |
20 | 18, 19 | isrhm 19465 | . 2 ⊢ (𝑇 ∈ (𝐶 RingHom 𝑄) ↔ ((𝐶 ∈ Ring ∧ 𝑄 ∈ Ring) ∧ (𝑇 ∈ (𝐶 GrpHom 𝑄) ∧ 𝑇 ∈ ((mulGrp‘𝐶) MndHom (mulGrp‘𝑄))))) |
21 | 3, 8, 17, 20 | syl21anbrc 1339 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐶 RingHom 𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ‘cfv 6348 (class class class)co 7148 Fincfn 8501 Basecbs 16475 ·𝑠 cvsca 16561 MndHom cmhm 17946 .gcmg 18216 GrpHom cghm 18347 mulGrpcmgp 19231 Ringcrg 19289 RingHom crh 19456 var1cv1 20336 Poly1cpl1 20337 Mat cmat 21008 pMatToMatPoly cpm2mp 21392 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-fal 1544 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-ot 4568 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-of 7401 df-ofr 7402 df-om 7573 df-1st 7681 df-2nd 7682 df-supp 7823 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-oadd 8098 df-er 8281 df-map 8400 df-pm 8401 df-ixp 8454 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-fsupp 8826 df-sup 8898 df-oi 8966 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-z 11974 df-dec 12091 df-uz 12236 df-fz 12885 df-fzo 13026 df-seq 13362 df-hash 13683 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-hom 16581 df-cco 16582 df-0g 16707 df-gsum 16708 df-prds 16713 df-pws 16715 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-ghm 18348 df-cntz 18439 df-cmn 18900 df-abl 18901 df-mgp 19232 df-ur 19244 df-srg 19248 df-ring 19291 df-rnghom 19459 df-subrg 19525 df-lmod 19628 df-lss 19696 df-sra 19936 df-rgmod 19937 df-ascl 20079 df-psr 20128 df-mvr 20129 df-mpl 20130 df-opsr 20132 df-psr1 20340 df-vr1 20341 df-ply1 20342 df-coe1 20343 df-dsmm 20868 df-frlm 20883 df-mamu 20987 df-mat 21009 df-decpmat 21363 df-pm2mp 21393 |
This theorem is referenced by: pm2mprngiso 21422 |
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