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Mirrors > Home > MPE Home > Th. List > mat2pmatrhm | Structured version Visualization version GIF version |
Description: The transformation of matrices into polynomial matrices is a ring homomorphism. (Contributed by AV, 29-Oct-2019.) |
Ref | Expression |
---|---|
mat2pmatbas.t | ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) |
mat2pmatbas.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
mat2pmatbas.b | ⊢ 𝐵 = (Base‘𝐴) |
mat2pmatbas.p | ⊢ 𝑃 = (Poly1‘𝑅) |
mat2pmatbas.c | ⊢ 𝐶 = (𝑁 Mat 𝑃) |
mat2pmatbas0.h | ⊢ 𝐻 = (Base‘𝐶) |
Ref | Expression |
---|---|
mat2pmatrhm | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingHom 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crngring 19793 | . . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
2 | mat2pmatbas.a | . . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
3 | 2 | matring 21590 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
4 | 1, 3 | sylan2 593 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring) |
5 | mat2pmatbas.p | . . . . 5 ⊢ 𝑃 = (Poly1‘𝑅) | |
6 | 5 | ply1ring 21417 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
7 | 1, 6 | syl 17 | . . 3 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring) |
8 | mat2pmatbas.c | . . . 4 ⊢ 𝐶 = (𝑁 Mat 𝑃) | |
9 | 8 | matring 21590 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝐶 ∈ Ring) |
10 | 7, 9 | sylan2 593 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐶 ∈ Ring) |
11 | mat2pmatbas.t | . . . . 5 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) | |
12 | mat2pmatbas.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐴) | |
13 | mat2pmatbas0.h | . . . . 5 ⊢ 𝐻 = (Base‘𝐶) | |
14 | 11, 2, 12, 5, 8, 13 | mat2pmatghm 21877 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐴 GrpHom 𝐶)) |
15 | 1, 14 | sylan2 593 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 GrpHom 𝐶)) |
16 | 11, 2, 12, 5, 8, 13 | mat2pmatmhm 21880 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝐶))) |
17 | 15, 16 | jca 512 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑇 ∈ (𝐴 GrpHom 𝐶) ∧ 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝐶)))) |
18 | eqid 2740 | . . 3 ⊢ (mulGrp‘𝐴) = (mulGrp‘𝐴) | |
19 | eqid 2740 | . . 3 ⊢ (mulGrp‘𝐶) = (mulGrp‘𝐶) | |
20 | 18, 19 | isrhm 19963 | . 2 ⊢ (𝑇 ∈ (𝐴 RingHom 𝐶) ↔ ((𝐴 ∈ Ring ∧ 𝐶 ∈ Ring) ∧ (𝑇 ∈ (𝐴 GrpHom 𝐶) ∧ 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝐶))))) |
21 | 4, 10, 17, 20 | syl21anbrc 1343 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingHom 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ‘cfv 6432 (class class class)co 7271 Fincfn 8716 Basecbs 16910 MndHom cmhm 18426 GrpHom cghm 18829 mulGrpcmgp 19718 Ringcrg 19781 CRingccrg 19782 RingHom crh 19954 Poly1cpl1 21346 Mat cmat 21552 matToPolyMat cmat2pmat 21851 |
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 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-ot 4576 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-of 7527 df-ofr 7528 df-om 7707 df-1st 7824 df-2nd 7825 df-supp 7969 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-map 8600 df-pm 8601 df-ixp 8669 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-fsupp 9107 df-sup 9179 df-oi 9247 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12437 df-uz 12582 df-fz 13239 df-fzo 13382 df-seq 13720 df-hash 14043 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-sca 16976 df-vsca 16977 df-ip 16978 df-tset 16979 df-ple 16980 df-ds 16982 df-hom 16984 df-cco 16985 df-0g 17150 df-gsum 17151 df-prds 17156 df-pws 17158 df-mre 17293 df-mrc 17294 df-acs 17296 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-mhm 18428 df-submnd 18429 df-grp 18578 df-minusg 18579 df-sbg 18580 df-mulg 18699 df-subg 18750 df-ghm 18830 df-cntz 18921 df-cmn 19386 df-abl 19387 df-mgp 19719 df-ur 19736 df-ring 19783 df-cring 19784 df-rnghom 19957 df-subrg 20020 df-lmod 20123 df-lss 20192 df-sra 20432 df-rgmod 20433 df-dsmm 20937 df-frlm 20952 df-assa 21058 df-ascl 21060 df-psr 21110 df-mpl 21112 df-opsr 21114 df-psr1 21349 df-ply1 21351 df-mamu 21531 df-mat 21553 df-mat2pmat 21854 |
This theorem is referenced by: cpmidgsumm2pm 22016 cayhamlem4 22035 |
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