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Mirrors > Home > MPE Home > Th. List > pm2mpmhm | Structured version Visualization version GIF version |
Description: The transformation of polynomial matrices into polynomials over matrices is a homomorphism of multiplicative monoids. (Contributed by AV, 22-Oct-2019.) (Revised by AV, 6-Dec-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 |
---|---|
pm2mpmhm | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ ((mulGrp‘𝐶) MndHom (mulGrp‘𝑄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2mpmhm.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | pm2mpmhm.c | . . . 4 ⊢ 𝐶 = (𝑁 Mat 𝑃) | |
3 | 1, 2 | pmatring 22685 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring) |
4 | eqid 2726 | . . . 4 ⊢ (mulGrp‘𝐶) = (mulGrp‘𝐶) | |
5 | 4 | ringmgp 20222 | . . 3 ⊢ (𝐶 ∈ Ring → (mulGrp‘𝐶) ∈ Mnd) |
6 | 3, 5 | syl 17 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (mulGrp‘𝐶) ∈ Mnd) |
7 | pm2mpmhm.a | . . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
8 | 7 | matring 22436 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
9 | pm2mpmhm.q | . . . 4 ⊢ 𝑄 = (Poly1‘𝐴) | |
10 | 9 | ply1ring 22237 | . . 3 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ Ring) |
11 | eqid 2726 | . . . 4 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄) | |
12 | 11 | ringmgp 20222 | . . 3 ⊢ (𝑄 ∈ Ring → (mulGrp‘𝑄) ∈ Mnd) |
13 | 8, 10, 12 | 3syl 18 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (mulGrp‘𝑄) ∈ Mnd) |
14 | eqid 2726 | . . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
15 | 4, 14 | mgpbas 20123 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘(mulGrp‘𝐶)) |
16 | 15 | eqcomi 2735 | . . . 4 ⊢ (Base‘(mulGrp‘𝐶)) = (Base‘𝐶) |
17 | eqid 2726 | . . . 4 ⊢ ( ·𝑠 ‘𝑄) = ( ·𝑠 ‘𝑄) | |
18 | eqid 2726 | . . . 4 ⊢ (.g‘(mulGrp‘𝑄)) = (.g‘(mulGrp‘𝑄)) | |
19 | eqid 2726 | . . . 4 ⊢ (var1‘𝐴) = (var1‘𝐴) | |
20 | pm2mpmhm.t | . . . 4 ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅) | |
21 | eqid 2726 | . . . . . 6 ⊢ (Base‘𝑄) = (Base‘𝑄) | |
22 | 11, 21 | mgpbas 20123 | . . . . 5 ⊢ (Base‘𝑄) = (Base‘(mulGrp‘𝑄)) |
23 | 22 | eqcomi 2735 | . . . 4 ⊢ (Base‘(mulGrp‘𝑄)) = (Base‘𝑄) |
24 | 1, 2, 16, 17, 18, 19, 7, 9, 20, 23 | pm2mpf 22791 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:(Base‘(mulGrp‘𝐶))⟶(Base‘(mulGrp‘𝑄))) |
25 | 1, 2, 7, 9, 20, 16 | pm2mpmhmlem2 22812 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ (Base‘(mulGrp‘𝐶))∀𝑦 ∈ (Base‘(mulGrp‘𝐶))(𝑇‘(𝑥(.r‘𝐶)𝑦)) = ((𝑇‘𝑥)(.r‘𝑄)(𝑇‘𝑦))) |
26 | 1, 2, 14, 17, 18, 19, 7, 9, 20 | idpm2idmp 22794 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇‘(1r‘𝐶)) = (1r‘𝑄)) |
27 | 24, 25, 26 | 3jca 1125 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇:(Base‘(mulGrp‘𝐶))⟶(Base‘(mulGrp‘𝑄)) ∧ ∀𝑥 ∈ (Base‘(mulGrp‘𝐶))∀𝑦 ∈ (Base‘(mulGrp‘𝐶))(𝑇‘(𝑥(.r‘𝐶)𝑦)) = ((𝑇‘𝑥)(.r‘𝑄)(𝑇‘𝑦)) ∧ (𝑇‘(1r‘𝐶)) = (1r‘𝑄))) |
28 | eqid 2726 | . . 3 ⊢ (Base‘(mulGrp‘𝐶)) = (Base‘(mulGrp‘𝐶)) | |
29 | eqid 2726 | . . 3 ⊢ (Base‘(mulGrp‘𝑄)) = (Base‘(mulGrp‘𝑄)) | |
30 | eqid 2726 | . . . 4 ⊢ (.r‘𝐶) = (.r‘𝐶) | |
31 | 4, 30 | mgpplusg 20121 | . . 3 ⊢ (.r‘𝐶) = (+g‘(mulGrp‘𝐶)) |
32 | eqid 2726 | . . . 4 ⊢ (.r‘𝑄) = (.r‘𝑄) | |
33 | 11, 32 | mgpplusg 20121 | . . 3 ⊢ (.r‘𝑄) = (+g‘(mulGrp‘𝑄)) |
34 | eqid 2726 | . . . 4 ⊢ (1r‘𝐶) = (1r‘𝐶) | |
35 | 4, 34 | ringidval 20166 | . . 3 ⊢ (1r‘𝐶) = (0g‘(mulGrp‘𝐶)) |
36 | eqid 2726 | . . . 4 ⊢ (1r‘𝑄) = (1r‘𝑄) | |
37 | 11, 36 | ringidval 20166 | . . 3 ⊢ (1r‘𝑄) = (0g‘(mulGrp‘𝑄)) |
38 | 28, 29, 31, 33, 35, 37 | ismhm 18775 | . 2 ⊢ (𝑇 ∈ ((mulGrp‘𝐶) MndHom (mulGrp‘𝑄)) ↔ (((mulGrp‘𝐶) ∈ Mnd ∧ (mulGrp‘𝑄) ∈ Mnd) ∧ (𝑇:(Base‘(mulGrp‘𝐶))⟶(Base‘(mulGrp‘𝑄)) ∧ ∀𝑥 ∈ (Base‘(mulGrp‘𝐶))∀𝑦 ∈ (Base‘(mulGrp‘𝐶))(𝑇‘(𝑥(.r‘𝐶)𝑦)) = ((𝑇‘𝑥)(.r‘𝑄)(𝑇‘𝑦)) ∧ (𝑇‘(1r‘𝐶)) = (1r‘𝑄)))) |
39 | 6, 13, 27, 38 | syl21anbrc 1341 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ ((mulGrp‘𝐶) MndHom (mulGrp‘𝑄))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ⟶wf 6550 ‘cfv 6554 (class class class)co 7424 Fincfn 8974 Basecbs 17213 .rcmulr 17267 ·𝑠 cvsca 17270 Mndcmnd 18727 MndHom cmhm 18771 .gcmg 19061 mulGrpcmgp 20117 1rcur 20164 Ringcrg 20216 var1cv1 22165 Poly1cpl1 22166 Mat cmat 22398 pMatToMatPoly cpm2mp 22785 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-ot 4642 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-ofr 7691 df-om 7877 df-1st 8003 df-2nd 8004 df-supp 8175 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-er 8734 df-map 8857 df-pm 8858 df-ixp 8927 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fsupp 9406 df-sup 9485 df-oi 9553 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12611 df-dec 12730 df-uz 12875 df-fz 13539 df-fzo 13682 df-seq 14022 df-hash 14348 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-sca 17282 df-vsca 17283 df-ip 17284 df-tset 17285 df-ple 17286 df-ds 17288 df-hom 17290 df-cco 17291 df-0g 17456 df-gsum 17457 df-prds 17462 df-pws 17464 df-mre 17599 df-mrc 17600 df-acs 17602 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-mhm 18773 df-submnd 18774 df-grp 18931 df-minusg 18932 df-sbg 18933 df-mulg 19062 df-subg 19117 df-ghm 19207 df-cntz 19311 df-cmn 19780 df-abl 19781 df-mgp 20118 df-rng 20136 df-ur 20165 df-srg 20170 df-ring 20218 df-subrng 20528 df-subrg 20553 df-lmod 20838 df-lss 20909 df-sra 21151 df-rgmod 21152 df-dsmm 21730 df-frlm 21745 df-ascl 21853 df-psr 21906 df-mvr 21907 df-mpl 21908 df-opsr 21910 df-psr1 22169 df-vr1 22170 df-ply1 22171 df-coe1 22172 df-mamu 22382 df-mat 22399 df-decpmat 22756 df-pm2mp 22786 |
This theorem is referenced by: pm2mprhm 22814 |
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