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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 21839 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring) |
4 | eqid 2738 | . . . 4 ⊢ (mulGrp‘𝐶) = (mulGrp‘𝐶) | |
5 | 4 | ringmgp 19787 | . . 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 21590 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
9 | pm2mpmhm.q | . . . 4 ⊢ 𝑄 = (Poly1‘𝐴) | |
10 | 9 | ply1ring 21417 | . . 3 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ Ring) |
11 | eqid 2738 | . . . 4 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄) | |
12 | 11 | ringmgp 19787 | . . 3 ⊢ (𝑄 ∈ Ring → (mulGrp‘𝑄) ∈ Mnd) |
13 | 8, 10, 12 | 3syl 18 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (mulGrp‘𝑄) ∈ Mnd) |
14 | eqid 2738 | . . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
15 | 4, 14 | mgpbas 19724 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘(mulGrp‘𝐶)) |
16 | 15 | eqcomi 2747 | . . . 4 ⊢ (Base‘(mulGrp‘𝐶)) = (Base‘𝐶) |
17 | eqid 2738 | . . . 4 ⊢ ( ·𝑠 ‘𝑄) = ( ·𝑠 ‘𝑄) | |
18 | eqid 2738 | . . . 4 ⊢ (.g‘(mulGrp‘𝑄)) = (.g‘(mulGrp‘𝑄)) | |
19 | eqid 2738 | . . . 4 ⊢ (var1‘𝐴) = (var1‘𝐴) | |
20 | pm2mpmhm.t | . . . 4 ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅) | |
21 | eqid 2738 | . . . . . 6 ⊢ (Base‘𝑄) = (Base‘𝑄) | |
22 | 11, 21 | mgpbas 19724 | . . . . 5 ⊢ (Base‘𝑄) = (Base‘(mulGrp‘𝑄)) |
23 | 22 | eqcomi 2747 | . . . 4 ⊢ (Base‘(mulGrp‘𝑄)) = (Base‘𝑄) |
24 | 1, 2, 16, 17, 18, 19, 7, 9, 20, 23 | pm2mpf 21945 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:(Base‘(mulGrp‘𝐶))⟶(Base‘(mulGrp‘𝑄))) |
25 | 1, 2, 7, 9, 20, 16 | pm2mpmhmlem2 21966 | . . 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 21948 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇‘(1r‘𝐶)) = (1r‘𝑄)) |
27 | 24, 25, 26 | 3jca 1127 | . 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 2738 | . . 3 ⊢ (Base‘(mulGrp‘𝐶)) = (Base‘(mulGrp‘𝐶)) | |
29 | eqid 2738 | . . 3 ⊢ (Base‘(mulGrp‘𝑄)) = (Base‘(mulGrp‘𝑄)) | |
30 | eqid 2738 | . . . 4 ⊢ (.r‘𝐶) = (.r‘𝐶) | |
31 | 4, 30 | mgpplusg 19722 | . . 3 ⊢ (.r‘𝐶) = (+g‘(mulGrp‘𝐶)) |
32 | eqid 2738 | . . . 4 ⊢ (.r‘𝑄) = (.r‘𝑄) | |
33 | 11, 32 | mgpplusg 19722 | . . 3 ⊢ (.r‘𝑄) = (+g‘(mulGrp‘𝑄)) |
34 | eqid 2738 | . . . 4 ⊢ (1r‘𝐶) = (1r‘𝐶) | |
35 | 4, 34 | ringidval 19737 | . . 3 ⊢ (1r‘𝐶) = (0g‘(mulGrp‘𝐶)) |
36 | eqid 2738 | . . . 4 ⊢ (1r‘𝑄) = (1r‘𝑄) | |
37 | 11, 36 | ringidval 19737 | . . 3 ⊢ (1r‘𝑄) = (0g‘(mulGrp‘𝑄)) |
38 | 28, 29, 31, 33, 35, 37 | ismhm 18430 | . 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 1343 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ ((mulGrp‘𝐶) MndHom (mulGrp‘𝑄))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ⟶wf 6431 ‘cfv 6435 (class class class)co 7277 Fincfn 8731 Basecbs 16910 .rcmulr 16961 ·𝑠 cvsca 16964 Mndcmnd 18383 MndHom cmhm 18426 .gcmg 18698 mulGrpcmgp 19718 1rcur 19735 Ringcrg 19781 var1cv1 21345 Poly1cpl1 21346 Mat cmat 21552 pMatToMatPoly cpm2mp 21939 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5211 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-cnex 10925 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-pre-mulgt0 10946 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-tp 4568 df-op 4570 df-ot 4572 df-uni 4842 df-int 4882 df-iun 4928 df-iin 4929 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-se 5547 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6204 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-isom 6444 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-of 7533 df-ofr 7534 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7976 df-frecs 8095 df-wrecs 8126 df-recs 8200 df-rdg 8239 df-1o 8295 df-er 8496 df-map 8615 df-pm 8616 df-ixp 8684 df-en 8732 df-dom 8733 df-sdom 8734 df-fin 8735 df-fsupp 9127 df-sup 9199 df-oi 9267 df-card 9695 df-pnf 11009 df-mnf 11010 df-xr 11011 df-ltxr 11012 df-le 11013 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-5 12037 df-6 12038 df-7 12039 df-8 12040 df-9 12041 df-n0 12232 df-z 12318 df-dec 12436 df-uz 12581 df-fz 13238 df-fzo 13381 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-srg 19740 df-ring 19783 df-subrg 20020 df-lmod 20123 df-lss 20192 df-sra 20432 df-rgmod 20433 df-dsmm 20937 df-frlm 20952 df-ascl 21060 df-psr 21110 df-mvr 21111 df-mpl 21112 df-opsr 21114 df-psr1 21349 df-vr1 21350 df-ply1 21351 df-coe1 21352 df-mamu 21531 df-mat 21553 df-decpmat 21910 df-pm2mp 21940 |
This theorem is referenced by: pm2mprhm 21968 |
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