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Mirrors > Home > MPE Home > Th. List > mat2pmatmhm | Structured version Visualization version GIF version |
Description: The transformation of matrices into polynomial matrices is a homomorphism of multiplicative monoids. (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 |
---|---|
mat2pmatmhm | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crngring 19302 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
2 | mat2pmatbas.a | . . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
3 | 2 | matring 21046 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
4 | 1, 3 | sylan2 594 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring) |
5 | eqid 2821 | . . . 4 ⊢ (mulGrp‘𝐴) = (mulGrp‘𝐴) | |
6 | 5 | ringmgp 19297 | . . 3 ⊢ (𝐴 ∈ Ring → (mulGrp‘𝐴) ∈ Mnd) |
7 | 4, 6 | syl 17 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (mulGrp‘𝐴) ∈ Mnd) |
8 | mat2pmatbas.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
9 | 8 | ply1ring 20410 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
10 | 1, 9 | syl 17 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring) |
11 | mat2pmatbas.c | . . . . 5 ⊢ 𝐶 = (𝑁 Mat 𝑃) | |
12 | 11 | matring 21046 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝐶 ∈ Ring) |
13 | 10, 12 | sylan2 594 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐶 ∈ Ring) |
14 | eqid 2821 | . . . 4 ⊢ (mulGrp‘𝐶) = (mulGrp‘𝐶) | |
15 | 14 | ringmgp 19297 | . . 3 ⊢ (𝐶 ∈ Ring → (mulGrp‘𝐶) ∈ Mnd) |
16 | 13, 15 | syl 17 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (mulGrp‘𝐶) ∈ Mnd) |
17 | mat2pmatbas.t | . . . . 5 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) | |
18 | mat2pmatbas.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐴) | |
19 | mat2pmatbas0.h | . . . . 5 ⊢ 𝐻 = (Base‘𝐶) | |
20 | 17, 2, 18, 8, 11, 19 | mat2pmatf 21330 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵⟶𝐻) |
21 | 1, 20 | sylan2 594 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇:𝐵⟶𝐻) |
22 | 17, 2, 18, 8, 11, 19 | mat2pmatmul 21333 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑇‘(𝑥(.r‘𝐴)𝑦)) = ((𝑇‘𝑥)(.r‘𝐶)(𝑇‘𝑦))) |
23 | 17, 2, 18, 8, 11, 19 | mat2pmat1 21334 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇‘(1r‘𝐴)) = (1r‘𝐶)) |
24 | 1, 23 | sylan2 594 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑇‘(1r‘𝐴)) = (1r‘𝐶)) |
25 | 21, 22, 24 | 3jca 1124 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑇:𝐵⟶𝐻 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑇‘(𝑥(.r‘𝐴)𝑦)) = ((𝑇‘𝑥)(.r‘𝐶)(𝑇‘𝑦)) ∧ (𝑇‘(1r‘𝐴)) = (1r‘𝐶))) |
26 | 5, 18 | mgpbas 19239 | . . 3 ⊢ 𝐵 = (Base‘(mulGrp‘𝐴)) |
27 | 14, 19 | mgpbas 19239 | . . 3 ⊢ 𝐻 = (Base‘(mulGrp‘𝐶)) |
28 | eqid 2821 | . . . 4 ⊢ (.r‘𝐴) = (.r‘𝐴) | |
29 | 5, 28 | mgpplusg 19237 | . . 3 ⊢ (.r‘𝐴) = (+g‘(mulGrp‘𝐴)) |
30 | eqid 2821 | . . . 4 ⊢ (.r‘𝐶) = (.r‘𝐶) | |
31 | 14, 30 | mgpplusg 19237 | . . 3 ⊢ (.r‘𝐶) = (+g‘(mulGrp‘𝐶)) |
32 | eqid 2821 | . . . 4 ⊢ (1r‘𝐴) = (1r‘𝐴) | |
33 | 5, 32 | ringidval 19247 | . . 3 ⊢ (1r‘𝐴) = (0g‘(mulGrp‘𝐴)) |
34 | eqid 2821 | . . . 4 ⊢ (1r‘𝐶) = (1r‘𝐶) | |
35 | 14, 34 | ringidval 19247 | . . 3 ⊢ (1r‘𝐶) = (0g‘(mulGrp‘𝐶)) |
36 | 26, 27, 29, 31, 33, 35 | ismhm 17952 | . 2 ⊢ (𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝐶)) ↔ (((mulGrp‘𝐴) ∈ Mnd ∧ (mulGrp‘𝐶) ∈ Mnd) ∧ (𝑇:𝐵⟶𝐻 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑇‘(𝑥(.r‘𝐴)𝑦)) = ((𝑇‘𝑥)(.r‘𝐶)(𝑇‘𝑦)) ∧ (𝑇‘(1r‘𝐴)) = (1r‘𝐶)))) |
37 | 7, 16, 25, 36 | syl21anbrc 1340 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 Fincfn 8503 Basecbs 16477 .rcmulr 16560 Mndcmnd 17905 MndHom cmhm 17948 mulGrpcmgp 19233 1rcur 19245 Ringcrg 19291 CRingccrg 19292 Poly1cpl1 20339 Mat cmat 21010 matToPolyMat cmat2pmat 21306 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-ot 4570 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-ofr 7404 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-sup 8900 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-fzo 13028 df-seq 13364 df-hash 13685 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-hom 16583 df-cco 16584 df-0g 16709 df-gsum 16710 df-prds 16715 df-pws 16717 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-mulg 18219 df-subg 18270 df-ghm 18350 df-cntz 18441 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-rnghom 19461 df-subrg 19527 df-lmod 19630 df-lss 19698 df-sra 19938 df-rgmod 19939 df-assa 20079 df-ascl 20081 df-psr 20130 df-mpl 20132 df-opsr 20134 df-psr1 20342 df-ply1 20344 df-dsmm 20870 df-frlm 20885 df-mamu 20989 df-mat 21011 df-mat2pmat 21309 |
This theorem is referenced by: mat2pmatrhm 21336 m2cpmmhm 21347 cayhamlem4 21490 |
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