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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 19440 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
2 | mat2pmatbas.a | . . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
3 | 2 | matring 21206 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
4 | 1, 3 | sylan2 596 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring) |
5 | eqid 2739 | . . . 4 ⊢ (mulGrp‘𝐴) = (mulGrp‘𝐴) | |
6 | 5 | ringmgp 19434 | . . 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 21035 | . . . . 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 21206 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝐶 ∈ Ring) |
13 | 10, 12 | sylan2 596 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐶 ∈ Ring) |
14 | eqid 2739 | . . . 4 ⊢ (mulGrp‘𝐶) = (mulGrp‘𝐶) | |
15 | 14 | ringmgp 19434 | . . 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 21491 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵⟶𝐻) |
21 | 1, 20 | sylan2 596 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇:𝐵⟶𝐻) |
22 | 17, 2, 18, 8, 11, 19 | mat2pmatmul 21494 | . . 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 21495 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇‘(1r‘𝐴)) = (1r‘𝐶)) |
24 | 1, 23 | sylan2 596 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑇‘(1r‘𝐴)) = (1r‘𝐶)) |
25 | 21, 22, 24 | 3jca 1129 | . 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 19376 | . . 3 ⊢ 𝐵 = (Base‘(mulGrp‘𝐴)) |
27 | 14, 19 | mgpbas 19376 | . . 3 ⊢ 𝐻 = (Base‘(mulGrp‘𝐶)) |
28 | eqid 2739 | . . . 4 ⊢ (.r‘𝐴) = (.r‘𝐴) | |
29 | 5, 28 | mgpplusg 19374 | . . 3 ⊢ (.r‘𝐴) = (+g‘(mulGrp‘𝐴)) |
30 | eqid 2739 | . . . 4 ⊢ (.r‘𝐶) = (.r‘𝐶) | |
31 | 14, 30 | mgpplusg 19374 | . . 3 ⊢ (.r‘𝐶) = (+g‘(mulGrp‘𝐶)) |
32 | eqid 2739 | . . . 4 ⊢ (1r‘𝐴) = (1r‘𝐴) | |
33 | 5, 32 | ringidval 19384 | . . 3 ⊢ (1r‘𝐴) = (0g‘(mulGrp‘𝐴)) |
34 | eqid 2739 | . . . 4 ⊢ (1r‘𝐶) = (1r‘𝐶) | |
35 | 14, 34 | ringidval 19384 | . . 3 ⊢ (1r‘𝐶) = (0g‘(mulGrp‘𝐶)) |
36 | 26, 27, 29, 31, 33, 35 | ismhm 18086 | . 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 1345 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ∀wral 3054 ⟶wf 6345 ‘cfv 6349 (class class class)co 7182 Fincfn 8567 Basecbs 16598 .rcmulr 16681 Mndcmnd 18039 MndHom cmhm 18082 mulGrpcmgp 19370 1rcur 19382 Ringcrg 19428 CRingccrg 19429 Poly1cpl1 20964 Mat cmat 21170 matToPolyMat cmat2pmat 21467 |
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 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-ot 4535 df-uni 4807 df-int 4847 df-iun 4893 df-iin 4894 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-se 5494 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-of 7437 df-ofr 7438 df-om 7612 df-1st 7726 df-2nd 7727 df-supp 7869 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-1o 8143 df-er 8332 df-map 8451 df-pm 8452 df-ixp 8520 df-en 8568 df-dom 8569 df-sdom 8570 df-fin 8571 df-fsupp 8919 df-sup 8991 df-oi 9059 df-card 9453 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-nn 11729 df-2 11791 df-3 11792 df-4 11793 df-5 11794 df-6 11795 df-7 11796 df-8 11797 df-9 11798 df-n0 11989 df-z 12075 df-dec 12192 df-uz 12337 df-fz 12994 df-fzo 13137 df-seq 13473 df-hash 13795 df-struct 16600 df-ndx 16601 df-slot 16602 df-base 16604 df-sets 16605 df-ress 16606 df-plusg 16693 df-mulr 16694 df-sca 16696 df-vsca 16697 df-ip 16698 df-tset 16699 df-ple 16700 df-ds 16702 df-hom 16704 df-cco 16705 df-0g 16830 df-gsum 16831 df-prds 16836 df-pws 16838 df-mre 16972 df-mrc 16973 df-acs 16975 df-mgm 17980 df-sgrp 18029 df-mnd 18040 df-mhm 18084 df-submnd 18085 df-grp 18234 df-minusg 18235 df-sbg 18236 df-mulg 18355 df-subg 18406 df-ghm 18486 df-cntz 18577 df-cmn 19038 df-abl 19039 df-mgp 19371 df-ur 19383 df-ring 19430 df-cring 19431 df-rnghom 19601 df-subrg 19664 df-lmod 19767 df-lss 19835 df-sra 20075 df-rgmod 20076 df-dsmm 20560 df-frlm 20575 df-assa 20681 df-ascl 20683 df-psr 20734 df-mpl 20736 df-opsr 20738 df-psr1 20967 df-ply1 20969 df-mamu 21149 df-mat 21171 df-mat2pmat 21470 |
This theorem is referenced by: mat2pmatrhm 21497 m2cpmmhm 21508 cayhamlem4 21651 |
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