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Mirrors > Home > MPE Home > Th. List > m2cpmmhm | Structured version Visualization version GIF version |
Description: The transformation of matrices into constant polynomial matrices is a homomorphism of multiplicative monoids. (Contributed by AV, 18-Nov-2019.) |
Ref | Expression |
---|---|
m2cpm.s | ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) |
m2cpm.t | ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) |
m2cpm.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
m2cpm.b | ⊢ 𝐵 = (Base‘𝐴) |
m2cpmghm.p | ⊢ 𝑃 = (Poly1‘𝑅) |
m2cpmghm.c | ⊢ 𝐶 = (𝑁 Mat 𝑃) |
m2cpmghm.u | ⊢ 𝑈 = (𝐶 ↾s 𝑆) |
Ref | Expression |
---|---|
m2cpmmhm | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑈))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | m2cpm.t | . . 3 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) | |
2 | m2cpm.a | . . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
3 | m2cpm.b | . . 3 ⊢ 𝐵 = (Base‘𝐴) | |
4 | m2cpmghm.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
5 | m2cpmghm.c | . . 3 ⊢ 𝐶 = (𝑁 Mat 𝑃) | |
6 | eqid 2823 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
7 | 1, 2, 3, 4, 5, 6 | mat2pmatmhm 21343 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝐶))) |
8 | crngring 19310 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
9 | 8 | anim2i 618 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring)) |
10 | m2cpm.s | . . . . 5 ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) | |
11 | 10, 4, 5 | cpmatsrgpmat 21331 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubRing‘𝐶)) |
12 | eqid 2823 | . . . . 5 ⊢ (mulGrp‘𝐶) = (mulGrp‘𝐶) | |
13 | 12 | subrgsubm 19550 | . . . 4 ⊢ (𝑆 ∈ (SubRing‘𝐶) → 𝑆 ∈ (SubMnd‘(mulGrp‘𝐶))) |
14 | 9, 11, 13 | 3syl 18 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑆 ∈ (SubMnd‘(mulGrp‘𝐶))) |
15 | 10, 1, 2, 3 | m2cpmf 21352 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵⟶𝑆) |
16 | frn 6522 | . . . 4 ⊢ (𝑇:𝐵⟶𝑆 → ran 𝑇 ⊆ 𝑆) | |
17 | 9, 15, 16 | 3syl 18 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → ran 𝑇 ⊆ 𝑆) |
18 | 5 | ovexi 7192 | . . . . . 6 ⊢ 𝐶 ∈ V |
19 | 10 | ovexi 7192 | . . . . . 6 ⊢ 𝑆 ∈ V |
20 | m2cpmghm.u | . . . . . . 7 ⊢ 𝑈 = (𝐶 ↾s 𝑆) | |
21 | 20, 12 | mgpress 19252 | . . . . . 6 ⊢ ((𝐶 ∈ V ∧ 𝑆 ∈ V) → ((mulGrp‘𝐶) ↾s 𝑆) = (mulGrp‘𝑈)) |
22 | 18, 19, 21 | mp2an 690 | . . . . 5 ⊢ ((mulGrp‘𝐶) ↾s 𝑆) = (mulGrp‘𝑈) |
23 | 22 | eqcomi 2832 | . . . 4 ⊢ (mulGrp‘𝑈) = ((mulGrp‘𝐶) ↾s 𝑆) |
24 | 23 | resmhm2b 17989 | . . 3 ⊢ ((𝑆 ∈ (SubMnd‘(mulGrp‘𝐶)) ∧ ran 𝑇 ⊆ 𝑆) → (𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝐶)) ↔ 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑈)))) |
25 | 14, 17, 24 | syl2anc 586 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝐶)) ↔ 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑈)))) |
26 | 7, 25 | mpbid 234 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑈))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ⊆ wss 3938 ran crn 5558 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 Fincfn 8511 Basecbs 16485 ↾s cress 16486 MndHom cmhm 17956 SubMndcsubmnd 17957 mulGrpcmgp 19241 Ringcrg 19299 CRingccrg 19300 SubRingcsubrg 19533 Poly1cpl1 20347 Mat cmat 21018 ConstPolyMat ccpmat 21313 matToPolyMat cmat2pmat 21314 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-ot 4578 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-ofr 7412 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-sup 8908 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-hom 16591 df-cco 16592 df-0g 16717 df-gsum 16718 df-prds 16723 df-pws 16725 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-subg 18278 df-ghm 18358 df-cntz 18449 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-srg 19258 df-ring 19301 df-cring 19302 df-rnghom 19469 df-subrg 19535 df-lmod 19638 df-lss 19706 df-sra 19946 df-rgmod 19947 df-assa 20087 df-ascl 20089 df-psr 20138 df-mvr 20139 df-mpl 20140 df-opsr 20142 df-psr1 20350 df-vr1 20351 df-ply1 20352 df-coe1 20353 df-dsmm 20878 df-frlm 20893 df-mamu 20997 df-mat 21019 df-cpmat 21316 df-mat2pmat 21317 |
This theorem is referenced by: m2cpmrhm 21356 |
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