Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > m2cpmrhm | Structured version Visualization version GIF version |
Description: The transformation of matrices into constant polynomial matrices is a ring homomorphism. (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 |
---|---|
m2cpmrhm | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingHom 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crngring 19311 | . . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
2 | m2cpm.a | . . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
3 | 2 | matring 21055 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
4 | 1, 3 | sylan2 594 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring) |
5 | m2cpm.s | . . . . 5 ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) | |
6 | m2cpmghm.p | . . . . 5 ⊢ 𝑃 = (Poly1‘𝑅) | |
7 | m2cpmghm.c | . . . . 5 ⊢ 𝐶 = (𝑁 Mat 𝑃) | |
8 | 5, 6, 7 | cpmatsrgpmat 21332 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubRing‘𝐶)) |
9 | 1, 8 | sylan2 594 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑆 ∈ (SubRing‘𝐶)) |
10 | m2cpmghm.u | . . . 4 ⊢ 𝑈 = (𝐶 ↾s 𝑆) | |
11 | 10 | subrgring 19541 | . . 3 ⊢ (𝑆 ∈ (SubRing‘𝐶) → 𝑈 ∈ Ring) |
12 | 9, 11 | syl 17 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑈 ∈ Ring) |
13 | m2cpm.t | . . . . 5 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) | |
14 | m2cpm.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐴) | |
15 | 5, 13, 2, 14, 6, 7, 10 | m2cpmghm 21355 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐴 GrpHom 𝑈)) |
16 | 1, 15 | sylan2 594 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 GrpHom 𝑈)) |
17 | 5, 13, 2, 14, 6, 7, 10 | m2cpmmhm 21356 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑈))) |
18 | 16, 17 | jca 514 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑇 ∈ (𝐴 GrpHom 𝑈) ∧ 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑈)))) |
19 | eqid 2824 | . . 3 ⊢ (mulGrp‘𝐴) = (mulGrp‘𝐴) | |
20 | eqid 2824 | . . 3 ⊢ (mulGrp‘𝑈) = (mulGrp‘𝑈) | |
21 | 19, 20 | isrhm 19476 | . 2 ⊢ (𝑇 ∈ (𝐴 RingHom 𝑈) ↔ ((𝐴 ∈ Ring ∧ 𝑈 ∈ Ring) ∧ (𝑇 ∈ (𝐴 GrpHom 𝑈) ∧ 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑈))))) |
22 | 4, 12, 18, 21 | syl21anbrc 1340 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingHom 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ‘cfv 6358 (class class class)co 7159 Fincfn 8512 Basecbs 16486 ↾s cress 16487 MndHom cmhm 17957 GrpHom cghm 18358 mulGrpcmgp 19242 Ringcrg 19300 CRingccrg 19301 RingHom crh 19467 SubRingcsubrg 19534 Poly1cpl1 20348 Mat cmat 21019 ConstPolyMat ccpmat 21314 matToPolyMat cmat2pmat 21315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-ot 4579 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-ofr 7413 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-ixp 8465 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-sup 8909 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 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 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-sca 16584 df-vsca 16585 df-ip 16586 df-tset 16587 df-ple 16588 df-ds 16590 df-hom 16592 df-cco 16593 df-0g 16718 df-gsum 16719 df-prds 16724 df-pws 16726 df-mre 16860 df-mrc 16861 df-acs 16863 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-mhm 17959 df-submnd 17960 df-grp 18109 df-minusg 18110 df-sbg 18111 df-mulg 18228 df-subg 18279 df-ghm 18359 df-cntz 18450 df-cmn 18911 df-abl 18912 df-mgp 19243 df-ur 19255 df-srg 19259 df-ring 19302 df-cring 19303 df-rnghom 19470 df-subrg 19536 df-lmod 19639 df-lss 19707 df-sra 19947 df-rgmod 19948 df-assa 20088 df-ascl 20090 df-psr 20139 df-mvr 20140 df-mpl 20141 df-opsr 20143 df-psr1 20351 df-vr1 20352 df-ply1 20353 df-coe1 20354 df-dsmm 20879 df-frlm 20894 df-mamu 20998 df-mat 21020 df-cpmat 21317 df-mat2pmat 21318 |
This theorem is referenced by: m2cpmrngiso 21369 |
Copyright terms: Public domain | W3C validator |