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Mirrors > Home > MPE Home > Th. List > 0mat2pmat | Structured version Visualization version GIF version |
Description: The transformed zero matrix is the zero polynomial matrix. (Contributed by AV, 5-Aug-2019.) (Proof shortened by AV, 19-Nov-2019.) |
Ref | Expression |
---|---|
idmatidpmat.t | ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) |
idmatidpmat.p | ⊢ 𝑃 = (Poly1‘𝑅) |
0mat2pmat.0 | ⊢ 0 = (0g‘(𝑁 Mat 𝑅)) |
0mat2pmat.z | ⊢ 𝑍 = (0g‘(𝑁 Mat 𝑃)) |
Ref | Expression |
---|---|
0mat2pmat | ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → (𝑇‘ 0 ) = 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idmatidpmat.t | . . . 4 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) | |
2 | eqid 2736 | . . . 4 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅) | |
3 | eqid 2736 | . . . 4 ⊢ (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅)) | |
4 | idmatidpmat.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
5 | eqid 2736 | . . . 4 ⊢ (𝑁 Mat 𝑃) = (𝑁 Mat 𝑃) | |
6 | eqid 2736 | . . . 4 ⊢ (Base‘(𝑁 Mat 𝑃)) = (Base‘(𝑁 Mat 𝑃)) | |
7 | 1, 2, 3, 4, 5, 6 | mat2pmatghm 21581 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ ((𝑁 Mat 𝑅) GrpHom (𝑁 Mat 𝑃))) |
8 | 7 | ancoms 462 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝑇 ∈ ((𝑁 Mat 𝑅) GrpHom (𝑁 Mat 𝑃))) |
9 | 0mat2pmat.0 | . . 3 ⊢ 0 = (0g‘(𝑁 Mat 𝑅)) | |
10 | 0mat2pmat.z | . . 3 ⊢ 𝑍 = (0g‘(𝑁 Mat 𝑃)) | |
11 | 9, 10 | ghmid 18582 | . 2 ⊢ (𝑇 ∈ ((𝑁 Mat 𝑅) GrpHom (𝑁 Mat 𝑃)) → (𝑇‘ 0 ) = 𝑍) |
12 | 8, 11 | syl 17 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → (𝑇‘ 0 ) = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ‘cfv 6358 (class class class)co 7191 Fincfn 8604 Basecbs 16666 0gc0g 16898 GrpHom cghm 18573 Ringcrg 19516 Poly1cpl1 21052 Mat cmat 21258 matToPolyMat cmat2pmat 21555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-ot 4536 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 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 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-ofr 7448 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-pm 8489 df-ixp 8557 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-sup 9036 df-oi 9104 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-fz 13061 df-fzo 13204 df-seq 13540 df-hash 13862 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-sca 16765 df-vsca 16766 df-ip 16767 df-tset 16768 df-ple 16769 df-ds 16771 df-hom 16773 df-cco 16774 df-0g 16900 df-gsum 16901 df-prds 16906 df-pws 16908 df-mre 17043 df-mrc 17044 df-acs 17046 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-mhm 18172 df-submnd 18173 df-grp 18322 df-minusg 18323 df-sbg 18324 df-mulg 18443 df-subg 18494 df-ghm 18574 df-cntz 18665 df-cmn 19126 df-abl 19127 df-mgp 19459 df-ur 19471 df-ring 19518 df-subrg 19752 df-lmod 19855 df-lss 19923 df-sra 20163 df-rgmod 20164 df-dsmm 20648 df-frlm 20663 df-ascl 20771 df-psr 20822 df-mpl 20824 df-opsr 20826 df-psr1 21055 df-ply1 21057 df-mamu 21237 df-mat 21259 df-mat2pmat 21558 |
This theorem is referenced by: m2cpminv0 21612 pmatcollpwfi 21633 pmatcollpw3fi1lem1 21637 |
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