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Mirrors > Home > MPE Home > Th. List > m2cpminv0 | Structured version Visualization version GIF version |
Description: The inverse matrix transformation applied to the zero polynomial matrix results in the zero of the matrices over the base ring of the polynomials. (Contributed by AV, 24-Nov-2019.) (Revised by AV, 15-Dec-2019.) |
Ref | Expression |
---|---|
m2cpminv0.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
m2cpminv0.i | ⊢ 𝐼 = (𝑁 cPolyMatToMat 𝑅) |
m2cpminv0.p | ⊢ 𝑃 = (Poly1‘𝑅) |
m2cpminv0.c | ⊢ 𝐶 = (𝑁 Mat 𝑃) |
m2cpminv0.0 | ⊢ 0 = (0g‘𝐴) |
m2cpminv0.z | ⊢ 𝑍 = (0g‘𝐶) |
Ref | Expression |
---|---|
m2cpminv0 | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐼‘𝑍) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . . . . 6 ⊢ (𝑁 matToPolyMat 𝑅) = (𝑁 matToPolyMat 𝑅) | |
2 | m2cpminv0.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | m2cpminv0.0 | . . . . . . 7 ⊢ 0 = (0g‘𝐴) | |
4 | m2cpminv0.a | . . . . . . . 8 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
5 | 4 | fveq2i 6841 | . . . . . . 7 ⊢ (0g‘𝐴) = (0g‘(𝑁 Mat 𝑅)) |
6 | 3, 5 | eqtri 2766 | . . . . . 6 ⊢ 0 = (0g‘(𝑁 Mat 𝑅)) |
7 | m2cpminv0.z | . . . . . . 7 ⊢ 𝑍 = (0g‘𝐶) | |
8 | m2cpminv0.c | . . . . . . . 8 ⊢ 𝐶 = (𝑁 Mat 𝑃) | |
9 | 8 | fveq2i 6841 | . . . . . . 7 ⊢ (0g‘𝐶) = (0g‘(𝑁 Mat 𝑃)) |
10 | 7, 9 | eqtri 2766 | . . . . . 6 ⊢ 𝑍 = (0g‘(𝑁 Mat 𝑃)) |
11 | 1, 2, 6, 10 | 0mat2pmat 22013 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → ((𝑁 matToPolyMat 𝑅)‘ 0 ) = 𝑍) |
12 | 11 | ancoms 460 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑁 matToPolyMat 𝑅)‘ 0 ) = 𝑍) |
13 | 12 | eqcomd 2744 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑍 = ((𝑁 matToPolyMat 𝑅)‘ 0 )) |
14 | 13 | fveq2d 6842 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐼‘𝑍) = (𝐼‘((𝑁 matToPolyMat 𝑅)‘ 0 ))) |
15 | 4 | matring 21720 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
16 | eqid 2738 | . . . . 5 ⊢ (Base‘𝐴) = (Base‘𝐴) | |
17 | 16, 3 | ring0cl 19920 | . . . 4 ⊢ (𝐴 ∈ Ring → 0 ∈ (Base‘𝐴)) |
18 | 15, 17 | syl 17 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 0 ∈ (Base‘𝐴)) |
19 | m2cpminv0.i | . . . 4 ⊢ 𝐼 = (𝑁 cPolyMatToMat 𝑅) | |
20 | 19, 4, 16, 1 | m2cpminvid 22030 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 0 ∈ (Base‘𝐴)) → (𝐼‘((𝑁 matToPolyMat 𝑅)‘ 0 )) = 0 ) |
21 | 18, 20 | mpd3an3 1463 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐼‘((𝑁 matToPolyMat 𝑅)‘ 0 )) = 0 ) |
22 | 14, 21 | eqtrd 2778 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐼‘𝑍) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ‘cfv 6492 (class class class)co 7350 Fincfn 8817 Basecbs 17019 0gc0g 17257 Ringcrg 19894 Poly1cpl1 21476 Mat cmat 21682 matToPolyMat cmat2pmat 21981 cPolyMatToMat ccpmat2mat 21982 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-ot 4594 df-uni 4865 df-int 4907 df-iun 4955 df-iin 4956 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-of 7608 df-ofr 7609 df-om 7794 df-1st 7912 df-2nd 7913 df-supp 8061 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-er 8582 df-map 8701 df-pm 8702 df-ixp 8770 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-fsupp 9240 df-sup 9312 df-oi 9380 df-card 9809 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-nn 12088 df-2 12150 df-3 12151 df-4 12152 df-5 12153 df-6 12154 df-7 12155 df-8 12156 df-9 12157 df-n0 12348 df-z 12434 df-dec 12553 df-uz 12698 df-fz 13355 df-fzo 13498 df-seq 13837 df-hash 14160 df-struct 16955 df-sets 16972 df-slot 16990 df-ndx 17002 df-base 17020 df-ress 17049 df-plusg 17082 df-mulr 17083 df-sca 17085 df-vsca 17086 df-ip 17087 df-tset 17088 df-ple 17089 df-ds 17091 df-hom 17093 df-cco 17094 df-0g 17259 df-gsum 17260 df-prds 17265 df-pws 17267 df-mre 17402 df-mrc 17403 df-acs 17405 df-mgm 18433 df-sgrp 18482 df-mnd 18493 df-mhm 18537 df-submnd 18538 df-grp 18687 df-minusg 18688 df-sbg 18689 df-mulg 18808 df-subg 18860 df-ghm 18941 df-cntz 19032 df-cmn 19499 df-abl 19500 df-mgp 19832 df-ur 19849 df-ring 19896 df-subrg 20149 df-lmod 20253 df-lss 20322 df-sra 20562 df-rgmod 20563 df-dsmm 21067 df-frlm 21082 df-ascl 21190 df-psr 21240 df-mvr 21241 df-mpl 21242 df-opsr 21244 df-psr1 21479 df-vr1 21480 df-ply1 21481 df-coe1 21482 df-mamu 21661 df-mat 21683 df-cpmat 21983 df-mat2pmat 21984 df-cpmat2mat 21985 |
This theorem is referenced by: cpmadumatpolylem2 22159 cayhamlem4 22165 cayleyhamilton0 22166 cayleyhamiltonALT 22168 |
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