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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 2732 | . . . . . 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 6891 | . . . . . . 7 ⊢ (0g‘𝐴) = (0g‘(𝑁 Mat 𝑅)) |
6 | 3, 5 | eqtri 2760 | . . . . . 6 ⊢ 0 = (0g‘(𝑁 Mat 𝑅)) |
7 | m2cpminv0.z | . . . . . . 7 ⊢ 𝑍 = (0g‘𝐶) | |
8 | m2cpminv0.c | . . . . . . . 8 ⊢ 𝐶 = (𝑁 Mat 𝑃) | |
9 | 8 | fveq2i 6891 | . . . . . . 7 ⊢ (0g‘𝐶) = (0g‘(𝑁 Mat 𝑃)) |
10 | 7, 9 | eqtri 2760 | . . . . . 6 ⊢ 𝑍 = (0g‘(𝑁 Mat 𝑃)) |
11 | 1, 2, 6, 10 | 0mat2pmat 22229 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → ((𝑁 matToPolyMat 𝑅)‘ 0 ) = 𝑍) |
12 | 11 | ancoms 459 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑁 matToPolyMat 𝑅)‘ 0 ) = 𝑍) |
13 | 12 | eqcomd 2738 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑍 = ((𝑁 matToPolyMat 𝑅)‘ 0 )) |
14 | 13 | fveq2d 6892 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐼‘𝑍) = (𝐼‘((𝑁 matToPolyMat 𝑅)‘ 0 ))) |
15 | 4 | matring 21936 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
16 | eqid 2732 | . . . . 5 ⊢ (Base‘𝐴) = (Base‘𝐴) | |
17 | 16, 3 | ring0cl 20077 | . . . 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 22246 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 0 ∈ (Base‘𝐴)) → (𝐼‘((𝑁 matToPolyMat 𝑅)‘ 0 )) = 0 ) |
21 | 18, 20 | mpd3an3 1462 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐼‘((𝑁 matToPolyMat 𝑅)‘ 0 )) = 0 ) |
22 | 14, 21 | eqtrd 2772 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐼‘𝑍) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ‘cfv 6540 (class class class)co 7405 Fincfn 8935 Basecbs 17140 0gc0g 17381 Ringcrg 20049 Poly1cpl1 21692 Mat cmat 21898 matToPolyMat cmat2pmat 22197 cPolyMatToMat ccpmat2mat 22198 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-ot 4636 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-ofr 7667 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-fzo 13624 df-seq 13963 df-hash 14287 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-hom 17217 df-cco 17218 df-0g 17383 df-gsum 17384 df-prds 17389 df-pws 17391 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-submnd 18668 df-grp 18818 df-minusg 18819 df-sbg 18820 df-mulg 18945 df-subg 18997 df-ghm 19084 df-cntz 19175 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-ring 20051 df-subrg 20353 df-lmod 20465 df-lss 20535 df-sra 20777 df-rgmod 20778 df-dsmm 21278 df-frlm 21293 df-ascl 21401 df-psr 21453 df-mvr 21454 df-mpl 21455 df-opsr 21457 df-psr1 21695 df-vr1 21696 df-ply1 21697 df-coe1 21698 df-mamu 21877 df-mat 21899 df-cpmat 22199 df-mat2pmat 22200 df-cpmat2mat 22201 |
This theorem is referenced by: cpmadumatpolylem2 22375 cayhamlem4 22381 cayleyhamilton0 22382 cayleyhamiltonALT 22384 |
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