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| Mirrors > Home > MPE Home > Th. List > cpmadurid | Structured version Visualization version GIF version | ||
| Description: The right-hand fundamental relation of the adjugate (see madurid 21701) applied to the characteristic matrix of a matrix. (Contributed by AV, 25-Oct-2019.) |
| Ref | Expression |
|---|---|
| cpmadurid.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
| cpmadurid.b | ⊢ 𝐵 = (Base‘𝐴) |
| cpmadurid.c | ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) |
| cpmadurid.p | ⊢ 𝑃 = (Poly1‘𝑅) |
| cpmadurid.y | ⊢ 𝑌 = (𝑁 Mat 𝑃) |
| cpmadurid.x | ⊢ 𝑋 = (var1‘𝑅) |
| cpmadurid.t | ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) |
| cpmadurid.s | ⊢ − = (-g‘𝑌) |
| cpmadurid.m1 | ⊢ · = ( ·𝑠 ‘𝑌) |
| cpmadurid.1 | ⊢ 1 = (1r‘𝑌) |
| cpmadurid.i | ⊢ 𝐼 = ((𝑋 · 1 ) − (𝑇‘𝑀)) |
| cpmadurid.j | ⊢ 𝐽 = (𝑁 maAdju 𝑃) |
| cpmadurid.m2 | ⊢ × = (.r‘𝑌) |
| Ref | Expression |
|---|---|
| cpmadurid | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐼 × (𝐽‘𝐼)) = ((𝐶‘𝑀) · 1 )) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | crngring 19710 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
| 2 | cpmadurid.a | . . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
| 3 | cpmadurid.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐴) | |
| 4 | cpmadurid.p | . . . . 5 ⊢ 𝑃 = (Poly1‘𝑅) | |
| 5 | cpmadurid.y | . . . . 5 ⊢ 𝑌 = (𝑁 Mat 𝑃) | |
| 6 | cpmadurid.x | . . . . 5 ⊢ 𝑋 = (var1‘𝑅) | |
| 7 | cpmadurid.t | . . . . 5 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) | |
| 8 | cpmadurid.s | . . . . 5 ⊢ − = (-g‘𝑌) | |
| 9 | cpmadurid.m1 | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑌) | |
| 10 | cpmadurid.1 | . . . . 5 ⊢ 1 = (1r‘𝑌) | |
| 11 | cpmadurid.i | . . . . 5 ⊢ 𝐼 = ((𝑋 · 1 ) − (𝑇‘𝑀)) | |
| 12 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | chmatcl 21885 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝐼 ∈ (Base‘𝑌)) |
| 13 | 1, 12 | syl3an2 1162 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐼 ∈ (Base‘𝑌)) |
| 14 | 4 | ply1crng 21279 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing) |
| 15 | 14 | 3ad2ant2 1132 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ CRing) |
| 16 | eqid 2738 | . . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
| 17 | cpmadurid.j | . . . 4 ⊢ 𝐽 = (𝑁 maAdju 𝑃) | |
| 18 | eqid 2738 | . . . 4 ⊢ (𝑁 maDet 𝑃) = (𝑁 maDet 𝑃) | |
| 19 | cpmadurid.m2 | . . . 4 ⊢ × = (.r‘𝑌) | |
| 20 | 5, 16, 17, 18, 10, 19, 9 | madurid 21701 | . . 3 ⊢ ((𝐼 ∈ (Base‘𝑌) ∧ 𝑃 ∈ CRing) → (𝐼 × (𝐽‘𝐼)) = (((𝑁 maDet 𝑃)‘𝐼) · 1 )) |
| 21 | 13, 15, 20 | syl2anc 583 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐼 × (𝐽‘𝐼)) = (((𝑁 maDet 𝑃)‘𝐼) · 1 )) |
| 22 | cpmadurid.c | . . . . 5 ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) | |
| 23 | 22, 2, 3, 4, 5, 18, 8, 6, 9, 7, 10 | chpmatval 21888 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) = ((𝑁 maDet 𝑃)‘((𝑋 · 1 ) − (𝑇‘𝑀)))) |
| 24 | 11 | a1i 11 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐼 = ((𝑋 · 1 ) − (𝑇‘𝑀))) |
| 25 | 24 | eqcomd 2744 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑋 · 1 ) − (𝑇‘𝑀)) = 𝐼) |
| 26 | 25 | fveq2d 6760 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑁 maDet 𝑃)‘((𝑋 · 1 ) − (𝑇‘𝑀))) = ((𝑁 maDet 𝑃)‘𝐼)) |
| 27 | 23, 26 | eqtr2d 2779 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑁 maDet 𝑃)‘𝐼) = (𝐶‘𝑀)) |
| 28 | 27 | oveq1d 7270 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (((𝑁 maDet 𝑃)‘𝐼) · 1 ) = ((𝐶‘𝑀) · 1 )) |
| 29 | 21, 28 | eqtrd 2778 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐼 × (𝐽‘𝐼)) = ((𝐶‘𝑀) · 1 )) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ‘cfv 6418 (class class class)co 7255 Fincfn 8691 Basecbs 16840 .rcmulr 16889 ·𝑠 cvsca 16892 -gcsg 18494 1rcur 19652 Ringcrg 19698 CRingccrg 19699 var1cv1 21257 Poly1cpl1 21258 Mat cmat 21464 maDet cmdat 21641 maAdju cmadu 21689 matToPolyMat cmat2pmat 21761 CharPlyMat cchpmat 21883 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-addf 10881 ax-mulf 10882 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-xor 1504 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-ot 4567 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-ofr 7512 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-sup 9131 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-xnn0 12236 df-z 12250 df-dec 12367 df-uz 12512 df-rp 12660 df-fz 13169 df-fzo 13312 df-seq 13650 df-exp 13711 df-hash 13973 df-word 14146 df-lsw 14194 df-concat 14202 df-s1 14229 df-substr 14282 df-pfx 14312 df-splice 14391 df-reverse 14400 df-s2 14489 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-0g 17069 df-gsum 17070 df-prds 17075 df-pws 17077 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-submnd 18346 df-efmnd 18423 df-grp 18495 df-minusg 18496 df-sbg 18497 df-mulg 18616 df-subg 18667 df-ghm 18747 df-gim 18790 df-cntz 18838 df-oppg 18865 df-symg 18890 df-pmtr 18965 df-psgn 19014 df-evpm 19015 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-rnghom 19874 df-drng 19908 df-subrg 19937 df-lmod 20040 df-lss 20109 df-sra 20349 df-rgmod 20350 df-cnfld 20511 df-zring 20583 df-zrh 20617 df-dsmm 20849 df-frlm 20864 df-ascl 20972 df-psr 21022 df-mvr 21023 df-mpl 21024 df-opsr 21026 df-psr1 21261 df-vr1 21262 df-ply1 21263 df-mamu 21443 df-mat 21465 df-mdet 21642 df-madu 21691 df-mat2pmat 21764 df-chpmat 21884 |
| This theorem is referenced by: chcoeffeq 21943 |
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