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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 22609) 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 20226 | . . . 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 22793 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝐼 ∈ (Base‘𝑌)) |
| 13 | 1, 12 | syl3an2 1165 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐼 ∈ (Base‘𝑌)) |
| 14 | 4 | ply1crng 22162 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing) |
| 15 | 14 | 3ad2ant2 1135 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ CRing) |
| 16 | eqid 2736 | . . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
| 17 | cpmadurid.j | . . . 4 ⊢ 𝐽 = (𝑁 maAdju 𝑃) | |
| 18 | eqid 2736 | . . . 4 ⊢ (𝑁 maDet 𝑃) = (𝑁 maDet 𝑃) | |
| 19 | cpmadurid.m2 | . . . 4 ⊢ × = (.r‘𝑌) | |
| 20 | 5, 16, 17, 18, 10, 19, 9 | madurid 22609 | . . 3 ⊢ ((𝐼 ∈ (Base‘𝑌) ∧ 𝑃 ∈ CRing) → (𝐼 × (𝐽‘𝐼)) = (((𝑁 maDet 𝑃)‘𝐼) · 1 )) |
| 21 | 13, 15, 20 | syl2anc 585 | . 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 22796 | . . . 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 2742 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑋 · 1 ) − (𝑇‘𝑀)) = 𝐼) |
| 26 | 25 | fveq2d 6844 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑁 maDet 𝑃)‘((𝑋 · 1 ) − (𝑇‘𝑀))) = ((𝑁 maDet 𝑃)‘𝐼)) |
| 27 | 23, 26 | eqtr2d 2772 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑁 maDet 𝑃)‘𝐼) = (𝐶‘𝑀)) |
| 28 | 27 | oveq1d 7382 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (((𝑁 maDet 𝑃)‘𝐼) · 1 ) = ((𝐶‘𝑀) · 1 )) |
| 29 | 21, 28 | eqtrd 2771 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐼 × (𝐽‘𝐼)) = ((𝐶‘𝑀) · 1 )) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ‘cfv 6498 (class class class)co 7367 Fincfn 8893 Basecbs 17179 .rcmulr 17221 ·𝑠 cvsca 17224 -gcsg 18911 1rcur 20162 Ringcrg 20214 CRingccrg 20215 var1cv1 22139 Poly1cpl1 22140 Mat cmat 22372 maDet cmdat 22549 maAdju cmadu 22597 matToPolyMat cmat2pmat 22669 CharPlyMat cchpmat 22791 |
| 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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-addf 11117 ax-mulf 11118 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-xor 1514 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-ot 4576 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 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 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-ofr 7632 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-tpos 8176 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-sup 9355 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-xnn0 12511 df-z 12525 df-dec 12645 df-uz 12789 df-rp 12943 df-fz 13462 df-fzo 13609 df-seq 13964 df-exp 14024 df-hash 14293 df-word 14476 df-lsw 14525 df-concat 14533 df-s1 14559 df-substr 14604 df-pfx 14634 df-splice 14712 df-reverse 14721 df-s2 14810 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-0g 17404 df-gsum 17405 df-prds 17410 df-pws 17412 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-mhm 18751 df-submnd 18752 df-efmnd 18837 df-grp 18912 df-minusg 18913 df-sbg 18914 df-mulg 19044 df-subg 19099 df-ghm 19188 df-gim 19234 df-cntz 19292 df-oppg 19321 df-symg 19345 df-pmtr 19417 df-psgn 19466 df-evpm 19467 df-cmn 19757 df-abl 19758 df-mgp 20122 df-rng 20134 df-ur 20163 df-ring 20216 df-cring 20217 df-oppr 20317 df-dvdsr 20337 df-unit 20338 df-invr 20368 df-dvr 20381 df-rhm 20452 df-subrng 20523 df-subrg 20547 df-drng 20708 df-lmod 20857 df-lss 20927 df-sra 21168 df-rgmod 21169 df-cnfld 21353 df-zring 21427 df-zrh 21483 df-dsmm 21712 df-frlm 21727 df-ascl 21835 df-psr 21889 df-mvr 21890 df-mpl 21891 df-opsr 21893 df-psr1 22143 df-vr1 22144 df-ply1 22145 df-mamu 22356 df-mat 22373 df-mdet 22550 df-madu 22599 df-mat2pmat 22672 df-chpmat 22792 |
| This theorem is referenced by: chcoeffeq 22851 |
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