Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > cpmadurid | Structured version Visualization version GIF version | ||
| Description: The right-hand fundamental relation of the adjugate (see madurid 21793) 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 19795 | . . . 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 21977 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝐼 ∈ (Base‘𝑌)) |
| 13 | 1, 12 | syl3an2 1163 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐼 ∈ (Base‘𝑌)) |
| 14 | 4 | ply1crng 21369 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing) |
| 15 | 14 | 3ad2ant2 1133 | . . 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 21793 | . . 3 ⊢ ((𝐼 ∈ (Base‘𝑌) ∧ 𝑃 ∈ CRing) → (𝐼 × (𝐽‘𝐼)) = (((𝑁 maDet 𝑃)‘𝐼) · 1 )) |
| 21 | 13, 15, 20 | syl2anc 584 | . 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 21980 | . . . 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 6778 | . . . 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 7290 | . 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 1086 = wceq 1539 ∈ wcel 2106 ‘cfv 6433 (class class class)co 7275 Fincfn 8733 Basecbs 16912 .rcmulr 16963 ·𝑠 cvsca 16966 -gcsg 18579 1rcur 19737 Ringcrg 19783 CRingccrg 19784 var1cv1 21347 Poly1cpl1 21348 Mat cmat 21554 maDet cmdat 21733 maAdju cmadu 21781 matToPolyMat cmat2pmat 21853 CharPlyMat cchpmat 21975 |
| 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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-addf 10950 ax-mulf 10951 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-xor 1507 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-ot 4570 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-ofr 7534 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-tpos 8042 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-sup 9201 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-xnn0 12306 df-z 12320 df-dec 12438 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-hash 14045 df-word 14218 df-lsw 14266 df-concat 14274 df-s1 14301 df-substr 14354 df-pfx 14384 df-splice 14463 df-reverse 14472 df-s2 14561 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-0g 17152 df-gsum 17153 df-prds 17158 df-pws 17160 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-submnd 18431 df-efmnd 18508 df-grp 18580 df-minusg 18581 df-sbg 18582 df-mulg 18701 df-subg 18752 df-ghm 18832 df-gim 18875 df-cntz 18923 df-oppg 18950 df-symg 18975 df-pmtr 19050 df-psgn 19099 df-evpm 19100 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-cring 19786 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-dvr 19925 df-rnghom 19959 df-drng 19993 df-subrg 20022 df-lmod 20125 df-lss 20194 df-sra 20434 df-rgmod 20435 df-cnfld 20598 df-zring 20671 df-zrh 20705 df-dsmm 20939 df-frlm 20954 df-ascl 21062 df-psr 21112 df-mvr 21113 df-mpl 21114 df-opsr 21116 df-psr1 21351 df-vr1 21352 df-ply1 21353 df-mamu 21533 df-mat 21555 df-mdet 21734 df-madu 21783 df-mat2pmat 21856 df-chpmat 21976 |
| This theorem is referenced by: chcoeffeq 22035 |
| Copyright terms: Public domain | W3C validator |