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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 22684) 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 20274 | . . . 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 22868 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝐼 ∈ (Base‘𝑌)) |
| 13 | 1, 12 | syl3an2 1176 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐼 ∈ (Base‘𝑌)) |
| 14 | 4 | ply1crng 22240 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing) |
| 15 | 14 | 3ad2ant2 1146 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ CRing) |
| 16 | eqid 2761 | . . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
| 17 | cpmadurid.j | . . . 4 ⊢ 𝐽 = (𝑁 maAdju 𝑃) | |
| 18 | eqid 2761 | . . . 4 ⊢ (𝑁 maDet 𝑃) = (𝑁 maDet 𝑃) | |
| 19 | cpmadurid.m2 | . . . 4 ⊢ × = (.r‘𝑌) | |
| 20 | 5, 16, 17, 18, 10, 19, 9 | madurid 22684 | . . 3 ⊢ ((𝐼 ∈ (Base‘𝑌) ∧ 𝑃 ∈ CRing) → (𝐼 × (𝐽‘𝐼)) = (((𝑁 maDet 𝑃)‘𝐼) · 1 )) |
| 21 | 13, 15, 20 | syl2anc 593 | . 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 22871 | . . . 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 2767 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑋 · 1 ) − (𝑇‘𝑀)) = 𝐼) |
| 26 | 25 | fveq2d 6867 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑁 maDet 𝑃)‘((𝑋 · 1 ) − (𝑇‘𝑀))) = ((𝑁 maDet 𝑃)‘𝐼)) |
| 27 | 23, 26 | eqtr2d 2797 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑁 maDet 𝑃)‘𝐼) = (𝐶‘𝑀)) |
| 28 | 27 | oveq1d 7407 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (((𝑁 maDet 𝑃)‘𝐼) · 1 ) = ((𝐶‘𝑀) · 1 )) |
| 29 | 21, 28 | eqtrd 2796 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐼 × (𝐽‘𝐼)) = ((𝐶‘𝑀) · 1 )) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ‘cfv 6517 (class class class)co 7392 Fincfn 8923 Basecbs 17228 .rcmulr 17270 ·𝑠 cvsca 17273 -gcsg 18960 1rcur 20210 Ringcrg 20262 CRingccrg 20263 var1cv1 22218 Poly1cpl1 22219 Mat cmat 22447 maDet cmdat 22624 maAdju cmadu 22672 matToPolyMat cmat2pmat 22744 CharPlyMat cchpmat 22866 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147 ax-addf 11149 ax-mulf 11150 |
| This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-xor 1531 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-ot 4590 df-uni 4865 df-int 4905 df-iun 4950 df-iin 4951 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-se 5599 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-isom 6526 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-of 7656 df-ofr 7657 df-om 7843 df-1st 7966 df-2nd 7967 df-supp 8136 df-tpos 8201 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-2o 8433 df-er 8673 df-map 8805 df-pm 8806 df-ixp 8876 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-fsupp 9305 df-sup 9385 df-oi 9455 df-card 9894 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12208 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12479 df-xnn0 12552 df-z 12566 df-dec 12686 df-uz 12837 df-rp 12991 df-fz 13510 df-fzo 13657 df-seq 14012 df-exp 14072 df-hash 14341 df-word 14524 df-lsw 14573 df-concat 14581 df-s1 14607 df-substr 14652 df-pfx 14682 df-splice 14760 df-reverse 14769 df-s2 14858 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17250 df-plusg 17282 df-mulr 17283 df-starv 17284 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ds 17291 df-unif 17292 df-hom 17293 df-cco 17294 df-0g 17453 df-gsum 17454 df-prds 17459 df-pws 17461 df-mre 17597 df-mrc 17598 df-acs 17600 df-mgm 18657 df-sgrp 18736 df-mnd 18752 df-mhm 18800 df-submnd 18801 df-efmnd 18886 df-grp 18961 df-minusg 18962 df-sbg 18963 df-mulg 19093 df-subg 19148 df-ghm 19237 df-gim 19282 df-cntz 19340 df-oppg 19369 df-symg 19393 df-pmtr 19465 df-psgn 19514 df-evpm 19515 df-cmn 19805 df-abl 19806 df-mgp 20170 df-rng 20182 df-ur 20211 df-ring 20264 df-cring 20265 df-oppr 20365 df-dvdsr 20385 df-unit 20386 df-invr 20416 df-dvr 20429 df-rhm 20500 df-subrng 20575 df-subrg 20599 df-drng 20760 df-lmod 20909 df-lss 20979 df-sra 21220 df-rgmod 21221 df-cnfld 21405 df-zring 21479 df-zrh 21535 df-dsmm 21764 df-frlm 21779 df-ascl 21887 df-psr 21941 df-mvr 21942 df-mpl 21943 df-opsr 21945 df-psr1 22222 df-vr1 22223 df-ply1 22224 df-mamu 22431 df-mat 22448 df-mdet 22625 df-madu 22674 df-mat2pmat 22747 df-chpmat 22867 |
| This theorem is referenced by: chcoeffeq 22926 |
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