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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 21256) 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 19311 | . . . 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 21439 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝐼 ∈ (Base‘𝑌)) |
| 13 | 1, 12 | syl3an2 1160 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐼 ∈ (Base‘𝑌)) |
| 14 | 4 | ply1crng 20369 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing) |
| 15 | 14 | 3ad2ant2 1130 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ CRing) |
| 16 | eqid 2824 | . . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
| 17 | cpmadurid.j | . . . 4 ⊢ 𝐽 = (𝑁 maAdju 𝑃) | |
| 18 | eqid 2824 | . . . 4 ⊢ (𝑁 maDet 𝑃) = (𝑁 maDet 𝑃) | |
| 19 | cpmadurid.m2 | . . . 4 ⊢ × = (.r‘𝑌) | |
| 20 | 5, 16, 17, 18, 10, 19, 9 | madurid 21256 | . . 3 ⊢ ((𝐼 ∈ (Base‘𝑌) ∧ 𝑃 ∈ CRing) → (𝐼 × (𝐽‘𝐼)) = (((𝑁 maDet 𝑃)‘𝐼) · 1 )) |
| 21 | 13, 15, 20 | syl2anc 586 | . 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 21442 | . . . 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 2830 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑋 · 1 ) − (𝑇‘𝑀)) = 𝐼) |
| 26 | 25 | fveq2d 6677 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑁 maDet 𝑃)‘((𝑋 · 1 ) − (𝑇‘𝑀))) = ((𝑁 maDet 𝑃)‘𝐼)) |
| 27 | 23, 26 | eqtr2d 2860 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑁 maDet 𝑃)‘𝐼) = (𝐶‘𝑀)) |
| 28 | 27 | oveq1d 7174 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (((𝑁 maDet 𝑃)‘𝐼) · 1 ) = ((𝐶‘𝑀) · 1 )) |
| 29 | 21, 28 | eqtrd 2859 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐼 × (𝐽‘𝐼)) = ((𝐶‘𝑀) · 1 )) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ‘cfv 6358 (class class class)co 7159 Fincfn 8512 Basecbs 16486 .rcmulr 16569 ·𝑠 cvsca 16572 -gcsg 18108 1rcur 19254 Ringcrg 19300 CRingccrg 19301 var1cv1 20347 Poly1cpl1 20348 Mat cmat 21019 maDet cmdat 21196 maAdju cmadu 21244 matToPolyMat cmat2pmat 21315 CharPlyMat cchpmat 21437 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-addf 10619 ax-mulf 10620 |
| This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-xor 1502 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-ot 4579 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-ofr 7413 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-tpos 7895 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-ixp 8465 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-sup 8909 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-xnn0 11971 df-z 11985 df-dec 12102 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-word 13865 df-lsw 13918 df-concat 13926 df-s1 13953 df-substr 14006 df-pfx 14036 df-splice 14115 df-reverse 14124 df-s2 14213 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-starv 16583 df-sca 16584 df-vsca 16585 df-ip 16586 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-hom 16592 df-cco 16593 df-0g 16718 df-gsum 16719 df-prds 16724 df-pws 16726 df-mre 16860 df-mrc 16861 df-acs 16863 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-mhm 17959 df-submnd 17960 df-efmnd 18037 df-grp 18109 df-minusg 18110 df-sbg 18111 df-mulg 18228 df-subg 18279 df-ghm 18359 df-gim 18402 df-cntz 18450 df-oppg 18477 df-symg 18499 df-pmtr 18573 df-psgn 18622 df-evpm 18623 df-cmn 18911 df-abl 18912 df-mgp 19243 df-ur 19255 df-ring 19302 df-cring 19303 df-oppr 19376 df-dvdsr 19394 df-unit 19395 df-invr 19425 df-dvr 19436 df-rnghom 19470 df-drng 19507 df-subrg 19536 df-lmod 19639 df-lss 19707 df-sra 19947 df-rgmod 19948 df-ascl 20090 df-psr 20139 df-mvr 20140 df-mpl 20141 df-opsr 20143 df-psr1 20351 df-vr1 20352 df-ply1 20353 df-cnfld 20549 df-zring 20621 df-zrh 20654 df-dsmm 20879 df-frlm 20894 df-mamu 20998 df-mat 21020 df-mdet 21197 df-madu 21246 df-mat2pmat 21318 df-chpmat 21438 |
| This theorem is referenced by: chcoeffeq 21497 |
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