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Mirrors > Home > MPE Home > Th. List > madulid | Structured version Visualization version GIF version |
Description: Multiplying the adjunct of a matrix with the matrix results in the identity matrix multiplied with the determinant of the matrix. See Proposition 4.16 in [Lang] p. 518. (Contributed by Stefan O'Rear, 17-Jul-2018.) |
Ref | Expression |
---|---|
madurid.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
madurid.b | ⊢ 𝐵 = (Base‘𝐴) |
madurid.j | ⊢ 𝐽 = (𝑁 maAdju 𝑅) |
madurid.d | ⊢ 𝐷 = (𝑁 maDet 𝑅) |
madurid.i | ⊢ 1 = (1r‘𝐴) |
madurid.t | ⊢ · = (.r‘𝐴) |
madurid.s | ⊢ ∙ = ( ·𝑠 ‘𝐴) |
Ref | Expression |
---|---|
madulid | ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → ((𝐽‘𝑀) · 𝑀) = ((𝐷‘𝑀) ∙ 1 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 479 | . . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 𝑅 ∈ CRing) | |
2 | madurid.a | . . . . . . . 8 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
3 | madurid.j | . . . . . . . 8 ⊢ 𝐽 = (𝑁 maAdju 𝑅) | |
4 | madurid.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝐴) | |
5 | 2, 3, 4 | maduf 20822 | . . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝐽:𝐵⟶𝐵) |
6 | 5 | ffvelrnda 6613 | . . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐽‘𝑀) ∈ 𝐵) |
7 | 6 | ancoms 452 | . . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝐽‘𝑀) ∈ 𝐵) |
8 | simpl 476 | . . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 𝑀 ∈ 𝐵) | |
9 | madurid.t | . . . . . 6 ⊢ · = (.r‘𝐴) | |
10 | 2, 4, 9 | mattposm 20640 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ (𝐽‘𝑀) ∈ 𝐵 ∧ 𝑀 ∈ 𝐵) → tpos ((𝐽‘𝑀) · 𝑀) = (tpos 𝑀 · tpos (𝐽‘𝑀))) |
11 | 1, 7, 8, 10 | syl3anc 1494 | . . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → tpos ((𝐽‘𝑀) · 𝑀) = (tpos 𝑀 · tpos (𝐽‘𝑀))) |
12 | 2, 3, 4 | madutpos 20823 | . . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐽‘tpos 𝑀) = tpos (𝐽‘𝑀)) |
13 | 12 | ancoms 452 | . . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝐽‘tpos 𝑀) = tpos (𝐽‘𝑀)) |
14 | 13 | oveq2d 6926 | . . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (tpos 𝑀 · (𝐽‘tpos 𝑀)) = (tpos 𝑀 · tpos (𝐽‘𝑀))) |
15 | 2, 4 | mattposcl 20634 | . . . . 5 ⊢ (𝑀 ∈ 𝐵 → tpos 𝑀 ∈ 𝐵) |
16 | madurid.d | . . . . . 6 ⊢ 𝐷 = (𝑁 maDet 𝑅) | |
17 | madurid.i | . . . . . 6 ⊢ 1 = (1r‘𝐴) | |
18 | madurid.s | . . . . . 6 ⊢ ∙ = ( ·𝑠 ‘𝐴) | |
19 | 2, 4, 3, 16, 17, 9, 18 | madurid 20825 | . . . . 5 ⊢ ((tpos 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (tpos 𝑀 · (𝐽‘tpos 𝑀)) = ((𝐷‘tpos 𝑀) ∙ 1 )) |
20 | 15, 19 | sylan 575 | . . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (tpos 𝑀 · (𝐽‘tpos 𝑀)) = ((𝐷‘tpos 𝑀) ∙ 1 )) |
21 | 11, 14, 20 | 3eqtr2d 2867 | . . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → tpos ((𝐽‘𝑀) · 𝑀) = ((𝐷‘tpos 𝑀) ∙ 1 )) |
22 | 21 | tposeqd 7625 | . 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → tpos tpos ((𝐽‘𝑀) · 𝑀) = tpos ((𝐷‘tpos 𝑀) ∙ 1 )) |
23 | 2, 4 | matrcl 20592 | . . . . . 6 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
24 | 23 | simpld 490 | . . . . 5 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin) |
25 | crngring 18919 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
26 | 2 | matring 20623 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
27 | 24, 25, 26 | syl2an 589 | . . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring) |
28 | 4, 9 | ringcl 18922 | . . . 4 ⊢ ((𝐴 ∈ Ring ∧ (𝐽‘𝑀) ∈ 𝐵 ∧ 𝑀 ∈ 𝐵) → ((𝐽‘𝑀) · 𝑀) ∈ 𝐵) |
29 | 27, 7, 8, 28 | syl3anc 1494 | . . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → ((𝐽‘𝑀) · 𝑀) ∈ 𝐵) |
30 | 2, 4 | mattpostpos 20635 | . . 3 ⊢ (((𝐽‘𝑀) · 𝑀) ∈ 𝐵 → tpos tpos ((𝐽‘𝑀) · 𝑀) = ((𝐽‘𝑀) · 𝑀)) |
31 | 29, 30 | syl 17 | . 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → tpos tpos ((𝐽‘𝑀) · 𝑀) = ((𝐽‘𝑀) · 𝑀)) |
32 | eqid 2825 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
33 | 16, 2, 4, 32 | mdetf 20776 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝐷:𝐵⟶(Base‘𝑅)) |
34 | 33 | adantl 475 | . . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 𝐷:𝐵⟶(Base‘𝑅)) |
35 | 15 | adantr 474 | . . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → tpos 𝑀 ∈ 𝐵) |
36 | 34, 35 | ffvelrnd 6614 | . . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝐷‘tpos 𝑀) ∈ (Base‘𝑅)) |
37 | 4, 17 | ringidcl 18929 | . . . . 5 ⊢ (𝐴 ∈ Ring → 1 ∈ 𝐵) |
38 | 27, 37 | syl 17 | . . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 1 ∈ 𝐵) |
39 | 2, 4, 32, 18 | mattposvs 20636 | . . . 4 ⊢ (((𝐷‘tpos 𝑀) ∈ (Base‘𝑅) ∧ 1 ∈ 𝐵) → tpos ((𝐷‘tpos 𝑀) ∙ 1 ) = ((𝐷‘tpos 𝑀) ∙ tpos 1 )) |
40 | 36, 38, 39 | syl2anc 579 | . . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → tpos ((𝐷‘tpos 𝑀) ∙ 1 ) = ((𝐷‘tpos 𝑀) ∙ tpos 1 )) |
41 | 16, 2, 4 | mdettpos 20792 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐷‘tpos 𝑀) = (𝐷‘𝑀)) |
42 | 41 | ancoms 452 | . . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝐷‘tpos 𝑀) = (𝐷‘𝑀)) |
43 | 2, 17 | mattpos1 20637 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → tpos 1 = 1 ) |
44 | 24, 25, 43 | syl2an 589 | . . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → tpos 1 = 1 ) |
45 | 42, 44 | oveq12d 6928 | . . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → ((𝐷‘tpos 𝑀) ∙ tpos 1 ) = ((𝐷‘𝑀) ∙ 1 )) |
46 | 40, 45 | eqtrd 2861 | . 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → tpos ((𝐷‘tpos 𝑀) ∙ 1 ) = ((𝐷‘𝑀) ∙ 1 )) |
47 | 22, 31, 46 | 3eqtr3d 2869 | 1 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → ((𝐽‘𝑀) · 𝑀) = ((𝐷‘𝑀) ∙ 1 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 Vcvv 3414 ⟶wf 6123 ‘cfv 6127 (class class class)co 6910 tpos ctpos 7621 Fincfn 8228 Basecbs 16229 .rcmulr 16313 ·𝑠 cvsca 16316 1rcur 18862 Ringcrg 18908 CRingccrg 18909 Mat cmat 20587 maDet cmdat 20765 maAdju cmadu 20813 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-addf 10338 ax-mulf 10339 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-xor 1638 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-ot 4408 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-of 7162 df-om 7332 df-1st 7433 df-2nd 7434 df-supp 7565 df-tpos 7622 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-2o 7832 df-oadd 7835 df-er 8014 df-map 8129 df-pm 8130 df-ixp 8182 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-fsupp 8551 df-sup 8623 df-oi 8691 df-card 9085 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-xnn0 11698 df-z 11712 df-dec 11829 df-uz 11976 df-rp 12120 df-fz 12627 df-fzo 12768 df-seq 13103 df-exp 13162 df-hash 13418 df-word 13582 df-lsw 13630 df-concat 13638 df-s1 13663 df-substr 13708 df-pfx 13757 df-splice 13864 df-reverse 13882 df-s2 13976 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-starv 16327 df-sca 16328 df-vsca 16329 df-ip 16330 df-tset 16331 df-ple 16332 df-ds 16334 df-unif 16335 df-hom 16336 df-cco 16337 df-0g 16462 df-gsum 16463 df-prds 16468 df-pws 16470 df-mre 16606 df-mrc 16607 df-acs 16609 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-mhm 17695 df-submnd 17696 df-grp 17786 df-minusg 17787 df-sbg 17788 df-mulg 17902 df-subg 17949 df-ghm 18016 df-gim 18059 df-cntz 18107 df-oppg 18133 df-symg 18155 df-pmtr 18219 df-psgn 18268 df-evpm 18269 df-cmn 18555 df-abl 18556 df-mgp 18851 df-ur 18863 df-ring 18910 df-cring 18911 df-oppr 18984 df-dvdsr 19002 df-unit 19003 df-invr 19033 df-dvr 19044 df-rnghom 19078 df-drng 19112 df-subrg 19141 df-lmod 19228 df-lss 19296 df-sra 19540 df-rgmod 19541 df-cnfld 20114 df-zring 20186 df-zrh 20219 df-dsmm 20446 df-frlm 20461 df-mamu 20564 df-mat 20588 df-mdet 20766 df-madu 20815 |
This theorem is referenced by: matinv 20859 |
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