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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 483 | . . . . 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 22631 | . . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝐽:𝐵⟶𝐵) |
6 | 5 | ffvelcdmda 7090 | . . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐽‘𝑀) ∈ 𝐵) |
7 | 6 | ancoms 457 | . . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝐽‘𝑀) ∈ 𝐵) |
8 | simpl 481 | . . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 𝑀 ∈ 𝐵) | |
9 | madurid.t | . . . . . 6 ⊢ · = (.r‘𝐴) | |
10 | 2, 4, 9 | mattposm 22449 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ (𝐽‘𝑀) ∈ 𝐵 ∧ 𝑀 ∈ 𝐵) → tpos ((𝐽‘𝑀) · 𝑀) = (tpos 𝑀 · tpos (𝐽‘𝑀))) |
11 | 1, 7, 8, 10 | syl3anc 1368 | . . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → tpos ((𝐽‘𝑀) · 𝑀) = (tpos 𝑀 · tpos (𝐽‘𝑀))) |
12 | 2, 3, 4 | madutpos 22632 | . . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐽‘tpos 𝑀) = tpos (𝐽‘𝑀)) |
13 | 12 | ancoms 457 | . . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝐽‘tpos 𝑀) = tpos (𝐽‘𝑀)) |
14 | 13 | oveq2d 7432 | . . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (tpos 𝑀 · (𝐽‘tpos 𝑀)) = (tpos 𝑀 · tpos (𝐽‘𝑀))) |
15 | 2, 4 | mattposcl 22443 | . . . . 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 22634 | . . . . 5 ⊢ ((tpos 𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (tpos 𝑀 · (𝐽‘tpos 𝑀)) = ((𝐷‘tpos 𝑀) ∙ 1 )) |
20 | 15, 19 | sylan 578 | . . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (tpos 𝑀 · (𝐽‘tpos 𝑀)) = ((𝐷‘tpos 𝑀) ∙ 1 )) |
21 | 11, 14, 20 | 3eqtr2d 2772 | . . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → tpos ((𝐽‘𝑀) · 𝑀) = ((𝐷‘tpos 𝑀) ∙ 1 )) |
22 | 21 | tposeqd 8236 | . 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → tpos tpos ((𝐽‘𝑀) · 𝑀) = tpos ((𝐷‘tpos 𝑀) ∙ 1 )) |
23 | 2, 4 | matrcl 22400 | . . . . . 6 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
24 | 23 | simpld 493 | . . . . 5 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin) |
25 | crngring 20224 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
26 | 2 | matring 22433 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring) |
27 | 24, 25, 26 | syl2an 594 | . . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring) |
28 | 4, 9 | ringcl 20229 | . . . 4 ⊢ ((𝐴 ∈ Ring ∧ (𝐽‘𝑀) ∈ 𝐵 ∧ 𝑀 ∈ 𝐵) → ((𝐽‘𝑀) · 𝑀) ∈ 𝐵) |
29 | 27, 7, 8, 28 | syl3anc 1368 | . . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → ((𝐽‘𝑀) · 𝑀) ∈ 𝐵) |
30 | 2, 4 | mattpostpos 22444 | . . 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 2726 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
33 | 16, 2, 4, 32 | mdetf 22585 | . . . . . 6 ⊢ (𝑅 ∈ CRing → 𝐷:𝐵⟶(Base‘𝑅)) |
34 | 33 | adantl 480 | . . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 𝐷:𝐵⟶(Base‘𝑅)) |
35 | 15 | adantr 479 | . . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → tpos 𝑀 ∈ 𝐵) |
36 | 34, 35 | ffvelcdmd 7091 | . . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝐷‘tpos 𝑀) ∈ (Base‘𝑅)) |
37 | 4, 17 | ringidcl 20241 | . . . . 5 ⊢ (𝐴 ∈ Ring → 1 ∈ 𝐵) |
38 | 27, 37 | syl 17 | . . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → 1 ∈ 𝐵) |
39 | 2, 4, 32, 18 | mattposvs 22445 | . . . 4 ⊢ (((𝐷‘tpos 𝑀) ∈ (Base‘𝑅) ∧ 1 ∈ 𝐵) → tpos ((𝐷‘tpos 𝑀) ∙ 1 ) = ((𝐷‘tpos 𝑀) ∙ tpos 1 )) |
40 | 36, 38, 39 | syl2anc 582 | . . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → tpos ((𝐷‘tpos 𝑀) ∙ 1 ) = ((𝐷‘tpos 𝑀) ∙ tpos 1 )) |
41 | 16, 2, 4 | mdettpos 22601 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐷‘tpos 𝑀) = (𝐷‘𝑀)) |
42 | 41 | ancoms 457 | . . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝐷‘tpos 𝑀) = (𝐷‘𝑀)) |
43 | 2, 17 | mattpos1 22446 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → tpos 1 = 1 ) |
44 | 24, 25, 43 | syl2an 594 | . . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → tpos 1 = 1 ) |
45 | 42, 44 | oveq12d 7434 | . . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → ((𝐷‘tpos 𝑀) ∙ tpos 1 ) = ((𝐷‘𝑀) ∙ 1 )) |
46 | 40, 45 | eqtrd 2766 | . 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → tpos ((𝐷‘tpos 𝑀) ∙ 1 ) = ((𝐷‘𝑀) ∙ 1 )) |
47 | 22, 31, 46 | 3eqtr3d 2774 | 1 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → ((𝐽‘𝑀) · 𝑀) = ((𝐷‘𝑀) ∙ 1 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ⟶wf 6542 ‘cfv 6546 (class class class)co 7416 tpos ctpos 8232 Fincfn 8966 Basecbs 17208 .rcmulr 17262 ·𝑠 cvsca 17265 1rcur 20160 Ringcrg 20212 CRingccrg 20213 Mat cmat 22395 maDet cmdat 22574 maAdju cmadu 22622 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-addf 11228 ax-mulf 11229 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-xor 1506 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-ot 4632 df-uni 4906 df-int 4947 df-iun 4995 df-iin 4996 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7995 df-2nd 7996 df-supp 8167 df-tpos 8233 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-er 8726 df-map 8849 df-pm 8850 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9399 df-sup 9478 df-oi 9546 df-card 9975 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12519 df-xnn0 12591 df-z 12605 df-dec 12724 df-uz 12869 df-rp 13023 df-fz 13533 df-fzo 13676 df-seq 14016 df-exp 14076 df-hash 14343 df-word 14518 df-lsw 14566 df-concat 14574 df-s1 14599 df-substr 14644 df-pfx 14674 df-splice 14753 df-reverse 14762 df-s2 14852 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-starv 17276 df-sca 17277 df-vsca 17278 df-ip 17279 df-tset 17280 df-ple 17281 df-ds 17283 df-unif 17284 df-hom 17285 df-cco 17286 df-0g 17451 df-gsum 17452 df-prds 17457 df-pws 17459 df-mre 17594 df-mrc 17595 df-acs 17597 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-mhm 18768 df-submnd 18769 df-efmnd 18854 df-grp 18926 df-minusg 18927 df-sbg 18928 df-mulg 19058 df-subg 19113 df-ghm 19203 df-gim 19249 df-cntz 19307 df-oppg 19336 df-symg 19361 df-pmtr 19436 df-psgn 19485 df-evpm 19486 df-cmn 19776 df-abl 19777 df-mgp 20114 df-rng 20132 df-ur 20161 df-ring 20214 df-cring 20215 df-oppr 20312 df-dvdsr 20335 df-unit 20336 df-invr 20366 df-dvr 20379 df-rhm 20450 df-subrng 20524 df-subrg 20549 df-drng 20705 df-lmod 20834 df-lss 20905 df-sra 21147 df-rgmod 21148 df-cnfld 21340 df-zring 21433 df-zrh 21489 df-dsmm 21726 df-frlm 21741 df-mamu 22379 df-mat 22396 df-mdet 22575 df-madu 22624 |
This theorem is referenced by: matinv 22667 |
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