Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > madjusmdet | Structured version Visualization version GIF version |
Description: Express the cofactor of the matrix, i.e. the entries of its adjunct matrix, using determinant of submatrices. (Contributed by Thierry Arnoux, 23-Aug-2020.) |
Ref | Expression |
---|---|
madjusmdet.b | ⊢ 𝐵 = (Base‘𝐴) |
madjusmdet.a | ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) |
madjusmdet.d | ⊢ 𝐷 = ((1...𝑁) maDet 𝑅) |
madjusmdet.k | ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅) |
madjusmdet.t | ⊢ · = (.r‘𝑅) |
madjusmdet.z | ⊢ 𝑍 = (ℤRHom‘𝑅) |
madjusmdet.e | ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅) |
madjusmdet.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
madjusmdet.r | ⊢ (𝜑 → 𝑅 ∈ CRing) |
madjusmdet.i | ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) |
madjusmdet.j | ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) |
madjusmdet.m | ⊢ (𝜑 → 𝑀 ∈ 𝐵) |
Ref | Expression |
---|---|
madjusmdet | ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘(-1↑(𝐼 + 𝐽))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | madjusmdet.b | . 2 ⊢ 𝐵 = (Base‘𝐴) | |
2 | madjusmdet.a | . 2 ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) | |
3 | madjusmdet.d | . 2 ⊢ 𝐷 = ((1...𝑁) maDet 𝑅) | |
4 | madjusmdet.k | . 2 ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅) | |
5 | madjusmdet.t | . 2 ⊢ · = (.r‘𝑅) | |
6 | madjusmdet.z | . 2 ⊢ 𝑍 = (ℤRHom‘𝑅) | |
7 | madjusmdet.e | . 2 ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅) | |
8 | madjusmdet.n | . 2 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
9 | madjusmdet.r | . 2 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
10 | madjusmdet.i | . 2 ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) | |
11 | madjusmdet.j | . 2 ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) | |
12 | madjusmdet.m | . 2 ⊢ (𝜑 → 𝑀 ∈ 𝐵) | |
13 | eqeq1 2822 | . . . 4 ⊢ (𝑘 = 𝑖 → (𝑘 = 1 ↔ 𝑖 = 1)) | |
14 | breq1 5060 | . . . . 5 ⊢ (𝑘 = 𝑖 → (𝑘 ≤ 𝐼 ↔ 𝑖 ≤ 𝐼)) | |
15 | oveq1 7152 | . . . . 5 ⊢ (𝑘 = 𝑖 → (𝑘 − 1) = (𝑖 − 1)) | |
16 | id 22 | . . . . 5 ⊢ (𝑘 = 𝑖 → 𝑘 = 𝑖) | |
17 | 14, 15, 16 | ifbieq12d 4490 | . . . 4 ⊢ (𝑘 = 𝑖 → if(𝑘 ≤ 𝐼, (𝑘 − 1), 𝑘) = if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)) |
18 | 13, 17 | ifbieq2d 4488 | . . 3 ⊢ (𝑘 = 𝑖 → if(𝑘 = 1, 𝐼, if(𝑘 ≤ 𝐼, (𝑘 − 1), 𝑘)) = if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) |
19 | 18 | cbvmptv 5160 | . 2 ⊢ (𝑘 ∈ (1...𝑁) ↦ if(𝑘 = 1, 𝐼, if(𝑘 ≤ 𝐼, (𝑘 − 1), 𝑘))) = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) |
20 | breq1 5060 | . . . . 5 ⊢ (𝑘 = 𝑖 → (𝑘 ≤ 𝑁 ↔ 𝑖 ≤ 𝑁)) | |
21 | 20, 15, 16 | ifbieq12d 4490 | . . . 4 ⊢ (𝑘 = 𝑖 → if(𝑘 ≤ 𝑁, (𝑘 − 1), 𝑘) = if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖)) |
22 | 13, 21 | ifbieq2d 4488 | . . 3 ⊢ (𝑘 = 𝑖 → if(𝑘 = 1, 𝑁, if(𝑘 ≤ 𝑁, (𝑘 − 1), 𝑘)) = if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖))) |
23 | 22 | cbvmptv 5160 | . 2 ⊢ (𝑘 ∈ (1...𝑁) ↦ if(𝑘 = 1, 𝑁, if(𝑘 ≤ 𝑁, (𝑘 − 1), 𝑘))) = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖))) |
24 | eqeq1 2822 | . . . 4 ⊢ (𝑙 = 𝑗 → (𝑙 = 1 ↔ 𝑗 = 1)) | |
25 | breq1 5060 | . . . . 5 ⊢ (𝑙 = 𝑗 → (𝑙 ≤ 𝐽 ↔ 𝑗 ≤ 𝐽)) | |
26 | oveq1 7152 | . . . . 5 ⊢ (𝑙 = 𝑗 → (𝑙 − 1) = (𝑗 − 1)) | |
27 | id 22 | . . . . 5 ⊢ (𝑙 = 𝑗 → 𝑙 = 𝑗) | |
28 | 25, 26, 27 | ifbieq12d 4490 | . . . 4 ⊢ (𝑙 = 𝑗 → if(𝑙 ≤ 𝐽, (𝑙 − 1), 𝑙) = if(𝑗 ≤ 𝐽, (𝑗 − 1), 𝑗)) |
29 | 24, 28 | ifbieq2d 4488 | . . 3 ⊢ (𝑙 = 𝑗 → if(𝑙 = 1, 𝐽, if(𝑙 ≤ 𝐽, (𝑙 − 1), 𝑙)) = if(𝑗 = 1, 𝐽, if(𝑗 ≤ 𝐽, (𝑗 − 1), 𝑗))) |
30 | 29 | cbvmptv 5160 | . 2 ⊢ (𝑙 ∈ (1...𝑁) ↦ if(𝑙 = 1, 𝐽, if(𝑙 ≤ 𝐽, (𝑙 − 1), 𝑙))) = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝐽, if(𝑗 ≤ 𝐽, (𝑗 − 1), 𝑗))) |
31 | breq1 5060 | . . . . 5 ⊢ (𝑙 = 𝑗 → (𝑙 ≤ 𝑁 ↔ 𝑗 ≤ 𝑁)) | |
32 | 31, 26, 27 | ifbieq12d 4490 | . . . 4 ⊢ (𝑙 = 𝑗 → if(𝑙 ≤ 𝑁, (𝑙 − 1), 𝑙) = if(𝑗 ≤ 𝑁, (𝑗 − 1), 𝑗)) |
33 | 24, 32 | ifbieq2d 4488 | . . 3 ⊢ (𝑙 = 𝑗 → if(𝑙 = 1, 𝑁, if(𝑙 ≤ 𝑁, (𝑙 − 1), 𝑙)) = if(𝑗 = 1, 𝑁, if(𝑗 ≤ 𝑁, (𝑗 − 1), 𝑗))) |
34 | 33 | cbvmptv 5160 | . 2 ⊢ (𝑙 ∈ (1...𝑁) ↦ if(𝑙 = 1, 𝑁, if(𝑙 ≤ 𝑁, (𝑙 − 1), 𝑙))) = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝑁, if(𝑗 ≤ 𝑁, (𝑗 − 1), 𝑗))) |
35 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 19, 23, 30, 34 | madjusmdetlem4 30994 | 1 ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘(-1↑(𝐼 + 𝐽))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ifcif 4463 class class class wbr 5057 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7145 1c1 10526 + caddc 10528 ≤ cle 10664 − cmin 10858 -cneg 10859 ℕcn 11626 ...cfz 12880 ↑cexp 13417 Basecbs 16471 .rcmulr 16554 CRingccrg 19227 ℤRHomczrh 20575 Mat cmat 20944 maDet cmdat 21121 maAdju cmadu 21169 subMat1csmat 30957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-xor 1496 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-ot 4566 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-sup 8894 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-xnn0 11956 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-fz 12881 df-fzo 13022 df-seq 13358 df-exp 13418 df-hash 13679 df-word 13850 df-lsw 13903 df-concat 13911 df-s1 13938 df-substr 13991 df-pfx 14021 df-splice 14100 df-reverse 14109 df-s2 14198 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-0g 16703 df-gsum 16704 df-prds 16709 df-pws 16711 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-submnd 17945 df-grp 18044 df-minusg 18045 df-mulg 18163 df-subg 18214 df-ghm 18294 df-gim 18337 df-cntz 18385 df-oppg 18412 df-symg 18434 df-pmtr 18499 df-psgn 18548 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-rnghom 19396 df-drng 19433 df-subrg 19462 df-sra 19873 df-rgmod 19874 df-cnfld 20474 df-zring 20546 df-zrh 20579 df-dsmm 20804 df-frlm 20819 df-mat 20945 df-marrep 21095 df-subma 21114 df-mdet 21122 df-madu 21171 df-minmar1 21172 df-smat 30958 |
This theorem is referenced by: mdetlap 30996 |
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