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Mirrors > Home > MPE Home > Th. List > smadiadetg | Structured version Visualization version GIF version |
Description: The determinant of a square matrix with one row replaced with 0's and an arbitrary element of the underlying ring at the diagonal position equals the ring element multiplied with the determinant of a submatrix of the square matrix obtained by removing the row and the column at the same index. (Contributed by AV, 14-Feb-2019.) |
Ref | Expression |
---|---|
smadiadet.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
smadiadet.b | ⊢ 𝐵 = (Base‘𝐴) |
smadiadet.r | ⊢ 𝑅 ∈ CRing |
smadiadet.d | ⊢ 𝐷 = (𝑁 maDet 𝑅) |
smadiadet.h | ⊢ 𝐸 = ((𝑁 ∖ {𝐾}) maDet 𝑅) |
smadiadetg.x | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
smadiadetg | ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝐷‘(𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾)) = (𝑆 · (𝐸‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smadiadet.d | . . 3 ⊢ 𝐷 = (𝑁 maDet 𝑅) | |
2 | smadiadet.a | . . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
3 | smadiadet.b | . . 3 ⊢ 𝐵 = (Base‘𝐴) | |
4 | eqid 2738 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
5 | smadiadetg.x | . . 3 ⊢ · = (.r‘𝑅) | |
6 | smadiadet.r | . . . 4 ⊢ 𝑅 ∈ CRing | |
7 | 6 | a1i 11 | . . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → 𝑅 ∈ CRing) |
8 | crngring 19428 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
9 | 6, 8 | mp1i 13 | . . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → 𝑅 ∈ Ring) |
10 | simp1 1137 | . . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → 𝑀 ∈ 𝐵) | |
11 | simp3 1139 | . . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → 𝑆 ∈ (Base‘𝑅)) | |
12 | simp2 1138 | . . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → 𝐾 ∈ 𝑁) | |
13 | 2, 3 | marrepcl 21315 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐾 ∈ 𝑁)) → (𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) ∈ 𝐵) |
14 | 9, 10, 11, 12, 12, 13 | syl32anc 1379 | . . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) ∈ 𝐵) |
15 | 2, 3 | minmar1cl 21402 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝐾 ∈ 𝑁 ∧ 𝐾 ∈ 𝑁)) → (𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) ∈ 𝐵) |
16 | 9, 10, 12, 12, 15 | syl22anc 838 | . . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) ∈ 𝐵) |
17 | smadiadet.h | . . . 4 ⊢ 𝐸 = ((𝑁 ∖ {𝐾}) maDet 𝑅) | |
18 | 2, 3, 6, 1, 17, 5 | smadiadetglem2 21423 | . . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → ((𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) ↾ ({𝐾} × 𝑁)) = ((({𝐾} × 𝑁) × {𝑆}) ∘f · ((𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) ↾ ({𝐾} × 𝑁)))) |
19 | 2, 3, 6, 1, 17 | smadiadetglem1 21422 | . . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → ((𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) ↾ ((𝑁 ∖ {𝐾}) × 𝑁)) = ((𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) ↾ ((𝑁 ∖ {𝐾}) × 𝑁))) |
20 | 1, 2, 3, 4, 5, 7, 14, 11, 16, 12, 18, 19 | mdetrsca 21354 | . 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝐷‘(𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾)) = (𝑆 · (𝐷‘(𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾)))) |
21 | 2, 3, 6, 1, 17 | smadiadet 21421 | . . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) → (𝐸‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾)) = (𝐷‘(𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾))) |
22 | 21 | 3adant3 1133 | . . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝐸‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾)) = (𝐷‘(𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾))) |
23 | 22 | eqcomd 2744 | . . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝐷‘(𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾)) = (𝐸‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾))) |
24 | 23 | oveq2d 7186 | . 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑆 · (𝐷‘(𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾))) = (𝑆 · (𝐸‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾)))) |
25 | 20, 24 | eqtrd 2773 | 1 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝐷‘(𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾)) = (𝑆 · (𝐸‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ∖ cdif 3840 {csn 4516 ‘cfv 6339 (class class class)co 7170 Basecbs 16586 .rcmulr 16669 Ringcrg 19416 CRingccrg 19417 Mat cmat 21158 matRRep cmarrep 21307 subMat csubma 21327 maDet cmdat 21335 minMatR1 cminmar1 21384 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-addf 10694 ax-mulf 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-xor 1507 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-ot 4525 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-of 7425 df-om 7600 df-1st 7714 df-2nd 7715 df-supp 7857 df-tpos 7921 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-2o 8132 df-er 8320 df-map 8439 df-pm 8440 df-ixp 8508 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-fsupp 8907 df-sup 8979 df-oi 9047 df-card 9441 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-xnn0 12049 df-z 12063 df-dec 12180 df-uz 12325 df-rp 12473 df-fz 12982 df-fzo 13125 df-seq 13461 df-exp 13522 df-hash 13783 df-word 13956 df-lsw 14004 df-concat 14012 df-s1 14039 df-substr 14092 df-pfx 14122 df-splice 14201 df-reverse 14210 df-s2 14299 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-starv 16683 df-sca 16684 df-vsca 16685 df-ip 16686 df-tset 16687 df-ple 16688 df-ds 16690 df-unif 16691 df-hom 16692 df-cco 16693 df-0g 16818 df-gsum 16819 df-prds 16824 df-pws 16826 df-mre 16960 df-mrc 16961 df-acs 16963 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-mhm 18072 df-submnd 18073 df-efmnd 18150 df-grp 18222 df-minusg 18223 df-mulg 18343 df-subg 18394 df-ghm 18474 df-gim 18517 df-cntz 18565 df-oppg 18592 df-symg 18614 df-pmtr 18688 df-psgn 18737 df-cmn 19026 df-abl 19027 df-mgp 19359 df-ur 19371 df-ring 19418 df-cring 19419 df-oppr 19495 df-dvdsr 19513 df-unit 19514 df-invr 19544 df-dvr 19555 df-rnghom 19589 df-drng 19623 df-subrg 19652 df-sra 20063 df-rgmod 20064 df-cnfld 20218 df-zring 20290 df-zrh 20324 df-dsmm 20548 df-frlm 20563 df-mat 21159 df-marrep 21309 df-subma 21328 df-mdet 21336 df-minmar1 21386 |
This theorem is referenced by: smadiadetg0 21425 |
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