Theorem smadiadetr 20857

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. Closed form of smadiadetg 20855. Special case of the "Laplace expansion", see definition in [Lang] p. 515. (Contributed by AV, 15-Feb-2019.)

Assertion

Ref	Expression
smadiadetr	⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝑁 Mat 𝑅))) ∧ (𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅))) → ((𝑁 maDet 𝑅)‘(𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾)) = (𝑆(.r‘𝑅)(((𝑁 ∖ {𝐾}) maDet 𝑅)‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾))))

Proof of Theorem smadiadetr

Step	Hyp	Ref	Expression
1		3anass 1120	. . . . 5 ⊢ ((𝑀 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ↔ (𝑀 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅))))
2		oveq2 6918	. . . . . . . 8 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝑁 Mat 𝑅) = (𝑁 Mat if(𝑅 ∈ CRing, 𝑅, ℂfld)))
3	2	fveq2d 6441	. . . . . . 7 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat if(𝑅 ∈ CRing, 𝑅, ℂfld))))
4	3	eleq2d 2892	. . . . . 6 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝑀 ∈ (Base‘(𝑁 Mat 𝑅)) ↔ 𝑀 ∈ (Base‘(𝑁 Mat if(𝑅 ∈ CRing, 𝑅, ℂfld)))))
5		fveq2 6437	. . . . . . 7 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (Base‘𝑅) = (Base‘if(𝑅 ∈ CRing, 𝑅, ℂfld)))
6	5	eleq2d 2892	. . . . . 6 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝑆 ∈ (Base‘𝑅) ↔ 𝑆 ∈ (Base‘if(𝑅 ∈ CRing, 𝑅, ℂfld))))
7	4, 6	3anbi13d 1566	. . . . 5 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → ((𝑀 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ↔ (𝑀 ∈ (Base‘(𝑁 Mat if(𝑅 ∈ CRing, 𝑅, ℂfld))) ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘if(𝑅 ∈ CRing, 𝑅, ℂfld)))))
8	1, 7	syl5bbr 277	. . . 4 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → ((𝑀 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅))) ↔ (𝑀 ∈ (Base‘(𝑁 Mat if(𝑅 ∈ CRing, 𝑅, ℂfld))) ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘if(𝑅 ∈ CRing, 𝑅, ℂfld)))))
9		oveq2 6918	. . . . . 6 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝑁 maDet 𝑅) = (𝑁 maDet if(𝑅 ∈ CRing, 𝑅, ℂfld)))
10		oveq2 6918	. . . . . . . 8 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝑁 matRRep 𝑅) = (𝑁 matRRep if(𝑅 ∈ CRing, 𝑅, ℂfld)))
11	10	oveqd 6927	. . . . . . 7 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝑀(𝑁 matRRep 𝑅)𝑆) = (𝑀(𝑁 matRRep if(𝑅 ∈ CRing, 𝑅, ℂfld))𝑆))
12	11	oveqd 6927	. . . . . 6 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) = (𝐾(𝑀(𝑁 matRRep if(𝑅 ∈ CRing, 𝑅, ℂfld))𝑆)𝐾))
13	9, 12	fveq12d 6444	. . . . 5 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → ((𝑁 maDet 𝑅)‘(𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾)) = ((𝑁 maDet if(𝑅 ∈ CRing, 𝑅, ℂfld))‘(𝐾(𝑀(𝑁 matRRep if(𝑅 ∈ CRing, 𝑅, ℂfld))𝑆)𝐾)))
14		fveq2 6437	. . . . . 6 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (.r‘𝑅) = (.r‘if(𝑅 ∈ CRing, 𝑅, ℂfld)))
15		eqidd 2826	. . . . . 6 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → 𝑆 = 𝑆)
16		oveq2 6918	. . . . . . 7 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → ((𝑁 ∖ {𝐾}) maDet 𝑅) = ((𝑁 ∖ {𝐾}) maDet if(𝑅 ∈ CRing, 𝑅, ℂfld)))
17		oveq2 6918	. . . . . . . . 9 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝑁 subMat 𝑅) = (𝑁 subMat if(𝑅 ∈ CRing, 𝑅, ℂfld)))
18	17	fveq1d 6439	. . . . . . . 8 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → ((𝑁 subMat 𝑅)‘𝑀) = ((𝑁 subMat if(𝑅 ∈ CRing, 𝑅, ℂfld))‘𝑀))
19	18	oveqd 6927	. . . . . . 7 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾) = (𝐾((𝑁 subMat if(𝑅 ∈ CRing, 𝑅, ℂfld))‘𝑀)𝐾))
20	16, 19	fveq12d 6444	. . . . . 6 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (((𝑁 ∖ {𝐾}) maDet 𝑅)‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾)) = (((𝑁 ∖ {𝐾}) maDet if(𝑅 ∈ CRing, 𝑅, ℂfld))‘(𝐾((𝑁 subMat if(𝑅 ∈ CRing, 𝑅, ℂfld))‘𝑀)𝐾)))
21	14, 15, 20	oveq123d 6931	. . . . 5 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝑆(.r‘𝑅)(((𝑁 ∖ {𝐾}) maDet 𝑅)‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾))) = (𝑆(.r‘if(𝑅 ∈ CRing, 𝑅, ℂfld))(((𝑁 ∖ {𝐾}) maDet if(𝑅 ∈ CRing, 𝑅, ℂfld))‘(𝐾((𝑁 subMat if(𝑅 ∈ CRing, 𝑅, ℂfld))‘𝑀)𝐾))))
22	13, 21	eqeq12d 2840	. . . 4 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (((𝑁 maDet 𝑅)‘(𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾)) = (𝑆(.r‘𝑅)(((𝑁 ∖ {𝐾}) maDet 𝑅)‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾))) ↔ ((𝑁 maDet if(𝑅 ∈ CRing, 𝑅, ℂfld))‘(𝐾(𝑀(𝑁 matRRep if(𝑅 ∈ CRing, 𝑅, ℂfld))𝑆)𝐾)) = (𝑆(.r‘if(𝑅 ∈ CRing, 𝑅, ℂfld))(((𝑁 ∖ {𝐾}) maDet if(𝑅 ∈ CRing, 𝑅, ℂfld))‘(𝐾((𝑁 subMat if(𝑅 ∈ CRing, 𝑅, ℂfld))‘𝑀)𝐾)))))
23	8, 22	imbi12d 336	. . 3 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (((𝑀 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅))) → ((𝑁 maDet 𝑅)‘(𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾)) = (𝑆(.r‘𝑅)(((𝑁 ∖ {𝐾}) maDet 𝑅)‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾)))) ↔ ((𝑀 ∈ (Base‘(𝑁 Mat if(𝑅 ∈ CRing, 𝑅, ℂfld))) ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘if(𝑅 ∈ CRing, 𝑅, ℂfld))) → ((𝑁 maDet if(𝑅 ∈ CRing, 𝑅, ℂfld))‘(𝐾(𝑀(𝑁 matRRep if(𝑅 ∈ CRing, 𝑅, ℂfld))𝑆)𝐾)) = (𝑆(.r‘if(𝑅 ∈ CRing, 𝑅, ℂfld))(((𝑁 ∖ {𝐾}) maDet if(𝑅 ∈ CRing, 𝑅, ℂfld))‘(𝐾((𝑁 subMat if(𝑅 ∈ CRing, 𝑅, ℂfld))‘𝑀)𝐾))))))
24		cncrng 20134	. . . . 5 ⊢ ℂfld ∈ CRing
25	24	elimel 4375	. . . 4 ⊢ if(𝑅 ∈ CRing, 𝑅, ℂfld) ∈ CRing
26	25	smadiadetg0 20856	. . 3 ⊢ ((𝑀 ∈ (Base‘(𝑁 Mat if(𝑅 ∈ CRing, 𝑅, ℂfld))) ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘if(𝑅 ∈ CRing, 𝑅, ℂfld))) → ((𝑁 maDet if(𝑅 ∈ CRing, 𝑅, ℂfld))‘(𝐾(𝑀(𝑁 matRRep if(𝑅 ∈ CRing, 𝑅, ℂfld))𝑆)𝐾)) = (𝑆(.r‘if(𝑅 ∈ CRing, 𝑅, ℂfld))(((𝑁 ∖ {𝐾}) maDet if(𝑅 ∈ CRing, 𝑅, ℂfld))‘(𝐾((𝑁 subMat if(𝑅 ∈ CRing, 𝑅, ℂfld))‘𝑀)𝐾))))
27	23, 26	dedth 4364	. 2 ⊢ (𝑅 ∈ CRing → ((𝑀 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅))) → ((𝑁 maDet 𝑅)‘(𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾)) = (𝑆(.r‘𝑅)(((𝑁 ∖ {𝐾}) maDet 𝑅)‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾)))))
28	27	impl 449	1 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝑁 Mat 𝑅))) ∧ (𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅))) → ((𝑁 maDet 𝑅)‘(𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾)) = (𝑆(.r‘𝑅)(((𝑁 ∖ {𝐾}) maDet 𝑅)‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾))))

Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164   ∖ cdif 3795  ifcif 4308  {csn 4399  ‘cfv 6127  (class class class)co 6910  Basecbs 16229  ._rcmulr 16313  CRingccrg 18909  ℂ_fldccnfld 20113   Mat cmat 20587   matRRep cmarrep 20737   subMat csubma 20757   maDet cmdat 20765

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-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-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-sra 19540  df-rgmod 19541  df-cnfld 20114  df-zring 20186  df-zrh 20219  df-dsmm 20446  df-frlm 20461  df-mat 20588  df-marrep 20739  df-subma 20758  df-mdet 20766  df-minmar1 20816

This theorem is referenced by:  cramerimplem1  20865  cramerimplem1OLD  20866  madjusmdetlem1  30434