Theorem smadiadetr 21572

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 21570. 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 1097	. . . . 5 ⊢ ((𝑀 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ↔ (𝑀 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅))))
2		oveq2 7221	. . . . . . . 8 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝑁 Mat 𝑅) = (𝑁 Mat if(𝑅 ∈ CRing, 𝑅, ℂfld)))
3	2	fveq2d 6721	. . . . . . 7 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat if(𝑅 ∈ CRing, 𝑅, ℂfld))))
4	3	eleq2d 2823	. . . . . 6 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝑀 ∈ (Base‘(𝑁 Mat 𝑅)) ↔ 𝑀 ∈ (Base‘(𝑁 Mat if(𝑅 ∈ CRing, 𝑅, ℂfld)))))
5		fveq2 6717	. . . . . . 7 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (Base‘𝑅) = (Base‘if(𝑅 ∈ CRing, 𝑅, ℂfld)))
6	5	eleq2d 2823	. . . . . 6 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝑆 ∈ (Base‘𝑅) ↔ 𝑆 ∈ (Base‘if(𝑅 ∈ CRing, 𝑅, ℂfld))))
7	4, 6	3anbi13d 1440	. . . . 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	bitr3id 288	. . . 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 7221	. . . . . 6 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝑁 maDet 𝑅) = (𝑁 maDet if(𝑅 ∈ CRing, 𝑅, ℂfld)))
10		oveq2 7221	. . . . . . . 8 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝑁 matRRep 𝑅) = (𝑁 matRRep if(𝑅 ∈ CRing, 𝑅, ℂfld)))
11	10	oveqd 7230	. . . . . . 7 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝑀(𝑁 matRRep 𝑅)𝑆) = (𝑀(𝑁 matRRep if(𝑅 ∈ CRing, 𝑅, ℂfld))𝑆))
12	11	oveqd 7230	. . . . . 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 6724	. . . . 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 6717	. . . . . 6 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (.r‘𝑅) = (.r‘if(𝑅 ∈ CRing, 𝑅, ℂfld)))
15		eqidd 2738	. . . . . 6 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → 𝑆 = 𝑆)
16		oveq2 7221	. . . . . . 7 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → ((𝑁 ∖ {𝐾}) maDet 𝑅) = ((𝑁 ∖ {𝐾}) maDet if(𝑅 ∈ CRing, 𝑅, ℂfld)))
17		oveq2 7221	. . . . . . . . 9 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝑁 subMat 𝑅) = (𝑁 subMat if(𝑅 ∈ CRing, 𝑅, ℂfld)))
18	17	fveq1d 6719	. . . . . . . 8 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → ((𝑁 subMat 𝑅)‘𝑀) = ((𝑁 subMat if(𝑅 ∈ CRing, 𝑅, ℂfld))‘𝑀))
19	18	oveqd 7230	. . . . . . 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 6724	. . . . . 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 7234	. . . . 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 2753	. . . 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 348	. . 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 20384	. . . . 5 ⊢ ℂfld ∈ CRing
25	24	elimel 4508	. . . 4 ⊢ if(𝑅 ∈ CRing, 𝑅, ℂfld) ∈ CRing
26	25	smadiadetg0 21571	. . 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 4497	. 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 459	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 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110   ∖ cdif 3863  ifcif 4439  {csn 4541  ‘cfv 6380  (class class class)co 7213  Basecbs 16760  ._rcmulr 16803  CRingccrg 19563  ℂ_fldccnfld 20363   Mat cmat 21304   matRRep cmarrep 21453   subMat csubma 21473   maDet cmdat 21481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-addf 10808  ax-mulf 10809

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-xor 1508  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-ot 4550  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-of 7469  df-om 7645  df-1st 7761  df-2nd 7762  df-supp 7904  df-tpos 7968  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-2o 8203  df-er 8391  df-map 8510  df-pm 8511  df-ixp 8579  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-fsupp 8986  df-sup 9058  df-oi 9126  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-5 11896  df-6 11897  df-7 11898  df-8 11899  df-9 11900  df-n0 12091  df-xnn0 12163  df-z 12177  df-dec 12294  df-uz 12439  df-rp 12587  df-fz 13096  df-fzo 13239  df-seq 13575  df-exp 13636  df-hash 13897  df-word 14070  df-lsw 14118  df-concat 14126  df-s1 14153  df-substr 14206  df-pfx 14236  df-splice 14315  df-reverse 14324  df-s2 14413  df-struct 16700  df-sets 16717  df-slot 16735  df-ndx 16745  df-base 16761  df-ress 16785  df-plusg 16815  df-mulr 16816  df-starv 16817  df-sca 16818  df-vsca 16819  df-ip 16820  df-tset 16821  df-ple 16822  df-ds 16824  df-unif 16825  df-hom 16826  df-cco 16827  df-0g 16946  df-gsum 16947  df-prds 16952  df-pws 16954  df-mre 17089  df-mrc 17090  df-acs 17092  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-mhm 18218  df-submnd 18219  df-efmnd 18296  df-grp 18368  df-minusg 18369  df-mulg 18489  df-subg 18540  df-ghm 18620  df-gim 18663  df-cntz 18711  df-oppg 18738  df-symg 18760  df-pmtr 18834  df-psgn 18883  df-cmn 19172  df-abl 19173  df-mgp 19505  df-ur 19517  df-ring 19564  df-cring 19565  df-oppr 19641  df-dvdsr 19659  df-unit 19660  df-invr 19690  df-dvr 19701  df-rnghom 19735  df-drng 19769  df-subrg 19798  df-sra 20209  df-rgmod 20210  df-cnfld 20364  df-zring 20436  df-zrh 20470  df-dsmm 20694  df-frlm 20709  df-mat 21305  df-marrep 21455  df-subma 21474  df-mdet 21482  df-minmar1 21532

This theorem is referenced by:  cramerimplem1  21580  madjusmdetlem1  31491