Theorem smadiadetr 22789

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 22787. 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 1109	. . . . 5 ⊢ ((𝑀 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) ↔ (𝑀 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅))))
2		oveq2 7408	. . . . . . . 8 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝑁 Mat 𝑅) = (𝑁 Mat if(𝑅 ∈ CRing, 𝑅, ℂfld)))
3	2	fveq2d 6875	. . . . . . 7 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat if(𝑅 ∈ CRing, 𝑅, ℂfld))))
4	3	eleq2d 2851	. . . . . 6 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝑀 ∈ (Base‘(𝑁 Mat 𝑅)) ↔ 𝑀 ∈ (Base‘(𝑁 Mat if(𝑅 ∈ CRing, 𝑅, ℂfld)))))
5		fveq2 6871	. . . . . . 7 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (Base‘𝑅) = (Base‘if(𝑅 ∈ CRing, 𝑅, ℂfld)))
6	5	eleq2d 2851	. . . . . 6 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝑆 ∈ (Base‘𝑅) ↔ 𝑆 ∈ (Base‘if(𝑅 ∈ CRing, 𝑅, ℂfld))))
7	4, 6	3anbi13d 1462	. . . . 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 7408	. . . . . 6 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝑁 maDet 𝑅) = (𝑁 maDet if(𝑅 ∈ CRing, 𝑅, ℂfld)))
10		oveq2 7408	. . . . . . . 8 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝑁 matRRep 𝑅) = (𝑁 matRRep if(𝑅 ∈ CRing, 𝑅, ℂfld)))
11	10	oveqd 7417	. . . . . . 7 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝑀(𝑁 matRRep 𝑅)𝑆) = (𝑀(𝑁 matRRep if(𝑅 ∈ CRing, 𝑅, ℂfld))𝑆))
12	11	oveqd 7417	. . . . . 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 6878	. . . . 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 6871	. . . . . 6 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (.r‘𝑅) = (.r‘if(𝑅 ∈ CRing, 𝑅, ℂfld)))
15		eqidd 2766	. . . . . 6 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → 𝑆 = 𝑆)
16		oveq2 7408	. . . . . . 7 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → ((𝑁 ∖ {𝐾}) maDet 𝑅) = ((𝑁 ∖ {𝐾}) maDet if(𝑅 ∈ CRing, 𝑅, ℂfld)))
17		oveq2 7408	. . . . . . . . 9 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → (𝑁 subMat 𝑅) = (𝑁 subMat if(𝑅 ∈ CRing, 𝑅, ℂfld)))
18	17	fveq1d 6873	. . . . . . . 8 ⊢ (𝑅 = if(𝑅 ∈ CRing, 𝑅, ℂfld) → ((𝑁 subMat 𝑅)‘𝑀) = ((𝑁 subMat if(𝑅 ∈ CRing, 𝑅, ℂfld))‘𝑀))
19	18	oveqd 7417	. . . . . . 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 6878	. . . . . 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 7421	. . . . 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 2781	. . . 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 347	. . 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 21500	. . . . 5 ⊢ ℂfld ∈ CRing
25	24	elimel 4553	. . . 4 ⊢ if(𝑅 ∈ CRing, 𝑅, ℂfld) ∈ CRing
26	25	smadiadetg0 22788	. . 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 4542	. 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 460	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 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145   ∖ cdif 3904  ifcif 4483  {csn 4585  ‘cfv 6525  (class class class)co 7400  Basecbs 17257  ._rcmulr 17299  CRingccrg 20304  ℂ_fldccnfld 21479   Mat cmat 22521   matRRep cmarrep 22670   subMat csubma 22690   maDet cmdat 22698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165  ax-addf 11167  ax-mulf 11168

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-xor 1535  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5105  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6291  df-ord 6352  df-on 6353  df-lim 6354  df-suc 6355  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-isom 6534  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-of 7664  df-om 7851  df-1st 7974  df-2nd 7975  df-supp 8145  df-tpos 8210  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-2o 8442  df-er 8682  df-map 8814  df-pm 8815  df-ixp 8884  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-fsupp 9310  df-sup 9390  df-oi 9460  df-card 9913  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-div 11860  df-nn 12222  df-2 12291  df-3 12292  df-4 12293  df-5 12294  df-6 12295  df-7 12296  df-8 12297  df-9 12298  df-n0 12493  df-xnn0 12566  df-z 12580  df-dec 12700  df-uz 12851  df-rp 13005  df-fz 13524  df-fzo 13671  df-seq 14026  df-exp 14086  df-hash 14355  df-word 14539  df-lsw 14588  df-concat 14596  df-s1 14622  df-substr 14667  df-pfx 14697  df-splice 14775  df-reverse 14784  df-s2 14873  df-struct 17195  df-sets 17212  df-slot 17230  df-ndx 17242  df-base 17258  df-ress 17279  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-0g 17482  df-gsum 17483  df-prds 17488  df-pws 17490  df-mre 17626  df-mrc 17627  df-acs 17629  df-mgm 18686  df-sgrp 18765  df-mnd 18781  df-mhm 18829  df-submnd 18830  df-efmnd 18916  df-grp 18991  df-minusg 18992  df-mulg 19122  df-subg 19177  df-ghm 19272  df-gim 19317  df-cntz 19375  df-oppg 19404  df-symg 19428  df-pmtr 19500  df-psgn 19549  df-cmn 19840  df-abl 19841  df-mgp 20205  df-rng 20219  df-ur 20252  df-ring 20305  df-cring 20306  df-oppr 20407  df-dvdsr 20427  df-unit 20428  df-invr 20458  df-dvr 20471  df-rhm 20542  df-subrng 20619  df-subrg 20643  df-drng 20803  df-sra 21260  df-rgmod 21261  df-cnfld 21480  df-zring 21554  df-zrh 21610  df-dsmm 21839  df-frlm 21854  df-mat 22522  df-marrep 22672  df-subma 22691  df-mdet 22699  df-minmar1 22749

This theorem is referenced by:  cramerimplem1  22797  madjusmdetlem1  34129