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Theorem smadiadetr 19262
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 19260. Special case of the "Laplace expansion", see definition in [Lang] p. 515. (Contributed by AV, 15-Feb-2019.)
Assertion
Ref Expression
smadiadetr  |-  ( ( ( R  e.  CRing  /\  M  e.  ( Base `  ( N Mat  R ) ) )  /\  ( K  e.  N  /\  S  e.  ( 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
StepHypRef Expression
1 3anass 975 . . . . 5  |-  ( ( M  e.  ( Base `  ( N Mat  R ) )  /\  K  e.  N  /\  S  e.  ( Base `  R
) )  <->  ( M  e.  ( Base `  ( N Mat  R ) )  /\  ( K  e.  N  /\  S  e.  ( Base `  R ) ) ) )
2 oveq2 6204 . . . . . . . 8  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  ( N Mat  R )  =  ( N Mat  if ( R  e.  CRing ,  R ,fld ) ) )
32fveq2d 5778 . . . . . . 7  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  ( Base `  ( N Mat  R ) )  =  ( Base `  ( N Mat  if ( R  e. 
CRing ,  R ,fld )
) ) )
43eleq2d 2452 . . . . . 6  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  ( M  e.  (
Base `  ( N Mat  R ) )  <->  M  e.  ( Base `  ( N Mat  if ( R  e.  CRing ,  R ,fld ) ) ) ) )
5 fveq2 5774 . . . . . . 7  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  ( Base `  R
)  =  ( Base `  if ( R  e. 
CRing ,  R ,fld )
) )
65eleq2d 2452 . . . . . 6  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  ( S  e.  (
Base `  R )  <->  S  e.  ( Base `  if ( R  e.  CRing ,  R ,fld ) ) ) )
74, 63anbi13d 1299 . . . . 5  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  ( ( M  e.  ( Base `  ( N Mat  R ) )  /\  K  e.  N  /\  S  e.  ( Base `  R ) )  <->  ( M  e.  ( Base `  ( N Mat  if ( R  e. 
CRing ,  R ,fld )
) )  /\  K  e.  N  /\  S  e.  ( Base `  if ( R  e.  CRing ,  R ,fld ) ) ) ) )
81, 7syl5bbr 259 . . . 4  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  ( ( M  e.  ( Base `  ( N Mat  R ) )  /\  ( K  e.  N  /\  S  e.  ( Base `  R ) ) )  <->  ( M  e.  ( Base `  ( N Mat  if ( R  e. 
CRing ,  R ,fld )
) )  /\  K  e.  N  /\  S  e.  ( Base `  if ( R  e.  CRing ,  R ,fld ) ) ) ) )
9 oveq2 6204 . . . . . 6  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  ( N maDet  R )  =  ( N maDet  if ( R  e.  CRing ,  R ,fld ) ) )
10 oveq2 6204 . . . . . . . 8  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  ( N matRRep  R )  =  ( N matRRep  if ( R  e.  CRing ,  R ,fld ) ) )
1110oveqd 6213 . . . . . . 7  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  ( M ( N matRRep  R ) S )  =  ( M ( N matRRep  if ( R  e. 
CRing ,  R ,fld )
) S ) )
1211oveqd 6213 . . . . . 6  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  ( K ( M ( N matRRep  R ) S ) K )  =  ( K ( M ( N matRRep  if ( R  e.  CRing ,  R ,fld ) ) S ) K ) )
139, 12fveq12d 5780 . . . . 5  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  ( ( N maDet  R
) `  ( K
( M ( N matRRep  R ) S ) K ) )  =  ( ( N maDet  if ( R  e.  CRing ,  R ,fld ) ) `  ( K ( M ( N matRRep  if ( R  e. 
CRing ,  R ,fld )
) S ) K ) ) )
14 fveq2 5774 . . . . . 6  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  ( .r `  R
)  =  ( .r
`  if ( R  e.  CRing ,  R ,fld )
) )
15 eqidd 2383 . . . . . 6  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  S  =  S )
16 oveq2 6204 . . . . . . 7  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  ( ( N  \  { K } ) maDet  R
)  =  ( ( N  \  { K } ) maDet  if ( R  e.  CRing ,  R ,fld )
) )
17 oveq2 6204 . . . . . . . . 9  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  ( N subMat  R )  =  ( N subMat  if ( R  e.  CRing ,  R ,fld ) ) )
1817fveq1d 5776 . . . . . . . 8  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  ( ( N subMat  R
) `  M )  =  ( ( N subMat  if ( R  e.  CRing ,  R ,fld ) ) `  M
) )
1918oveqd 6213 . . . . . . 7  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  ( K ( ( N subMat  R ) `  M ) K )  =  ( K ( ( N subMat  if ( R  e.  CRing ,  R ,fld ) ) `  M
) K ) )
2016, 19fveq12d 5780 . . . . . 6  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  ( ( ( N 
\  { K }
) maDet  R ) `  ( K ( ( N subMat  R ) `  M
) K ) )  =  ( ( ( N  \  { K } ) maDet  if ( R  e.  CRing ,  R ,fld )
) `  ( K
( ( N subMat  if ( R  e.  CRing ,  R ,fld ) ) `  M
) K ) ) )
2114, 15, 20oveq123d 6217 . . . . 5  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  ( S ( .r
`  R ) ( ( ( N  \  { K } ) maDet  R
) `  ( K
( ( N subMat  R
) `  M ) K ) ) )  =  ( S ( .r `  if ( R  e.  CRing ,  R ,fld ) ) ( ( ( N  \  { K } ) maDet  if ( R  e.  CRing ,  R ,fld ) ) `  ( K ( ( N subMat  if ( R  e.  CRing ,  R ,fld ) ) `  M
) K ) ) ) )
2213, 21eqeq12d 2404 . . . 4  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  ( ( ( 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  e.  CRing ,  R ,fld ) ) `  ( K ( M ( N matRRep  if ( R  e. 
CRing ,  R ,fld )
) S ) K ) )  =  ( S ( .r `  if ( R  e.  CRing ,  R ,fld ) ) ( ( ( N  \  { K } ) maDet  if ( R  e.  CRing ,  R ,fld ) ) `  ( K ( ( N subMat  if ( R  e.  CRing ,  R ,fld ) ) `  M
) K ) ) ) ) )
238, 22imbi12d 318 . . 3  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  ( ( ( M  e.  ( Base `  ( N Mat  R ) )  /\  ( K  e.  N  /\  S  e.  ( 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  e.  ( Base `  ( N Mat  if ( R  e. 
CRing ,  R ,fld )
) )  /\  K  e.  N  /\  S  e.  ( Base `  if ( R  e.  CRing ,  R ,fld ) ) )  -> 
( ( N maDet  if ( R  e.  CRing ,  R ,fld ) ) `  ( K ( M ( N matRRep  if ( R  e. 
CRing ,  R ,fld )
) S ) K ) )  =  ( S ( .r `  if ( R  e.  CRing ,  R ,fld ) ) ( ( ( N  \  { K } ) maDet  if ( R  e.  CRing ,  R ,fld ) ) `  ( K ( ( N subMat  if ( R  e.  CRing ,  R ,fld ) ) `  M
) K ) ) ) ) ) )
24 cncrng 18552 . . . . 5  |-fld  e.  CRing
2524elimel 3919 . . . 4  |-  if ( R  e.  CRing ,  R ,fld )  e.  CRing
2625smadiadetg0 19261 . . 3  |-  ( ( M  e.  ( Base `  ( N Mat  if ( R  e.  CRing ,  R ,fld ) ) )  /\  K  e.  N  /\  S  e.  ( Base `  if ( R  e. 
CRing ,  R ,fld )
) )  ->  (
( N maDet  if ( R  e.  CRing ,  R ,fld ) ) `  ( K ( M ( N matRRep  if ( R  e. 
CRing ,  R ,fld )
) S ) K ) )  =  ( S ( .r `  if ( R  e.  CRing ,  R ,fld ) ) ( ( ( N  \  { K } ) maDet  if ( R  e.  CRing ,  R ,fld ) ) `  ( K ( ( N subMat  if ( R  e.  CRing ,  R ,fld ) ) `  M
) K ) ) ) )
2723, 26dedth 3908 . 2  |-  ( R  e.  CRing  ->  ( ( M  e.  ( Base `  ( N Mat  R ) )  /\  ( K  e.  N  /\  S  e.  ( 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 ) ) ) ) )
2827impl 618 1  |-  ( ( ( R  e.  CRing  /\  M  e.  ( Base `  ( N Mat  R ) ) )  /\  ( K  e.  N  /\  S  e.  ( 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 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826    \ cdif 3386   ifcif 3857   {csn 3944   ` cfv 5496  (class class class)co 6196   Basecbs 14634   .rcmulr 14703   CRingccrg 17312  ℂfldccnfld 18533   Mat cmat 18994   matRRep cmarrep 19143   subMat csubma 19163   maDet cmdat 19171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-addf 9482  ax-mulf 9483
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-xor 1363  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-ot 3953  df-uni 4164  df-int 4200  df-iun 4245  df-iin 4246  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-of 6439  df-om 6600  df-1st 6699  df-2nd 6700  df-supp 6818  df-tpos 6873  df-recs 6960  df-rdg 6994  df-1o 7048  df-2o 7049  df-oadd 7052  df-er 7229  df-map 7340  df-pm 7341  df-ixp 7389  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-fsupp 7745  df-sup 7816  df-oi 7850  df-card 8233  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-7 10516  df-8 10517  df-9 10518  df-10 10519  df-n0 10713  df-z 10782  df-dec 10896  df-uz 11002  df-rp 11140  df-fz 11594  df-fzo 11718  df-seq 12011  df-exp 12070  df-hash 12308  df-word 12446  df-lsw 12447  df-concat 12448  df-s1 12449  df-substr 12450  df-splice 12451  df-reverse 12452  df-s2 12724  df-struct 14636  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-mulr 14716  df-starv 14717  df-sca 14718  df-vsca 14719  df-ip 14720  df-tset 14721  df-ple 14722  df-ds 14724  df-unif 14725  df-hom 14726  df-cco 14727  df-0g 14849  df-gsum 14850  df-prds 14855  df-pws 14857  df-mre 14993  df-mrc 14994  df-acs 14996  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-mhm 16083  df-submnd 16084  df-grp 16174  df-minusg 16175  df-mulg 16177  df-subg 16315  df-ghm 16382  df-gim 16424  df-cntz 16472  df-oppg 16498  df-symg 16520  df-pmtr 16584  df-psgn 16633  df-cmn 16917  df-abl 16918  df-mgp 17255  df-ur 17267  df-ring 17313  df-cring 17314  df-oppr 17385  df-dvdsr 17403  df-unit 17404  df-invr 17434  df-dvr 17445  df-rnghom 17477  df-drng 17511  df-subrg 17540  df-sra 17931  df-rgmod 17932  df-cnfld 18534  df-zring 18602  df-zrh 18634  df-dsmm 18854  df-frlm 18869  df-mat 18995  df-marrep 19145  df-subma 19164  df-mdet 19172  df-minmar1 19222
This theorem is referenced by:  cramerimplem1  19270
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