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Theorem smadiadetr 18503
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 18501. 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 969 . . . . 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 6120 . . . . . . . 8  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  ( N Mat  R )  =  ( N Mat  if ( R  e.  CRing ,  R ,fld ) ) )
32fveq2d 5716 . . . . . . 7  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  ( Base `  ( N Mat  R ) )  =  ( Base `  ( N Mat  if ( R  e. 
CRing ,  R ,fld )
) ) )
43eleq2d 2510 . . . . . 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 5712 . . . . . . 7  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  ( Base `  R
)  =  ( Base `  if ( R  e. 
CRing ,  R ,fld )
) )
65eleq2d 2510 . . . . . 6  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  ( S  e.  (
Base `  R )  <->  S  e.  ( Base `  if ( R  e.  CRing ,  R ,fld ) ) ) )
74, 63anbi13d 1291 . . . . 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 6120 . . . . . 6  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  ( N maDet  R )  =  ( N maDet  if ( R  e.  CRing ,  R ,fld ) ) )
10 oveq2 6120 . . . . . . . 8  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  ( N matRRep  R )  =  ( N matRRep  if ( R  e.  CRing ,  R ,fld ) ) )
1110oveqd 6129 . . . . . . 7  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  ( M ( N matRRep  R ) S )  =  ( M ( N matRRep  if ( R  e. 
CRing ,  R ,fld )
) S ) )
1211oveqd 6129 . . . . . 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 5718 . . . . 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 5712 . . . . . 6  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  ( .r `  R
)  =  ( .r
`  if ( R  e.  CRing ,  R ,fld )
) )
15 eqidd 2444 . . . . . 6  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  S  =  S )
16 oveq2 6120 . . . . . . 7  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  ( ( N  \  { K } ) maDet  R
)  =  ( ( N  \  { K } ) maDet  if ( R  e.  CRing ,  R ,fld )
) )
17 oveq2 6120 . . . . . . . . 9  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  ( N subMat  R )  =  ( N subMat  if ( R  e.  CRing ,  R ,fld ) ) )
1817fveq1d 5714 . . . . . . . 8  |-  ( R  =  if ( R  e.  CRing ,  R ,fld )  ->  ( ( N subMat  R
) `  M )  =  ( ( N subMat  if ( R  e.  CRing ,  R ,fld ) ) `  M
) )
1918oveqd 6129 . . . . . . 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 5718 . . . . . 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 6133 . . . . 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 2457 . . . 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 320 . . 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 17859 . . . . 5  |-fld  e.  CRing
2524elimel 3873 . . . 4  |-  if ( R  e.  CRing ,  R ,fld )  e.  CRing
2625smadiadetg0 18502 . . 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 3862 . 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 620 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756    \ cdif 3346   ifcif 3812   {csn 3898   ` cfv 5439  (class class class)co 6112   Basecbs 14195   .rcmulr 14260   CRingccrg 16668  ℂfldccnfld 17840   Mat cmat 18302   matRRep cmarrep 18389   subMat csubma 18409   maDet cmdat 18417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-addf 9382  ax-mulf 9383
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-xor 1351  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-ot 3907  df-uni 4113  df-int 4150  df-iun 4194  df-iin 4195  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-of 6341  df-om 6498  df-1st 6598  df-2nd 6599  df-supp 6712  df-tpos 6766  df-recs 6853  df-rdg 6887  df-1o 6941  df-2o 6942  df-oadd 6945  df-er 7122  df-map 7237  df-pm 7238  df-ixp 7285  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-fsupp 7642  df-sup 7712  df-oi 7745  df-card 8130  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-7 10406  df-8 10407  df-9 10408  df-10 10409  df-n0 10601  df-z 10668  df-dec 10777  df-uz 10883  df-rp 11013  df-fz 11459  df-fzo 11570  df-seq 11828  df-exp 11887  df-hash 12125  df-word 12250  df-concat 12252  df-s1 12253  df-substr 12254  df-splice 12255  df-reverse 12256  df-s2 12496  df-struct 14197  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-mulr 14273  df-starv 14274  df-sca 14275  df-vsca 14276  df-ip 14277  df-tset 14278  df-ple 14279  df-ds 14281  df-unif 14282  df-hom 14283  df-cco 14284  df-0g 14401  df-gsum 14402  df-prds 14407  df-pws 14409  df-mre 14545  df-mrc 14546  df-acs 14548  df-mnd 15436  df-mhm 15485  df-submnd 15486  df-grp 15566  df-minusg 15567  df-mulg 15569  df-subg 15699  df-ghm 15766  df-gim 15808  df-cntz 15856  df-oppg 15882  df-symg 15904  df-pmtr 15969  df-psgn 16018  df-cmn 16300  df-abl 16301  df-mgp 16614  df-ur 16626  df-rng 16669  df-cring 16670  df-oppr 16737  df-dvdsr 16755  df-unit 16756  df-invr 16786  df-dvr 16797  df-rnghom 16828  df-drng 16856  df-subrg 16885  df-sra 17275  df-rgmod 17276  df-cnfld 17841  df-zring 17906  df-zrh 17957  df-dsmm 18179  df-frlm 18194  df-mat 18304  df-marrep 18391  df-subma 18410  df-mdet 18418  df-minmar1 18463
This theorem is referenced by:  cramerimplem1  18511
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