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Theorem smadiadetglem1 18980
Description: Lemma 1 for smadiadetg 18982. (Contributed by AV, 13-Feb-2019.)
Hypotheses
Ref Expression
smadiadet.a  |-  A  =  ( N Mat  R )
smadiadet.b  |-  B  =  ( Base `  A
)
smadiadet.r  |-  R  e. 
CRing
smadiadet.d  |-  D  =  ( N maDet  R )
smadiadet.h  |-  E  =  ( ( N  \  { K } ) maDet  R
)
Assertion
Ref Expression
smadiadetglem1  |-  ( ( M  e.  B  /\  K  e.  N  /\  S  e.  ( Base `  R ) )  -> 
( ( K ( M ( N matRRep  R
) S ) K )  |`  ( ( N  \  { K }
)  X.  N ) )  =  ( ( K ( ( N minMatR1  R ) `  M
) K )  |`  ( ( N  \  { K } )  X.  N ) ) )

Proof of Theorem smadiadetglem1
Dummy variables  i 
j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mpt2difsnif 6380 . . . . 5  |-  ( i  e.  ( N  \  { K } ) ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  K ,  S ,  ( 0g `  R ) ) ,  ( i M j ) ) )  =  ( i  e.  ( N  \  { K } ) ,  j  e.  N  |->  ( i M j ) )
2 mpt2difsnif 6380 . . . . 5  |-  ( i  e.  ( N  \  { K } ) ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  K ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( i M j ) ) )  =  ( i  e.  ( N  \  { K } ) ,  j  e.  N  |->  ( i M j ) )
31, 2eqtr4i 2499 . . . 4  |-  ( i  e.  ( N  \  { K } ) ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  K ,  S ,  ( 0g `  R ) ) ,  ( i M j ) ) )  =  ( i  e.  ( N  \  { K } ) ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  K ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( i M j ) ) )
4 difss 3631 . . . . . 6  |-  ( N 
\  { K }
)  C_  N
5 ssid 3523 . . . . . 6  |-  N  C_  N
64, 5pm3.2i 455 . . . . 5  |-  ( ( N  \  { K } )  C_  N  /\  N  C_  N )
7 resmpt2 6385 . . . . 5  |-  ( ( ( N  \  { K } )  C_  N  /\  N  C_  N )  ->  ( ( i  e.  N ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  K ,  S ,  ( 0g `  R ) ) ,  ( i M j ) ) )  |`  ( ( N  \  { K }
)  X.  N ) )  =  ( i  e.  ( N  \  { K } ) ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  K ,  S ,  ( 0g `  R ) ) ,  ( i M j ) ) ) )
86, 7mp1i 12 . . . 4  |-  ( ( M  e.  B  /\  K  e.  N  /\  S  e.  ( Base `  R ) )  -> 
( ( i  e.  N ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  K ,  S ,  ( 0g `  R ) ) ,  ( i M j ) ) )  |`  ( ( N  \  { K }
)  X.  N ) )  =  ( i  e.  ( N  \  { K } ) ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  K ,  S ,  ( 0g `  R ) ) ,  ( i M j ) ) ) )
9 resmpt2 6385 . . . . 5  |-  ( ( ( N  \  { K } )  C_  N  /\  N  C_  N )  ->  ( ( i  e.  N ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  K ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( i M j ) ) )  |`  ( ( N  \  { K }
)  X.  N ) )  =  ( i  e.  ( N  \  { K } ) ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  K ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( i M j ) ) ) )
106, 9mp1i 12 . . . 4  |-  ( ( M  e.  B  /\  K  e.  N  /\  S  e.  ( Base `  R ) )  -> 
( ( i  e.  N ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  K ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( i M j ) ) )  |`  ( ( N  \  { K }
)  X.  N ) )  =  ( i  e.  ( N  \  { K } ) ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  K ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( i M j ) ) ) )
113, 8, 103eqtr4a 2534 . . 3  |-  ( ( M  e.  B  /\  K  e.  N  /\  S  e.  ( Base `  R ) )  -> 
( ( i  e.  N ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  K ,  S ,  ( 0g `  R ) ) ,  ( i M j ) ) )  |`  ( ( N  \  { K }
)  X.  N ) )  =  ( ( i  e.  N , 
j  e.  N  |->  if ( i  =  K ,  if ( j  =  K ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( i M j ) ) )  |`  ( ( N  \  { K } )  X.  N ) ) )
12 simp1 996 . . . . 5  |-  ( ( M  e.  B  /\  K  e.  N  /\  S  e.  ( Base `  R ) )  ->  M  e.  B )
13 simp3 998 . . . . 5  |-  ( ( M  e.  B  /\  K  e.  N  /\  S  e.  ( Base `  R ) )  ->  S  e.  ( Base `  R ) )
14 simp2 997 . . . . 5  |-  ( ( M  e.  B  /\  K  e.  N  /\  S  e.  ( Base `  R ) )  ->  K  e.  N )
15 smadiadet.a . . . . . 6  |-  A  =  ( N Mat  R )
16 smadiadet.b . . . . . 6  |-  B  =  ( Base `  A
)
17 eqid 2467 . . . . . 6  |-  ( N matRRep  R )  =  ( N matRRep  R )
18 eqid 2467 . . . . . 6  |-  ( 0g
`  R )  =  ( 0g `  R
)
1915, 16, 17, 18marrepval 18871 . . . . 5  |-  ( ( ( M  e.  B  /\  S  e.  ( Base `  R ) )  /\  ( K  e.  N  /\  K  e.  N ) )  -> 
( K ( M ( N matRRep  R ) S ) K )  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  K ,  S ,  ( 0g `  R ) ) ,  ( i M j ) ) ) )
2012, 13, 14, 14, 19syl22anc 1229 . . . 4  |-  ( ( M  e.  B  /\  K  e.  N  /\  S  e.  ( Base `  R ) )  -> 
( K ( M ( N matRRep  R ) S ) K )  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  K ,  S ,  ( 0g `  R ) ) ,  ( i M j ) ) ) )
2120reseq1d 5272 . . 3  |-  ( ( M  e.  B  /\  K  e.  N  /\  S  e.  ( Base `  R ) )  -> 
( ( K ( M ( N matRRep  R
) S ) K )  |`  ( ( N  \  { K }
)  X.  N ) )  =  ( ( i  e.  N , 
j  e.  N  |->  if ( i  =  K ,  if ( j  =  K ,  S ,  ( 0g `  R ) ) ,  ( i M j ) ) )  |`  ( ( N  \  { K } )  X.  N ) ) )
22 smadiadet.r . . . . . 6  |-  R  e. 
CRing
23 crngrng 17022 . . . . . . 7  |-  ( R  e.  CRing  ->  R  e.  Ring )
24 eqid 2467 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
25 eqid 2467 . . . . . . . 8  |-  ( 1r
`  R )  =  ( 1r `  R
)
2624, 25rngidcl 17032 . . . . . . 7  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  ( Base `  R
) )
2723, 26syl 16 . . . . . 6  |-  ( R  e.  CRing  ->  ( 1r `  R )  e.  (
Base `  R )
)
2822, 27mp1i 12 . . . . 5  |-  ( ( M  e.  B  /\  K  e.  N  /\  S  e.  ( Base `  R ) )  -> 
( 1r `  R
)  e.  ( Base `  R ) )
2915, 16, 17, 18marrepval 18871 . . . . 5  |-  ( ( ( M  e.  B  /\  ( 1r `  R
)  e.  ( Base `  R ) )  /\  ( K  e.  N  /\  K  e.  N
) )  ->  ( K ( M ( N matRRep  R ) ( 1r
`  R ) ) K )  =  ( i  e.  N , 
j  e.  N  |->  if ( i  =  K ,  if ( j  =  K ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( i M j ) ) ) )
3012, 28, 14, 14, 29syl22anc 1229 . . . 4  |-  ( ( M  e.  B  /\  K  e.  N  /\  S  e.  ( Base `  R ) )  -> 
( K ( M ( N matRRep  R )
( 1r `  R
) ) K )  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  K ,  if ( j  =  K ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( i M j ) ) ) )
3130reseq1d 5272 . . 3  |-  ( ( M  e.  B  /\  K  e.  N  /\  S  e.  ( Base `  R ) )  -> 
( ( K ( M ( N matRRep  R
) ( 1r `  R ) ) K )  |`  ( ( N  \  { K }
)  X.  N ) )  =  ( ( i  e.  N , 
j  e.  N  |->  if ( i  =  K ,  if ( j  =  K ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( i M j ) ) )  |`  ( ( N  \  { K } )  X.  N ) ) )
3211, 21, 313eqtr4d 2518 . 2  |-  ( ( M  e.  B  /\  K  e.  N  /\  S  e.  ( Base `  R ) )  -> 
( ( K ( M ( N matRRep  R
) S ) K )  |`  ( ( N  \  { K }
)  X.  N ) )  =  ( ( K ( M ( N matRRep  R ) ( 1r
`  R ) ) K )  |`  (
( N  \  { K } )  X.  N
) ) )
3322, 23ax-mp 5 . . . . . 6  |-  R  e. 
Ring
3415, 16, 17, 25minmar1marrep 18959 . . . . . 6  |-  ( ( R  e.  Ring  /\  M  e.  B )  ->  (
( N minMatR1  R ) `  M )  =  ( M ( N matRRep  R
) ( 1r `  R ) ) )
3533, 12, 34sylancr 663 . . . . 5  |-  ( ( M  e.  B  /\  K  e.  N  /\  S  e.  ( Base `  R ) )  -> 
( ( N minMatR1  R ) `
 M )  =  ( M ( N matRRep  R ) ( 1r
`  R ) ) )
3635eqcomd 2475 . . . 4  |-  ( ( M  e.  B  /\  K  e.  N  /\  S  e.  ( Base `  R ) )  -> 
( M ( N matRRep  R ) ( 1r
`  R ) )  =  ( ( N minMatR1  R ) `  M
) )
3736oveqd 6302 . . 3  |-  ( ( M  e.  B  /\  K  e.  N  /\  S  e.  ( Base `  R ) )  -> 
( K ( M ( N matRRep  R )
( 1r `  R
) ) K )  =  ( K ( ( N minMatR1  R ) `  M ) K ) )
3837reseq1d 5272 . 2  |-  ( ( M  e.  B  /\  K  e.  N  /\  S  e.  ( Base `  R ) )  -> 
( ( K ( M ( N matRRep  R
) ( 1r `  R ) ) K )  |`  ( ( N  \  { K }
)  X.  N ) )  =  ( ( K ( ( N minMatR1  R ) `  M
) K )  |`  ( ( N  \  { K } )  X.  N ) ) )
3932, 38eqtrd 2508 1  |-  ( ( M  e.  B  /\  K  e.  N  /\  S  e.  ( Base `  R ) )  -> 
( ( K ( M ( N matRRep  R
) S ) K )  |`  ( ( N  \  { K }
)  X.  N ) )  =  ( ( K ( ( N minMatR1  R ) `  M
) K )  |`  ( ( N  \  { K } )  X.  N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    \ cdif 3473    C_ wss 3476   ifcif 3939   {csn 4027    X. cxp 4997    |` cres 5001   ` cfv 5588  (class class class)co 6285    |-> cmpt2 6287   Basecbs 14493   0gc0g 14698   1rcur 16967   Ringcrg 17012   CRingccrg 17013   Mat cmat 18716   matRRep cmarrep 18865   maDet cmdat 18893   minMatR1 cminmar1 18942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-plusg 14571  df-0g 14700  df-mnd 15735  df-mgp 16956  df-ur 16968  df-rng 17014  df-cring 17015  df-mat 18717  df-marrep 18867  df-minmar1 18944
This theorem is referenced by:  smadiadetg  18982
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