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Theorem smadiadetglem1 18489
Description: Lemma 1 for smadiadetg 18491. (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 6195 . . . . 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 6195 . . . . 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 2466 . . . 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 3495 . . . . . 6  |-  ( N 
\  { K }
)  C_  N
5 ssid 3387 . . . . . 6  |-  N  C_  N
64, 5pm3.2i 455 . . . . 5  |-  ( ( N  \  { K } )  C_  N  /\  N  C_  N )
7 resmpt2 6200 . . . . 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 6200 . . . . 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 2501 . . 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 988 . . . . 5  |-  ( ( M  e.  B  /\  K  e.  N  /\  S  e.  ( Base `  R ) )  ->  M  e.  B )
13 simp3 990 . . . . 5  |-  ( ( M  e.  B  /\  K  e.  N  /\  S  e.  ( Base `  R ) )  ->  S  e.  ( Base `  R ) )
14 simp2 989 . . . . 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 2443 . . . . . 6  |-  ( N matRRep  R )  =  ( N matRRep  R )
18 eqid 2443 . . . . . 6  |-  ( 0g
`  R )  =  ( 0g `  R
)
1915, 16, 17, 18marrepval 18385 . . . . 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 1219 . . . 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 5121 . . 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 16667 . . . . . . 7  |-  ( R  e.  CRing  ->  R  e.  Ring )
24 eqid 2443 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
25 eqid 2443 . . . . . . . 8  |-  ( 1r
`  R )  =  ( 1r `  R
)
2624, 25rngidcl 16677 . . . . . . 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 18385 . . . . 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 1219 . . . 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 5121 . . 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 2485 . 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 18468 . . . . . 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 2448 . . . 4  |-  ( ( M  e.  B  /\  K  e.  N  /\  S  e.  ( Base `  R ) )  -> 
( M ( N matRRep  R ) ( 1r
`  R ) )  =  ( ( N minMatR1  R ) `  M
) )
3736oveqd 6120 . . 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 5121 . 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 2475 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 965    = wceq 1369    e. wcel 1756    \ cdif 3337    C_ wss 3340   ifcif 3803   {csn 3889    X. cxp 4850    |` cres 4854   ` cfv 5430  (class class class)co 6103    e. cmpt2 6105   Basecbs 14186   0gc0g 14390   1rcur 16615   Ringcrg 16657   CRingccrg 16658   Mat cmat 18292   matRRep cmarrep 18379   maDet cmdat 18407   minMatR1 cminmar1 18451
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 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  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 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-2 10392  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-plusg 14263  df-0g 14392  df-mnd 15427  df-mgp 16604  df-ur 16616  df-rng 16659  df-cring 16660  df-mat 18294  df-marrep 18381  df-minmar1 18453
This theorem is referenced by:  smadiadetg  18491
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