MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  smadiadetglem1 Structured version   Unicode version

Theorem smadiadetglem1 19046
Description: Lemma 1 for smadiadetg 19048. (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 2475 . . . 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 3616 . . . . . 6  |-  ( N 
\  { K }
)  C_  N
5 ssid 3508 . . . . . 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 2510 . . 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 997 . . . . 5  |-  ( ( M  e.  B  /\  K  e.  N  /\  S  e.  ( Base `  R ) )  ->  M  e.  B )
13 simp3 999 . . . . 5  |-  ( ( M  e.  B  /\  K  e.  N  /\  S  e.  ( Base `  R ) )  ->  S  e.  ( Base `  R ) )
14 simp2 998 . . . . 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 18937 . . . . 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 1230 . . . 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 5262 . . 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 crngring 17083 . . . . . . 7  |-  ( R  e.  CRing  ->  R  e.  Ring )
24 eqid 2443 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
25 eqid 2443 . . . . . . . 8  |-  ( 1r
`  R )  =  ( 1r `  R
)
2624, 25ringidcl 17093 . . . . . . 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 18937 . . . . 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 1230 . . . 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 5262 . . 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 2494 . 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 19025 . . . . . 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 2451 . . . 4  |-  ( ( M  e.  B  /\  K  e.  N  /\  S  e.  ( Base `  R ) )  -> 
( M ( N matRRep  R ) ( 1r
`  R ) )  =  ( ( N minMatR1  R ) `  M
) )
3736oveqd 6298 . . 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 5262 . 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 2484 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 974    = wceq 1383    e. wcel 1804    \ cdif 3458    C_ wss 3461   ifcif 3926   {csn 4014    X. cxp 4987    |` cres 4991   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283   Basecbs 14509   0gc0g 14714   1rcur 17027   Ringcrg 17072   CRingccrg 17073   Mat cmat 18782   matRRep cmarrep 18931   maDet cmdat 18959   minMatR1 cminmar1 19008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-plusg 14587  df-0g 14716  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mgp 17016  df-ur 17028  df-ring 17074  df-cring 17075  df-mat 18783  df-marrep 18933  df-minmar1 19010
This theorem is referenced by:  smadiadetg  19048
  Copyright terms: Public domain W3C validator