MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  1marepvmarrepid Structured version   Unicode version

Theorem 1marepvmarrepid 19054
Description: Replacing the ith row by 0's and the ith component of a (column) vector at the diagonal position for the identity matrix with the ith column replaced by the vector results in the matrix itself. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 27-Feb-2019.)
Hypotheses
Ref Expression
marepvmarrep1.v  |-  V  =  ( ( Base `  R
)  ^m  N )
marepvmarrep1.o  |-  .1.  =  ( 1r `  ( N Mat 
R ) )
marepvmarrep1.x  |-  X  =  ( (  .1.  ( N matRepV  R ) Z ) `
 I )
Assertion
Ref Expression
1marepvmarrepid  |-  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V
) )  ->  (
I ( X ( N matRRep  R ) ( Z `
 I ) ) I )  =  X )

Proof of Theorem 1marepvmarrepid
Dummy variables  i 
j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 marepvmarrep1.x . . . 4  |-  X  =  ( (  .1.  ( N matRepV  R ) Z ) `
 I )
2 eqid 2443 . . . . . 6  |-  ( N Mat 
R )  =  ( N Mat  R )
3 eqid 2443 . . . . . 6  |-  ( Base `  ( N Mat  R ) )  =  ( Base `  ( N Mat  R ) )
4 marepvmarrep1.v . . . . . 6  |-  V  =  ( ( Base `  R
)  ^m  N )
5 marepvmarrep1.o . . . . . 6  |-  .1.  =  ( 1r `  ( N Mat 
R ) )
62, 3, 4, 5ma1repvcl 19049 . . . . 5  |-  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( Z  e.  V  /\  I  e.  N
) )  ->  (
(  .1.  ( N matRepV  R ) Z ) `
 I )  e.  ( Base `  ( N Mat  R ) ) )
76ancom2s 802 . . . 4  |-  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V
) )  ->  (
(  .1.  ( N matRepV  R ) Z ) `
 I )  e.  ( Base `  ( N Mat  R ) ) )
81, 7syl5eqel 2535 . . 3  |-  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V
) )  ->  X  e.  ( Base `  ( N Mat  R ) ) )
9 elmapi 7442 . . . . . . 7  |-  ( Z  e.  ( ( Base `  R )  ^m  N
)  ->  Z : N
--> ( Base `  R
) )
10 ffvelrn 6014 . . . . . . . 8  |-  ( ( Z : N --> ( Base `  R )  /\  I  e.  N )  ->  ( Z `  I )  e.  ( Base `  R
) )
1110ex 434 . . . . . . 7  |-  ( Z : N --> ( Base `  R )  ->  (
I  e.  N  -> 
( Z `  I
)  e.  ( Base `  R ) ) )
129, 11syl 16 . . . . . 6  |-  ( Z  e.  ( ( Base `  R )  ^m  N
)  ->  ( I  e.  N  ->  ( Z `
 I )  e.  ( Base `  R
) ) )
1312, 4eleq2s 2551 . . . . 5  |-  ( Z  e.  V  ->  (
I  e.  N  -> 
( Z `  I
)  e.  ( Base `  R ) ) )
1413impcom 430 . . . 4  |-  ( ( I  e.  N  /\  Z  e.  V )  ->  ( Z `  I
)  e.  ( Base `  R ) )
1514adantl 466 . . 3  |-  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V
) )  ->  ( Z `  I )  e.  ( Base `  R
) )
16 simpl 457 . . . 4  |-  ( ( I  e.  N  /\  Z  e.  V )  ->  I  e.  N )
1716adantl 466 . . 3  |-  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V
) )  ->  I  e.  N )
18 eqid 2443 . . . 4  |-  ( N matRRep  R )  =  ( N matRRep  R )
19 eqid 2443 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
202, 3, 18, 19marrepval 19041 . . 3  |-  ( ( ( X  e.  (
Base `  ( N Mat  R ) )  /\  ( Z `  I )  e.  ( Base `  R
) )  /\  (
I  e.  N  /\  I  e.  N )
)  ->  ( I
( X ( N matRRep  R ) ( Z `
 I ) ) I )  =  ( i  e.  N , 
j  e.  N  |->  if ( i  =  I ,  if ( j  =  I ,  ( Z `  I ) ,  ( 0g `  R ) ) ,  ( i X j ) ) ) )
218, 15, 17, 17, 20syl22anc 1230 . 2  |-  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V
) )  ->  (
I ( X ( N matRRep  R ) ( Z `
 I ) ) I )  =  ( i  e.  N , 
j  e.  N  |->  if ( i  =  I ,  if ( j  =  I ,  ( Z `  I ) ,  ( 0g `  R ) ) ,  ( i X j ) ) ) )
22 iftrue 3932 . . . . . 6  |-  ( i  =  I  ->  if ( i  =  I ,  if ( j  =  I ,  ( Z `  I ) ,  ( 0g `  R ) ) ,  ( i X j ) )  =  if ( j  =  I ,  ( Z `  I ) ,  ( 0g `  R ) ) )
2322adantr 465 . . . . 5  |-  ( ( i  =  I  /\  ( ( ( R  e.  Ring  /\  N  e. 
Fin )  /\  (
I  e.  N  /\  Z  e.  V )
)  /\  i  e.  N  /\  j  e.  N
) )  ->  if ( i  =  I ,  if ( j  =  I ,  ( Z `  I ) ,  ( 0g `  R ) ) ,  ( i X j ) )  =  if ( j  =  I ,  ( Z `  I ) ,  ( 0g `  R ) ) )
24 iftrue 3932 . . . . . . . 8  |-  ( j  =  I  ->  if ( j  =  I ,  ( Z `  I ) ,  ( 0g `  R ) )  =  ( Z `
 I ) )
2524adantr 465 . . . . . . 7  |-  ( ( j  =  I  /\  ( i  =  I  /\  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  (
I  e.  N  /\  Z  e.  V )
)  /\  i  e.  N  /\  j  e.  N
) ) )  ->  if ( j  =  I ,  ( Z `  I ) ,  ( 0g `  R ) )  =  ( Z `
 I ) )
26 iftrue 3932 . . . . . . . 8  |-  ( j  =  I  ->  if ( j  =  I ,  ( Z `  i ) ,  ( i  .1.  j ) )  =  ( Z `
 i ) )
27 fveq2 5856 . . . . . . . . 9  |-  ( i  =  I  ->  ( Z `  i )  =  ( Z `  I ) )
2827adantr 465 . . . . . . . 8  |-  ( ( i  =  I  /\  ( ( ( R  e.  Ring  /\  N  e. 
Fin )  /\  (
I  e.  N  /\  Z  e.  V )
)  /\  i  e.  N  /\  j  e.  N
) )  ->  ( Z `  i )  =  ( Z `  I ) )
2926, 28sylan9eq 2504 . . . . . . 7  |-  ( ( j  =  I  /\  ( i  =  I  /\  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  (
I  e.  N  /\  Z  e.  V )
)  /\  i  e.  N  /\  j  e.  N
) ) )  ->  if ( j  =  I ,  ( Z `  i ) ,  ( i  .1.  j ) )  =  ( Z `
 I ) )
3025, 29eqtr4d 2487 . . . . . 6  |-  ( ( j  =  I  /\  ( i  =  I  /\  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  (
I  e.  N  /\  Z  e.  V )
)  /\  i  e.  N  /\  j  e.  N
) ) )  ->  if ( j  =  I ,  ( Z `  I ) ,  ( 0g `  R ) )  =  if ( j  =  I ,  ( Z `  i
) ,  ( i  .1.  j ) ) )
31 eqid 2443 . . . . . . . . . . 11  |-  ( 1r
`  R )  =  ( 1r `  R
)
32 simpr 461 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  N  e.  Fin )  ->  N  e.  Fin )
3332adantr 465 . . . . . . . . . . . 12  |-  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V
) )  ->  N  e.  Fin )
34333ad2ant1 1018 . . . . . . . . . . 11  |-  ( ( ( ( R  e. 
Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V ) )  /\  i  e.  N  /\  j  e.  N )  ->  N  e.  Fin )
35 simpl 457 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  N  e.  Fin )  ->  R  e.  Ring )
3635adantr 465 . . . . . . . . . . . 12  |-  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V
) )  ->  R  e.  Ring )
37363ad2ant1 1018 . . . . . . . . . . 11  |-  ( ( ( ( R  e. 
Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V ) )  /\  i  e.  N  /\  j  e.  N )  ->  R  e.  Ring )
38 simp2 998 . . . . . . . . . . 11  |-  ( ( ( ( R  e. 
Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V ) )  /\  i  e.  N  /\  j  e.  N )  ->  i  e.  N )
39 simp3 999 . . . . . . . . . . 11  |-  ( ( ( ( R  e. 
Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V ) )  /\  i  e.  N  /\  j  e.  N )  ->  j  e.  N )
402, 31, 19, 34, 37, 38, 39, 5mat1ov 18927 . . . . . . . . . 10  |-  ( ( ( ( R  e. 
Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V ) )  /\  i  e.  N  /\  j  e.  N )  ->  ( i  .1.  j
)  =  if ( i  =  j ,  ( 1r `  R
) ,  ( 0g
`  R ) ) )
4140adantl 466 . . . . . . . . 9  |-  ( ( i  =  I  /\  ( ( ( R  e.  Ring  /\  N  e. 
Fin )  /\  (
I  e.  N  /\  Z  e.  V )
)  /\  i  e.  N  /\  j  e.  N
) )  ->  (
i  .1.  j )  =  if ( i  =  j ,  ( 1r `  R ) ,  ( 0g `  R ) ) )
4241adantl 466 . . . . . . . 8  |-  ( ( -.  j  =  I  /\  ( i  =  I  /\  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V
) )  /\  i  e.  N  /\  j  e.  N ) ) )  ->  ( i  .1.  j )  =  if ( i  =  j ,  ( 1r `  R ) ,  ( 0g `  R ) ) )
43 eqtr2 2470 . . . . . . . . . . . . . 14  |-  ( ( i  =  I  /\  i  =  j )  ->  I  =  j )
4443eqcomd 2451 . . . . . . . . . . . . 13  |-  ( ( i  =  I  /\  i  =  j )  ->  j  =  I )
4544ex 434 . . . . . . . . . . . 12  |-  ( i  =  I  ->  (
i  =  j  -> 
j  =  I ) )
4645con3d 133 . . . . . . . . . . 11  |-  ( i  =  I  ->  ( -.  j  =  I  ->  -.  i  =  j ) )
4746adantr 465 . . . . . . . . . 10  |-  ( ( i  =  I  /\  ( ( ( R  e.  Ring  /\  N  e. 
Fin )  /\  (
I  e.  N  /\  Z  e.  V )
)  /\  i  e.  N  /\  j  e.  N
) )  ->  ( -.  j  =  I  ->  -.  i  =  j ) )
4847impcom 430 . . . . . . . . 9  |-  ( ( -.  j  =  I  /\  ( i  =  I  /\  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V
) )  /\  i  e.  N  /\  j  e.  N ) ) )  ->  -.  i  =  j )
49 iffalse 3935 . . . . . . . . 9  |-  ( -.  i  =  j  ->  if ( i  =  j ,  ( 1r `  R ) ,  ( 0g `  R ) )  =  ( 0g
`  R ) )
5048, 49syl 16 . . . . . . . 8  |-  ( ( -.  j  =  I  /\  ( i  =  I  /\  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V
) )  /\  i  e.  N  /\  j  e.  N ) ) )  ->  if ( i  =  j ,  ( 1r `  R ) ,  ( 0g `  R ) )  =  ( 0g `  R
) )
5142, 50eqtrd 2484 . . . . . . 7  |-  ( ( -.  j  =  I  /\  ( i  =  I  /\  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V
) )  /\  i  e.  N  /\  j  e.  N ) ) )  ->  ( i  .1.  j )  =  ( 0g `  R ) )
52 iffalse 3935 . . . . . . . 8  |-  ( -.  j  =  I  ->  if ( j  =  I ,  ( Z `  i ) ,  ( i  .1.  j ) )  =  ( i  .1.  j ) )
5352adantr 465 . . . . . . 7  |-  ( ( -.  j  =  I  /\  ( i  =  I  /\  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V
) )  /\  i  e.  N  /\  j  e.  N ) ) )  ->  if ( j  =  I ,  ( Z `  i ) ,  ( i  .1.  j ) )  =  ( i  .1.  j
) )
54 iffalse 3935 . . . . . . . 8  |-  ( -.  j  =  I  ->  if ( j  =  I ,  ( Z `  I ) ,  ( 0g `  R ) )  =  ( 0g
`  R ) )
5554adantr 465 . . . . . . 7  |-  ( ( -.  j  =  I  /\  ( i  =  I  /\  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V
) )  /\  i  e.  N  /\  j  e.  N ) ) )  ->  if ( j  =  I ,  ( Z `  I ) ,  ( 0g `  R ) )  =  ( 0g `  R
) )
5651, 53, 553eqtr4rd 2495 . . . . . 6  |-  ( ( -.  j  =  I  /\  ( i  =  I  /\  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V
) )  /\  i  e.  N  /\  j  e.  N ) ) )  ->  if ( j  =  I ,  ( Z `  I ) ,  ( 0g `  R ) )  =  if ( j  =  I ,  ( Z `
 i ) ,  ( i  .1.  j
) ) )
5730, 56pm2.61ian 790 . . . . 5  |-  ( ( i  =  I  /\  ( ( ( R  e.  Ring  /\  N  e. 
Fin )  /\  (
I  e.  N  /\  Z  e.  V )
)  /\  i  e.  N  /\  j  e.  N
) )  ->  if ( j  =  I ,  ( Z `  I ) ,  ( 0g `  R ) )  =  if ( j  =  I ,  ( Z `  i
) ,  ( i  .1.  j ) ) )
5823, 57eqtrd 2484 . . . 4  |-  ( ( i  =  I  /\  ( ( ( R  e.  Ring  /\  N  e. 
Fin )  /\  (
I  e.  N  /\  Z  e.  V )
)  /\  i  e.  N  /\  j  e.  N
) )  ->  if ( i  =  I ,  if ( j  =  I ,  ( Z `  I ) ,  ( 0g `  R ) ) ,  ( i X j ) )  =  if ( j  =  I ,  ( Z `  i ) ,  ( i  .1.  j ) ) )
59 iffalse 3935 . . . . . 6  |-  ( -.  i  =  I  ->  if ( i  =  I ,  if ( j  =  I ,  ( Z `  I ) ,  ( 0g `  R ) ) ,  ( i X j ) )  =  ( i X j ) )
6059adantr 465 . . . . 5  |-  ( ( -.  i  =  I  /\  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  (
I  e.  N  /\  Z  e.  V )
)  /\  i  e.  N  /\  j  e.  N
) )  ->  if ( i  =  I ,  if ( j  =  I ,  ( Z `  I ) ,  ( 0g `  R ) ) ,  ( i X j ) )  =  ( i X j ) )
612, 3, 5mat1bas 18928 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  N  e.  Fin )  ->  .1.  e.  ( Base `  ( N Mat  R ) ) )
6261adantr 465 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V
) )  ->  .1.  e.  ( Base `  ( N Mat  R ) ) )
63 simpr 461 . . . . . . . . . . 11  |-  ( ( I  e.  N  /\  Z  e.  V )  ->  Z  e.  V )
6463adantl 466 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V
) )  ->  Z  e.  V )
6562, 64, 173jca 1177 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V
) )  ->  (  .1.  e.  ( Base `  ( N Mat  R ) )  /\  Z  e.  V  /\  I  e.  N )
)
66653ad2ant1 1018 . . . . . . . 8  |-  ( ( ( ( R  e. 
Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V ) )  /\  i  e.  N  /\  j  e.  N )  ->  (  .1.  e.  (
Base `  ( N Mat  R ) )  /\  Z  e.  V  /\  I  e.  N ) )
67 3simpc 996 . . . . . . . 8  |-  ( ( ( ( R  e. 
Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V ) )  /\  i  e.  N  /\  j  e.  N )  ->  ( i  e.  N  /\  j  e.  N
) )
6837, 66, 673jca 1177 . . . . . . 7  |-  ( ( ( ( R  e. 
Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V ) )  /\  i  e.  N  /\  j  e.  N )  ->  ( R  e.  Ring  /\  (  .1.  e.  (
Base `  ( N Mat  R ) )  /\  Z  e.  V  /\  I  e.  N )  /\  (
i  e.  N  /\  j  e.  N )
) )
6968adantl 466 . . . . . 6  |-  ( ( -.  i  =  I  /\  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  (
I  e.  N  /\  Z  e.  V )
)  /\  i  e.  N  /\  j  e.  N
) )  ->  ( R  e.  Ring  /\  (  .1.  e.  ( Base `  ( N Mat  R ) )  /\  Z  e.  V  /\  I  e.  N )  /\  ( i  e.  N  /\  j  e.  N
) ) )
702, 3, 4, 5, 19, 1ma1repveval 19050 . . . . . 6  |-  ( ( R  e.  Ring  /\  (  .1.  e.  ( Base `  ( N Mat  R ) )  /\  Z  e.  V  /\  I  e.  N )  /\  ( i  e.  N  /\  j  e.  N
) )  ->  (
i X j )  =  if ( j  =  I ,  ( Z `  i ) ,  if ( j  =  i ,  ( 1r `  R ) ,  ( 0g `  R ) ) ) )
7169, 70syl 16 . . . . 5  |-  ( ( -.  i  =  I  /\  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  (
I  e.  N  /\  Z  e.  V )
)  /\  i  e.  N  /\  j  e.  N
) )  ->  (
i X j )  =  if ( j  =  I ,  ( Z `  i ) ,  if ( j  =  i ,  ( 1r `  R ) ,  ( 0g `  R ) ) ) )
7234ad2antlr 726 . . . . . . . 8  |-  ( ( ( -.  i  =  I  /\  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V
) )  /\  i  e.  N  /\  j  e.  N ) )  /\  -.  j  =  I
)  ->  N  e.  Fin )
7337ad2antlr 726 . . . . . . . 8  |-  ( ( ( -.  i  =  I  /\  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V
) )  /\  i  e.  N  /\  j  e.  N ) )  /\  -.  j  =  I
)  ->  R  e.  Ring )
7438ad2antlr 726 . . . . . . . 8  |-  ( ( ( -.  i  =  I  /\  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V
) )  /\  i  e.  N  /\  j  e.  N ) )  /\  -.  j  =  I
)  ->  i  e.  N )
7539ad2antlr 726 . . . . . . . 8  |-  ( ( ( -.  i  =  I  /\  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V
) )  /\  i  e.  N  /\  j  e.  N ) )  /\  -.  j  =  I
)  ->  j  e.  N )
762, 31, 19, 72, 73, 74, 75, 5mat1ov 18927 . . . . . . 7  |-  ( ( ( -.  i  =  I  /\  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V
) )  /\  i  e.  N  /\  j  e.  N ) )  /\  -.  j  =  I
)  ->  ( i  .1.  j )  =  if ( i  =  j ,  ( 1r `  R ) ,  ( 0g `  R ) ) )
77 equcom 1780 . . . . . . . . 9  |-  ( i  =  j  <->  j  =  i )
7877a1i 11 . . . . . . . 8  |-  ( ( ( -.  i  =  I  /\  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V
) )  /\  i  e.  N  /\  j  e.  N ) )  /\  -.  j  =  I
)  ->  ( i  =  j  <->  j  =  i ) )
7978ifbid 3948 . . . . . . 7  |-  ( ( ( -.  i  =  I  /\  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V
) )  /\  i  e.  N  /\  j  e.  N ) )  /\  -.  j  =  I
)  ->  if (
i  =  j ,  ( 1r `  R
) ,  ( 0g
`  R ) )  =  if ( j  =  i ,  ( 1r `  R ) ,  ( 0g `  R ) ) )
8076, 79eqtr2d 2485 . . . . . 6  |-  ( ( ( -.  i  =  I  /\  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V
) )  /\  i  e.  N  /\  j  e.  N ) )  /\  -.  j  =  I
)  ->  if (
j  =  i ,  ( 1r `  R
) ,  ( 0g
`  R ) )  =  ( i  .1.  j ) )
8180ifeq2da 3957 . . . . 5  |-  ( ( -.  i  =  I  /\  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  (
I  e.  N  /\  Z  e.  V )
)  /\  i  e.  N  /\  j  e.  N
) )  ->  if ( j  =  I ,  ( Z `  i ) ,  if ( j  =  i ,  ( 1r `  R ) ,  ( 0g `  R ) ) )  =  if ( j  =  I ,  ( Z `  i ) ,  ( i  .1.  j ) ) )
8260, 71, 813eqtrd 2488 . . . 4  |-  ( ( -.  i  =  I  /\  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  (
I  e.  N  /\  Z  e.  V )
)  /\  i  e.  N  /\  j  e.  N
) )  ->  if ( i  =  I ,  if ( j  =  I ,  ( Z `  I ) ,  ( 0g `  R ) ) ,  ( i X j ) )  =  if ( j  =  I ,  ( Z `  i ) ,  ( i  .1.  j ) ) )
8358, 82pm2.61ian 790 . . 3  |-  ( ( ( ( R  e. 
Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V ) )  /\  i  e.  N  /\  j  e.  N )  ->  if ( i  =  I ,  if ( j  =  I ,  ( Z `  I
) ,  ( 0g
`  R ) ) ,  ( i X j ) )  =  if ( j  =  I ,  ( Z `
 i ) ,  ( i  .1.  j
) ) )
8483mpt2eq3dva 6346 . 2  |-  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V
) )  ->  (
i  e.  N , 
j  e.  N  |->  if ( i  =  I ,  if ( j  =  I ,  ( Z `  I ) ,  ( 0g `  R ) ) ,  ( i X j ) ) )  =  ( i  e.  N ,  j  e.  N  |->  if ( j  =  I ,  ( Z `
 i ) ,  ( i  .1.  j
) ) ) )
85 eqid 2443 . . . . 5  |-  ( N matRepV  R )  =  ( N matRepV  R )
862, 3, 85, 4marepvval 19046 . . . 4  |-  ( (  .1.  e.  ( Base `  ( N Mat  R ) )  /\  Z  e.  V  /\  I  e.  N )  ->  (
(  .1.  ( N matRepV  R ) Z ) `
 I )  =  ( i  e.  N ,  j  e.  N  |->  if ( j  =  I ,  ( Z `
 i ) ,  ( i  .1.  j
) ) ) )
8765, 86syl 16 . . 3  |-  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V
) )  ->  (
(  .1.  ( N matRepV  R ) Z ) `
 I )  =  ( i  e.  N ,  j  e.  N  |->  if ( j  =  I ,  ( Z `
 i ) ,  ( i  .1.  j
) ) ) )
881, 87syl5req 2497 . 2  |-  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V
) )  ->  (
i  e.  N , 
j  e.  N  |->  if ( j  =  I ,  ( Z `  i ) ,  ( i  .1.  j ) ) )  =  X )
8921, 84, 883eqtrd 2488 1  |-  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V
) )  ->  (
I ( X ( N matRRep  R ) ( Z `
 I ) ) I )  =  X )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   ifcif 3926   -->wf 5574   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283    ^m cmap 7422   Fincfn 7518   Basecbs 14613   0gc0g 14818   1rcur 17131   Ringcrg 17176   Mat cmat 18886   matRRep cmarrep 19035   matRepV cmatrepV 19036
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-inf2 8061  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-ot 4023  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  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-se 4829  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-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-sup 7903  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10986  df-uz 11092  df-fz 11683  df-fzo 11806  df-seq 12089  df-hash 12387  df-struct 14615  df-ndx 14616  df-slot 14617  df-base 14618  df-sets 14619  df-ress 14620  df-plusg 14691  df-mulr 14692  df-sca 14694  df-vsca 14695  df-ip 14696  df-tset 14697  df-ple 14698  df-ds 14700  df-hom 14702  df-cco 14703  df-0g 14820  df-gsum 14821  df-prds 14826  df-pws 14828  df-mre 14964  df-mrc 14965  df-acs 14967  df-mgm 15850  df-sgrp 15889  df-mnd 15899  df-mhm 15944  df-submnd 15945  df-grp 16035  df-minusg 16036  df-sbg 16037  df-mulg 16038  df-subg 16176  df-ghm 16243  df-cntz 16333  df-cmn 16778  df-abl 16779  df-mgp 17120  df-ur 17132  df-ring 17178  df-subrg 17405  df-lmod 17492  df-lss 17557  df-sra 17796  df-rgmod 17797  df-dsmm 18740  df-frlm 18755  df-mamu 18863  df-mat 18887  df-marrep 19037  df-marepv 19038
This theorem is referenced by:  cramerimplem1  19162
  Copyright terms: Public domain W3C validator