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

Theorem 1marepvmarrepid 18361
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 2438 . . . . . 6  |-  ( N Mat 
R )  =  ( N Mat  R )
3 eqid 2438 . . . . . 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 18356 . . . . 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 800 . . . 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 2522 . . 3  |-  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V
) )  ->  X  e.  ( Base `  ( N Mat  R ) ) )
9 elmapi 7226 . . . . . . 7  |-  ( Z  e.  ( ( Base `  R )  ^m  N
)  ->  Z : N
--> ( Base `  R
) )
10 ffvelrn 5836 . . . . . . . 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 2530 . . . . 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 2438 . . . 4  |-  ( N matRRep  R )  =  ( N matRRep  R )
19 eqid 2438 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
202, 3, 18, 19marrepval 18348 . . 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 1219 . 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 3792 . . . . . 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 3792 . . . . . . . 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 3792 . . . . . . . 8  |-  ( j  =  I  ->  if ( j  =  I ,  ( Z `  i ) ,  ( i  .1.  j ) )  =  ( Z `
 i ) )
27 fveq2 5686 . . . . . . . . 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 2490 . . . . . . 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 2473 . . . . . 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 2438 . . . . . . . . . . 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 1009 . . . . . . . . . . 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 1009 . . . . . . . . . . 11  |-  ( ( ( ( R  e. 
Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V ) )  /\  i  e.  N  /\  j  e.  N )  ->  R  e.  Ring )
38 simp2 989 . . . . . . . . . . 11  |-  ( ( ( ( R  e. 
Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V ) )  /\  i  e.  N  /\  j  e.  N )  ->  i  e.  N )
39 simp3 990 . . . . . . . . . . 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 18310 . . . . . . . . . 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 2456 . . . . . . . . . . . . . 14  |-  ( ( i  =  I  /\  i  =  j )  ->  I  =  j )
4443eqcomd 2443 . . . . . . . . . . . . 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 3794 . . . . . . . . 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 2470 . . . . . . 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 3794 . . . . . . . 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 3794 . . . . . . . 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 2481 . . . . . 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 788 . . . . 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 2470 . . . 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 3794 . . . . . 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 18311 . . . . . . . . . . 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 1168 . . . . . . . . 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 1009 . . . . . . . 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 987 . . . . . . . 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 1168 . . . . . . 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 18357 . . . . . 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 18310 . . . . . . 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 1732 . . . . . . . . 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 3806 . . . . . . 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 2471 . . . . . 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 3815 . . . . 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 2474 . . . 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 788 . . 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 6145 . 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 2438 . . . . 5  |-  ( N matRepV  R )  =  ( N matRepV  R )
862, 3, 85, 4marepvval 18353 . . . 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 2483 . 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 2474 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 965    = wceq 1369    e. wcel 1756   ifcif 3786   -->wf 5409   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088    ^m cmap 7206   Fincfn 7302   Basecbs 14166   0gc0g 14370   1rcur 16591   Ringcrg 16633   Mat cmat 18255   matRRep cmarrep 18342   matRepV cmatrepV 18343
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-ot 3881  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-sup 7683  df-oi 7716  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-fz 11430  df-fzo 11541  df-seq 11799  df-hash 12096  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-hom 14254  df-cco 14255  df-0g 14372  df-gsum 14373  df-prds 14378  df-pws 14380  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-mhm 15456  df-submnd 15457  df-grp 15536  df-minusg 15537  df-sbg 15538  df-mulg 15539  df-subg 15669  df-ghm 15736  df-cntz 15826  df-cmn 16270  df-abl 16271  df-mgp 16580  df-ur 16592  df-rng 16635  df-subrg 16841  df-lmod 16928  df-lss 16991  df-sra 17230  df-rgmod 17231  df-dsmm 18132  df-frlm 18147  df-mamu 18256  df-mat 18257  df-marrep 18344  df-marepv 18345
This theorem is referenced by:  cramerimplem1  18464
  Copyright terms: Public domain W3C validator