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Theorem 1marepvmarrepid 18506
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 2451 . . . . . 6  |-  ( N Mat 
R )  =  ( N Mat  R )
3 eqid 2451 . . . . . 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 18501 . . . . 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 2543 . . 3  |-  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V
) )  ->  X  e.  ( Base `  ( N Mat  R ) ) )
9 elmapi 7337 . . . . . . 7  |-  ( Z  e.  ( ( Base `  R )  ^m  N
)  ->  Z : N
--> ( Base `  R
) )
10 ffvelrn 5943 . . . . . . . 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 2559 . . . . 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 2451 . . . 4  |-  ( N matRRep  R )  =  ( N matRRep  R )
19 eqid 2451 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
202, 3, 18, 19marrepval 18493 . . 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 1220 . 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 3898 . . . . . 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 3898 . . . . . . . 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 3898 . . . . . . . 8  |-  ( j  =  I  ->  if ( j  =  I ,  ( Z `  i ) ,  ( i  .1.  j ) )  =  ( Z `
 i ) )
27 fveq2 5792 . . . . . . . . 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 2512 . . . . . . 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 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 ) ) )
31 eqid 2451 . . . . . . . . . . 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 18455 . . . . . . . . . 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 2478 . . . . . . . . . . . . . 14  |-  ( ( i  =  I  /\  i  =  j )  ->  I  =  j )
4443eqcomd 2459 . . . . . . . . . . . . 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 3900 . . . . . . . . 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 2492 . . . . . . 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 3900 . . . . . . . 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 3900 . . . . . . . 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 2503 . . . . . 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 2492 . . . 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 3900 . . . . . 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 18456 . . . . . . . . . . 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 18502 . . . . . 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 18455 . . . . . . 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 1734 . . . . . . . . 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 3912 . . . . . . 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 2493 . . . . . 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 3921 . . . . 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 2496 . . . 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 6252 . 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 2451 . . . . 5  |-  ( N matRepV  R )  =  ( N matRepV  R )
862, 3, 85, 4marepvval 18498 . . . 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 2505 . 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 2496 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 1370    e. wcel 1758   ifcif 3892   -->wf 5515   ` cfv 5519  (class class class)co 6193    |-> cmpt2 6195    ^m cmap 7317   Fincfn 7413   Basecbs 14285   0gc0g 14489   1rcur 16717   Ringcrg 16760   Mat cmat 18398   matRRep cmarrep 18487   matRepV cmatrepV 18488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-ot 3987  df-uni 4193  df-int 4230  df-iun 4274  df-iin 4275  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-of 6423  df-om 6580  df-1st 6680  df-2nd 6681  df-supp 6794  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-map 7319  df-ixp 7367  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-fsupp 7725  df-sup 7795  df-oi 7828  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-7 10489  df-8 10490  df-9 10491  df-10 10492  df-n0 10684  df-z 10751  df-dec 10860  df-uz 10966  df-fz 11548  df-fzo 11659  df-seq 11917  df-hash 12214  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-mulr 14363  df-sca 14365  df-vsca 14366  df-ip 14367  df-tset 14368  df-ple 14369  df-ds 14371  df-hom 14373  df-cco 14374  df-0g 14491  df-gsum 14492  df-prds 14497  df-pws 14499  df-mre 14635  df-mrc 14636  df-acs 14638  df-mnd 15526  df-mhm 15575  df-submnd 15576  df-grp 15656  df-minusg 15657  df-sbg 15658  df-mulg 15659  df-subg 15789  df-ghm 15856  df-cntz 15946  df-cmn 16392  df-abl 16393  df-mgp 16706  df-ur 16718  df-rng 16762  df-subrg 16978  df-lmod 17065  df-lss 17129  df-sra 17368  df-rgmod 17369  df-dsmm 18275  df-frlm 18290  df-mamu 18399  df-mat 18400  df-marrep 18489  df-marepv 18490
This theorem is referenced by:  cramerimplem1  18614
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