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Theorem 1marepvmarrepid 18844
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 2467 . . . . . 6  |-  ( N Mat 
R )  =  ( N Mat  R )
3 eqid 2467 . . . . . 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 18839 . . . . 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 2559 . . 3  |-  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V
) )  ->  X  e.  ( Base `  ( N Mat  R ) ) )
9 elmapi 7437 . . . . . . 7  |-  ( Z  e.  ( ( Base `  R )  ^m  N
)  ->  Z : N
--> ( Base `  R
) )
10 ffvelrn 6017 . . . . . . . 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 2575 . . . . 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 2467 . . . 4  |-  ( N matRRep  R )  =  ( N matRRep  R )
19 eqid 2467 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
202, 3, 18, 19marrepval 18831 . . 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 1229 . 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 3945 . . . . . 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 3945 . . . . . . . 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 3945 . . . . . . . 8  |-  ( j  =  I  ->  if ( j  =  I ,  ( Z `  i ) ,  ( i  .1.  j ) )  =  ( Z `
 i ) )
27 fveq2 5864 . . . . . . . . 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 2528 . . . . . . 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 2511 . . . . . 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 2467 . . . . . . . . . . 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 1017 . . . . . . . . . . 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 1017 . . . . . . . . . . 11  |-  ( ( ( ( R  e. 
Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V ) )  /\  i  e.  N  /\  j  e.  N )  ->  R  e.  Ring )
38 simp2 997 . . . . . . . . . . 11  |-  ( ( ( ( R  e. 
Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V ) )  /\  i  e.  N  /\  j  e.  N )  ->  i  e.  N )
39 simp3 998 . . . . . . . . . . 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 18717 . . . . . . . . . 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 2494 . . . . . . . . . . . . . 14  |-  ( ( i  =  I  /\  i  =  j )  ->  I  =  j )
4443eqcomd 2475 . . . . . . . . . . . . 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 3948 . . . . . . . . 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 2508 . . . . . . 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 3948 . . . . . . . 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 3948 . . . . . . . 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 2519 . . . . . 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 2508 . . . 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 3948 . . . . . 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 18718 . . . . . . . . . . 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 1176 . . . . . . . . 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 1017 . . . . . . . 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 995 . . . . . . . 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 1176 . . . . . . 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 18840 . . . . . 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 18717 . . . . . . 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 1743 . . . . . . . . 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 3961 . . . . . . 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 2509 . . . . . 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 3970 . . . . 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 2512 . . . 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 6343 . 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 2467 . . . . 5  |-  ( N matRepV  R )  =  ( N matRepV  R )
862, 3, 85, 4marepvval 18836 . . . 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 2521 . 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 2512 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 973    = wceq 1379    e. wcel 1767   ifcif 3939   -->wf 5582   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284    ^m cmap 7417   Fincfn 7513   Basecbs 14486   0gc0g 14691   1rcur 16943   Ringcrg 16986   Mat cmat 18676   matRRep cmarrep 18825   matRepV cmatrepV 18826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-ot 4036  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-fz 11669  df-fzo 11789  df-seq 12072  df-hash 12370  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-hom 14575  df-cco 14576  df-0g 14693  df-gsum 14694  df-prds 14699  df-pws 14701  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-mhm 15777  df-submnd 15778  df-grp 15858  df-minusg 15859  df-sbg 15860  df-mulg 15861  df-subg 15993  df-ghm 16060  df-cntz 16150  df-cmn 16596  df-abl 16597  df-mgp 16932  df-ur 16944  df-rng 16988  df-subrg 17210  df-lmod 17297  df-lss 17362  df-sra 17601  df-rgmod 17602  df-dsmm 18530  df-frlm 18545  df-mamu 18653  df-mat 18677  df-marrep 18827  df-marepv 18828
This theorem is referenced by:  cramerimplem1  18952
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