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Theorem mulmarep1gsum2 19648
Description: The sum of element by element multiplications of a matrix with an identity matrix with a column replaced by a vector. (Contributed by AV, 18-Feb-2019.) (Revised by AV, 26-Feb-2019.)
Hypotheses
Ref Expression
marepvcl.a  |-  A  =  ( N Mat  R )
marepvcl.b  |-  B  =  ( Base `  A
)
marepvcl.v  |-  V  =  ( ( Base `  R
)  ^m  N )
ma1repvcl.1  |-  .1.  =  ( 1r `  A )
mulmarep1el.0  |-  .0.  =  ( 0g `  R )
mulmarep1el.e  |-  E  =  ( (  .1.  ( N matRepV  R ) C ) `
 K )
mulmarep1gsum2.x  |-  .X.  =  ( R maVecMul  <. N ,  N >. )
Assertion
Ref Expression
mulmarep1gsum2  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N  /\  ( X  .X.  C
)  =  Z ) )  ->  ( R  gsumg  ( l  e.  N  |->  ( ( I X l ) ( .r `  R ) ( l E J ) ) ) )  =  if ( J  =  K ,  ( Z `  I ) ,  ( I X J ) ) )
Distinct variable groups:    B, l    C, l    I, l    J, l    K, l    N, l    R, l    V, l    X, l    .0. , l    A, l    Z, l    .X. , l
Allowed substitution hints:    .1. ( l)    E( l)

Proof of Theorem mulmarep1gsum2
StepHypRef Expression
1 simp1 1014 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N  /\  ( X  .X.  C
)  =  Z ) )  ->  R  e.  Ring )
21adantr 471 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  ( X 
.X.  C )  =  Z ) )  /\  l  e.  N )  ->  R  e.  Ring )
3 simpl2 1018 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  ( X 
.X.  C )  =  Z ) )  /\  l  e.  N )  ->  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
) )
4 simp1 1014 . . . . . . . . . . . . 13  |-  ( ( I  e.  N  /\  J  e.  N  /\  ( X  .X.  C )  =  Z )  ->  I  e.  N )
543ad2ant3 1037 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N  /\  ( X  .X.  C
)  =  Z ) )  ->  I  e.  N )
65adantr 471 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  ( X 
.X.  C )  =  Z ) )  /\  l  e.  N )  ->  I  e.  N )
7 simpl32 1096 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  ( X 
.X.  C )  =  Z ) )  /\  l  e.  N )  ->  J  e.  N )
8 simpr 467 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  ( X 
.X.  C )  =  Z ) )  /\  l  e.  N )  ->  l  e.  N )
96, 7, 83jca 1194 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  ( X 
.X.  C )  =  Z ) )  /\  l  e.  N )  ->  ( I  e.  N  /\  J  e.  N  /\  l  e.  N
) )
102, 3, 93jca 1194 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  ( X 
.X.  C )  =  Z ) )  /\  l  e.  N )  ->  ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  l  e.  N ) ) )
1110adantll 725 . . . . . . . 8  |-  ( ( ( J  =  K  /\  ( R  e. 
Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  (
I  e.  N  /\  J  e.  N  /\  ( X  .X.  C )  =  Z ) ) )  /\  l  e.  N )  ->  ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N  /\  l  e.  N
) ) )
12 marepvcl.a . . . . . . . . 9  |-  A  =  ( N Mat  R )
13 marepvcl.b . . . . . . . . 9  |-  B  =  ( Base `  A
)
14 marepvcl.v . . . . . . . . 9  |-  V  =  ( ( Base `  R
)  ^m  N )
15 ma1repvcl.1 . . . . . . . . 9  |-  .1.  =  ( 1r `  A )
16 mulmarep1el.0 . . . . . . . . 9  |-  .0.  =  ( 0g `  R )
17 mulmarep1el.e . . . . . . . . 9  |-  E  =  ( (  .1.  ( N matRepV  R ) C ) `
 K )
1812, 13, 14, 15, 16, 17mulmarep1el 19646 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N  /\  l  e.  N
) )  ->  (
( I X l ) ( .r `  R ) ( l E J ) )  =  if ( J  =  K ,  ( ( I X l ) ( .r `  R ) ( C `
 l ) ) ,  if ( J  =  l ,  ( I X l ) ,  .0.  ) ) )
1911, 18syl 17 . . . . . . 7  |-  ( ( ( J  =  K  /\  ( R  e. 
Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  (
I  e.  N  /\  J  e.  N  /\  ( X  .X.  C )  =  Z ) ) )  /\  l  e.  N )  ->  (
( I X l ) ( .r `  R ) ( l E J ) )  =  if ( J  =  K ,  ( ( I X l ) ( .r `  R ) ( C `
 l ) ) ,  if ( J  =  l ,  ( I X l ) ,  .0.  ) ) )
20 iftrue 3899 . . . . . . . . 9  |-  ( J  =  K  ->  if ( J  =  K ,  ( ( I X l ) ( .r `  R ) ( C `  l
) ) ,  if ( J  =  l ,  ( I X l ) ,  .0.  ) )  =  ( ( I X l ) ( .r `  R ) ( C `
 l ) ) )
2120adantr 471 . . . . . . . 8  |-  ( ( J  =  K  /\  ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  ( X 
.X.  C )  =  Z ) ) )  ->  if ( J  =  K ,  ( ( I X l ) ( .r `  R ) ( C `
 l ) ) ,  if ( J  =  l ,  ( I X l ) ,  .0.  ) )  =  ( ( I X l ) ( .r `  R ) ( C `  l
) ) )
2221adantr 471 . . . . . . 7  |-  ( ( ( J  =  K  /\  ( R  e. 
Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  (
I  e.  N  /\  J  e.  N  /\  ( X  .X.  C )  =  Z ) ) )  /\  l  e.  N )  ->  if ( J  =  K ,  ( ( I X l ) ( .r `  R ) ( C `  l
) ) ,  if ( J  =  l ,  ( I X l ) ,  .0.  ) )  =  ( ( I X l ) ( .r `  R ) ( C `
 l ) ) )
2319, 22eqtrd 2496 . . . . . 6  |-  ( ( ( J  =  K  /\  ( R  e. 
Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  (
I  e.  N  /\  J  e.  N  /\  ( X  .X.  C )  =  Z ) ) )  /\  l  e.  N )  ->  (
( I X l ) ( .r `  R ) ( l E J ) )  =  ( ( I X l ) ( .r `  R ) ( C `  l
) ) )
2423mpteq2dva 4503 . . . . 5  |-  ( ( J  =  K  /\  ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  ( X 
.X.  C )  =  Z ) ) )  ->  ( l  e.  N  |->  ( ( I X l ) ( .r `  R ) ( l E J ) ) )  =  ( l  e.  N  |->  ( ( I X l ) ( .r
`  R ) ( C `  l ) ) ) )
2524oveq2d 6331 . . . 4  |-  ( ( J  =  K  /\  ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  ( X 
.X.  C )  =  Z ) ) )  ->  ( R  gsumg  ( l  e.  N  |->  ( ( I X l ) ( .r `  R
) ( l E J ) ) ) )  =  ( R 
gsumg  ( l  e.  N  |->  ( ( I X l ) ( .r
`  R ) ( C `  l ) ) ) ) )
26 fveq1 5887 . . . . . . . . 9  |-  ( ( X  .X.  C )  =  Z  ->  ( ( X  .X.  C ) `  I )  =  ( Z `  I ) )
2726eqcomd 2468 . . . . . . . 8  |-  ( ( X  .X.  C )  =  Z  ->  ( Z `
 I )  =  ( ( X  .X.  C ) `  I
) )
28273ad2ant3 1037 . . . . . . 7  |-  ( ( I  e.  N  /\  J  e.  N  /\  ( X  .X.  C )  =  Z )  -> 
( Z `  I
)  =  ( ( X  .X.  C ) `  I ) )
29283ad2ant3 1037 . . . . . 6  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N  /\  ( X  .X.  C
)  =  Z ) )  ->  ( Z `  I )  =  ( ( X  .X.  C
) `  I )
)
3029adantl 472 . . . . 5  |-  ( ( J  =  K  /\  ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  ( X 
.X.  C )  =  Z ) ) )  ->  ( Z `  I )  =  ( ( X  .X.  C
) `  I )
)
31 mulmarep1gsum2.x . . . . . 6  |-  .X.  =  ( R maVecMul  <. N ,  N >. )
32 eqid 2462 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
33 eqid 2462 . . . . . 6  |-  ( .r
`  R )  =  ( .r `  R
)
341adantl 472 . . . . . 6  |-  ( ( J  =  K  /\  ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  ( X 
.X.  C )  =  Z ) ) )  ->  R  e.  Ring )
3512, 13matrcl 19486 . . . . . . . . . 10  |-  ( X  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
3635simpld 465 . . . . . . . . 9  |-  ( X  e.  B  ->  N  e.  Fin )
37363ad2ant1 1035 . . . . . . . 8  |-  ( ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  ->  N  e.  Fin )
38373ad2ant2 1036 . . . . . . 7  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N  /\  ( X  .X.  C
)  =  Z ) )  ->  N  e.  Fin )
3938adantl 472 . . . . . 6  |-  ( ( J  =  K  /\  ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  ( X 
.X.  C )  =  Z ) ) )  ->  N  e.  Fin )
4013eleq2i 2532 . . . . . . . . . 10  |-  ( X  e.  B  <->  X  e.  ( Base `  A )
)
4140biimpi 199 . . . . . . . . 9  |-  ( X  e.  B  ->  X  e.  ( Base `  A
) )
42413ad2ant1 1035 . . . . . . . 8  |-  ( ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  ->  X  e.  ( Base `  A ) )
43423ad2ant2 1036 . . . . . . 7  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N  /\  ( X  .X.  C
)  =  Z ) )  ->  X  e.  ( Base `  A )
)
4443adantl 472 . . . . . 6  |-  ( ( J  =  K  /\  ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  ( X 
.X.  C )  =  Z ) ) )  ->  X  e.  (
Base `  A )
)
4514eleq2i 2532 . . . . . . . . . 10  |-  ( C  e.  V  <->  C  e.  ( ( Base `  R
)  ^m  N )
)
4645biimpi 199 . . . . . . . . 9  |-  ( C  e.  V  ->  C  e.  ( ( Base `  R
)  ^m  N )
)
47463ad2ant2 1036 . . . . . . . 8  |-  ( ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  ->  C  e.  ( (
Base `  R )  ^m  N ) )
48473ad2ant2 1036 . . . . . . 7  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N  /\  ( X  .X.  C
)  =  Z ) )  ->  C  e.  ( ( Base `  R
)  ^m  N )
)
4948adantl 472 . . . . . 6  |-  ( ( J  =  K  /\  ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  ( X 
.X.  C )  =  Z ) ) )  ->  C  e.  ( ( Base `  R
)  ^m  N )
)
505adantl 472 . . . . . 6  |-  ( ( J  =  K  /\  ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  ( X 
.X.  C )  =  Z ) ) )  ->  I  e.  N
)
5112, 31, 32, 33, 34, 39, 44, 49, 50mavmulfv 19620 . . . . 5  |-  ( ( J  =  K  /\  ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  ( X 
.X.  C )  =  Z ) ) )  ->  ( ( X 
.X.  C ) `  I )  =  ( R  gsumg  ( l  e.  N  |->  ( ( I X l ) ( .r
`  R ) ( C `  l ) ) ) ) )
5230, 51eqtrd 2496 . . . 4  |-  ( ( J  =  K  /\  ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  ( X 
.X.  C )  =  Z ) ) )  ->  ( Z `  I )  =  ( R  gsumg  ( l  e.  N  |->  ( ( I X l ) ( .r
`  R ) ( C `  l ) ) ) ) )
53 iftrue 3899 . . . . . 6  |-  ( J  =  K  ->  if ( J  =  K ,  ( Z `  I ) ,  ( I X J ) )  =  ( Z `
 I ) )
5453eqcomd 2468 . . . . 5  |-  ( J  =  K  ->  ( Z `  I )  =  if ( J  =  K ,  ( Z `
 I ) ,  ( I X J ) ) )
5554adantr 471 . . . 4  |-  ( ( J  =  K  /\  ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  ( X 
.X.  C )  =  Z ) ) )  ->  ( Z `  I )  =  if ( J  =  K ,  ( Z `  I ) ,  ( I X J ) ) )
5625, 52, 553eqtr2d 2502 . . 3  |-  ( ( J  =  K  /\  ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  ( X 
.X.  C )  =  Z ) ) )  ->  ( R  gsumg  ( l  e.  N  |->  ( ( I X l ) ( .r `  R
) ( l E J ) ) ) )  =  if ( J  =  K , 
( Z `  I
) ,  ( I X J ) ) )
5756ex 440 . 2  |-  ( J  =  K  ->  (
( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  ( X 
.X.  C )  =  Z ) )  -> 
( R  gsumg  ( l  e.  N  |->  ( ( I X l ) ( .r
`  R ) ( l E J ) ) ) )  =  if ( J  =  K ,  ( Z `
 I ) ,  ( I X J ) ) ) )
581adantr 471 . . . . 5  |-  ( ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  ( X 
.X.  C )  =  Z ) )  /\  J  =/=  K )  ->  R  e.  Ring )
59 simpl2 1018 . . . . 5  |-  ( ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  ( X 
.X.  C )  =  Z ) )  /\  J  =/=  K )  -> 
( X  e.  B  /\  C  e.  V  /\  K  e.  N
) )
605adantr 471 . . . . 5  |-  ( ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  ( X 
.X.  C )  =  Z ) )  /\  J  =/=  K )  ->  I  e.  N )
61 simpl32 1096 . . . . 5  |-  ( ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  ( X 
.X.  C )  =  Z ) )  /\  J  =/=  K )  ->  J  e.  N )
62 simpr 467 . . . . 5  |-  ( ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  ( X 
.X.  C )  =  Z ) )  /\  J  =/=  K )  ->  J  =/=  K )
6312, 13, 14, 15, 16, 17mulmarep1gsum1 19647 . . . . 5  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N  /\  J  =/=  K
) )  ->  ( R  gsumg  ( l  e.  N  |->  ( ( I X l ) ( .r
`  R ) ( l E J ) ) ) )  =  ( I X J ) )
6458, 59, 60, 61, 62, 63syl113anc 1288 . . . 4  |-  ( ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  ( X 
.X.  C )  =  Z ) )  /\  J  =/=  K )  -> 
( R  gsumg  ( l  e.  N  |->  ( ( I X l ) ( .r
`  R ) ( l E J ) ) ) )  =  ( I X J ) )
65 df-ne 2635 . . . . . 6  |-  ( J  =/=  K  <->  -.  J  =  K )
66 iffalse 3902 . . . . . . 7  |-  ( -.  J  =  K  ->  if ( J  =  K ,  ( Z `  I ) ,  ( I X J ) )  =  ( I X J ) )
6766eqcomd 2468 . . . . . 6  |-  ( -.  J  =  K  -> 
( I X J )  =  if ( J  =  K , 
( Z `  I
) ,  ( I X J ) ) )
6865, 67sylbi 200 . . . . 5  |-  ( J  =/=  K  ->  (
I X J )  =  if ( J  =  K ,  ( Z `  I ) ,  ( I X J ) ) )
6968adantl 472 . . . 4  |-  ( ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  ( X 
.X.  C )  =  Z ) )  /\  J  =/=  K )  -> 
( I X J )  =  if ( J  =  K , 
( Z `  I
) ,  ( I X J ) ) )
7064, 69eqtrd 2496 . . 3  |-  ( ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  ( X 
.X.  C )  =  Z ) )  /\  J  =/=  K )  -> 
( R  gsumg  ( l  e.  N  |->  ( ( I X l ) ( .r
`  R ) ( l E J ) ) ) )  =  if ( J  =  K ,  ( Z `
 I ) ,  ( I X J ) ) )
7170expcom 441 . 2  |-  ( J  =/=  K  ->  (
( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N
)  /\  ( I  e.  N  /\  J  e.  N  /\  ( X 
.X.  C )  =  Z ) )  -> 
( R  gsumg  ( l  e.  N  |->  ( ( I X l ) ( .r
`  R ) ( l E J ) ) ) )  =  if ( J  =  K ,  ( Z `
 I ) ,  ( I X J ) ) ) )
7257, 71pm2.61ine 2719 1  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N  /\  ( X  .X.  C
)  =  Z ) )  ->  ( R  gsumg  ( l  e.  N  |->  ( ( I X l ) ( .r `  R ) ( l E J ) ) ) )  =  if ( J  =  K ,  ( Z `  I ) ,  ( I X J ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898    =/= wne 2633   _Vcvv 3057   ifcif 3893   <.cop 3986    |-> cmpt 4475   ` cfv 5601  (class class class)co 6315    ^m cmap 7498   Fincfn 7595   Basecbs 15170   .rcmulr 15240   0gc0g 15387    gsumg cgsu 15388   1rcur 17784   Ringcrg 17829   Mat cmat 19481   maVecMul cmvmul 19614   matRepV cmatrepV 19631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-inf2 8172  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-ot 3989  df-uni 4213  df-int 4249  df-iun 4294  df-iin 4295  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-of 6558  df-om 6720  df-1st 6820  df-2nd 6821  df-supp 6942  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-oadd 7212  df-er 7389  df-map 7500  df-ixp 7549  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-fsupp 7910  df-sup 7982  df-oi 8051  df-card 8399  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-nn 10638  df-2 10696  df-3 10697  df-4 10698  df-5 10699  df-6 10700  df-7 10701  df-8 10702  df-9 10703  df-10 10704  df-n0 10899  df-z 10967  df-dec 11081  df-uz 11189  df-fz 11814  df-fzo 11947  df-seq 12246  df-hash 12548  df-struct 15172  df-ndx 15173  df-slot 15174  df-base 15175  df-sets 15176  df-ress 15177  df-plusg 15252  df-mulr 15253  df-sca 15255  df-vsca 15256  df-ip 15257  df-tset 15258  df-ple 15259  df-ds 15261  df-hom 15263  df-cco 15264  df-0g 15389  df-gsum 15390  df-prds 15395  df-pws 15397  df-mre 15541  df-mrc 15542  df-acs 15544  df-mgm 16537  df-sgrp 16576  df-mnd 16586  df-mhm 16631  df-submnd 16632  df-grp 16722  df-minusg 16723  df-sbg 16724  df-mulg 16725  df-subg 16863  df-ghm 16930  df-cntz 17020  df-cmn 17481  df-abl 17482  df-mgp 17773  df-ur 17785  df-ring 17831  df-subrg 18055  df-lmod 18142  df-lss 18205  df-sra 18444  df-rgmod 18445  df-dsmm 19344  df-frlm 19359  df-mamu 19458  df-mat 19482  df-mvmul 19615  df-marepv 19633
This theorem is referenced by:  cramerimplem2  19758
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