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Theorem mdetunilem4 19244
Description: Lemma for mdetuni 19251. (Contributed by SO, 15-Jul-2018.)
Hypotheses
Ref Expression
mdetuni.a  |-  A  =  ( N Mat  R )
mdetuni.b  |-  B  =  ( Base `  A
)
mdetuni.k  |-  K  =  ( Base `  R
)
mdetuni.0g  |-  .0.  =  ( 0g `  R )
mdetuni.1r  |-  .1.  =  ( 1r `  R )
mdetuni.pg  |-  .+  =  ( +g  `  R )
mdetuni.tg  |-  .x.  =  ( .r `  R )
mdetuni.n  |-  ( ph  ->  N  e.  Fin )
mdetuni.r  |-  ( ph  ->  R  e.  Ring )
mdetuni.ff  |-  ( ph  ->  D : B --> K )
mdetuni.al  |-  ( ph  ->  A. x  e.  B  A. y  e.  N  A. z  e.  N  ( ( y  =/=  z  /\  A. w  e.  N  ( y
x w )  =  ( z x w ) )  ->  ( D `  x )  =  .0.  ) )
mdetuni.li  |-  ( ph  ->  A. x  e.  B  A. y  e.  B  A. z  e.  B  A. w  e.  N  ( ( ( x  |`  ( { w }  X.  N ) )  =  ( ( y  |`  ( { w }  X.  N ) )  oF  .+  ( z  |`  ( { w }  X.  N ) ) )  /\  ( x  |`  ( ( N  \  { w } )  X.  N ) )  =  ( y  |`  ( ( N  \  { w } )  X.  N ) )  /\  ( x  |`  ( ( N  \  { w } )  X.  N ) )  =  ( z  |`  ( ( N  \  { w } )  X.  N ) ) )  ->  ( D `  x )  =  ( ( D `  y
)  .+  ( D `  z ) ) ) )
mdetuni.sc  |-  ( ph  ->  A. x  e.  B  A. y  e.  K  A. z  e.  B  A. w  e.  N  ( ( ( x  |`  ( { w }  X.  N ) )  =  ( ( ( { w }  X.  N
)  X.  { y } )  oF  .x.  ( z  |`  ( { w }  X.  N ) ) )  /\  ( x  |`  ( ( N  \  { w } )  X.  N ) )  =  ( z  |`  ( ( N  \  { w } )  X.  N ) ) )  ->  ( D `  x )  =  ( y  .x.  ( D `
 z ) ) ) )
Assertion
Ref Expression
mdetunilem4  |-  ( (
ph  /\  ( E  e.  B  /\  F  e.  K  /\  G  e.  B )  /\  ( H  e.  N  /\  ( E  |`  ( { H }  X.  N
) )  =  ( ( ( { H }  X.  N )  X. 
{ F } )  oF  .x.  ( G  |`  ( { H }  X.  N ) ) )  /\  ( E  |`  ( ( N  \  { H } )  X.  N ) )  =  ( G  |`  (
( N  \  { H } )  X.  N
) ) ) )  ->  ( D `  E )  =  ( F  .x.  ( D `
 G ) ) )
Distinct variable groups:    ph, x, y, z, w    x, B, y, z, w    x, K, y, z, w    x, N, y, z, w    x, D, y, z, w    x,  .x. , y, z, w    x,  .+ , y, z, w    x,  .0. , y, z, w    x,  .1. , y, z, w    x, R, y, z, w    x, A, y, z, w    x, E, y, z, w    x, F, y, z, w    x, G, y, z, w    x, H, y, z, w

Proof of Theorem mdetunilem4
StepHypRef Expression
1 simp32 1033 . 2  |-  ( (
ph  /\  ( E  e.  B  /\  F  e.  K  /\  G  e.  B )  /\  ( H  e.  N  /\  ( E  |`  ( { H }  X.  N
) )  =  ( ( ( { H }  X.  N )  X. 
{ F } )  oF  .x.  ( G  |`  ( { H }  X.  N ) ) )  /\  ( E  |`  ( ( N  \  { H } )  X.  N ) )  =  ( G  |`  (
( N  \  { H } )  X.  N
) ) ) )  ->  ( E  |`  ( { H }  X.  N ) )  =  ( ( ( { H }  X.  N
)  X.  { F } )  oF  .x.  ( G  |`  ( { H }  X.  N ) ) ) )
2 simp33 1034 . 2  |-  ( (
ph  /\  ( E  e.  B  /\  F  e.  K  /\  G  e.  B )  /\  ( H  e.  N  /\  ( E  |`  ( { H }  X.  N
) )  =  ( ( ( { H }  X.  N )  X. 
{ F } )  oF  .x.  ( G  |`  ( { H }  X.  N ) ) )  /\  ( E  |`  ( ( N  \  { H } )  X.  N ) )  =  ( G  |`  (
( N  \  { H } )  X.  N
) ) ) )  ->  ( E  |`  ( ( N  \  { H } )  X.  N ) )  =  ( G  |`  (
( N  \  { H } )  X.  N
) ) )
3 simp1 996 . . 3  |-  ( ( H  e.  N  /\  ( E  |`  ( { H }  X.  N
) )  =  ( ( ( { H }  X.  N )  X. 
{ F } )  oF  .x.  ( G  |`  ( { H }  X.  N ) ) )  /\  ( E  |`  ( ( N  \  { H } )  X.  N ) )  =  ( G  |`  (
( N  \  { H } )  X.  N
) ) )  ->  H  e.  N )
4 simp23 1031 . . . 4  |-  ( (
ph  /\  ( E  e.  B  /\  F  e.  K  /\  G  e.  B )  /\  H  e.  N )  ->  G  e.  B )
5 simp3 998 . . . 4  |-  ( (
ph  /\  ( E  e.  B  /\  F  e.  K  /\  G  e.  B )  /\  H  e.  N )  ->  H  e.  N )
6 simp21 1029 . . . . 5  |-  ( (
ph  /\  ( E  e.  B  /\  F  e.  K  /\  G  e.  B )  /\  H  e.  N )  ->  E  e.  B )
7 simp22 1030 . . . . 5  |-  ( (
ph  /\  ( E  e.  B  /\  F  e.  K  /\  G  e.  B )  /\  H  e.  N )  ->  F  e.  K )
8 mdetuni.sc . . . . . 6  |-  ( ph  ->  A. x  e.  B  A. y  e.  K  A. z  e.  B  A. w  e.  N  ( ( ( x  |`  ( { w }  X.  N ) )  =  ( ( ( { w }  X.  N
)  X.  { y } )  oF  .x.  ( z  |`  ( { w }  X.  N ) ) )  /\  ( x  |`  ( ( N  \  { w } )  X.  N ) )  =  ( z  |`  ( ( N  \  { w } )  X.  N ) ) )  ->  ( D `  x )  =  ( y  .x.  ( D `
 z ) ) ) )
983ad2ant1 1017 . . . . 5  |-  ( (
ph  /\  ( E  e.  B  /\  F  e.  K  /\  G  e.  B )  /\  H  e.  N )  ->  A. x  e.  B  A. y  e.  K  A. z  e.  B  A. w  e.  N  ( (
( x  |`  ( { w }  X.  N ) )  =  ( ( ( { w }  X.  N
)  X.  { y } )  oF  .x.  ( z  |`  ( { w }  X.  N ) ) )  /\  ( x  |`  ( ( N  \  { w } )  X.  N ) )  =  ( z  |`  ( ( N  \  { w } )  X.  N ) ) )  ->  ( D `  x )  =  ( y  .x.  ( D `
 z ) ) ) )
10 reseq1 5277 . . . . . . . . . 10  |-  ( x  =  E  ->  (
x  |`  ( { w }  X.  N ) )  =  ( E  |`  ( { w }  X.  N ) ) )
1110eqeq1d 2459 . . . . . . . . 9  |-  ( x  =  E  ->  (
( x  |`  ( { w }  X.  N ) )  =  ( ( ( { w }  X.  N
)  X.  { y } )  oF  .x.  ( z  |`  ( { w }  X.  N ) ) )  <-> 
( E  |`  ( { w }  X.  N ) )  =  ( ( ( { w }  X.  N
)  X.  { y } )  oF  .x.  ( z  |`  ( { w }  X.  N ) ) ) ) )
12 reseq1 5277 . . . . . . . . . 10  |-  ( x  =  E  ->  (
x  |`  ( ( N 
\  { w }
)  X.  N ) )  =  ( E  |`  ( ( N  \  { w } )  X.  N ) ) )
1312eqeq1d 2459 . . . . . . . . 9  |-  ( x  =  E  ->  (
( x  |`  (
( N  \  {
w } )  X.  N ) )  =  ( z  |`  (
( N  \  {
w } )  X.  N ) )  <->  ( E  |`  ( ( N  \  { w } )  X.  N ) )  =  ( z  |`  ( ( N  \  { w } )  X.  N ) ) ) )
1411, 13anbi12d 710 . . . . . . . 8  |-  ( x  =  E  ->  (
( ( x  |`  ( { w }  X.  N ) )  =  ( ( ( { w }  X.  N
)  X.  { y } )  oF  .x.  ( z  |`  ( { w }  X.  N ) ) )  /\  ( x  |`  ( ( N  \  { w } )  X.  N ) )  =  ( z  |`  ( ( N  \  { w } )  X.  N ) ) )  <->  ( ( E  |`  ( { w }  X.  N ) )  =  ( ( ( { w }  X.  N
)  X.  { y } )  oF  .x.  ( z  |`  ( { w }  X.  N ) ) )  /\  ( E  |`  ( ( N  \  { w } )  X.  N ) )  =  ( z  |`  ( ( N  \  { w } )  X.  N ) ) ) ) )
15 fveq2 5872 . . . . . . . . 9  |-  ( x  =  E  ->  ( D `  x )  =  ( D `  E ) )
1615eqeq1d 2459 . . . . . . . 8  |-  ( x  =  E  ->  (
( D `  x
)  =  ( y 
.x.  ( D `  z ) )  <->  ( D `  E )  =  ( y  .x.  ( D `
 z ) ) ) )
1714, 16imbi12d 320 . . . . . . 7  |-  ( x  =  E  ->  (
( ( ( x  |`  ( { w }  X.  N ) )  =  ( ( ( { w }  X.  N
)  X.  { y } )  oF  .x.  ( z  |`  ( { w }  X.  N ) ) )  /\  ( x  |`  ( ( N  \  { w } )  X.  N ) )  =  ( z  |`  ( ( N  \  { w } )  X.  N ) ) )  ->  ( D `  x )  =  ( y  .x.  ( D `
 z ) ) )  <->  ( ( ( E  |`  ( {
w }  X.  N
) )  =  ( ( ( { w }  X.  N )  X. 
{ y } )  oF  .x.  (
z  |`  ( { w }  X.  N ) ) )  /\  ( E  |`  ( ( N  \  { w } )  X.  N ) )  =  ( z  |`  ( ( N  \  { w } )  X.  N ) ) )  ->  ( D `  E )  =  ( y  .x.  ( D `
 z ) ) ) ) )
18172ralbidv 2901 . . . . . 6  |-  ( x  =  E  ->  ( A. z  e.  B  A. w  e.  N  ( ( ( x  |`  ( { w }  X.  N ) )  =  ( ( ( { w }  X.  N
)  X.  { y } )  oF  .x.  ( z  |`  ( { w }  X.  N ) ) )  /\  ( x  |`  ( ( N  \  { w } )  X.  N ) )  =  ( z  |`  ( ( N  \  { w } )  X.  N ) ) )  ->  ( D `  x )  =  ( y  .x.  ( D `
 z ) ) )  <->  A. z  e.  B  A. w  e.  N  ( ( ( E  |`  ( { w }  X.  N ) )  =  ( ( ( { w }  X.  N
)  X.  { y } )  oF  .x.  ( z  |`  ( { w }  X.  N ) ) )  /\  ( E  |`  ( ( N  \  { w } )  X.  N ) )  =  ( z  |`  ( ( N  \  { w } )  X.  N ) ) )  ->  ( D `  E )  =  ( y  .x.  ( D `
 z ) ) ) ) )
19 sneq 4042 . . . . . . . . . . . 12  |-  ( y  =  F  ->  { y }  =  { F } )
2019xpeq2d 5032 . . . . . . . . . . 11  |-  ( y  =  F  ->  (
( { w }  X.  N )  X.  {
y } )  =  ( ( { w }  X.  N )  X. 
{ F } ) )
2120oveq1d 6311 . . . . . . . . . 10  |-  ( y  =  F  ->  (
( ( { w }  X.  N )  X. 
{ y } )  oF  .x.  (
z  |`  ( { w }  X.  N ) ) )  =  ( ( ( { w }  X.  N )  X.  { F } )  oF  .x.  ( z  |`  ( { w }  X.  N ) ) ) )
2221eqeq2d 2471 . . . . . . . . 9  |-  ( y  =  F  ->  (
( E  |`  ( { w }  X.  N ) )  =  ( ( ( { w }  X.  N
)  X.  { y } )  oF  .x.  ( z  |`  ( { w }  X.  N ) ) )  <-> 
( E  |`  ( { w }  X.  N ) )  =  ( ( ( { w }  X.  N
)  X.  { F } )  oF  .x.  ( z  |`  ( { w }  X.  N ) ) ) ) )
2322anbi1d 704 . . . . . . . 8  |-  ( y  =  F  ->  (
( ( E  |`  ( { w }  X.  N ) )  =  ( ( ( { w }  X.  N
)  X.  { y } )  oF  .x.  ( z  |`  ( { w }  X.  N ) ) )  /\  ( E  |`  ( ( N  \  { w } )  X.  N ) )  =  ( z  |`  ( ( N  \  { w } )  X.  N ) ) )  <->  ( ( E  |`  ( { w }  X.  N ) )  =  ( ( ( { w }  X.  N
)  X.  { F } )  oF  .x.  ( z  |`  ( { w }  X.  N ) ) )  /\  ( E  |`  ( ( N  \  { w } )  X.  N ) )  =  ( z  |`  ( ( N  \  { w } )  X.  N ) ) ) ) )
24 oveq1 6303 . . . . . . . . 9  |-  ( y  =  F  ->  (
y  .x.  ( D `  z ) )  =  ( F  .x.  ( D `  z )
) )
2524eqeq2d 2471 . . . . . . . 8  |-  ( y  =  F  ->  (
( D `  E
)  =  ( y 
.x.  ( D `  z ) )  <->  ( D `  E )  =  ( F  .x.  ( D `
 z ) ) ) )
2623, 25imbi12d 320 . . . . . . 7  |-  ( y  =  F  ->  (
( ( ( E  |`  ( { w }  X.  N ) )  =  ( ( ( { w }  X.  N
)  X.  { y } )  oF  .x.  ( z  |`  ( { w }  X.  N ) ) )  /\  ( E  |`  ( ( N  \  { w } )  X.  N ) )  =  ( z  |`  ( ( N  \  { w } )  X.  N ) ) )  ->  ( D `  E )  =  ( y  .x.  ( D `
 z ) ) )  <->  ( ( ( E  |`  ( {
w }  X.  N
) )  =  ( ( ( { w }  X.  N )  X. 
{ F } )  oF  .x.  (
z  |`  ( { w }  X.  N ) ) )  /\  ( E  |`  ( ( N  \  { w } )  X.  N ) )  =  ( z  |`  ( ( N  \  { w } )  X.  N ) ) )  ->  ( D `  E )  =  ( F  .x.  ( D `
 z ) ) ) ) )
27262ralbidv 2901 . . . . . 6  |-  ( y  =  F  ->  ( A. z  e.  B  A. w  e.  N  ( ( ( E  |`  ( { w }  X.  N ) )  =  ( ( ( { w }  X.  N
)  X.  { y } )  oF  .x.  ( z  |`  ( { w }  X.  N ) ) )  /\  ( E  |`  ( ( N  \  { w } )  X.  N ) )  =  ( z  |`  ( ( N  \  { w } )  X.  N ) ) )  ->  ( D `  E )  =  ( y  .x.  ( D `
 z ) ) )  <->  A. z  e.  B  A. w  e.  N  ( ( ( E  |`  ( { w }  X.  N ) )  =  ( ( ( { w }  X.  N
)  X.  { F } )  oF  .x.  ( z  |`  ( { w }  X.  N ) ) )  /\  ( E  |`  ( ( N  \  { w } )  X.  N ) )  =  ( z  |`  ( ( N  \  { w } )  X.  N ) ) )  ->  ( D `  E )  =  ( F  .x.  ( D `
 z ) ) ) ) )
2818, 27rspc2va 3220 . . . . 5  |-  ( ( ( E  e.  B  /\  F  e.  K
)  /\  A. x  e.  B  A. y  e.  K  A. z  e.  B  A. w  e.  N  ( (
( x  |`  ( { w }  X.  N ) )  =  ( ( ( { w }  X.  N
)  X.  { y } )  oF  .x.  ( z  |`  ( { w }  X.  N ) ) )  /\  ( x  |`  ( ( N  \  { w } )  X.  N ) )  =  ( z  |`  ( ( N  \  { w } )  X.  N ) ) )  ->  ( D `  x )  =  ( y  .x.  ( D `
 z ) ) ) )  ->  A. z  e.  B  A. w  e.  N  ( (
( E  |`  ( { w }  X.  N ) )  =  ( ( ( { w }  X.  N
)  X.  { F } )  oF  .x.  ( z  |`  ( { w }  X.  N ) ) )  /\  ( E  |`  ( ( N  \  { w } )  X.  N ) )  =  ( z  |`  ( ( N  \  { w } )  X.  N ) ) )  ->  ( D `  E )  =  ( F  .x.  ( D `
 z ) ) ) )
296, 7, 9, 28syl21anc 1227 . . . 4  |-  ( (
ph  /\  ( E  e.  B  /\  F  e.  K  /\  G  e.  B )  /\  H  e.  N )  ->  A. z  e.  B  A. w  e.  N  ( (
( E  |`  ( { w }  X.  N ) )  =  ( ( ( { w }  X.  N
)  X.  { F } )  oF  .x.  ( z  |`  ( { w }  X.  N ) ) )  /\  ( E  |`  ( ( N  \  { w } )  X.  N ) )  =  ( z  |`  ( ( N  \  { w } )  X.  N ) ) )  ->  ( D `  E )  =  ( F  .x.  ( D `
 z ) ) ) )
30 reseq1 5277 . . . . . . . . 9  |-  ( z  =  G  ->  (
z  |`  ( { w }  X.  N ) )  =  ( G  |`  ( { w }  X.  N ) ) )
3130oveq2d 6312 . . . . . . . 8  |-  ( z  =  G  ->  (
( ( { w }  X.  N )  X. 
{ F } )  oF  .x.  (
z  |`  ( { w }  X.  N ) ) )  =  ( ( ( { w }  X.  N )  X.  { F } )  oF  .x.  ( G  |`  ( { w }  X.  N ) ) ) )
3231eqeq2d 2471 . . . . . . 7  |-  ( z  =  G  ->  (
( E  |`  ( { w }  X.  N ) )  =  ( ( ( { w }  X.  N
)  X.  { F } )  oF  .x.  ( z  |`  ( { w }  X.  N ) ) )  <-> 
( E  |`  ( { w }  X.  N ) )  =  ( ( ( { w }  X.  N
)  X.  { F } )  oF  .x.  ( G  |`  ( { w }  X.  N ) ) ) ) )
33 reseq1 5277 . . . . . . . 8  |-  ( z  =  G  ->  (
z  |`  ( ( N 
\  { w }
)  X.  N ) )  =  ( G  |`  ( ( N  \  { w } )  X.  N ) ) )
3433eqeq2d 2471 . . . . . . 7  |-  ( z  =  G  ->  (
( E  |`  (
( N  \  {
w } )  X.  N ) )  =  ( z  |`  (
( N  \  {
w } )  X.  N ) )  <->  ( E  |`  ( ( N  \  { w } )  X.  N ) )  =  ( G  |`  ( ( N  \  { w } )  X.  N ) ) ) )
3532, 34anbi12d 710 . . . . . 6  |-  ( z  =  G  ->  (
( ( E  |`  ( { w }  X.  N ) )  =  ( ( ( { w }  X.  N
)  X.  { F } )  oF  .x.  ( z  |`  ( { w }  X.  N ) ) )  /\  ( E  |`  ( ( N  \  { w } )  X.  N ) )  =  ( z  |`  ( ( N  \  { w } )  X.  N ) ) )  <->  ( ( E  |`  ( { w }  X.  N ) )  =  ( ( ( { w }  X.  N
)  X.  { F } )  oF  .x.  ( G  |`  ( { w }  X.  N ) ) )  /\  ( E  |`  ( ( N  \  { w } )  X.  N ) )  =  ( G  |`  ( ( N  \  { w } )  X.  N ) ) ) ) )
36 fveq2 5872 . . . . . . . 8  |-  ( z  =  G  ->  ( D `  z )  =  ( D `  G ) )
3736oveq2d 6312 . . . . . . 7  |-  ( z  =  G  ->  ( F  .x.  ( D `  z ) )  =  ( F  .x.  ( D `  G )
) )
3837eqeq2d 2471 . . . . . 6  |-  ( z  =  G  ->  (
( D `  E
)  =  ( F 
.x.  ( D `  z ) )  <->  ( D `  E )  =  ( F  .x.  ( D `
 G ) ) ) )
3935, 38imbi12d 320 . . . . 5  |-  ( z  =  G  ->  (
( ( ( E  |`  ( { w }  X.  N ) )  =  ( ( ( { w }  X.  N
)  X.  { F } )  oF  .x.  ( z  |`  ( { w }  X.  N ) ) )  /\  ( E  |`  ( ( N  \  { w } )  X.  N ) )  =  ( z  |`  ( ( N  \  { w } )  X.  N ) ) )  ->  ( D `  E )  =  ( F  .x.  ( D `
 z ) ) )  <->  ( ( ( E  |`  ( {
w }  X.  N
) )  =  ( ( ( { w }  X.  N )  X. 
{ F } )  oF  .x.  ( G  |`  ( { w }  X.  N ) ) )  /\  ( E  |`  ( ( N  \  { w } )  X.  N ) )  =  ( G  |`  ( ( N  \  { w } )  X.  N ) ) )  ->  ( D `  E )  =  ( F  .x.  ( D `
 G ) ) ) ) )
40 sneq 4042 . . . . . . . . . 10  |-  ( w  =  H  ->  { w }  =  { H } )
4140xpeq1d 5031 . . . . . . . . 9  |-  ( w  =  H  ->  ( { w }  X.  N )  =  ( { H }  X.  N ) )
4241reseq2d 5283 . . . . . . . 8  |-  ( w  =  H  ->  ( E  |`  ( { w }  X.  N ) )  =  ( E  |`  ( { H }  X.  N ) ) )
4341xpeq1d 5031 . . . . . . . . 9  |-  ( w  =  H  ->  (
( { w }  X.  N )  X.  { F } )  =  ( ( { H }  X.  N )  X.  { F } ) )
4441reseq2d 5283 . . . . . . . . 9  |-  ( w  =  H  ->  ( G  |`  ( { w }  X.  N ) )  =  ( G  |`  ( { H }  X.  N ) ) )
4543, 44oveq12d 6314 . . . . . . . 8  |-  ( w  =  H  ->  (
( ( { w }  X.  N )  X. 
{ F } )  oF  .x.  ( G  |`  ( { w }  X.  N ) ) )  =  ( ( ( { H }  X.  N )  X.  { F } )  oF  .x.  ( G  |`  ( { H }  X.  N ) ) ) )
4642, 45eqeq12d 2479 . . . . . . 7  |-  ( w  =  H  ->  (
( E  |`  ( { w }  X.  N ) )  =  ( ( ( { w }  X.  N
)  X.  { F } )  oF  .x.  ( G  |`  ( { w }  X.  N ) ) )  <-> 
( E  |`  ( { H }  X.  N
) )  =  ( ( ( { H }  X.  N )  X. 
{ F } )  oF  .x.  ( G  |`  ( { H }  X.  N ) ) ) ) )
4740difeq2d 3618 . . . . . . . . . 10  |-  ( w  =  H  ->  ( N  \  { w }
)  =  ( N 
\  { H }
) )
4847xpeq1d 5031 . . . . . . . . 9  |-  ( w  =  H  ->  (
( N  \  {
w } )  X.  N )  =  ( ( N  \  { H } )  X.  N
) )
4948reseq2d 5283 . . . . . . . 8  |-  ( w  =  H  ->  ( E  |`  ( ( N 
\  { w }
)  X.  N ) )  =  ( E  |`  ( ( N  \  { H } )  X.  N ) ) )
5048reseq2d 5283 . . . . . . . 8  |-  ( w  =  H  ->  ( G  |`  ( ( N 
\  { w }
)  X.  N ) )  =  ( G  |`  ( ( N  \  { H } )  X.  N ) ) )
5149, 50eqeq12d 2479 . . . . . . 7  |-  ( w  =  H  ->  (
( E  |`  (
( N  \  {
w } )  X.  N ) )  =  ( G  |`  (
( N  \  {
w } )  X.  N ) )  <->  ( E  |`  ( ( N  \  { H } )  X.  N ) )  =  ( G  |`  (
( N  \  { H } )  X.  N
) ) ) )
5246, 51anbi12d 710 . . . . . 6  |-  ( w  =  H  ->  (
( ( E  |`  ( { w }  X.  N ) )  =  ( ( ( { w }  X.  N
)  X.  { F } )  oF  .x.  ( G  |`  ( { w }  X.  N ) ) )  /\  ( E  |`  ( ( N  \  { w } )  X.  N ) )  =  ( G  |`  ( ( N  \  { w } )  X.  N ) ) )  <->  ( ( E  |`  ( { H }  X.  N ) )  =  ( ( ( { H }  X.  N
)  X.  { F } )  oF  .x.  ( G  |`  ( { H }  X.  N ) ) )  /\  ( E  |`  ( ( N  \  { H } )  X.  N ) )  =  ( G  |`  (
( N  \  { H } )  X.  N
) ) ) ) )
5352imbi1d 317 . . . . 5  |-  ( w  =  H  ->  (
( ( ( E  |`  ( { w }  X.  N ) )  =  ( ( ( { w }  X.  N
)  X.  { F } )  oF  .x.  ( G  |`  ( { w }  X.  N ) ) )  /\  ( E  |`  ( ( N  \  { w } )  X.  N ) )  =  ( G  |`  ( ( N  \  { w } )  X.  N ) ) )  ->  ( D `  E )  =  ( F  .x.  ( D `
 G ) ) )  <->  ( ( ( E  |`  ( { H }  X.  N
) )  =  ( ( ( { H }  X.  N )  X. 
{ F } )  oF  .x.  ( G  |`  ( { H }  X.  N ) ) )  /\  ( E  |`  ( ( N  \  { H } )  X.  N ) )  =  ( G  |`  (
( N  \  { H } )  X.  N
) ) )  -> 
( D `  E
)  =  ( F 
.x.  ( D `  G ) ) ) ) )
5439, 53rspc2va 3220 . . . 4  |-  ( ( ( G  e.  B  /\  H  e.  N
)  /\  A. z  e.  B  A. w  e.  N  ( (
( E  |`  ( { w }  X.  N ) )  =  ( ( ( { w }  X.  N
)  X.  { F } )  oF  .x.  ( z  |`  ( { w }  X.  N ) ) )  /\  ( E  |`  ( ( N  \  { w } )  X.  N ) )  =  ( z  |`  ( ( N  \  { w } )  X.  N ) ) )  ->  ( D `  E )  =  ( F  .x.  ( D `
 z ) ) ) )  ->  (
( ( E  |`  ( { H }  X.  N ) )  =  ( ( ( { H }  X.  N
)  X.  { F } )  oF  .x.  ( G  |`  ( { H }  X.  N ) ) )  /\  ( E  |`  ( ( N  \  { H } )  X.  N ) )  =  ( G  |`  (
( N  \  { H } )  X.  N
) ) )  -> 
( D `  E
)  =  ( F 
.x.  ( D `  G ) ) ) )
554, 5, 29, 54syl21anc 1227 . . 3  |-  ( (
ph  /\  ( E  e.  B  /\  F  e.  K  /\  G  e.  B )  /\  H  e.  N )  ->  (
( ( E  |`  ( { H }  X.  N ) )  =  ( ( ( { H }  X.  N
)  X.  { F } )  oF  .x.  ( G  |`  ( { H }  X.  N ) ) )  /\  ( E  |`  ( ( N  \  { H } )  X.  N ) )  =  ( G  |`  (
( N  \  { H } )  X.  N
) ) )  -> 
( D `  E
)  =  ( F 
.x.  ( D `  G ) ) ) )
563, 55syl3an3 1263 . 2  |-  ( (
ph  /\  ( E  e.  B  /\  F  e.  K  /\  G  e.  B )  /\  ( H  e.  N  /\  ( E  |`  ( { H }  X.  N
) )  =  ( ( ( { H }  X.  N )  X. 
{ F } )  oF  .x.  ( G  |`  ( { H }  X.  N ) ) )  /\  ( E  |`  ( ( N  \  { H } )  X.  N ) )  =  ( G  |`  (
( N  \  { H } )  X.  N
) ) ) )  ->  ( ( ( E  |`  ( { H }  X.  N
) )  =  ( ( ( { H }  X.  N )  X. 
{ F } )  oF  .x.  ( G  |`  ( { H }  X.  N ) ) )  /\  ( E  |`  ( ( N  \  { H } )  X.  N ) )  =  ( G  |`  (
( N  \  { H } )  X.  N
) ) )  -> 
( D `  E
)  =  ( F 
.x.  ( D `  G ) ) ) )
571, 2, 56mp2and 679 1  |-  ( (
ph  /\  ( E  e.  B  /\  F  e.  K  /\  G  e.  B )  /\  ( H  e.  N  /\  ( E  |`  ( { H }  X.  N
) )  =  ( ( ( { H }  X.  N )  X. 
{ F } )  oF  .x.  ( G  |`  ( { H }  X.  N ) ) )  /\  ( E  |`  ( ( N  \  { H } )  X.  N ) )  =  ( G  |`  (
( N  \  { H } )  X.  N
) ) ) )  ->  ( D `  E )  =  ( F  .x.  ( D `
 G ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807    \ cdif 3468   {csn 4032    X. cxp 5006    |` cres 5010   -->wf 5590   ` cfv 5594  (class class class)co 6296    oFcof 6537   Fincfn 7535   Basecbs 14644   +g cplusg 14712   .rcmulr 14713   0gc0g 14857   1rcur 17280   Ringcrg 17325   Mat cmat 19036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-xp 5014  df-res 5020  df-iota 5557  df-fv 5602  df-ov 6299
This theorem is referenced by:  mdetuni0  19250
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