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Theorem mplcoe3OLD 17976
Description: Decompose a monomial in one variable into a power of a variable. (Contributed by Mario Carneiro, 7-Jan-2015.) Obsolete version of mplcoe3 17975 as of 18-Jul-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
mplcoe1.p  |-  P  =  ( I mPoly  R )
mplcoe1.d  |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
mplcoe1.z  |-  .0.  =  ( 0g `  R )
mplcoe1.o  |-  .1.  =  ( 1r `  R )
mplcoe1.i  |-  ( ph  ->  I  e.  W )
mplcoe2.g  |-  G  =  (mulGrp `  P )
mplcoe2.m  |-  .^  =  (.g
`  G )
mplcoe2.v  |-  V  =  ( I mVar  R )
mplcoe3.r  |-  ( ph  ->  R  e.  Ring )
mplcoe3.x  |-  ( ph  ->  X  e.  I )
mplcoe3.n  |-  ( ph  ->  N  e.  NN0 )
Assertion
Ref Expression
mplcoe3OLD  |-  ( ph  ->  ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  N , 
0 ) ) ,  .1.  ,  .0.  )
)  =  ( N 
.^  ( V `  X ) ) )
Distinct variable groups:    .^ , k    y,
k,  .1.    k, G    f,
k, y, I    k, N, y    ph, k, y    R, f, y    D, k, y    P, k    k, V    .0. , f, k, y    f, X, k, y    k, W, y
Allowed substitution hints:    ph( f)    D( f)    P( y, f)    R( k)    .1. ( f)    .^ ( y, f)    G( y, f)    N( f)    V( y, f)    W( f)

Proof of Theorem mplcoe3OLD
Dummy variables  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mplcoe3.n . 2  |-  ( ph  ->  N  e.  NN0 )
2 ifeq1 3948 . . . . . . . . . . 11  |-  ( x  =  0  ->  if ( k  =  X ,  x ,  0 )  =  if ( k  =  X , 
0 ,  0 ) )
3 ifid 3981 . . . . . . . . . . 11  |-  if ( k  =  X , 
0 ,  0 )  =  0
42, 3syl6eq 2524 . . . . . . . . . 10  |-  ( x  =  0  ->  if ( k  =  X ,  x ,  0 )  =  0 )
54mpteq2dv 4539 . . . . . . . . 9  |-  ( x  =  0  ->  (
k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  =  ( k  e.  I  |->  0 ) )
6 fconstmpt 5048 . . . . . . . . 9  |-  ( I  X.  { 0 } )  =  ( k  e.  I  |->  0 )
75, 6syl6eqr 2526 . . . . . . . 8  |-  ( x  =  0  ->  (
k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  =  ( I  X.  { 0 } ) )
87eqeq2d 2481 . . . . . . 7  |-  ( x  =  0  ->  (
y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  <->  y  =  ( I  X.  { 0 } ) ) )
98ifbid 3966 . . . . . 6  |-  ( x  =  0  ->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) ) ,  .1.  ,  .0.  )  =  if ( y  =  ( I  X.  { 0 } ) ,  .1.  ,  .0.  ) )
109mpteq2dv 4539 . . . . 5  |-  ( x  =  0  ->  (
y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) ) ,  .1.  ,  .0.  ) )  =  ( y  e.  D  |->  if ( y  =  ( I  X.  {
0 } ) ,  .1.  ,  .0.  )
) )
11 oveq1 6301 . . . . 5  |-  ( x  =  0  ->  (
x  .^  ( V `  X ) )  =  ( 0  .^  ( V `  X )
) )
1210, 11eqeq12d 2489 . . . 4  |-  ( x  =  0  ->  (
( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) ) ,  .1.  ,  .0.  )
)  =  ( x 
.^  ( V `  X ) )  <->  ( y  e.  D  |->  if ( y  =  ( I  X.  { 0 } ) ,  .1.  ,  .0.  ) )  =  ( 0  .^  ( V `  X ) ) ) )
1312imbi2d 316 . . 3  |-  ( x  =  0  ->  (
( ph  ->  ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) ) ,  .1.  ,  .0.  ) )  =  ( x  .^  ( V `  X ) ) )  <-> 
( ph  ->  ( y  e.  D  |->  if ( y  =  ( I  X.  { 0 } ) ,  .1.  ,  .0.  ) )  =  ( 0  .^  ( V `  X ) ) ) ) )
14 ifeq1 3948 . . . . . . . . 9  |-  ( x  =  n  ->  if ( k  =  X ,  x ,  0 )  =  if ( k  =  X ,  n ,  0 ) )
1514mpteq2dv 4539 . . . . . . . 8  |-  ( x  =  n  ->  (
k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  =  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) )
1615eqeq2d 2481 . . . . . . 7  |-  ( x  =  n  ->  (
y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  <->  y  =  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) ) )
1716ifbid 3966 . . . . . 6  |-  ( x  =  n  ->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) ) ,  .1.  ,  .0.  )  =  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) ,  .1.  ,  .0.  ) )
1817mpteq2dv 4539 . . . . 5  |-  ( x  =  n  ->  (
y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) ) ,  .1.  ,  .0.  ) )  =  ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) ,  .1.  ,  .0.  )
) )
19 oveq1 6301 . . . . 5  |-  ( x  =  n  ->  (
x  .^  ( V `  X ) )  =  ( n  .^  ( V `  X )
) )
2018, 19eqeq12d 2489 . . . 4  |-  ( x  =  n  ->  (
( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) ) ,  .1.  ,  .0.  )
)  =  ( x 
.^  ( V `  X ) )  <->  ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) ,  .1.  ,  .0.  ) )  =  ( n  .^  ( V `  X ) ) ) )
2120imbi2d 316 . . 3  |-  ( x  =  n  ->  (
( ph  ->  ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) ) ,  .1.  ,  .0.  ) )  =  ( x  .^  ( V `  X ) ) )  <-> 
( ph  ->  ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) ,  .1.  ,  .0.  ) )  =  ( n  .^  ( V `  X ) ) ) ) )
22 ifeq1 3948 . . . . . . . . 9  |-  ( x  =  ( n  + 
1 )  ->  if ( k  =  X ,  x ,  0 )  =  if ( k  =  X , 
( n  +  1 ) ,  0 ) )
2322mpteq2dv 4539 . . . . . . . 8  |-  ( x  =  ( n  + 
1 )  ->  (
k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  =  ( k  e.  I  |->  if ( k  =  X ,  ( n  + 
1 ) ,  0 ) ) )
2423eqeq2d 2481 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  (
y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  <->  y  =  ( k  e.  I  |->  if ( k  =  X ,  ( n  + 
1 ) ,  0 ) ) ) )
2524ifbid 3966 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) ) ,  .1.  ,  .0.  )  =  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  ( n  + 
1 ) ,  0 ) ) ,  .1.  ,  .0.  ) )
2625mpteq2dv 4539 . . . . 5  |-  ( x  =  ( n  + 
1 )  ->  (
y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) ) ,  .1.  ,  .0.  ) )  =  ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  ( n  +  1 ) ,  0 ) ) ,  .1.  ,  .0.  )
) )
27 oveq1 6301 . . . . 5  |-  ( x  =  ( n  + 
1 )  ->  (
x  .^  ( V `  X ) )  =  ( ( n  + 
1 )  .^  ( V `  X )
) )
2826, 27eqeq12d 2489 . . . 4  |-  ( x  =  ( n  + 
1 )  ->  (
( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) ) ,  .1.  ,  .0.  )
)  =  ( x 
.^  ( V `  X ) )  <->  ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X , 
( n  +  1 ) ,  0 ) ) ,  .1.  ,  .0.  ) )  =  ( ( n  +  1 )  .^  ( V `  X ) ) ) )
2928imbi2d 316 . . 3  |-  ( x  =  ( n  + 
1 )  ->  (
( ph  ->  ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) ) ,  .1.  ,  .0.  ) )  =  ( x  .^  ( V `  X ) ) )  <-> 
( ph  ->  ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X , 
( n  +  1 ) ,  0 ) ) ,  .1.  ,  .0.  ) )  =  ( ( n  +  1 )  .^  ( V `  X ) ) ) ) )
30 ifeq1 3948 . . . . . . . . 9  |-  ( x  =  N  ->  if ( k  =  X ,  x ,  0 )  =  if ( k  =  X ,  N ,  0 ) )
3130mpteq2dv 4539 . . . . . . . 8  |-  ( x  =  N  ->  (
k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  =  ( k  e.  I  |->  if ( k  =  X ,  N ,  0 ) ) )
3231eqeq2d 2481 . . . . . . 7  |-  ( x  =  N  ->  (
y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  <->  y  =  ( k  e.  I  |->  if ( k  =  X ,  N ,  0 ) ) ) )
3332ifbid 3966 . . . . . 6  |-  ( x  =  N  ->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) ) ,  .1.  ,  .0.  )  =  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  N ,  0 ) ) ,  .1.  ,  .0.  ) )
3433mpteq2dv 4539 . . . . 5  |-  ( x  =  N  ->  (
y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) ) ,  .1.  ,  .0.  ) )  =  ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  N , 
0 ) ) ,  .1.  ,  .0.  )
) )
35 oveq1 6301 . . . . 5  |-  ( x  =  N  ->  (
x  .^  ( V `  X ) )  =  ( N  .^  ( V `  X )
) )
3634, 35eqeq12d 2489 . . . 4  |-  ( x  =  N  ->  (
( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) ) ,  .1.  ,  .0.  )
)  =  ( x 
.^  ( V `  X ) )  <->  ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  N ,  0 ) ) ,  .1.  ,  .0.  ) )  =  ( N  .^  ( V `  X ) ) ) )
3736imbi2d 316 . . 3  |-  ( x  =  N  ->  (
( ph  ->  ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) ) ,  .1.  ,  .0.  ) )  =  ( x  .^  ( V `  X ) ) )  <-> 
( ph  ->  ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  N ,  0 ) ) ,  .1.  ,  .0.  ) )  =  ( N  .^  ( V `  X ) ) ) ) )
38 mplcoe1.p . . . . . 6  |-  P  =  ( I mPoly  R )
39 mplcoe2.v . . . . . 6  |-  V  =  ( I mVar  R )
40 eqid 2467 . . . . . 6  |-  ( Base `  P )  =  (
Base `  P )
41 mplcoe1.i . . . . . 6  |-  ( ph  ->  I  e.  W )
42 mplcoe3.r . . . . . 6  |-  ( ph  ->  R  e.  Ring )
43 mplcoe3.x . . . . . 6  |-  ( ph  ->  X  e.  I )
4438, 39, 40, 41, 42, 43mvrcl 17958 . . . . 5  |-  ( ph  ->  ( V `  X
)  e.  ( Base `  P ) )
45 mplcoe2.g . . . . . . 7  |-  G  =  (mulGrp `  P )
4645, 40mgpbas 16996 . . . . . 6  |-  ( Base `  P )  =  (
Base `  G )
47 eqid 2467 . . . . . . 7  |-  ( 1r
`  P )  =  ( 1r `  P
)
4845, 47rngidval 17004 . . . . . 6  |-  ( 1r
`  P )  =  ( 0g `  G
)
49 mplcoe2.m . . . . . 6  |-  .^  =  (.g
`  G )
5046, 48, 49mulg0 15996 . . . . 5  |-  ( ( V `  X )  e.  ( Base `  P
)  ->  ( 0 
.^  ( V `  X ) )  =  ( 1r `  P
) )
5144, 50syl 16 . . . 4  |-  ( ph  ->  ( 0  .^  ( V `  X )
)  =  ( 1r
`  P ) )
52 mplcoe1.d . . . . 5  |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
53 mplcoe1.z . . . . 5  |-  .0.  =  ( 0g `  R )
54 mplcoe1.o . . . . 5  |-  .1.  =  ( 1r `  R )
5538, 52, 53, 54, 47, 41, 42mpl1 17953 . . . 4  |-  ( ph  ->  ( 1r `  P
)  =  ( y  e.  D  |->  if ( y  =  ( I  X.  { 0 } ) ,  .1.  ,  .0.  ) ) )
5651, 55eqtr2d 2509 . . 3  |-  ( ph  ->  ( y  e.  D  |->  if ( y  =  ( I  X.  {
0 } ) ,  .1.  ,  .0.  )
)  =  ( 0 
.^  ( V `  X ) ) )
57 oveq1 6301 . . . . . 6  |-  ( ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) ,  .1.  ,  .0.  ) )  =  ( n  .^  ( V `  X )
)  ->  ( (
y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) ,  .1.  ,  .0.  ) ) ( .r `  P ) ( V `  X
) )  =  ( ( n  .^  ( V `  X )
) ( .r `  P ) ( V `
 X ) ) )
5841adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN0 )  ->  I  e.  W )
5942adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN0 )  ->  R  e.  Ring )
60 simplr 754 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  k  e.  I )  ->  n  e.  NN0 )
61 0nn0 10820 . . . . . . . . . . . 12  |-  0  e.  NN0
62 ifcl 3986 . . . . . . . . . . . 12  |-  ( ( n  e.  NN0  /\  0  e.  NN0 )  ->  if ( k  =  X ,  n ,  0 )  e.  NN0 )
6360, 61, 62sylancl 662 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  k  e.  I )  ->  if ( k  =  X ,  n ,  0 )  e.  NN0 )
64 eqid 2467 . . . . . . . . . . 11  |-  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )  =  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )
6563, 64fmptd 6055 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) : I --> NN0 )
66 nn0suppOLD 10860 . . . . . . . . . . . 12  |-  ( ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) : I --> NN0  ->  ( `' ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) "
( _V  \  {
0 } ) )  =  ( `' ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) " NN ) )
6765, 66syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( `' ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) "
( _V  \  {
0 } ) )  =  ( `' ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) " NN ) )
68 snfi 7606 . . . . . . . . . . . 12  |-  { X }  e.  Fin
69 eldifsni 4158 . . . . . . . . . . . . . . . 16  |-  ( k  e.  ( I  \  { X } )  -> 
k  =/=  X )
7069adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  k  e.  ( I  \  { X } ) )  -> 
k  =/=  X )
7170neneqd 2669 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  k  e.  ( I  \  { X } ) )  ->  -.  k  =  X
)
72 iffalse 3953 . . . . . . . . . . . . . 14  |-  ( -.  k  =  X  ->  if ( k  =  X ,  n ,  0 )  =  0 )
7371, 72syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  k  e.  ( I  \  { X } ) )  ->  if ( k  =  X ,  n ,  0 )  =  0 )
7473suppss2OLD 6524 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( `' ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) "
( _V  \  {
0 } ) ) 
C_  { X }
)
75 ssfi 7750 . . . . . . . . . . . 12  |-  ( ( { X }  e.  Fin  /\  ( `' ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) " ( _V  \  { 0 } ) )  C_  { X } )  ->  ( `' ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )
" ( _V  \  { 0 } ) )  e.  Fin )
7668, 74, 75sylancr 663 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( `' ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) "
( _V  \  {
0 } ) )  e.  Fin )
7767, 76eqeltrrd 2556 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( `' ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) " NN )  e.  Fin )
7852psrbag 17860 . . . . . . . . . . 11  |-  ( I  e.  W  ->  (
( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )  e.  D  <->  ( ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) : I --> NN0  /\  ( `' ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )
" NN )  e. 
Fin ) ) )
7958, 78syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( (
k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )  e.  D  <->  ( ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) : I --> NN0  /\  ( `' ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )
" NN )  e. 
Fin ) ) )
8065, 77, 79mpbir2and 920 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )  e.  D )
81 eqid 2467 . . . . . . . . 9  |-  ( .r
`  P )  =  ( .r `  P
)
8252mvridlemOLD 17922 . . . . . . . . . 10  |-  ( I  e.  W  ->  (
k  e.  I  |->  if ( k  =  X ,  1 ,  0 ) )  e.  D
)
8358, 82syl 16 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( k  e.  I  |->  if ( k  =  X , 
1 ,  0 ) )  e.  D )
8438, 40, 53, 54, 52, 58, 59, 80, 81, 83mplmonmul 17973 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( (
y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) ,  .1.  ,  .0.  ) ) ( .r `  P ) ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  1 ,  0 ) ) ,  .1.  ,  .0.  )
) )  =  ( y  e.  D  |->  if ( y  =  ( ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )  oF  +  ( k  e.  I  |->  if ( k  =  X , 
1 ,  0 ) ) ) ,  .1.  ,  .0.  ) ) )
8543adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN0 )  ->  X  e.  I )
8639, 52, 53, 54, 58, 59, 85mvrval 17924 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( V `  X )  =  ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  1 ,  0 ) ) ,  .1.  ,  .0.  ) ) )
8786eqcomd 2475 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X , 
1 ,  0 ) ) ,  .1.  ,  .0.  ) )  =  ( V `  X ) )
8887oveq2d 6310 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( (
y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) ,  .1.  ,  .0.  ) ) ( .r `  P ) ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  1 ,  0 ) ) ,  .1.  ,  .0.  )
) )  =  ( ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) ,  .1.  ,  .0.  )
) ( .r `  P ) ( V `
 X ) ) )
89 1nn0 10821 . . . . . . . . . . . . . . 15  |-  1  e.  NN0
9089, 61keepel 4012 . . . . . . . . . . . . . 14  |-  if ( k  =  X , 
1 ,  0 )  e.  NN0
9190a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  k  e.  I )  ->  if ( k  =  X ,  1 ,  0 )  e.  NN0 )
92 eqidd 2468 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )  =  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) )
93 eqidd 2468 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( k  e.  I  |->  if ( k  =  X , 
1 ,  0 ) )  =  ( k  e.  I  |->  if ( k  =  X , 
1 ,  0 ) ) )
9458, 63, 91, 92, 93offval2 6550 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( (
k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )  oF  +  ( k  e.  I  |->  if ( k  =  X ,  1 ,  0 ) ) )  =  ( k  e.  I  |->  ( if ( k  =  X ,  n ,  0 )  +  if ( k  =  X , 
1 ,  0 ) ) ) )
95 iftrue 3950 . . . . . . . . . . . . . . . 16  |-  ( k  =  X  ->  if ( k  =  X ,  n ,  0 )  =  n )
96 iftrue 3950 . . . . . . . . . . . . . . . 16  |-  ( k  =  X  ->  if ( k  =  X ,  1 ,  0 )  =  1 )
9795, 96oveq12d 6312 . . . . . . . . . . . . . . 15  |-  ( k  =  X  ->  ( if ( k  =  X ,  n ,  0 )  +  if ( k  =  X , 
1 ,  0 ) )  =  ( n  +  1 ) )
98 iftrue 3950 . . . . . . . . . . . . . . 15  |-  ( k  =  X  ->  if ( k  =  X ,  ( n  + 
1 ) ,  0 )  =  ( n  +  1 ) )
9997, 98eqtr4d 2511 . . . . . . . . . . . . . 14  |-  ( k  =  X  ->  ( if ( k  =  X ,  n ,  0 )  +  if ( k  =  X , 
1 ,  0 ) )  =  if ( k  =  X , 
( n  +  1 ) ,  0 ) )
100 00id 9764 . . . . . . . . . . . . . . 15  |-  ( 0  +  0 )  =  0
101 iffalse 3953 . . . . . . . . . . . . . . . 16  |-  ( -.  k  =  X  ->  if ( k  =  X ,  1 ,  0 )  =  0 )
10272, 101oveq12d 6312 . . . . . . . . . . . . . . 15  |-  ( -.  k  =  X  -> 
( if ( k  =  X ,  n ,  0 )  +  if ( k  =  X ,  1 ,  0 ) )  =  ( 0  +  0 ) )
103 iffalse 3953 . . . . . . . . . . . . . . 15  |-  ( -.  k  =  X  ->  if ( k  =  X ,  ( n  + 
1 ) ,  0 )  =  0 )
104100, 102, 1033eqtr4a 2534 . . . . . . . . . . . . . 14  |-  ( -.  k  =  X  -> 
( if ( k  =  X ,  n ,  0 )  +  if ( k  =  X ,  1 ,  0 ) )  =  if ( k  =  X ,  ( n  +  1 ) ,  0 ) )
10599, 104pm2.61i 164 . . . . . . . . . . . . 13  |-  ( if ( k  =  X ,  n ,  0 )  +  if ( k  =  X , 
1 ,  0 ) )  =  if ( k  =  X , 
( n  +  1 ) ,  0 )
106105mpteq2i 4535 . . . . . . . . . . . 12  |-  ( k  e.  I  |->  ( if ( k  =  X ,  n ,  0 )  +  if ( k  =  X , 
1 ,  0 ) ) )  =  ( k  e.  I  |->  if ( k  =  X ,  ( n  + 
1 ) ,  0 ) )
10794, 106syl6eq 2524 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( (
k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )  oF  +  ( k  e.  I  |->  if ( k  =  X ,  1 ,  0 ) ) )  =  ( k  e.  I  |->  if ( k  =  X , 
( n  +  1 ) ,  0 ) ) )
108107eqeq2d 2481 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( y  =  ( ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )  oF  +  ( k  e.  I  |->  if ( k  =  X ,  1 ,  0 ) ) )  <-> 
y  =  ( k  e.  I  |->  if ( k  =  X , 
( n  +  1 ) ,  0 ) ) ) )
109108ifbid 3966 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN0 )  ->  if (
y  =  ( ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )  oF  +  ( k  e.  I  |->  if ( k  =  X ,  1 ,  0 ) ) ) ,  .1.  ,  .0.  )  =  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  ( n  + 
1 ) ,  0 ) ) ,  .1.  ,  .0.  ) )
110109mpteq2dv 4539 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( y  e.  D  |->  if ( y  =  ( ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )  oF  +  ( k  e.  I  |->  if ( k  =  X ,  1 ,  0 ) ) ) ,  .1.  ,  .0.  ) )  =  ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  ( n  + 
1 ) ,  0 ) ) ,  .1.  ,  .0.  ) ) )
11184, 88, 1103eqtr3rd 2517 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X , 
( n  +  1 ) ,  0 ) ) ,  .1.  ,  .0.  ) )  =  ( ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) ,  .1.  ,  .0.  )
) ( .r `  P ) ( V `
 X ) ) )
11238mplring 17961 . . . . . . . . . . 11  |-  ( ( I  e.  W  /\  R  e.  Ring )  ->  P  e.  Ring )
11341, 42, 112syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  P  e.  Ring )
11445ringmgp 17053 . . . . . . . . . 10  |-  ( P  e.  Ring  ->  G  e. 
Mnd )
115113, 114syl 16 . . . . . . . . 9  |-  ( ph  ->  G  e.  Mnd )
116115adantr 465 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN0 )  ->  G  e.  Mnd )
117 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN0 )  ->  n  e.  NN0 )
11844adantr 465 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( V `  X )  e.  (
Base `  P )
)
11945, 81mgpplusg 16994 . . . . . . . . 9  |-  ( .r
`  P )  =  ( +g  `  G
)
12046, 49, 119mulgnn0p1 16002 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  n  e.  NN0  /\  ( V `  X )  e.  ( Base `  P
) )  ->  (
( n  +  1 )  .^  ( V `  X ) )  =  ( ( n  .^  ( V `  X ) ) ( .r `  P ) ( V `
 X ) ) )
121116, 117, 118, 120syl3anc 1228 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( (
n  +  1 ) 
.^  ( V `  X ) )  =  ( ( n  .^  ( V `  X ) ) ( .r `  P ) ( V `
 X ) ) )
122111, 121eqeq12d 2489 . . . . . 6  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( (
y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  ( n  + 
1 ) ,  0 ) ) ,  .1.  ,  .0.  ) )  =  ( ( n  + 
1 )  .^  ( V `  X )
)  <->  ( ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) ,  .1.  ,  .0.  ) ) ( .r
`  P ) ( V `  X ) )  =  ( ( n  .^  ( V `  X ) ) ( .r `  P ) ( V `  X
) ) ) )
12357, 122syl5ibr 221 . . . . 5  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( (
y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) ,  .1.  ,  .0.  ) )  =  ( n  .^  ( V `  X )
)  ->  ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X , 
( n  +  1 ) ,  0 ) ) ,  .1.  ,  .0.  ) )  =  ( ( n  +  1 )  .^  ( V `  X ) ) ) )
124123expcom 435 . . . 4  |-  ( n  e.  NN0  ->  ( ph  ->  ( ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) ,  .1.  ,  .0.  ) )  =  ( n  .^  ( V `  X ) )  -> 
( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  ( n  +  1 ) ,  0 ) ) ,  .1.  ,  .0.  )
)  =  ( ( n  +  1 ) 
.^  ( V `  X ) ) ) ) )
125124a2d 26 . . 3  |-  ( n  e.  NN0  ->  ( (
ph  ->  ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) ,  .1.  ,  .0.  ) )  =  ( n  .^  ( V `  X ) ) )  ->  ( ph  ->  ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  ( n  + 
1 ) ,  0 ) ) ,  .1.  ,  .0.  ) )  =  ( ( n  + 
1 )  .^  ( V `  X )
) ) ) )
12613, 21, 29, 37, 56, 125nn0ind 10967 . 2  |-  ( N  e.  NN0  ->  ( ph  ->  ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  N , 
0 ) ) ,  .1.  ,  .0.  )
)  =  ( N 
.^  ( V `  X ) ) ) )
1271, 126mpcom 36 1  |-  ( ph  ->  ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  N , 
0 ) ) ,  .1.  ,  .0.  )
)  =  ( N 
.^  ( V `  X ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   {crab 2821   _Vcvv 3118    \ cdif 3478    C_ wss 3481   ifcif 3944   {csn 4032    |-> cmpt 4510    X. cxp 5002   `'ccnv 5003   "cima 5007   -->wf 5589   ` cfv 5593  (class class class)co 6294    oFcof 6532    ^m cmap 7430   Fincfn 7526   0cc0 9502   1c1 9503    + caddc 9505   NNcn 10546   NN0cn0 10805   Basecbs 14502   .rcmulr 14568   0gc0g 14707   Mndcmnd 15772  .gcmg 15905  mulGrpcmgp 16990   1rcur 17002   Ringcrg 17047   mVar cmvr 17848   mPoly cmpl 17849
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 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-inf2 8068  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-iin 4333  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-se 4844  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-isom 5602  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-of 6534  df-ofr 6535  df-om 6695  df-1st 6794  df-2nd 6795  df-supp 6912  df-recs 7052  df-rdg 7086  df-1o 7140  df-2o 7141  df-oadd 7144  df-er 7321  df-map 7432  df-pm 7433  df-ixp 7480  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-fsupp 7840  df-oi 7945  df-card 8330  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-nn 10547  df-2 10604  df-3 10605  df-4 10606  df-5 10607  df-6 10608  df-7 10609  df-8 10610  df-9 10611  df-n0 10806  df-z 10875  df-uz 11093  df-fz 11683  df-fzo 11803  df-seq 12086  df-hash 12384  df-struct 14504  df-ndx 14505  df-slot 14506  df-base 14507  df-sets 14508  df-ress 14509  df-plusg 14580  df-mulr 14581  df-sca 14583  df-vsca 14584  df-tset 14586  df-0g 14709  df-gsum 14710  df-mre 14853  df-mrc 14854  df-acs 14856  df-mgm 15741  df-sgrp 15764  df-mnd 15774  df-mhm 15819  df-submnd 15820  df-grp 15906  df-minusg 15907  df-mulg 15909  df-subg 16047  df-ghm 16114  df-cntz 16204  df-cmn 16650  df-abl 16651  df-mgp 16991  df-ur 17003  df-ring 17049  df-subrg 17275  df-psr 17852  df-mvr 17853  df-mpl 17854
This theorem is referenced by: (None)
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