MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mplcoe3 Unicode version

Theorem mplcoe3 16210
Description: Decompose a monomial in one variable into a power of a variable. (Contributed by Mario Carneiro, 7-Jan-2015.)
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
mplcoe3  |-  ( 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   
k, W    .0. , f,
k, y    f, X, k, 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( y, f)

Proof of Theorem mplcoe3
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 3569 . . . . . . . . . . 11  |-  ( x  =  0  ->  if ( k  =  X ,  x ,  0 )  =  if ( k  =  X , 
0 ,  0 ) )
3 ifid 3597 . . . . . . . . . . 11  |-  if ( k  =  X , 
0 ,  0 )  =  0
42, 3syl6eq 2331 . . . . . . . . . 10  |-  ( x  =  0  ->  if ( k  =  X ,  x ,  0 )  =  0 )
54mpteq2dv 4107 . . . . . . . . 9  |-  ( x  =  0  ->  (
k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  =  ( k  e.  I  |->  0 ) )
6 fconstmpt 4732 . . . . . . . . 9  |-  ( I  X.  { 0 } )  =  ( k  e.  I  |->  0 )
75, 6syl6eqr 2333 . . . . . . . 8  |-  ( x  =  0  ->  (
k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  =  ( I  X.  { 0 } ) )
87eqeq2d 2294 . . . . . . 7  |-  ( x  =  0  ->  (
y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  <->  y  =  ( I  X.  { 0 } ) ) )
98ifbid 3583 . . . . . 6  |-  ( x  =  0  ->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) ) ,  .1.  ,  .0.  )  =  if ( y  =  ( I  X.  { 0 } ) ,  .1.  ,  .0.  ) )
109mpteq2dv 4107 . . . . 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 5865 . . . . 5  |-  ( x  =  0  ->  (
x  .^  ( V `  X ) )  =  ( 0  .^  ( V `  X )
) )
1210, 11eqeq12d 2297 . . . 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 307 . . 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 3569 . . . . . . . . 9  |-  ( x  =  n  ->  if ( k  =  X ,  x ,  0 )  =  if ( k  =  X ,  n ,  0 ) )
1514mpteq2dv 4107 . . . . . . . 8  |-  ( x  =  n  ->  (
k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  =  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) )
1615eqeq2d 2294 . . . . . . 7  |-  ( x  =  n  ->  (
y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  <->  y  =  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) ) )
1716ifbid 3583 . . . . . 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 4107 . . . . 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 5865 . . . . 5  |-  ( x  =  n  ->  (
x  .^  ( V `  X ) )  =  ( n  .^  ( V `  X )
) )
2018, 19eqeq12d 2297 . . . 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 307 . . 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 3569 . . . . . . . . 9  |-  ( x  =  ( n  + 
1 )  ->  if ( k  =  X ,  x ,  0 )  =  if ( k  =  X , 
( n  +  1 ) ,  0 ) )
2322mpteq2dv 4107 . . . . . . . 8  |-  ( x  =  ( n  + 
1 )  ->  (
k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  =  ( k  e.  I  |->  if ( k  =  X ,  ( n  + 
1 ) ,  0 ) ) )
2423eqeq2d 2294 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  (
y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  <->  y  =  ( k  e.  I  |->  if ( k  =  X ,  ( n  + 
1 ) ,  0 ) ) ) )
2524ifbid 3583 . . . . . 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 4107 . . . . 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 5865 . . . . 5  |-  ( x  =  ( n  + 
1 )  ->  (
x  .^  ( V `  X ) )  =  ( ( n  + 
1 )  .^  ( V `  X )
) )
2826, 27eqeq12d 2297 . . . 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 307 . . 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 3569 . . . . . . . . 9  |-  ( x  =  N  ->  if ( k  =  X ,  x ,  0 )  =  if ( k  =  X ,  N ,  0 ) )
3130mpteq2dv 4107 . . . . . . . 8  |-  ( x  =  N  ->  (
k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  =  ( k  e.  I  |->  if ( k  =  X ,  N ,  0 ) ) )
3231eqeq2d 2294 . . . . . . 7  |-  ( x  =  N  ->  (
y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  <->  y  =  ( k  e.  I  |->  if ( k  =  X ,  N ,  0 ) ) ) )
3332ifbid 3583 . . . . . 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 4107 . . . . 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 5865 . . . . 5  |-  ( x  =  N  ->  (
x  .^  ( V `  X ) )  =  ( N  .^  ( V `  X )
) )
3634, 35eqeq12d 2297 . . . 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 307 . . 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 2283 . . . . . 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 16193 . . . . 5  |-  ( ph  ->  ( V `  X
)  e.  ( Base `  P ) )
45 mplcoe2.g . . . . . . 7  |-  G  =  (mulGrp `  P )
4645, 40mgpbas 15331 . . . . . 6  |-  ( Base `  P )  =  (
Base `  G )
47 eqid 2283 . . . . . . 7  |-  ( 1r
`  P )  =  ( 1r `  P
)
4845, 47rngidval 15343 . . . . . 6  |-  ( 1r
`  P )  =  ( 0g `  G
)
49 mplcoe2.m . . . . . 6  |-  .^  =  (.g
`  G )
5046, 48, 49mulg0 14572 . . . . 5  |-  ( ( V `  X )  e.  ( Base `  P
)  ->  ( 0 
.^  ( V `  X ) )  =  ( 1r `  P
) )
5144, 50syl 15 . . . 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 16188 . . . 4  |-  ( ph  ->  ( 1r `  P
)  =  ( y  e.  D  |->  if ( y  =  ( I  X.  { 0 } ) ,  .1.  ,  .0.  ) ) )
5651, 55eqtr2d 2316 . . 3  |-  ( ph  ->  ( y  e.  D  |->  if ( y  =  ( I  X.  {
0 } ) ,  .1.  ,  .0.  )
)  =  ( 0 
.^  ( V `  X ) ) )
57 oveq1 5865 . . . . . 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 451 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN0 )  ->  I  e.  W )
5942adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN0 )  ->  R  e.  Ring )
60 simplr 731 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  k  e.  I )  ->  n  e.  NN0 )
61 0nn0 9980 . . . . . . . . . . . 12  |-  0  e.  NN0
62 ifcl 3601 . . . . . . . . . . . 12  |-  ( ( n  e.  NN0  /\  0  e.  NN0 )  ->  if ( k  =  X ,  n ,  0 )  e.  NN0 )
6360, 61, 62sylancl 643 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  k  e.  I )  ->  if ( k  =  X ,  n ,  0 )  e.  NN0 )
64 eqid 2283 . . . . . . . . . . 11  |-  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )  =  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )
6563, 64fmptd 5684 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) : I --> NN0 )
66 nn0supp 10017 . . . . . . . . . . . 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 15 . . . . . . . . . . 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 6941 . . . . . . . . . . . 12  |-  { X }  e.  Fin
69 eldifsni 3750 . . . . . . . . . . . . . . . 16  |-  ( k  e.  ( I  \  { X } )  -> 
k  =/=  X )
7069adantl 452 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  k  e.  ( I  \  { X } ) )  -> 
k  =/=  X )
7170neneqd 2462 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  k  e.  ( I  \  { X } ) )  ->  -.  k  =  X
)
72 iffalse 3572 . . . . . . . . . . . . . 14  |-  ( -.  k  =  X  ->  if ( k  =  X ,  n ,  0 )  =  0 )
7371, 72syl 15 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  k  e.  ( I  \  { X } ) )  ->  if ( k  =  X ,  n ,  0 )  =  0 )
7473suppss2 6073 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( `' ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) "
( _V  \  {
0 } ) ) 
C_  { X }
)
75 ssfi 7083 . . . . . . . . . . . 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 644 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( `' ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) "
( _V  \  {
0 } ) )  e.  Fin )
7767, 76eqeltrrd 2358 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( `' ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) " NN )  e.  Fin )
7852psrbag 16112 . . . . . . . . . . 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 15 . . . . . . . . . 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 888 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )  e.  D )
81 eqid 2283 . . . . . . . . 9  |-  ( .r
`  P )  =  ( .r `  P
)
8252mvridlem 16164 . . . . . . . . . 10  |-  ( I  e.  W  ->  (
k  e.  I  |->  if ( k  =  X ,  1 ,  0 ) )  e.  D
)
8358, 82syl 15 . . . . . . . . 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 16208 . . . . . . . 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 ) )  o F  +  ( k  e.  I  |->  if ( k  =  X , 
1 ,  0 ) ) ) ,  .1.  ,  .0.  ) ) )
8543adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN0 )  ->  X  e.  I )
8639, 52, 53, 54, 58, 59, 85mvrval 16166 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( V `  X )  =  ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  1 ,  0 ) ) ,  .1.  ,  .0.  ) ) )
8786eqcomd 2288 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X , 
1 ,  0 ) ) ,  .1.  ,  .0.  ) )  =  ( V `  X ) )
8887oveq2d 5874 . . . . . . . 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 9981 . . . . . . . . . . . . . . 15  |-  1  e.  NN0
9089, 61keepel 3622 . . . . . . . . . . . . . 14  |-  if ( k  =  X , 
1 ,  0 )  e.  NN0
9190a1i 10 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  k  e.  I )  ->  if ( k  =  X ,  1 ,  0 )  e.  NN0 )
92 eqidd 2284 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )  =  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) )
93 eqidd 2284 . . . . . . . . . . . . 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 6095 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( (
k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )  o F  +  ( k  e.  I  |->  if ( k  =  X ,  1 ,  0 ) ) )  =  ( k  e.  I  |->  ( if ( k  =  X ,  n ,  0 )  +  if ( k  =  X , 
1 ,  0 ) ) ) )
95 iftrue 3571 . . . . . . . . . . . . . . . 16  |-  ( k  =  X  ->  if ( k  =  X ,  n ,  0 )  =  n )
96 iftrue 3571 . . . . . . . . . . . . . . . 16  |-  ( k  =  X  ->  if ( k  =  X ,  1 ,  0 )  =  1 )
9795, 96oveq12d 5876 . . . . . . . . . . . . . . 15  |-  ( k  =  X  ->  ( if ( k  =  X ,  n ,  0 )  +  if ( k  =  X , 
1 ,  0 ) )  =  ( n  +  1 ) )
98 iftrue 3571 . . . . . . . . . . . . . . 15  |-  ( k  =  X  ->  if ( k  =  X ,  ( n  + 
1 ) ,  0 )  =  ( n  +  1 ) )
9997, 98eqtr4d 2318 . . . . . . . . . . . . . 14  |-  ( k  =  X  ->  ( if ( k  =  X ,  n ,  0 )  +  if ( k  =  X , 
1 ,  0 ) )  =  if ( k  =  X , 
( n  +  1 ) ,  0 ) )
100 00id 8987 . . . . . . . . . . . . . . 15  |-  ( 0  +  0 )  =  0
101 iffalse 3572 . . . . . . . . . . . . . . . 16  |-  ( -.  k  =  X  ->  if ( k  =  X ,  1 ,  0 )  =  0 )
10272, 101oveq12d 5876 . . . . . . . . . . . . . . 15  |-  ( -.  k  =  X  -> 
( if ( k  =  X ,  n ,  0 )  +  if ( k  =  X ,  1 ,  0 ) )  =  ( 0  +  0 ) )
103 iffalse 3572 . . . . . . . . . . . . . . 15  |-  ( -.  k  =  X  ->  if ( k  =  X ,  ( n  + 
1 ) ,  0 )  =  0 )
104100, 102, 1033eqtr4a 2341 . . . . . . . . . . . . . 14  |-  ( -.  k  =  X  -> 
( if ( k  =  X ,  n ,  0 )  +  if ( k  =  X ,  1 ,  0 ) )  =  if ( k  =  X ,  ( n  +  1 ) ,  0 ) )
10599, 104pm2.61i 156 . . . . . . . . . . . . 13  |-  ( if ( k  =  X ,  n ,  0 )  +  if ( k  =  X , 
1 ,  0 ) )  =  if ( k  =  X , 
( n  +  1 ) ,  0 )
106105mpteq2i 4103 . . . . . . . . . . . 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 2331 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( (
k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )  o F  +  ( k  e.  I  |->  if ( k  =  X ,  1 ,  0 ) ) )  =  ( k  e.  I  |->  if ( k  =  X , 
( n  +  1 ) ,  0 ) ) )
108107eqeq2d 2294 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( y  =  ( ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )  o F  +  ( k  e.  I  |->  if ( k  =  X ,  1 ,  0 ) ) )  <-> 
y  =  ( k  e.  I  |->  if ( k  =  X , 
( n  +  1 ) ,  0 ) ) ) )
109108ifbid 3583 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN0 )  ->  if (
y  =  ( ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )  o F  +  ( k  e.  I  |->  if ( k  =  X ,  1 ,  0 ) ) ) ,  .1.  ,  .0.  )  =  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  ( n  + 
1 ) ,  0 ) ) ,  .1.  ,  .0.  ) )
110109mpteq2dv 4107 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( y  e.  D  |->  if ( y  =  ( ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )  o F  +  ( 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 2324 . . . . . . 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 ) ) )
11238mplrng 16196 . . . . . . . . . . 11  |-  ( ( I  e.  W  /\  R  e.  Ring )  ->  P  e.  Ring )
11341, 42, 112syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  P  e.  Ring )
11445rngmgp 15347 . . . . . . . . . 10  |-  ( P  e.  Ring  ->  G  e. 
Mnd )
115113, 114syl 15 . . . . . . . . 9  |-  ( ph  ->  G  e.  Mnd )
116115adantr 451 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN0 )  ->  G  e.  Mnd )
117 simpr 447 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN0 )  ->  n  e.  NN0 )
11844adantr 451 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( V `  X )  e.  (
Base `  P )
)
11945, 81mgpplusg 15329 . . . . . . . . 9  |-  ( .r
`  P )  =  ( +g  `  G
)
12046, 49, 119mulgnn0p1 14578 . . . . . . . 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 1182 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( (
n  +  1 ) 
.^  ( V `  X ) )  =  ( ( n  .^  ( V `  X ) ) ( .r `  P ) ( V `
 X ) ) )
122111, 121eqeq12d 2297 . . . . . 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 212 . . . . 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 424 . . . 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 23 . . 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 10108 . 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 32 1  |-  ( ph  ->  ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  N , 
0 ) ) ,  .1.  ,  .0.  )
)  =  ( N 
.^  ( V `  X ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547   _Vcvv 2788    \ cdif 3149    C_ wss 3152   ifcif 3565   {csn 3640    e. cmpt 4077    X. cxp 4687   `'ccnv 4688   "cima 4692   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076    ^m cmap 6772   Fincfn 6863   0cc0 8737   1c1 8738    + caddc 8740   NNcn 9746   NN0cn0 9965   Basecbs 13148   .rcmulr 13209   0gc0g 13400   Mndcmnd 14361  .gcmg 14366  mulGrpcmgp 15325   Ringcrg 15337   1rcur 15339   mVar cmvr 16088   mPoly cmpl 16089
This theorem is referenced by:  mplcoe2  16211  coe1tm  16349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-ofr 6079  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-mulg 14492  df-subg 14618  df-ghm 14681  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-subrg 15543  df-psr 16098  df-mvr 16099  df-mpl 16100
  Copyright terms: Public domain W3C validator