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Theorem mplcoe3 16226
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 3582 . . . . . . . . . . 11  |-  ( x  =  0  ->  if ( k  =  X ,  x ,  0 )  =  if ( k  =  X , 
0 ,  0 ) )
3 ifid 3610 . . . . . . . . . . 11  |-  if ( k  =  X , 
0 ,  0 )  =  0
42, 3syl6eq 2344 . . . . . . . . . 10  |-  ( x  =  0  ->  if ( k  =  X ,  x ,  0 )  =  0 )
54mpteq2dv 4123 . . . . . . . . 9  |-  ( x  =  0  ->  (
k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  =  ( k  e.  I  |->  0 ) )
6 fconstmpt 4748 . . . . . . . . 9  |-  ( I  X.  { 0 } )  =  ( k  e.  I  |->  0 )
75, 6syl6eqr 2346 . . . . . . . 8  |-  ( x  =  0  ->  (
k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  =  ( I  X.  { 0 } ) )
87eqeq2d 2307 . . . . . . 7  |-  ( x  =  0  ->  (
y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  <->  y  =  ( I  X.  { 0 } ) ) )
98ifbid 3596 . . . . . 6  |-  ( x  =  0  ->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) ) ,  .1.  ,  .0.  )  =  if ( y  =  ( I  X.  { 0 } ) ,  .1.  ,  .0.  ) )
109mpteq2dv 4123 . . . . 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 5881 . . . . 5  |-  ( x  =  0  ->  (
x  .^  ( V `  X ) )  =  ( 0  .^  ( V `  X )
) )
1210, 11eqeq12d 2310 . . . 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 3582 . . . . . . . . 9  |-  ( x  =  n  ->  if ( k  =  X ,  x ,  0 )  =  if ( k  =  X ,  n ,  0 ) )
1514mpteq2dv 4123 . . . . . . . 8  |-  ( x  =  n  ->  (
k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  =  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) )
1615eqeq2d 2307 . . . . . . 7  |-  ( x  =  n  ->  (
y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  <->  y  =  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) ) )
1716ifbid 3596 . . . . . 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 4123 . . . . 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 5881 . . . . 5  |-  ( x  =  n  ->  (
x  .^  ( V `  X ) )  =  ( n  .^  ( V `  X )
) )
2018, 19eqeq12d 2310 . . . 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 3582 . . . . . . . . 9  |-  ( x  =  ( n  + 
1 )  ->  if ( k  =  X ,  x ,  0 )  =  if ( k  =  X , 
( n  +  1 ) ,  0 ) )
2322mpteq2dv 4123 . . . . . . . 8  |-  ( x  =  ( n  + 
1 )  ->  (
k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  =  ( k  e.  I  |->  if ( k  =  X ,  ( n  + 
1 ) ,  0 ) ) )
2423eqeq2d 2307 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  (
y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  <->  y  =  ( k  e.  I  |->  if ( k  =  X ,  ( n  + 
1 ) ,  0 ) ) ) )
2524ifbid 3596 . . . . . 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 4123 . . . . 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 5881 . . . . 5  |-  ( x  =  ( n  + 
1 )  ->  (
x  .^  ( V `  X ) )  =  ( ( n  + 
1 )  .^  ( V `  X )
) )
2826, 27eqeq12d 2310 . . . 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 3582 . . . . . . . . 9  |-  ( x  =  N  ->  if ( k  =  X ,  x ,  0 )  =  if ( k  =  X ,  N ,  0 ) )
3130mpteq2dv 4123 . . . . . . . 8  |-  ( x  =  N  ->  (
k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  =  ( k  e.  I  |->  if ( k  =  X ,  N ,  0 ) ) )
3231eqeq2d 2307 . . . . . . 7  |-  ( x  =  N  ->  (
y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  <->  y  =  ( k  e.  I  |->  if ( k  =  X ,  N ,  0 ) ) ) )
3332ifbid 3596 . . . . . 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 4123 . . . . 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 5881 . . . . 5  |-  ( x  =  N  ->  (
x  .^  ( V `  X ) )  =  ( N  .^  ( V `  X )
) )
3634, 35eqeq12d 2310 . . . 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 2296 . . . . . 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 16209 . . . . 5  |-  ( ph  ->  ( V `  X
)  e.  ( Base `  P ) )
45 mplcoe2.g . . . . . . 7  |-  G  =  (mulGrp `  P )
4645, 40mgpbas 15347 . . . . . 6  |-  ( Base `  P )  =  (
Base `  G )
47 eqid 2296 . . . . . . 7  |-  ( 1r
`  P )  =  ( 1r `  P
)
4845, 47rngidval 15359 . . . . . 6  |-  ( 1r
`  P )  =  ( 0g `  G
)
49 mplcoe2.m . . . . . 6  |-  .^  =  (.g
`  G )
5046, 48, 49mulg0 14588 . . . . 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 16204 . . . 4  |-  ( ph  ->  ( 1r `  P
)  =  ( y  e.  D  |->  if ( y  =  ( I  X.  { 0 } ) ,  .1.  ,  .0.  ) ) )
5651, 55eqtr2d 2329 . . 3  |-  ( ph  ->  ( y  e.  D  |->  if ( y  =  ( I  X.  {
0 } ) ,  .1.  ,  .0.  )
)  =  ( 0 
.^  ( V `  X ) ) )
57 oveq1 5881 . . . . . 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 9996 . . . . . . . . . . . 12  |-  0  e.  NN0
62 ifcl 3614 . . . . . . . . . . . 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 2296 . . . . . . . . . . 11  |-  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )  =  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )
6563, 64fmptd 5700 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) : I --> NN0 )
66 nn0supp 10033 . . . . . . . . . . . 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 6957 . . . . . . . . . . . 12  |-  { X }  e.  Fin
69 eldifsni 3763 . . . . . . . . . . . . . . . 16  |-  ( k  e.  ( I  \  { X } )  -> 
k  =/=  X )
7069adantl 452 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  k  e.  ( I  \  { X } ) )  -> 
k  =/=  X )
7170neneqd 2475 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  k  e.  ( I  \  { X } ) )  ->  -.  k  =  X
)
72 iffalse 3585 . . . . . . . . . . . . . 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 6089 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( `' ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) "
( _V  \  {
0 } ) ) 
C_  { X }
)
75 ssfi 7099 . . . . . . . . . . . 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 2371 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( `' ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) " NN )  e.  Fin )
7852psrbag 16128 . . . . . . . . . . 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 2296 . . . . . . . . 9  |-  ( .r
`  P )  =  ( .r `  P
)
8252mvridlem 16180 . . . . . . . . . 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 16224 . . . . . . . 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 16182 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( V `  X )  =  ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  1 ,  0 ) ) ,  .1.  ,  .0.  ) ) )
8786eqcomd 2301 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X , 
1 ,  0 ) ) ,  .1.  ,  .0.  ) )  =  ( V `  X ) )
8887oveq2d 5890 . . . . . . . 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 9997 . . . . . . . . . . . . . . 15  |-  1  e.  NN0
9089, 61keepel 3635 . . . . . . . . . . . . . 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 2297 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )  =  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) )
93 eqidd 2297 . . . . . . . . . . . . 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 6111 . . . . . . . . . . . 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 3584 . . . . . . . . . . . . . . . 16  |-  ( k  =  X  ->  if ( k  =  X ,  n ,  0 )  =  n )
96 iftrue 3584 . . . . . . . . . . . . . . . 16  |-  ( k  =  X  ->  if ( k  =  X ,  1 ,  0 )  =  1 )
9795, 96oveq12d 5892 . . . . . . . . . . . . . . 15  |-  ( k  =  X  ->  ( if ( k  =  X ,  n ,  0 )  +  if ( k  =  X , 
1 ,  0 ) )  =  ( n  +  1 ) )
98 iftrue 3584 . . . . . . . . . . . . . . 15  |-  ( k  =  X  ->  if ( k  =  X ,  ( n  + 
1 ) ,  0 )  =  ( n  +  1 ) )
9997, 98eqtr4d 2331 . . . . . . . . . . . . . 14  |-  ( k  =  X  ->  ( if ( k  =  X ,  n ,  0 )  +  if ( k  =  X , 
1 ,  0 ) )  =  if ( k  =  X , 
( n  +  1 ) ,  0 ) )
100 00id 9003 . . . . . . . . . . . . . . 15  |-  ( 0  +  0 )  =  0
101 iffalse 3585 . . . . . . . . . . . . . . . 16  |-  ( -.  k  =  X  ->  if ( k  =  X ,  1 ,  0 )  =  0 )
10272, 101oveq12d 5892 . . . . . . . . . . . . . . 15  |-  ( -.  k  =  X  -> 
( if ( k  =  X ,  n ,  0 )  +  if ( k  =  X ,  1 ,  0 ) )  =  ( 0  +  0 ) )
103 iffalse 3585 . . . . . . . . . . . . . . 15  |-  ( -.  k  =  X  ->  if ( k  =  X ,  ( n  + 
1 ) ,  0 )  =  0 )
104100, 102, 1033eqtr4a 2354 . . . . . . . . . . . . . 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 4119 . . . . . . . . . . . 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 2344 . . . . . . . . . . 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 2307 . . . . . . . . . 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 3596 . . . . . . . . 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 4123 . . . . . . . 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 2337 . . . . . . 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 16212 . . . . . . . . . . 11  |-  ( ( I  e.  W  /\  R  e.  Ring )  ->  P  e.  Ring )
11341, 42, 112syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  P  e.  Ring )
11445rngmgp 15363 . . . . . . . . . 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 15345 . . . . . . . . 9  |-  ( .r
`  P )  =  ( +g  `  G
)
12046, 49, 119mulgnn0p1 14594 . . . . . . . 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 2310 . . . . . 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 10124 . 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 1632    e. wcel 1696    =/= wne 2459   {crab 2560   _Vcvv 2801    \ cdif 3162    C_ wss 3165   ifcif 3578   {csn 3653    e. cmpt 4093    X. cxp 4703   `'ccnv 4704   "cima 4708   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092    ^m cmap 6788   Fincfn 6879   0cc0 8753   1c1 8754    + caddc 8756   NNcn 9762   NN0cn0 9981   Basecbs 13164   .rcmulr 13225   0gc0g 13416   Mndcmnd 14377  .gcmg 14382  mulGrpcmgp 15341   Ringcrg 15353   1rcur 15355   mVar cmvr 16104   mPoly cmpl 16105
This theorem is referenced by:  mplcoe2  16227  coe1tm  16365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-ofr 6095  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-0g 13420  df-gsum 13421  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-grp 14505  df-minusg 14506  df-mulg 14508  df-subg 14634  df-ghm 14697  df-cntz 14809  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-subrg 15559  df-psr 16114  df-mvr 16115  df-mpl 16116
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