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Theorem mplcoe3 16484
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 3703 . . . . . . . . . . 11  |-  ( x  =  0  ->  if ( k  =  X ,  x ,  0 )  =  if ( k  =  X , 
0 ,  0 ) )
3 ifid 3731 . . . . . . . . . . 11  |-  if ( k  =  X , 
0 ,  0 )  =  0
42, 3syl6eq 2452 . . . . . . . . . 10  |-  ( x  =  0  ->  if ( k  =  X ,  x ,  0 )  =  0 )
54mpteq2dv 4256 . . . . . . . . 9  |-  ( x  =  0  ->  (
k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  =  ( k  e.  I  |->  0 ) )
6 fconstmpt 4880 . . . . . . . . 9  |-  ( I  X.  { 0 } )  =  ( k  e.  I  |->  0 )
75, 6syl6eqr 2454 . . . . . . . 8  |-  ( x  =  0  ->  (
k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  =  ( I  X.  { 0 } ) )
87eqeq2d 2415 . . . . . . 7  |-  ( x  =  0  ->  (
y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  <->  y  =  ( I  X.  { 0 } ) ) )
98ifbid 3717 . . . . . 6  |-  ( x  =  0  ->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) ) ,  .1.  ,  .0.  )  =  if ( y  =  ( I  X.  { 0 } ) ,  .1.  ,  .0.  ) )
109mpteq2dv 4256 . . . . 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 6047 . . . . 5  |-  ( x  =  0  ->  (
x  .^  ( V `  X ) )  =  ( 0  .^  ( V `  X )
) )
1210, 11eqeq12d 2418 . . . 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 308 . . 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 3703 . . . . . . . . 9  |-  ( x  =  n  ->  if ( k  =  X ,  x ,  0 )  =  if ( k  =  X ,  n ,  0 ) )
1514mpteq2dv 4256 . . . . . . . 8  |-  ( x  =  n  ->  (
k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  =  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) )
1615eqeq2d 2415 . . . . . . 7  |-  ( x  =  n  ->  (
y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  <->  y  =  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) ) )
1716ifbid 3717 . . . . . 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 4256 . . . . 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 6047 . . . . 5  |-  ( x  =  n  ->  (
x  .^  ( V `  X ) )  =  ( n  .^  ( V `  X )
) )
2018, 19eqeq12d 2418 . . . 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 308 . . 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 3703 . . . . . . . . 9  |-  ( x  =  ( n  + 
1 )  ->  if ( k  =  X ,  x ,  0 )  =  if ( k  =  X , 
( n  +  1 ) ,  0 ) )
2322mpteq2dv 4256 . . . . . . . 8  |-  ( x  =  ( n  + 
1 )  ->  (
k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  =  ( k  e.  I  |->  if ( k  =  X ,  ( n  + 
1 ) ,  0 ) ) )
2423eqeq2d 2415 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  (
y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  <->  y  =  ( k  e.  I  |->  if ( k  =  X ,  ( n  + 
1 ) ,  0 ) ) ) )
2524ifbid 3717 . . . . . 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 4256 . . . . 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 6047 . . . . 5  |-  ( x  =  ( n  + 
1 )  ->  (
x  .^  ( V `  X ) )  =  ( ( n  + 
1 )  .^  ( V `  X )
) )
2826, 27eqeq12d 2418 . . . 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 308 . . 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 3703 . . . . . . . . 9  |-  ( x  =  N  ->  if ( k  =  X ,  x ,  0 )  =  if ( k  =  X ,  N ,  0 ) )
3130mpteq2dv 4256 . . . . . . . 8  |-  ( x  =  N  ->  (
k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  =  ( k  e.  I  |->  if ( k  =  X ,  N ,  0 ) ) )
3231eqeq2d 2415 . . . . . . 7  |-  ( x  =  N  ->  (
y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  <->  y  =  ( k  e.  I  |->  if ( k  =  X ,  N ,  0 ) ) ) )
3332ifbid 3717 . . . . . 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 4256 . . . . 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 6047 . . . . 5  |-  ( x  =  N  ->  (
x  .^  ( V `  X ) )  =  ( N  .^  ( V `  X )
) )
3634, 35eqeq12d 2418 . . . 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 308 . . 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 2404 . . . . . 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 16467 . . . . 5  |-  ( ph  ->  ( V `  X
)  e.  ( Base `  P ) )
45 mplcoe2.g . . . . . . 7  |-  G  =  (mulGrp `  P )
4645, 40mgpbas 15609 . . . . . 6  |-  ( Base `  P )  =  (
Base `  G )
47 eqid 2404 . . . . . . 7  |-  ( 1r
`  P )  =  ( 1r `  P
)
4845, 47rngidval 15621 . . . . . 6  |-  ( 1r
`  P )  =  ( 0g `  G
)
49 mplcoe2.m . . . . . 6  |-  .^  =  (.g
`  G )
5046, 48, 49mulg0 14850 . . . . 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 16462 . . . 4  |-  ( ph  ->  ( 1r `  P
)  =  ( y  e.  D  |->  if ( y  =  ( I  X.  { 0 } ) ,  .1.  ,  .0.  ) ) )
5651, 55eqtr2d 2437 . . 3  |-  ( ph  ->  ( y  e.  D  |->  if ( y  =  ( I  X.  {
0 } ) ,  .1.  ,  .0.  )
)  =  ( 0 
.^  ( V `  X ) ) )
57 oveq1 6047 . . . . . 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 452 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN0 )  ->  I  e.  W )
5942adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN0 )  ->  R  e.  Ring )
60 simplr 732 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  k  e.  I )  ->  n  e.  NN0 )
61 0nn0 10192 . . . . . . . . . . . 12  |-  0  e.  NN0
62 ifcl 3735 . . . . . . . . . . . 12  |-  ( ( n  e.  NN0  /\  0  e.  NN0 )  ->  if ( k  =  X ,  n ,  0 )  e.  NN0 )
6360, 61, 62sylancl 644 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  k  e.  I )  ->  if ( k  =  X ,  n ,  0 )  e.  NN0 )
64 eqid 2404 . . . . . . . . . . 11  |-  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )  =  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )
6563, 64fmptd 5852 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) : I --> NN0 )
66 nn0supp 10229 . . . . . . . . . . . 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 7146 . . . . . . . . . . . 12  |-  { X }  e.  Fin
69 eldifsni 3888 . . . . . . . . . . . . . . . 16  |-  ( k  e.  ( I  \  { X } )  -> 
k  =/=  X )
7069adantl 453 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  k  e.  ( I  \  { X } ) )  -> 
k  =/=  X )
7170neneqd 2583 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  k  e.  ( I  \  { X } ) )  ->  -.  k  =  X
)
72 iffalse 3706 . . . . . . . . . . . . . 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 )
7473suppss2 6259 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( `' ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) "
( _V  \  {
0 } ) ) 
C_  { X }
)
75 ssfi 7288 . . . . . . . . . . . 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 645 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( `' ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) "
( _V  \  {
0 } ) )  e.  Fin )
7767, 76eqeltrrd 2479 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( `' ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) " NN )  e.  Fin )
7852psrbag 16386 . . . . . . . . . . 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 889 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )  e.  D )
81 eqid 2404 . . . . . . . . 9  |-  ( .r
`  P )  =  ( .r `  P
)
8252mvridlem 16438 . . . . . . . . . 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 16482 . . . . . . . 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 452 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN0 )  ->  X  e.  I )
8639, 52, 53, 54, 58, 59, 85mvrval 16440 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( V `  X )  =  ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  1 ,  0 ) ) ,  .1.  ,  .0.  ) ) )
8786eqcomd 2409 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X , 
1 ,  0 ) ) ,  .1.  ,  .0.  ) )  =  ( V `  X ) )
8887oveq2d 6056 . . . . . . . 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 10193 . . . . . . . . . . . . . . 15  |-  1  e.  NN0
9089, 61keepel 3756 . . . . . . . . . . . . . 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 2405 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )  =  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) )
93 eqidd 2405 . . . . . . . . . . . . 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 6281 . . . . . . . . . . . 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 3705 . . . . . . . . . . . . . . . 16  |-  ( k  =  X  ->  if ( k  =  X ,  n ,  0 )  =  n )
96 iftrue 3705 . . . . . . . . . . . . . . . 16  |-  ( k  =  X  ->  if ( k  =  X ,  1 ,  0 )  =  1 )
9795, 96oveq12d 6058 . . . . . . . . . . . . . . 15  |-  ( k  =  X  ->  ( if ( k  =  X ,  n ,  0 )  +  if ( k  =  X , 
1 ,  0 ) )  =  ( n  +  1 ) )
98 iftrue 3705 . . . . . . . . . . . . . . 15  |-  ( k  =  X  ->  if ( k  =  X ,  ( n  + 
1 ) ,  0 )  =  ( n  +  1 ) )
9997, 98eqtr4d 2439 . . . . . . . . . . . . . 14  |-  ( k  =  X  ->  ( if ( k  =  X ,  n ,  0 )  +  if ( k  =  X , 
1 ,  0 ) )  =  if ( k  =  X , 
( n  +  1 ) ,  0 ) )
100 00id 9197 . . . . . . . . . . . . . . 15  |-  ( 0  +  0 )  =  0
101 iffalse 3706 . . . . . . . . . . . . . . . 16  |-  ( -.  k  =  X  ->  if ( k  =  X ,  1 ,  0 )  =  0 )
10272, 101oveq12d 6058 . . . . . . . . . . . . . . 15  |-  ( -.  k  =  X  -> 
( if ( k  =  X ,  n ,  0 )  +  if ( k  =  X ,  1 ,  0 ) )  =  ( 0  +  0 ) )
103 iffalse 3706 . . . . . . . . . . . . . . 15  |-  ( -.  k  =  X  ->  if ( k  =  X ,  ( n  + 
1 ) ,  0 )  =  0 )
104100, 102, 1033eqtr4a 2462 . . . . . . . . . . . . . 14  |-  ( -.  k  =  X  -> 
( if ( k  =  X ,  n ,  0 )  +  if ( k  =  X ,  1 ,  0 ) )  =  if ( k  =  X ,  ( n  +  1 ) ,  0 ) )
10599, 104pm2.61i 158 . . . . . . . . . . . . 13  |-  ( if ( k  =  X ,  n ,  0 )  +  if ( k  =  X , 
1 ,  0 ) )  =  if ( k  =  X , 
( n  +  1 ) ,  0 )
106105mpteq2i 4252 . . . . . . . . . . . 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 2452 . . . . . . . . . . 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 2415 . . . . . . . . . 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 3717 . . . . . . . . 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 4256 . . . . . . . 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 2445 . . . . . . 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 16470 . . . . . . . . . . 11  |-  ( ( I  e.  W  /\  R  e.  Ring )  ->  P  e.  Ring )
11341, 42, 112syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  P  e.  Ring )
11445rngmgp 15625 . . . . . . . . . 10  |-  ( P  e.  Ring  ->  G  e. 
Mnd )
115113, 114syl 16 . . . . . . . . 9  |-  ( ph  ->  G  e.  Mnd )
116115adantr 452 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN0 )  ->  G  e.  Mnd )
117 simpr 448 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN0 )  ->  n  e.  NN0 )
11844adantr 452 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( V `  X )  e.  (
Base `  P )
)
11945, 81mgpplusg 15607 . . . . . . . . 9  |-  ( .r
`  P )  =  ( +g  `  G
)
12046, 49, 119mulgnn0p1 14856 . . . . . . . 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 1184 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( (
n  +  1 ) 
.^  ( V `  X ) )  =  ( ( n  .^  ( V `  X ) ) ( .r `  P ) ( V `
 X ) ) )
122111, 121eqeq12d 2418 . . . . . 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 213 . . . . 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 425 . . . 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 24 . . 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 10322 . 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 34 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 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   {crab 2670   _Vcvv 2916    \ cdif 3277    C_ wss 3280   ifcif 3699   {csn 3774    e. cmpt 4226    X. cxp 4835   `'ccnv 4836   "cima 4840   -->wf 5409   ` cfv 5413  (class class class)co 6040    o Fcof 6262    ^m cmap 6977   Fincfn 7068   0cc0 8946   1c1 8947    + caddc 8949   NNcn 9956   NN0cn0 10177   Basecbs 13424   .rcmulr 13485   0gc0g 13678   Mndcmnd 14639  .gcmg 14644  mulGrpcmgp 15603   Ringcrg 15615   1rcur 15617   mVar cmvr 16362   mPoly cmpl 16363
This theorem is referenced by:  mplcoe2  16485  coe1tm  16620
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-ofr 6265  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-seq 11279  df-hash 11574  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-tset 13503  df-0g 13682  df-gsum 13683  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-mhm 14693  df-submnd 14694  df-grp 14767  df-minusg 14768  df-mulg 14770  df-subg 14896  df-ghm 14959  df-cntz 15071  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-ur 15620  df-subrg 15821  df-psr 16372  df-mvr 16373  df-mpl 16374
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