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Theorem mplcoe3 17668
Description: Decompose a monomial in one variable into a power of a variable. (Contributed by Mario Carneiro, 7-Jan-2015.) (Proof shortened by AV, 18-Jul-2019.)
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    .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 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 3902 . . . . . . . . . . 11  |-  ( x  =  0  ->  if ( k  =  X ,  x ,  0 )  =  if ( k  =  X , 
0 ,  0 ) )
3 ifid 3933 . . . . . . . . . . 11  |-  if ( k  =  X , 
0 ,  0 )  =  0
42, 3syl6eq 2511 . . . . . . . . . 10  |-  ( x  =  0  ->  if ( k  =  X ,  x ,  0 )  =  0 )
54mpteq2dv 4486 . . . . . . . . 9  |-  ( x  =  0  ->  (
k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  =  ( k  e.  I  |->  0 ) )
6 fconstmpt 4989 . . . . . . . . 9  |-  ( I  X.  { 0 } )  =  ( k  e.  I  |->  0 )
75, 6syl6eqr 2513 . . . . . . . 8  |-  ( x  =  0  ->  (
k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  =  ( I  X.  { 0 } ) )
87eqeq2d 2468 . . . . . . 7  |-  ( x  =  0  ->  (
y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  <->  y  =  ( I  X.  { 0 } ) ) )
98ifbid 3918 . . . . . 6  |-  ( x  =  0  ->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) ) ,  .1.  ,  .0.  )  =  if ( y  =  ( I  X.  { 0 } ) ,  .1.  ,  .0.  ) )
109mpteq2dv 4486 . . . . 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 6206 . . . . 5  |-  ( x  =  0  ->  (
x  .^  ( V `  X ) )  =  ( 0  .^  ( V `  X )
) )
1210, 11eqeq12d 2476 . . . 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 3902 . . . . . . . . 9  |-  ( x  =  n  ->  if ( k  =  X ,  x ,  0 )  =  if ( k  =  X ,  n ,  0 ) )
1514mpteq2dv 4486 . . . . . . . 8  |-  ( x  =  n  ->  (
k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  =  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) )
1615eqeq2d 2468 . . . . . . 7  |-  ( x  =  n  ->  (
y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  <->  y  =  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) ) )
1716ifbid 3918 . . . . . 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 4486 . . . . 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 6206 . . . . 5  |-  ( x  =  n  ->  (
x  .^  ( V `  X ) )  =  ( n  .^  ( V `  X )
) )
2018, 19eqeq12d 2476 . . . 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 3902 . . . . . . . . 9  |-  ( x  =  ( n  + 
1 )  ->  if ( k  =  X ,  x ,  0 )  =  if ( k  =  X , 
( n  +  1 ) ,  0 ) )
2322mpteq2dv 4486 . . . . . . . 8  |-  ( x  =  ( n  + 
1 )  ->  (
k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  =  ( k  e.  I  |->  if ( k  =  X ,  ( n  + 
1 ) ,  0 ) ) )
2423eqeq2d 2468 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  (
y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  <->  y  =  ( k  e.  I  |->  if ( k  =  X ,  ( n  + 
1 ) ,  0 ) ) ) )
2524ifbid 3918 . . . . . 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 4486 . . . . 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 6206 . . . . 5  |-  ( x  =  ( n  + 
1 )  ->  (
x  .^  ( V `  X ) )  =  ( ( n  + 
1 )  .^  ( V `  X )
) )
2826, 27eqeq12d 2476 . . . 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 3902 . . . . . . . . 9  |-  ( x  =  N  ->  if ( k  =  X ,  x ,  0 )  =  if ( k  =  X ,  N ,  0 ) )
3130mpteq2dv 4486 . . . . . . . 8  |-  ( x  =  N  ->  (
k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  =  ( k  e.  I  |->  if ( k  =  X ,  N ,  0 ) ) )
3231eqeq2d 2468 . . . . . . 7  |-  ( x  =  N  ->  (
y  =  ( k  e.  I  |->  if ( k  =  X ,  x ,  0 ) )  <->  y  =  ( k  e.  I  |->  if ( k  =  X ,  N ,  0 ) ) ) )
3332ifbid 3918 . . . . . 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 4486 . . . . 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 6206 . . . . 5  |-  ( x  =  N  ->  (
x  .^  ( V `  X ) )  =  ( N  .^  ( V `  X )
) )
3634, 35eqeq12d 2476 . . . 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 2454 . . . . . 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 17651 . . . . 5  |-  ( ph  ->  ( V `  X
)  e.  ( Base `  P ) )
45 mplcoe2.g . . . . . . 7  |-  G  =  (mulGrp `  P )
4645, 40mgpbas 16718 . . . . . 6  |-  ( Base `  P )  =  (
Base `  G )
47 eqid 2454 . . . . . . 7  |-  ( 1r
`  P )  =  ( 1r `  P
)
4845, 47rngidval 16726 . . . . . 6  |-  ( 1r
`  P )  =  ( 0g `  G
)
49 mplcoe2.m . . . . . 6  |-  .^  =  (.g
`  G )
5046, 48, 49mulg0 15750 . . . . 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 17646 . . . 4  |-  ( ph  ->  ( 1r `  P
)  =  ( y  e.  D  |->  if ( y  =  ( I  X.  { 0 } ) ,  .1.  ,  .0.  ) ) )
5651, 55eqtr2d 2496 . . 3  |-  ( ph  ->  ( y  e.  D  |->  if ( y  =  ( I  X.  {
0 } ) ,  .1.  ,  .0.  )
)  =  ( 0 
.^  ( V `  X ) ) )
57 oveq1 6206 . . . . . 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 )
6052snifpsrbag 17555 . . . . . . . . . 10  |-  ( ( I  e.  W  /\  n  e.  NN0 )  -> 
( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )  e.  D )
6141, 60sylan 471 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )  e.  D )
62 eqid 2454 . . . . . . . . 9  |-  ( .r
`  P )  =  ( .r `  P
)
63 1nn0 10705 . . . . . . . . . . 11  |-  1  e.  NN0
6463a1i 11 . . . . . . . . . 10  |-  ( n  e.  NN0  ->  1  e. 
NN0 )
6552snifpsrbag 17555 . . . . . . . . . 10  |-  ( ( I  e.  W  /\  1  e.  NN0 )  -> 
( k  e.  I  |->  if ( k  =  X ,  1 ,  0 ) )  e.  D )
6641, 64, 65syl2an 477 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( k  e.  I  |->  if ( k  =  X , 
1 ,  0 ) )  e.  D )
6738, 40, 53, 54, 52, 58, 59, 61, 62, 66mplmonmul 17666 . . . . . . . 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.  ) ) )
6843adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN0 )  ->  X  e.  I )
6939, 52, 53, 54, 58, 59, 68mvrval 17617 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( V `  X )  =  ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  1 ,  0 ) ) ,  .1.  ,  .0.  ) ) )
7069eqcomd 2462 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X , 
1 ,  0 ) ) ,  .1.  ,  .0.  ) )  =  ( V `  X ) )
7170oveq2d 6215 . . . . . . . 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 ) ) )
72 simplr 754 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  k  e.  I )  ->  n  e.  NN0 )
73 0nn0 10704 . . . . . . . . . . . . . 14  |-  0  e.  NN0
74 ifcl 3938 . . . . . . . . . . . . . 14  |-  ( ( n  e.  NN0  /\  0  e.  NN0 )  ->  if ( k  =  X ,  n ,  0 )  e.  NN0 )
7572, 73, 74sylancl 662 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  k  e.  I )  ->  if ( k  =  X ,  n ,  0 )  e.  NN0 )
7663, 73keepel 3964 . . . . . . . . . . . . . 14  |-  if ( k  =  X , 
1 ,  0 )  e.  NN0
7776a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  k  e.  I )  ->  if ( k  =  X ,  1 ,  0 )  e.  NN0 )
78 eqidd 2455 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) )  =  ( k  e.  I  |->  if ( k  =  X ,  n ,  0 ) ) )
79 eqidd 2455 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( k  e.  I  |->  if ( k  =  X , 
1 ,  0 ) )  =  ( k  e.  I  |->  if ( k  =  X , 
1 ,  0 ) ) )
8058, 75, 77, 78, 79offval2 6445 . . . . . . . . . . . 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 ) ) ) )
81 iftrue 3904 . . . . . . . . . . . . . . . 16  |-  ( k  =  X  ->  if ( k  =  X ,  n ,  0 )  =  n )
82 iftrue 3904 . . . . . . . . . . . . . . . 16  |-  ( k  =  X  ->  if ( k  =  X ,  1 ,  0 )  =  1 )
8381, 82oveq12d 6217 . . . . . . . . . . . . . . 15  |-  ( k  =  X  ->  ( if ( k  =  X ,  n ,  0 )  +  if ( k  =  X , 
1 ,  0 ) )  =  ( n  +  1 ) )
84 iftrue 3904 . . . . . . . . . . . . . . 15  |-  ( k  =  X  ->  if ( k  =  X ,  ( n  + 
1 ) ,  0 )  =  ( n  +  1 ) )
8583, 84eqtr4d 2498 . . . . . . . . . . . . . 14  |-  ( k  =  X  ->  ( if ( k  =  X ,  n ,  0 )  +  if ( k  =  X , 
1 ,  0 ) )  =  if ( k  =  X , 
( n  +  1 ) ,  0 ) )
86 00id 9654 . . . . . . . . . . . . . . 15  |-  ( 0  +  0 )  =  0
87 iffalse 3906 . . . . . . . . . . . . . . . 16  |-  ( -.  k  =  X  ->  if ( k  =  X ,  n ,  0 )  =  0 )
88 iffalse 3906 . . . . . . . . . . . . . . . 16  |-  ( -.  k  =  X  ->  if ( k  =  X ,  1 ,  0 )  =  0 )
8987, 88oveq12d 6217 . . . . . . . . . . . . . . 15  |-  ( -.  k  =  X  -> 
( if ( k  =  X ,  n ,  0 )  +  if ( k  =  X ,  1 ,  0 ) )  =  ( 0  +  0 ) )
90 iffalse 3906 . . . . . . . . . . . . . . 15  |-  ( -.  k  =  X  ->  if ( k  =  X ,  ( n  + 
1 ) ,  0 )  =  0 )
9186, 89, 903eqtr4a 2521 . . . . . . . . . . . . . 14  |-  ( -.  k  =  X  -> 
( if ( k  =  X ,  n ,  0 )  +  if ( k  =  X ,  1 ,  0 ) )  =  if ( k  =  X ,  ( n  +  1 ) ,  0 ) )
9285, 91pm2.61i 164 . . . . . . . . . . . . 13  |-  ( if ( k  =  X ,  n ,  0 )  +  if ( k  =  X , 
1 ,  0 ) )  =  if ( k  =  X , 
( n  +  1 ) ,  0 )
9392mpteq2i 4482 . . . . . . . . . . . 12  |-  ( k  e.  I  |->  ( if ( k  =  X ,  n ,  0 )  +  if ( k  =  X , 
1 ,  0 ) ) )  =  ( k  e.  I  |->  if ( k  =  X ,  ( n  + 
1 ) ,  0 ) )
9480, 93syl6eq 2511 . . . . . . . . . . 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 ) ) )
9594eqeq2d 2468 . . . . . . . . . 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 ) ) ) )
9695ifbid 3918 . . . . . . . . 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.  ) )
9796mpteq2dv 4486 . . . . . . . 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.  ) ) )
9867, 71, 973eqtr3rd 2504 . . . . . . 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 ) ) )
9938mplrng 17654 . . . . . . . . . . 11  |-  ( ( I  e.  W  /\  R  e.  Ring )  ->  P  e.  Ring )
10041, 42, 99syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  P  e.  Ring )
10145rngmgp 16773 . . . . . . . . . 10  |-  ( P  e.  Ring  ->  G  e. 
Mnd )
102100, 101syl 16 . . . . . . . . 9  |-  ( ph  ->  G  e.  Mnd )
103102adantr 465 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN0 )  ->  G  e.  Mnd )
104 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN0 )  ->  n  e.  NN0 )
10544adantr 465 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( V `  X )  e.  (
Base `  P )
)
10645, 62mgpplusg 16716 . . . . . . . . 9  |-  ( .r
`  P )  =  ( +g  `  G
)
10746, 49, 106mulgnn0p1 15756 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  n  e.  NN0  /\  ( V `  X )  e.  ( Base `  P
) )  ->  (
( n  +  1 )  .^  ( V `  X ) )  =  ( ( n  .^  ( V `  X ) ) ( .r `  P ) ( V `
 X ) ) )
108103, 104, 105, 107syl3anc 1219 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( (
n  +  1 ) 
.^  ( V `  X ) )  =  ( ( n  .^  ( V `  X ) ) ( .r `  P ) ( V `
 X ) ) )
10998, 108eqeq12d 2476 . . . . . 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
) ) ) )
11057, 109syl5ibr 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 ) ) ) )
111110expcom 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 ) ) ) ) )
112111a2d 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 )
) ) ) )
11313, 21, 29, 37, 56, 112nn0ind 10848 . 2  |-  ( N  e.  NN0  ->  ( ph  ->  ( y  e.  D  |->  if ( y  =  ( k  e.  I  |->  if ( k  =  X ,  N , 
0 ) ) ,  .1.  ,  .0.  )
)  =  ( N 
.^  ( V `  X ) ) ) )
1141, 113mpcom 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    /\ wa 369    = wceq 1370    e. wcel 1758   {crab 2802   ifcif 3898   {csn 3984    |-> cmpt 4457    X. cxp 4945   `'ccnv 4946   "cima 4950   ` cfv 5525  (class class class)co 6199    oFcof 6427    ^m cmap 7323   Fincfn 7419   0cc0 9392   1c1 9393    + caddc 9395   NNcn 10432   NN0cn0 10689   Basecbs 14291   .rcmulr 14357   0gc0g 14496   Mndcmnd 15527  .gcmg 15532  mulGrpcmgp 16712   1rcur 16724   Ringcrg 16767   mVar cmvr 17541   mPoly cmpl 17542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-inf2 7957  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-of 6429  df-ofr 6430  df-om 6586  df-1st 6686  df-2nd 6687  df-supp 6800  df-recs 6941  df-rdg 6975  df-1o 7029  df-2o 7030  df-oadd 7033  df-er 7210  df-map 7325  df-pm 7326  df-ixp 7373  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-fsupp 7731  df-oi 7834  df-card 8219  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-2 10490  df-3 10491  df-4 10492  df-5 10493  df-6 10494  df-7 10495  df-8 10496  df-9 10497  df-n0 10690  df-z 10757  df-uz 10972  df-fz 11554  df-fzo 11665  df-seq 11923  df-hash 12220  df-struct 14293  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-mulr 14370  df-sca 14372  df-vsca 14373  df-tset 14375  df-0g 14498  df-gsum 14499  df-mre 14642  df-mrc 14643  df-acs 14645  df-mnd 15533  df-mhm 15582  df-submnd 15583  df-grp 15663  df-minusg 15664  df-mulg 15666  df-subg 15796  df-ghm 15863  df-cntz 15953  df-cmn 16399  df-abl 16400  df-mgp 16713  df-ur 16725  df-rng 16769  df-subrg 16985  df-psr 17545  df-mvr 17546  df-mpl 17547
This theorem is referenced by:  mplcoe5  17671  mplcoe2OLD  17673  coe1tm  17849
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