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Theorem pserval 19802
Description: Value of the function  G that gives the sequence of monomials of a power series. (Contributed by Mario Carneiro, 26-Feb-2015.)
Hypothesis
Ref Expression
pser.g  |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  (
x ^ n ) ) ) )
Assertion
Ref Expression
pserval  |-  ( X  e.  CC  ->  ( G `  X )  =  ( m  e. 
NN0  |->  ( ( A `
 m )  x.  ( X ^ m
) ) ) )
Distinct variable groups:    m, n, x, A    m, X    m, G
Allowed substitution hints:    G( x, n)    X( x, n)

Proof of Theorem pserval
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 oveq1 5881 . . . 4  |-  ( y  =  X  ->  (
y ^ m )  =  ( X ^
m ) )
21oveq2d 5890 . . 3  |-  ( y  =  X  ->  (
( A `  m
)  x.  ( y ^ m ) )  =  ( ( A `
 m )  x.  ( X ^ m
) ) )
32mpteq2dv 4123 . 2  |-  ( y  =  X  ->  (
m  e.  NN0  |->  ( ( A `  m )  x.  ( y ^
m ) ) )  =  ( m  e. 
NN0  |->  ( ( A `
 m )  x.  ( X ^ m
) ) ) )
4 pser.g . . 3  |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  (
x ^ n ) ) ) )
5 fveq2 5541 . . . . . . 7  |-  ( n  =  m  ->  ( A `  n )  =  ( A `  m ) )
6 oveq2 5882 . . . . . . 7  |-  ( n  =  m  ->  (
x ^ n )  =  ( x ^
m ) )
75, 6oveq12d 5892 . . . . . 6  |-  ( n  =  m  ->  (
( A `  n
)  x.  ( x ^ n ) )  =  ( ( A `
 m )  x.  ( x ^ m
) ) )
87cbvmptv 4127 . . . . 5  |-  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^
n ) ) )  =  ( m  e. 
NN0  |->  ( ( A `
 m )  x.  ( x ^ m
) ) )
9 oveq1 5881 . . . . . . 7  |-  ( x  =  y  ->  (
x ^ m )  =  ( y ^
m ) )
109oveq2d 5890 . . . . . 6  |-  ( x  =  y  ->  (
( A `  m
)  x.  ( x ^ m ) )  =  ( ( A `
 m )  x.  ( y ^ m
) ) )
1110mpteq2dv 4123 . . . . 5  |-  ( x  =  y  ->  (
m  e.  NN0  |->  ( ( A `  m )  x.  ( x ^
m ) ) )  =  ( m  e. 
NN0  |->  ( ( A `
 m )  x.  ( y ^ m
) ) ) )
128, 11syl5eq 2340 . . . 4  |-  ( x  =  y  ->  (
n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^
n ) ) )  =  ( m  e. 
NN0  |->  ( ( A `
 m )  x.  ( y ^ m
) ) ) )
1312cbvmptv 4127 . . 3  |-  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^
n ) ) ) )  =  ( y  e.  CC  |->  ( m  e.  NN0  |->  ( ( A `  m )  x.  ( y ^
m ) ) ) )
144, 13eqtri 2316 . 2  |-  G  =  ( y  e.  CC  |->  ( m  e.  NN0  |->  ( ( A `  m )  x.  (
y ^ m ) ) ) )
15 nn0ex 9987 . . 3  |-  NN0  e.  _V
1615mptex 5762 . 2  |-  ( m  e.  NN0  |->  ( ( A `  m )  x.  ( X ^
m ) ) )  e.  _V
173, 14, 16fvmpt 5618 1  |-  ( X  e.  CC  ->  ( G `  X )  =  ( m  e. 
NN0  |->  ( ( A `
 m )  x.  ( X ^ m
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   CCcc 8751    x. cmul 8758   NN0cn0 9981   ^cexp 11120
This theorem is referenced by:  pserval2  19803  psergf  19804
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-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-i2m1 8821  ax-1ne0 8822  ax-rrecex 8825  ax-cnre 8826
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-ral 2561  df-rex 2562  df-reu 2563  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-iun 3923  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-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-ov 5877  df-recs 6404  df-rdg 6439  df-nn 9763  df-n0 9982
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