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Theorem pserval2 19888
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
pserval2  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( ( G `  X ) `  N
)  =  ( ( A `  N )  x.  ( X ^ N ) ) )
Distinct variable groups:    x, n, A    n, N
Allowed substitution hints:    G( x, n)    N( x)    X( x, n)

Proof of Theorem pserval2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 pser.g . . . 4  |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  (
x ^ n ) ) ) )
21pserval 19887 . . 3  |-  ( X  e.  CC  ->  ( G `  X )  =  ( y  e. 
NN0  |->  ( ( A `
 y )  x.  ( X ^ y
) ) ) )
32fveq1d 5607 . 2  |-  ( X  e.  CC  ->  (
( G `  X
) `  N )  =  ( ( y  e.  NN0  |->  ( ( A `  y )  x.  ( X ^
y ) ) ) `
 N ) )
4 fveq2 5605 . . . 4  |-  ( y  =  N  ->  ( A `  y )  =  ( A `  N ) )
5 oveq2 5950 . . . 4  |-  ( y  =  N  ->  ( X ^ y )  =  ( X ^ N
) )
64, 5oveq12d 5960 . . 3  |-  ( y  =  N  ->  (
( A `  y
)  x.  ( X ^ y ) )  =  ( ( A `
 N )  x.  ( X ^ N
) ) )
7 eqid 2358 . . 3  |-  ( y  e.  NN0  |->  ( ( A `  y )  x.  ( X ^
y ) ) )  =  ( y  e. 
NN0  |->  ( ( A `
 y )  x.  ( X ^ y
) ) )
8 ovex 5967 . . 3  |-  ( ( A `  N )  x.  ( X ^ N ) )  e. 
_V
96, 7, 8fvmpt 5682 . 2  |-  ( N  e.  NN0  ->  ( ( y  e.  NN0  |->  ( ( A `  y )  x.  ( X ^
y ) ) ) `
 N )  =  ( ( A `  N )  x.  ( X ^ N ) ) )
103, 9sylan9eq 2410 1  |-  ( ( X  e.  CC  /\  N  e.  NN0 )  -> 
( ( G `  X ) `  N
)  =  ( ( A `  N )  x.  ( X ^ N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710    e. cmpt 4156   ` cfv 5334  (class class class)co 5942   CCcc 8822    x. cmul 8829   NN0cn0 10054   ^cexp 11194
This theorem is referenced by:  radcnvlem1  19890  radcnv0  19893  dvradcnv  19898  pserulm  19899  psercn2  19900  pserdvlem2  19905  abelth  19918
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-i2m1 8892  ax-1ne0 8893  ax-rrecex 8896  ax-cnre 8897
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-recs 6472  df-rdg 6507  df-nn 9834  df-n0 10055
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