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Theorem pserval 19780
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
StepHypRef Expression
1 oveq1 5826 . . . 4  |-  ( y  =  X  ->  (
y ^ m )  =  ( X ^
m ) )
21oveq2d 5835 . . 3  |-  ( y  =  X  ->  (
( A `  m
)  x.  ( y ^ m ) )  =  ( ( A `
 m )  x.  ( X ^ m
) ) )
32mpteq2dv 4108 . 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 5485 . . . . . . 7  |-  ( n  =  m  ->  ( A `  n )  =  ( A `  m ) )
6 oveq2 5827 . . . . . . 7  |-  ( n  =  m  ->  (
x ^ n )  =  ( x ^
m ) )
75, 6oveq12d 5837 . . . . . 6  |-  ( n  =  m  ->  (
( A `  n
)  x.  ( x ^ n ) )  =  ( ( A `
 m )  x.  ( x ^ m
) ) )
87cbvmptv 4112 . . . . 5  |-  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^
n ) ) )  =  ( m  e. 
NN0  |->  ( ( A `
 m )  x.  ( x ^ m
) ) )
9 oveq1 5826 . . . . . . 7  |-  ( x  =  y  ->  (
x ^ m )  =  ( y ^
m ) )
109oveq2d 5835 . . . . . 6  |-  ( x  =  y  ->  (
( A `  m
)  x.  ( x ^ m ) )  =  ( ( A `
 m )  x.  ( y ^ m
) ) )
1110mpteq2dv 4108 . . . . 5  |-  ( x  =  y  ->  (
m  e.  NN0  |->  ( ( A `  m )  x.  ( x ^
m ) ) )  =  ( m  e. 
NN0  |->  ( ( A `
 m )  x.  ( y ^ m
) ) ) )
128, 11syl5eq 2328 . . . 4  |-  ( x  =  y  ->  (
n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^
n ) ) )  =  ( m  e. 
NN0  |->  ( ( A `
 m )  x.  ( y ^ m
) ) ) )
1312cbvmptv 4112 . . 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 2304 . 2  |-  G  =  ( y  e.  CC  |->  ( m  e.  NN0  |->  ( ( A `  m )  x.  (
y ^ m ) ) ) )
15 nn0ex 9966 . . 3  |-  NN0  e.  _V
1615mptex 5707 . 2  |-  ( m  e.  NN0  |->  ( ( A `  m )  x.  ( X ^
m ) ) )  e.  _V
173, 14, 16fvmpt 5563 1  |-  ( X  e.  CC  ->  ( G `  X )  =  ( m  e. 
NN0  |->  ( ( A `
 m )  x.  ( X ^ m
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1628    e. wcel 1688    e. cmpt 4078   ` cfv 5221  (class class class)co 5819   CCcc 8730    x. cmul 8737   NN0cn0 9960   ^cexp 11098
This theorem is referenced by:  pserval2  19781  psergf  19782
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-i2m1 8800  ax-1ne0 8801  ax-rrecex 8804  ax-cnre 8805
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-recs 6383  df-rdg 6418  df-nn 9742  df-n0 9961
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