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Theorem pserval 20279
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 6047 . . . 4  |-  ( y  =  X  ->  (
y ^ m )  =  ( X ^
m ) )
21oveq2d 6056 . . 3  |-  ( y  =  X  ->  (
( A `  m
)  x.  ( y ^ m ) )  =  ( ( A `
 m )  x.  ( X ^ m
) ) )
32mpteq2dv 4256 . 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 5687 . . . . . . 7  |-  ( n  =  m  ->  ( A `  n )  =  ( A `  m ) )
6 oveq2 6048 . . . . . . 7  |-  ( n  =  m  ->  (
x ^ n )  =  ( x ^
m ) )
75, 6oveq12d 6058 . . . . . 6  |-  ( n  =  m  ->  (
( A `  n
)  x.  ( x ^ n ) )  =  ( ( A `
 m )  x.  ( x ^ m
) ) )
87cbvmptv 4260 . . . . 5  |-  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^
n ) ) )  =  ( m  e. 
NN0  |->  ( ( A `
 m )  x.  ( x ^ m
) ) )
9 oveq1 6047 . . . . . . 7  |-  ( x  =  y  ->  (
x ^ m )  =  ( y ^
m ) )
109oveq2d 6056 . . . . . 6  |-  ( x  =  y  ->  (
( A `  m
)  x.  ( x ^ m ) )  =  ( ( A `
 m )  x.  ( y ^ m
) ) )
1110mpteq2dv 4256 . . . . 5  |-  ( x  =  y  ->  (
m  e.  NN0  |->  ( ( A `  m )  x.  ( x ^
m ) ) )  =  ( m  e. 
NN0  |->  ( ( A `
 m )  x.  ( y ^ m
) ) ) )
128, 11syl5eq 2448 . . . 4  |-  ( x  =  y  ->  (
n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^
n ) ) )  =  ( m  e. 
NN0  |->  ( ( A `
 m )  x.  ( y ^ m
) ) ) )
1312cbvmptv 4260 . . 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 2424 . 2  |-  G  =  ( y  e.  CC  |->  ( m  e.  NN0  |->  ( ( A `  m )  x.  (
y ^ m ) ) ) )
15 nn0ex 10183 . . 3  |-  NN0  e.  _V
1615mptex 5925 . 2  |-  ( m  e.  NN0  |->  ( ( A `  m )  x.  ( X ^
m ) ) )  e.  _V
173, 14, 16fvmpt 5765 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 1649    e. wcel 1721    e. cmpt 4226   ` cfv 5413  (class class class)co 6040   CCcc 8944    x. cmul 8951   NN0cn0 10177   ^cexp 11337
This theorem is referenced by:  pserval2  20280  psergf  20281
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-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-i2m1 9014  ax-1ne0 9015  ax-rrecex 9018  ax-cnre 9019
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-ral 2671  df-rex 2672  df-reu 2673  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-iun 4055  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-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-ov 6043  df-recs 6592  df-rdg 6627  df-nn 9957  df-n0 10178
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