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Theorem pserval 20326
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 6088 . . . 4  |-  ( y  =  X  ->  (
y ^ m )  =  ( X ^
m ) )
21oveq2d 6097 . . 3  |-  ( y  =  X  ->  (
( A `  m
)  x.  ( y ^ m ) )  =  ( ( A `
 m )  x.  ( X ^ m
) ) )
32mpteq2dv 4296 . 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 5728 . . . . . . 7  |-  ( n  =  m  ->  ( A `  n )  =  ( A `  m ) )
6 oveq2 6089 . . . . . . 7  |-  ( n  =  m  ->  (
x ^ n )  =  ( x ^
m ) )
75, 6oveq12d 6099 . . . . . 6  |-  ( n  =  m  ->  (
( A `  n
)  x.  ( x ^ n ) )  =  ( ( A `
 m )  x.  ( x ^ m
) ) )
87cbvmptv 4300 . . . . 5  |-  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^
n ) ) )  =  ( m  e. 
NN0  |->  ( ( A `
 m )  x.  ( x ^ m
) ) )
9 oveq1 6088 . . . . . . 7  |-  ( x  =  y  ->  (
x ^ m )  =  ( y ^
m ) )
109oveq2d 6097 . . . . . 6  |-  ( x  =  y  ->  (
( A `  m
)  x.  ( x ^ m ) )  =  ( ( A `
 m )  x.  ( y ^ m
) ) )
1110mpteq2dv 4296 . . . . 5  |-  ( x  =  y  ->  (
m  e.  NN0  |->  ( ( A `  m )  x.  ( x ^
m ) ) )  =  ( m  e. 
NN0  |->  ( ( A `
 m )  x.  ( y ^ m
) ) ) )
128, 11syl5eq 2480 . . . 4  |-  ( x  =  y  ->  (
n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^
n ) ) )  =  ( m  e. 
NN0  |->  ( ( A `
 m )  x.  ( y ^ m
) ) ) )
1312cbvmptv 4300 . . 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 2456 . 2  |-  G  =  ( y  e.  CC  |->  ( m  e.  NN0  |->  ( ( A `  m )  x.  (
y ^ m ) ) ) )
15 nn0ex 10227 . . 3  |-  NN0  e.  _V
1615mptex 5966 . 2  |-  ( m  e.  NN0  |->  ( ( A `  m )  x.  ( X ^
m ) ) )  e.  _V
173, 14, 16fvmpt 5806 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 1652    e. wcel 1725    e. cmpt 4266   ` cfv 5454  (class class class)co 6081   CCcc 8988    x. cmul 8995   NN0cn0 10221   ^cexp 11382
This theorem is referenced by:  pserval2  20327  psergf  20328
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-i2m1 9058  ax-1ne0 9059  ax-rrecex 9062  ax-cnre 9063
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-recs 6633  df-rdg 6668  df-nn 10001  df-n0 10222
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