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Theorem pserval 19734
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 5785 . . . 4  |-  ( y  =  X  ->  (
y ^ m )  =  ( X ^
m ) )
21oveq2d 5794 . . 3  |-  ( y  =  X  ->  (
( A `  m
)  x.  ( y ^ m ) )  =  ( ( A `
 m )  x.  ( X ^ m
) ) )
32mpteq2dv 4067 . 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 5444 . . . . . . 7  |-  ( n  =  m  ->  ( A `  n )  =  ( A `  m ) )
6 oveq2 5786 . . . . . . 7  |-  ( n  =  m  ->  (
x ^ n )  =  ( x ^
m ) )
75, 6oveq12d 5796 . . . . . 6  |-  ( n  =  m  ->  (
( A `  n
)  x.  ( x ^ n ) )  =  ( ( A `
 m )  x.  ( x ^ m
) ) )
87cbvmptv 4071 . . . . 5  |-  ( n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^
n ) ) )  =  ( m  e. 
NN0  |->  ( ( A `
 m )  x.  ( x ^ m
) ) )
9 oveq1 5785 . . . . . . 7  |-  ( x  =  y  ->  (
x ^ m )  =  ( y ^
m ) )
109oveq2d 5794 . . . . . 6  |-  ( x  =  y  ->  (
( A `  m
)  x.  ( x ^ m ) )  =  ( ( A `
 m )  x.  ( y ^ m
) ) )
1110mpteq2dv 4067 . . . . 5  |-  ( x  =  y  ->  (
m  e.  NN0  |->  ( ( A `  m )  x.  ( x ^
m ) ) )  =  ( m  e. 
NN0  |->  ( ( A `
 m )  x.  ( y ^ m
) ) ) )
128, 11syl5eq 2300 . . . 4  |-  ( x  =  y  ->  (
n  e.  NN0  |->  ( ( A `  n )  x.  ( x ^
n ) ) )  =  ( m  e. 
NN0  |->  ( ( A `
 m )  x.  ( y ^ m
) ) ) )
1312cbvmptv 4071 . . 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 2276 . 2  |-  G  =  ( y  e.  CC  |->  ( m  e.  NN0  |->  ( ( A `  m )  x.  (
y ^ m ) ) ) )
15 nn0ex 9924 . . 3  |-  NN0  e.  _V
1615mptex 5666 . 2  |-  ( m  e.  NN0  |->  ( ( A `  m )  x.  ( X ^
m ) ) )  e.  _V
173, 14, 16fvmpt 5522 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 1619    e. wcel 1621    e. cmpt 4037   ` cfv 4659  (class class class)co 5778   CCcc 8689    x. cmul 8696   NN0cn0 9918   ^cexp 11056
This theorem is referenced by:  pserval2  19735  psergf  19736
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pr 4172  ax-un 4470  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-i2m1 8759  ax-1ne0 8760  ax-rrecex 8763  ax-cnre 8764
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 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-recs 6342  df-rdg 6377  df-n 9701  df-n0 9919
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