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Theorem coe1fval 16335
Description: Value of the univariate polynomial coefficient function. (Contributed by Stefan O'Rear, 21-Mar-2015.)
Hypothesis
Ref Expression
coe1fval.a  |-  A  =  (coe1 `  F )
Assertion
Ref Expression
coe1fval  |-  ( F  e.  V  ->  A  =  ( n  e. 
NN0  |->  ( F `  ( 1o  X.  { n } ) ) ) )
Distinct variable group:    n, F
Allowed substitution hints:    A( n)    V( n)

Proof of Theorem coe1fval
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 elex 2830 . 2  |-  ( F  e.  V  ->  F  e.  _V )
2 coe1fval.a . . 3  |-  A  =  (coe1 `  F )
3 fveq1 5562 . . . . 5  |-  ( f  =  F  ->  (
f `  ( 1o  X.  { n } ) )  =  ( F `
 ( 1o  X.  { n } ) ) )
43mpteq2dv 4144 . . . 4  |-  ( f  =  F  ->  (
n  e.  NN0  |->  ( f `
 ( 1o  X.  { n } ) ) )  =  ( n  e.  NN0  |->  ( F `
 ( 1o  X.  { n } ) ) ) )
5 df-coe1 16311 . . . 4  |- coe1  =  (
f  e.  _V  |->  ( n  e.  NN0  |->  ( f `
 ( 1o  X.  { n } ) ) ) )
6 nn0ex 10018 . . . . 5  |-  NN0  e.  _V
76mptex 5787 . . . 4  |-  ( n  e.  NN0  |->  ( F `
 ( 1o  X.  { n } ) ) )  e.  _V
84, 5, 7fvmpt 5640 . . 3  |-  ( F  e.  _V  ->  (coe1 `  F )  =  ( n  e.  NN0  |->  ( F `
 ( 1o  X.  { n } ) ) ) )
92, 8syl5eq 2360 . 2  |-  ( F  e.  _V  ->  A  =  ( n  e. 
NN0  |->  ( F `  ( 1o  X.  { n } ) ) ) )
101, 9syl 15 1  |-  ( F  e.  V  ->  A  =  ( n  e. 
NN0  |->  ( F `  ( 1o  X.  { n } ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1633    e. wcel 1701   _Vcvv 2822   {csn 3674    e. cmpt 4114    X. cxp 4724   ` cfv 5292   1oc1o 6514   NN0cn0 10012  coe1cco1 16304
This theorem is referenced by:  coe1fv  16336  coe1fval3  16338
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-i2m1 8850  ax-1ne0 8851  ax-rrecex 8854  ax-cnre 8855
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-recs 6430  df-rdg 6465  df-nn 9792  df-n0 10013  df-coe1 16311
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