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Theorem plyreres 19663
Description: Real-coefficient polynomials restrict to real functions. (Contributed by Stefan O'Rear, 16-Nov-2014.)
Assertion
Ref Expression
plyreres  |-  ( F  e.  (Poly `  RR )  ->  ( F  |`  RR ) : RR --> RR )

Proof of Theorem plyreres
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 plybss 19576 . . 3  |-  ( F  e.  (Poly `  RR )  ->  RR  C_  CC )
2 plyf 19580 . . . 4  |-  ( F  e.  (Poly `  RR )  ->  F : CC --> CC )
3 ffn 5389 . . . 4  |-  ( F : CC --> CC  ->  F  Fn  CC )
4 fnssresb 5356 . . . 4  |-  ( F  Fn  CC  ->  (
( F  |`  RR )  Fn  RR  <->  RR  C_  CC ) )
52, 3, 43syl 18 . . 3  |-  ( F  e.  (Poly `  RR )  ->  ( ( F  |`  RR )  Fn  RR  <->  RR  C_  CC ) )
61, 5mpbird 223 . 2  |-  ( F  e.  (Poly `  RR )  ->  ( F  |`  RR )  Fn  RR )
7 fvres 5542 . . . . . 6  |-  ( a  e.  RR  ->  (
( F  |`  RR ) `
 a )  =  ( F `  a
) )
87adantl 452 . . . . 5  |-  ( ( F  e.  (Poly `  RR )  /\  a  e.  RR )  ->  (
( F  |`  RR ) `
 a )  =  ( F `  a
) )
9 recn 8827 . . . . . . 7  |-  ( a  e.  RR  ->  a  e.  CC )
10 ffvelrn 5663 . . . . . . 7  |-  ( ( F : CC --> CC  /\  a  e.  CC )  ->  ( F `  a
)  e.  CC )
112, 9, 10syl2an 463 . . . . . 6  |-  ( ( F  e.  (Poly `  RR )  /\  a  e.  RR )  ->  ( F `  a )  e.  CC )
12 plyrecj 19660 . . . . . . . 8  |-  ( ( F  e.  (Poly `  RR )  /\  a  e.  CC )  ->  (
* `  ( F `  a ) )  =  ( F `  (
* `  a )
) )
139, 12sylan2 460 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  a  e.  RR )  ->  (
* `  ( F `  a ) )  =  ( F `  (
* `  a )
) )
14 cjre 11624 . . . . . . . . 9  |-  ( a  e.  RR  ->  (
* `  a )  =  a )
1514adantl 452 . . . . . . . 8  |-  ( ( F  e.  (Poly `  RR )  /\  a  e.  RR )  ->  (
* `  a )  =  a )
1615fveq2d 5529 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  a  e.  RR )  ->  ( F `  ( * `  a ) )  =  ( F `  a
) )
1713, 16eqtrd 2315 . . . . . 6  |-  ( ( F  e.  (Poly `  RR )  /\  a  e.  RR )  ->  (
* `  ( F `  a ) )  =  ( F `  a
) )
1811, 17cjrebd 11687 . . . . 5  |-  ( ( F  e.  (Poly `  RR )  /\  a  e.  RR )  ->  ( F `  a )  e.  RR )
198, 18eqeltrd 2357 . . . 4  |-  ( ( F  e.  (Poly `  RR )  /\  a  e.  RR )  ->  (
( F  |`  RR ) `
 a )  e.  RR )
2019ralrimiva 2626 . . 3  |-  ( F  e.  (Poly `  RR )  ->  A. a  e.  RR  ( ( F  |`  RR ) `  a )  e.  RR )
21 fnfvrnss 5687 . . 3  |-  ( ( ( F  |`  RR )  Fn  RR  /\  A. a  e.  RR  (
( F  |`  RR ) `
 a )  e.  RR )  ->  ran  ( F  |`  RR ) 
C_  RR )
226, 20, 21syl2anc 642 . 2  |-  ( F  e.  (Poly `  RR )  ->  ran  ( F  |`  RR )  C_  RR )
23 df-f 5259 . 2  |-  ( ( F  |`  RR ) : RR --> RR  <->  ( ( F  |`  RR )  Fn  RR  /\  ran  ( F  |`  RR )  C_  RR ) )
246, 22, 23sylanbrc 645 1  |-  ( F  e.  (Poly `  RR )  ->  ( F  |`  RR ) : RR --> RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   ran crn 4690    |` cres 4691    Fn wfn 5250   -->wf 5251   ` cfv 5255   CCcc 8735   RRcr 8736   *ccj 11581  Polycply 19566
This theorem is referenced by:  aalioulem3  19714  taylthlem2  19753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-0p 19025  df-ply 19570  df-coe 19572  df-dgr 19573
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