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Theorem plyreres 20200
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 20113 . . 3  |-  ( F  e.  (Poly `  RR )  ->  RR  C_  CC )
2 plyf 20117 . . . 4  |-  ( F  e.  (Poly `  RR )  ->  F : CC --> CC )
3 ffn 5591 . . . 4  |-  ( F : CC --> CC  ->  F  Fn  CC )
4 fnssresb 5557 . . . 4  |-  ( F  Fn  CC  ->  (
( F  |`  RR )  Fn  RR  <->  RR  C_  CC ) )
52, 3, 43syl 19 . . 3  |-  ( F  e.  (Poly `  RR )  ->  ( ( F  |`  RR )  Fn  RR  <->  RR  C_  CC ) )
61, 5mpbird 224 . 2  |-  ( F  e.  (Poly `  RR )  ->  ( F  |`  RR )  Fn  RR )
7 fvres 5745 . . . . . 6  |-  ( a  e.  RR  ->  (
( F  |`  RR ) `
 a )  =  ( F `  a
) )
87adantl 453 . . . . 5  |-  ( ( F  e.  (Poly `  RR )  /\  a  e.  RR )  ->  (
( F  |`  RR ) `
 a )  =  ( F `  a
) )
9 recn 9080 . . . . . . 7  |-  ( a  e.  RR  ->  a  e.  CC )
10 ffvelrn 5868 . . . . . . 7  |-  ( ( F : CC --> CC  /\  a  e.  CC )  ->  ( F `  a
)  e.  CC )
112, 9, 10syl2an 464 . . . . . 6  |-  ( ( F  e.  (Poly `  RR )  /\  a  e.  RR )  ->  ( F `  a )  e.  CC )
12 plyrecj 20197 . . . . . . . 8  |-  ( ( F  e.  (Poly `  RR )  /\  a  e.  CC )  ->  (
* `  ( F `  a ) )  =  ( F `  (
* `  a )
) )
139, 12sylan2 461 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  a  e.  RR )  ->  (
* `  ( F `  a ) )  =  ( F `  (
* `  a )
) )
14 cjre 11944 . . . . . . . . 9  |-  ( a  e.  RR  ->  (
* `  a )  =  a )
1514adantl 453 . . . . . . . 8  |-  ( ( F  e.  (Poly `  RR )  /\  a  e.  RR )  ->  (
* `  a )  =  a )
1615fveq2d 5732 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  a  e.  RR )  ->  ( F `  ( * `  a ) )  =  ( F `  a
) )
1713, 16eqtrd 2468 . . . . . 6  |-  ( ( F  e.  (Poly `  RR )  /\  a  e.  RR )  ->  (
* `  ( F `  a ) )  =  ( F `  a
) )
1811, 17cjrebd 12007 . . . . 5  |-  ( ( F  e.  (Poly `  RR )  /\  a  e.  RR )  ->  ( F `  a )  e.  RR )
198, 18eqeltrd 2510 . . . 4  |-  ( ( F  e.  (Poly `  RR )  /\  a  e.  RR )  ->  (
( F  |`  RR ) `
 a )  e.  RR )
2019ralrimiva 2789 . . 3  |-  ( F  e.  (Poly `  RR )  ->  A. a  e.  RR  ( ( F  |`  RR ) `  a )  e.  RR )
21 fnfvrnss 5896 . . 3  |-  ( ( ( F  |`  RR )  Fn  RR  /\  A. a  e.  RR  (
( F  |`  RR ) `
 a )  e.  RR )  ->  ran  ( F  |`  RR ) 
C_  RR )
226, 20, 21syl2anc 643 . 2  |-  ( F  e.  (Poly `  RR )  ->  ran  ( F  |`  RR )  C_  RR )
23 df-f 5458 . 2  |-  ( ( F  |`  RR ) : RR --> RR  <->  ( ( F  |`  RR )  Fn  RR  /\  ran  ( F  |`  RR )  C_  RR ) )
246, 22, 23sylanbrc 646 1  |-  ( F  e.  (Poly `  RR )  ->  ( F  |`  RR ) : RR --> RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705    C_ wss 3320   ran crn 4879    |` cres 4880    Fn wfn 5449   -->wf 5450   ` cfv 5454   CCcc 8988   RRcr 8989   *ccj 11901  Polycply 20103
This theorem is referenced by:  aalioulem3  20251  taylthlem2  20290
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-inf2 7596  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-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069
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-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  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-int 4051  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-se 4542  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-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-fzo 11136  df-fl 11202  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-rlim 12283  df-sum 12480  df-0p 19562  df-ply 20107  df-coe 20109  df-dgr 20110
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