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Theorem plyreres 21883
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 21796 . . 3  |-  ( F  e.  (Poly `  RR )  ->  RR  C_  CC )
2 plyf 21800 . . . 4  |-  ( F  e.  (Poly `  RR )  ->  F : CC --> CC )
3 ffn 5668 . . . 4  |-  ( F : CC --> CC  ->  F  Fn  CC )
4 fnssresb 5632 . . . 4  |-  ( F  Fn  CC  ->  (
( F  |`  RR )  Fn  RR  <->  RR  C_  CC ) )
52, 3, 43syl 20 . . 3  |-  ( F  e.  (Poly `  RR )  ->  ( ( F  |`  RR )  Fn  RR  <->  RR  C_  CC ) )
61, 5mpbird 232 . 2  |-  ( F  e.  (Poly `  RR )  ->  ( F  |`  RR )  Fn  RR )
7 fvres 5814 . . . . . 6  |-  ( a  e.  RR  ->  (
( F  |`  RR ) `
 a )  =  ( F `  a
) )
87adantl 466 . . . . 5  |-  ( ( F  e.  (Poly `  RR )  /\  a  e.  RR )  ->  (
( F  |`  RR ) `
 a )  =  ( F `  a
) )
9 recn 9484 . . . . . . 7  |-  ( a  e.  RR  ->  a  e.  CC )
10 ffvelrn 5951 . . . . . . 7  |-  ( ( F : CC --> CC  /\  a  e.  CC )  ->  ( F `  a
)  e.  CC )
112, 9, 10syl2an 477 . . . . . 6  |-  ( ( F  e.  (Poly `  RR )  /\  a  e.  RR )  ->  ( F `  a )  e.  CC )
12 plyrecj 21880 . . . . . . . 8  |-  ( ( F  e.  (Poly `  RR )  /\  a  e.  CC )  ->  (
* `  ( F `  a ) )  =  ( F `  (
* `  a )
) )
139, 12sylan2 474 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  a  e.  RR )  ->  (
* `  ( F `  a ) )  =  ( F `  (
* `  a )
) )
14 cjre 12747 . . . . . . . . 9  |-  ( a  e.  RR  ->  (
* `  a )  =  a )
1514adantl 466 . . . . . . . 8  |-  ( ( F  e.  (Poly `  RR )  /\  a  e.  RR )  ->  (
* `  a )  =  a )
1615fveq2d 5804 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  a  e.  RR )  ->  ( F `  ( * `  a ) )  =  ( F `  a
) )
1713, 16eqtrd 2495 . . . . . 6  |-  ( ( F  e.  (Poly `  RR )  /\  a  e.  RR )  ->  (
* `  ( F `  a ) )  =  ( F `  a
) )
1811, 17cjrebd 12810 . . . . 5  |-  ( ( F  e.  (Poly `  RR )  /\  a  e.  RR )  ->  ( F `  a )  e.  RR )
198, 18eqeltrd 2542 . . . 4  |-  ( ( F  e.  (Poly `  RR )  /\  a  e.  RR )  ->  (
( F  |`  RR ) `
 a )  e.  RR )
2019ralrimiva 2830 . . 3  |-  ( F  e.  (Poly `  RR )  ->  A. a  e.  RR  ( ( F  |`  RR ) `  a )  e.  RR )
21 fnfvrnss 5981 . . 3  |-  ( ( ( F  |`  RR )  Fn  RR  /\  A. a  e.  RR  (
( F  |`  RR ) `
 a )  e.  RR )  ->  ran  ( F  |`  RR ) 
C_  RR )
226, 20, 21syl2anc 661 . 2  |-  ( F  e.  (Poly `  RR )  ->  ran  ( F  |`  RR )  C_  RR )
23 df-f 5531 . 2  |-  ( ( F  |`  RR ) : RR --> RR  <->  ( ( F  |`  RR )  Fn  RR  /\  ran  ( F  |`  RR )  C_  RR ) )
246, 22, 23sylanbrc 664 1  |-  ( F  e.  (Poly `  RR )  ->  ( F  |`  RR ) : RR --> RR )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2799    C_ wss 3437   ran crn 4950    |` cres 4951    Fn wfn 5522   -->wf 5523   ` cfv 5527   CCcc 9392   RRcr 9393   *ccj 12704  Polycply 21786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-inf2 7959  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471  ax-pre-sup 9472  ax-addf 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-se 4789  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-of 6431  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-map 7327  df-pm 7328  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-sup 7803  df-oi 7836  df-card 8221  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-div 10106  df-nn 10435  df-2 10492  df-3 10493  df-n0 10692  df-z 10759  df-uz 10974  df-rp 11104  df-fz 11556  df-fzo 11667  df-fl 11760  df-seq 11925  df-exp 11984  df-hash 12222  df-cj 12707  df-re 12708  df-im 12709  df-sqr 12843  df-abs 12844  df-clim 13085  df-rlim 13086  df-sum 13283  df-0p 21282  df-ply 21790  df-coe 21792  df-dgr 21793
This theorem is referenced by:  aalioulem3  21934  taylthlem2  21973  plyrecld  27095
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