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Theorem plyreres 22845
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 22757 . . 3  |-  ( F  e.  (Poly `  RR )  ->  RR  C_  CC )
2 plyf 22761 . . . 4  |-  ( F  e.  (Poly `  RR )  ->  F : CC --> CC )
3 ffn 5713 . . . 4  |-  ( F : CC --> CC  ->  F  Fn  CC )
4 fnssresb 5675 . . . 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 5862 . . . . . 6  |-  ( a  e.  RR  ->  (
( F  |`  RR ) `
 a )  =  ( F `  a
) )
87adantl 464 . . . . 5  |-  ( ( F  e.  (Poly `  RR )  /\  a  e.  RR )  ->  (
( F  |`  RR ) `
 a )  =  ( F `  a
) )
9 recn 9571 . . . . . . 7  |-  ( a  e.  RR  ->  a  e.  CC )
10 ffvelrn 6005 . . . . . . 7  |-  ( ( F : CC --> CC  /\  a  e.  CC )  ->  ( F `  a
)  e.  CC )
112, 9, 10syl2an 475 . . . . . 6  |-  ( ( F  e.  (Poly `  RR )  /\  a  e.  RR )  ->  ( F `  a )  e.  CC )
12 plyrecj 22842 . . . . . . . 8  |-  ( ( F  e.  (Poly `  RR )  /\  a  e.  CC )  ->  (
* `  ( F `  a ) )  =  ( F `  (
* `  a )
) )
139, 12sylan2 472 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  a  e.  RR )  ->  (
* `  ( F `  a ) )  =  ( F `  (
* `  a )
) )
14 cjre 13054 . . . . . . . . 9  |-  ( a  e.  RR  ->  (
* `  a )  =  a )
1514adantl 464 . . . . . . . 8  |-  ( ( F  e.  (Poly `  RR )  /\  a  e.  RR )  ->  (
* `  a )  =  a )
1615fveq2d 5852 . . . . . . 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 13117 . . . . 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 2868 . . 3  |-  ( F  e.  (Poly `  RR )  ->  A. a  e.  RR  ( ( F  |`  RR ) `  a )  e.  RR )
21 fnfvrnss 6035 . . 3  |-  ( ( ( F  |`  RR )  Fn  RR  /\  A. a  e.  RR  (
( F  |`  RR ) `
 a )  e.  RR )  ->  ran  ( F  |`  RR ) 
C_  RR )
226, 20, 21syl2anc 659 . 2  |-  ( F  e.  (Poly `  RR )  ->  ran  ( F  |`  RR )  C_  RR )
23 df-f 5574 . 2  |-  ( ( F  |`  RR ) : RR --> RR  <->  ( ( F  |`  RR )  Fn  RR  /\  ran  ( F  |`  RR )  C_  RR ) )
246, 22, 23sylanbrc 662 1  |-  ( F  e.  (Poly `  RR )  ->  ( F  |`  RR ) : RR --> RR )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804    C_ wss 3461   ran crn 4989    |` cres 4990    Fn wfn 5565   -->wf 5566   ` cfv 5570   CCcc 9479   RRcr 9480   *ccj 13011  Polycply 22747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-fz 11676  df-fzo 11800  df-fl 11910  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-clim 13393  df-rlim 13394  df-sum 13591  df-0p 22243  df-ply 22751  df-coe 22753  df-dgr 22754
This theorem is referenced by:  aalioulem3  22896  taylthlem2  22935  plyrecld  28770
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