Theorem plyreres 14934

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.

Step	Hyp	Ref	Expression
1		plybss 14904	. . 3 |- ( F e. (Poly ` RR ) -> RR C_ CC )
2		plyf 14908	. . . 4 |- ( F e. (Poly ` RR ) -> F : CC --> CC )
3		ffn 5404	. . . 4 |- ( F : CC --> CC -> F Fn CC )
4		fnssresb 5367	. . . 4 |- ( F Fn CC -> ( ( F |` RR ) Fn RR <-> RR C_ CC ) )
5	2, 3, 4	3syl 17	. . 3 |- ( F e. (Poly ` RR ) -> ( ( F |` RR ) Fn RR <-> RR C_ CC ) )
6	1, 5	mpbird 167	. 2 |- ( F e. (Poly ` RR ) -> ( F |` RR ) Fn RR )
7		fvres 5579	. . . . . 6 |- ( a e. RR -> ( ( F |` RR ) ` a ) = ( F ` a ) )
8	7	adantl 277	. . . . 5 |- ( ( F e. (Poly ` RR ) /\ a e. RR ) -> ( ( F |` RR ) ` a ) = ( F ` a ) )
9		recn 8007	. . . . . . 7 |- ( a e. RR -> a e. CC )
10		ffvelcdm 5692	. . . . . . 7 |- ( ( F : CC --> CC /\ a e. CC ) -> ( F ` a ) e. CC )
11	2, 9, 10	syl2an 289	. . . . . 6 |- ( ( F e. (Poly ` RR ) /\ a e. RR ) -> ( F ` a ) e. CC )
12		plyrecj 14933	. . . . . . . 8 |- ( ( F e. (Poly ` RR ) /\ a e. CC ) -> ( * ` ( F ` a ) ) = ( F ` ( * ` a ) ) )
13	9, 12	sylan2 286	. . . . . . 7 |- ( ( F e. (Poly ` RR ) /\ a e. RR ) -> ( * ` ( F ` a ) ) = ( F ` ( * ` a ) ) )
14		cjre 11029	. . . . . . . . 9 |- ( a e. RR -> ( * ` a ) = a )
15	14	adantl 277	. . . . . . . 8 |- ( ( F e. (Poly ` RR ) /\ a e. RR ) -> ( * ` a ) = a )
16	15	fveq2d 5559	. . . . . . 7 |- ( ( F e. (Poly ` RR ) /\ a e. RR ) -> ( F ` ( * ` a ) ) = ( F ` a ) )
17	13, 16	eqtrd 2226	. . . . . 6 |- ( ( F e. (Poly ` RR ) /\ a e. RR ) -> ( * ` ( F ` a ) ) = ( F ` a ) )
18	11, 17	cjrebd 11093	. . . . 5 |- ( ( F e. (Poly ` RR ) /\ a e. RR ) -> ( F ` a ) e. RR )
19	8, 18	eqeltrd 2270	. . . 4 |- ( ( F e. (Poly ` RR ) /\ a e. RR ) -> ( ( F |` RR ) ` a ) e. RR )
20	19	ralrimiva 2567	. . 3 |- ( F e. (Poly ` RR ) -> A. a e. RR ( ( F |` RR ) ` a ) e. RR )
21		fnfvrnss 5719	. . 3 |- ( ( ( F |` RR ) Fn RR /\ A. a e. RR ( ( F |` RR ) ` a ) e. RR ) -> ran ( F |` RR ) C_ RR )
22	6, 20, 21	syl2anc 411	. 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 ) )
24	6, 22, 23	sylanbrc 417	1 |- ( F e. (Poly ` RR ) -> ( F |` RR ) : RR --> RR )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   A.wral 2472   C_ wss 3154   ran crn 4661   |` cres 4662   Fn wfn 5250   -->wf 5251   ` cfv 5255   CCcc 7872   RRcr 7873   *ccj 10986  Polycply 14899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993  ax-caucvg 7994

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  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-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-frec 6446  df-1o 6471  df-oadd 6475  df-er 6589  df-map 6706  df-en 6797  df-dom 6798  df-fin 6799  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-n0 9244  df-z 9321  df-uz 9596  df-q 9688  df-rp 9723  df-fz 10078  df-fzo 10212  df-seqfrec 10522  df-exp 10613  df-ihash 10850  df-cj 10989  df-re 10990  df-im 10991  df-rsqrt 11145  df-abs 11146  df-clim 11425  df-sumdc 11500  df-ply 14901

This theorem is referenced by: (None)