Theorem plyreres 15084

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 15053	. . 3 |- ( F e. (Poly ` RR ) -> RR C_ CC )
2		plyf 15057	. . . 4 |- ( F e. (Poly ` RR ) -> F : CC --> CC )
3		ffn 5410	. . . 4 |- ( F : CC --> CC -> F Fn CC )
4		fnssresb 5373	. . . 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 5585	. . . . . 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 8029	. . . . . . 7 |- ( a e. RR -> a e. CC )
10		ffvelcdm 5698	. . . . . . 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 15083	. . . . . . . 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 11064	. . . . . . . . 9 |- ( a e. RR -> ( * ` a ) = a )
15	14	adantl 277	. . . . . . . 8 |- ( ( F e. (Poly ` RR ) /\ a e. RR ) -> ( * ` a ) = a )
16	15	fveq2d 5565	. . . . . . 7 |- ( ( F e. (Poly ` RR ) /\ a e. RR ) -> ( F ` ( * ` a ) ) = ( F ` a ) )
17	13, 16	eqtrd 2229	. . . . . 6 |- ( ( F e. (Poly ` RR ) /\ a e. RR ) -> ( * ` ( F ` a ) ) = ( F ` a ) )
18	11, 17	cjrebd 11128	. . . . 5 |- ( ( F e. (Poly ` RR ) /\ a e. RR ) -> ( F ` a ) e. RR )
19	8, 18	eqeltrd 2273	. . . 4 |- ( ( F e. (Poly ` RR ) /\ a e. RR ) -> ( ( F |` RR ) ` a ) e. RR )
20	19	ralrimiva 2570	. . 3 |- ( F e. (Poly ` RR ) -> A. a e. RR ( ( F |` RR ) ` a ) e. RR )
21		fnfvrnss 5725	. . 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 5263	. 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 2167   A.wral 2475   C_ wss 3157   ran crn 4665   |` cres 4666   Fn wfn 5254   -->wf 5255   ` cfv 5259   CCcc 7894   RRcr 7895   *ccj 11021  Polycply 15048

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-frec 6458  df-1o 6483  df-oadd 6487  df-er 6601  df-map 6718  df-en 6809  df-dom 6810  df-fin 6811  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-fz 10101  df-fzo 10235  df-seqfrec 10557  df-exp 10648  df-ihash 10885  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-clim 11461  df-sumdc 11536  df-ply 15050

This theorem is referenced by: (None)