Theorem plyreres 15848

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 15817	. . 3 |- ( F e. (Poly ` RR ) -> RR C_ CC )
2		plyf 15821	. . . 4 |- ( F e. (Poly ` RR ) -> F : CC --> CC )
3		ffn 5531	. . . 4 |- ( F : CC --> CC -> F Fn CC )
4		fnssresb 5493	. . . 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 5717	. . . . . 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 8306	. . . . . . 7 |- ( a e. RR -> a e. CC )
10		ffvelcdm 5835	. . . . . . 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 15847	. . . . . . . 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 11630	. . . . . . . . 9 |- ( a e. RR -> ( * ` a ) = a )
15	14	adantl 277	. . . . . . . 8 |- ( ( F e. (Poly ` RR ) /\ a e. RR ) -> ( * ` a ) = a )
16	15	fveq2d 5697	. . . . . . 7 |- ( ( F e. (Poly ` RR ) /\ a e. RR ) -> ( F ` ( * ` a ) ) = ( F ` a ) )
17	13, 16	eqtrd 2271	. . . . . 6 |- ( ( F e. (Poly ` RR ) /\ a e. RR ) -> ( * ` ( F ` a ) ) = ( F ` a ) )
18	11, 17	cjrebd 11695	. . . . 5 |- ( ( F e. (Poly ` RR ) /\ a e. RR ) -> ( F ` a ) e. RR )
19	8, 18	eqeltrd 2315	. . . 4 |- ( ( F e. (Poly ` RR ) /\ a e. RR ) -> ( ( F |` RR ) ` a ) e. RR )
20	19	ralrimiva 2623	. . 3 |- ( F e. (Poly ` RR ) -> A. a e. RR ( ( F |` RR ) ` a ) e. RR )
21		fnfvrnss 5862	. . 3 |- ( ( ( F |` RR ) Fn RR /\ A. a e. RR ( ( F |` RR ) ` a ) e. RR ) -> ran ( F |` RR ) C_ RR )
22	6, 20, 21	syl2anc 415	. 2 |- ( F e. (Poly ` RR ) -> ran ( F |` RR ) C_ RR )
23		df-f 5379	. 2 |- ( ( F |` RR ) : RR --> RR <-> ( ( F |` RR ) Fn RR /\ ran ( F |` RR ) C_ RR ) )
24	6, 22, 23	sylanbrc 421	1 |- ( F e. (Poly ` RR ) -> ( F |` RR ) : RR --> RR )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   C_ wss 3220   ran crn 4773   |` cres 4774   Fn wfn 5370   -->wf 5371   ` cfv 5375   CCcc 8171   RRcr 8172   *ccj 11587  Polycply 15812

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291  ax-arch 8292  ax-caucvg 8293

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-isom 5384  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-frec 6656  df-1o 6681  df-oadd 6685  df-er 6801  df-map 6918  df-en 7017  df-dom 7018  df-fin 7019  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-n0 9547  df-z 9628  df-uz 9905  df-q 10003  df-rp 10038  df-fz 10395  df-fzo 10533  df-seqfrec 10868  df-exp 10959  df-ihash 11198  df-cj 11590  df-re 11591  df-im 11592  df-rsqrt 11747  df-abs 11748  df-clim 12028  df-sumdc 12103  df-ply 15814

This theorem is referenced by: (None)