Theorem plyreres 15423

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 15392	. . 3 |- ( F e. (Poly ` RR ) -> RR C_ CC )
2		plyf 15396	. . . 4 |- ( F e. (Poly ` RR ) -> F : CC --> CC )
3		ffn 5469	. . . 4 |- ( F : CC --> CC -> F Fn CC )
4		fnssresb 5431	. . . 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 5647	. . . . . 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 8120	. . . . . . 7 |- ( a e. RR -> a e. CC )
10		ffvelcdm 5761	. . . . . . 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 15422	. . . . . . . 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 11379	. . . . . . . . 9 |- ( a e. RR -> ( * ` a ) = a )
15	14	adantl 277	. . . . . . . 8 |- ( ( F e. (Poly ` RR ) /\ a e. RR ) -> ( * ` a ) = a )
16	15	fveq2d 5627	. . . . . . 7 |- ( ( F e. (Poly ` RR ) /\ a e. RR ) -> ( F ` ( * ` a ) ) = ( F ` a ) )
17	13, 16	eqtrd 2262	. . . . . 6 |- ( ( F e. (Poly ` RR ) /\ a e. RR ) -> ( * ` ( F ` a ) ) = ( F ` a ) )
18	11, 17	cjrebd 11443	. . . . 5 |- ( ( F e. (Poly ` RR ) /\ a e. RR ) -> ( F ` a ) e. RR )
19	8, 18	eqeltrd 2306	. . . 4 |- ( ( F e. (Poly ` RR ) /\ a e. RR ) -> ( ( F |` RR ) ` a ) e. RR )
20	19	ralrimiva 2603	. . 3 |- ( F e. (Poly ` RR ) -> A. a e. RR ( ( F |` RR ) ` a ) e. RR )
21		fnfvrnss 5788	. . 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 5318	. 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 1395   e. wcel 2200   A.wral 2508   C_ wss 3197   ran crn 4717   |` cres 4718   Fn wfn 5309   -->wf 5310   ` cfv 5314   CCcc 7985   RRcr 7986   *ccj 11336  Polycply 15387

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-mulrcl 8086  ax-addcom 8087  ax-mulcom 8088  ax-addass 8089  ax-mulass 8090  ax-distr 8091  ax-i2m1 8092  ax-0lt1 8093  ax-1rid 8094  ax-0id 8095  ax-rnegex 8096  ax-precex 8097  ax-cnre 8098  ax-pre-ltirr 8099  ax-pre-ltwlin 8100  ax-pre-lttrn 8101  ax-pre-apti 8102  ax-pre-ltadd 8103  ax-pre-mulgt0 8104  ax-pre-mulext 8105  ax-arch 8106  ax-caucvg 8107

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4381  df-po 4384  df-iso 4385  df-iord 4454  df-on 4456  df-ilim 4457  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-isom 5323  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-recs 6441  df-irdg 6506  df-frec 6527  df-1o 6552  df-oadd 6556  df-er 6670  df-map 6787  df-en 6878  df-dom 6879  df-fin 6880  df-pnf 8171  df-mnf 8172  df-xr 8173  df-ltxr 8174  df-le 8175  df-sub 8307  df-neg 8308  df-reap 8710  df-ap 8717  df-div 8808  df-inn 9099  df-2 9157  df-3 9158  df-4 9159  df-n0 9358  df-z 9435  df-uz 9711  df-q 9803  df-rp 9838  df-fz 10193  df-fzo 10327  df-seqfrec 10657  df-exp 10748  df-ihash 10985  df-cj 11339  df-re 11340  df-im 11341  df-rsqrt 11495  df-abs 11496  df-clim 11776  df-sumdc 11851  df-ply 15389

This theorem is referenced by: (None)