Theorem plyreres 15280

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 15249	. . 3 |- ( F e. (Poly ` RR ) -> RR C_ CC )
2		plyf 15253	. . . 4 |- ( F e. (Poly ` RR ) -> F : CC --> CC )
3		ffn 5431	. . . 4 |- ( F : CC --> CC -> F Fn CC )
4		fnssresb 5393	. . . 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 5607	. . . . . 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 8065	. . . . . . 7 |- ( a e. RR -> a e. CC )
10		ffvelcdm 5720	. . . . . . 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 15279	. . . . . . . 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 11237	. . . . . . . . 9 |- ( a e. RR -> ( * ` a ) = a )
15	14	adantl 277	. . . . . . . 8 |- ( ( F e. (Poly ` RR ) /\ a e. RR ) -> ( * ` a ) = a )
16	15	fveq2d 5587	. . . . . . 7 |- ( ( F e. (Poly ` RR ) /\ a e. RR ) -> ( F ` ( * ` a ) ) = ( F ` a ) )
17	13, 16	eqtrd 2239	. . . . . 6 |- ( ( F e. (Poly ` RR ) /\ a e. RR ) -> ( * ` ( F ` a ) ) = ( F ` a ) )
18	11, 17	cjrebd 11301	. . . . 5 |- ( ( F e. (Poly ` RR ) /\ a e. RR ) -> ( F ` a ) e. RR )
19	8, 18	eqeltrd 2283	. . . 4 |- ( ( F e. (Poly ` RR ) /\ a e. RR ) -> ( ( F |` RR ) ` a ) e. RR )
20	19	ralrimiva 2580	. . 3 |- ( F e. (Poly ` RR ) -> A. a e. RR ( ( F |` RR ) ` a ) e. RR )
21		fnfvrnss 5747	. . 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 5280	. 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 1373   e. wcel 2177   A.wral 2485   C_ wss 3167   ran crn 4680   |` cres 4681   Fn wfn 5271   -->wf 5272   ` cfv 5276   CCcc 7930   RRcr 7931   *ccj 11194  Polycply 15244

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-iinf 4640  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-mulrcl 8031  ax-addcom 8032  ax-mulcom 8033  ax-addass 8034  ax-mulass 8035  ax-distr 8036  ax-i2m1 8037  ax-0lt1 8038  ax-1rid 8039  ax-0id 8040  ax-rnegex 8041  ax-precex 8042  ax-cnre 8043  ax-pre-ltirr 8044  ax-pre-ltwlin 8045  ax-pre-lttrn 8046  ax-pre-apti 8047  ax-pre-ltadd 8048  ax-pre-mulgt0 8049  ax-pre-mulext 8050  ax-arch 8051  ax-caucvg 8052

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-if 3573  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-tr 4147  df-id 4344  df-po 4347  df-iso 4348  df-iord 4417  df-on 4419  df-ilim 4420  df-suc 4422  df-iom 4643  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-isom 5285  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-recs 6398  df-irdg 6463  df-frec 6484  df-1o 6509  df-oadd 6513  df-er 6627  df-map 6744  df-en 6835  df-dom 6836  df-fin 6837  df-pnf 8116  df-mnf 8117  df-xr 8118  df-ltxr 8119  df-le 8120  df-sub 8252  df-neg 8253  df-reap 8655  df-ap 8662  df-div 8753  df-inn 9044  df-2 9102  df-3 9103  df-4 9104  df-n0 9303  df-z 9380  df-uz 9656  df-q 9748  df-rp 9783  df-fz 10138  df-fzo 10272  df-seqfrec 10600  df-exp 10691  df-ihash 10928  df-cj 11197  df-re 11198  df-im 11199  df-rsqrt 11353  df-abs 11354  df-clim 11634  df-sumdc 11709  df-ply 15246

This theorem is referenced by: (None)