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Theorem cpnres 22208
Description: The restriction of a  C^n function is  C^n. (Contributed by Mario Carneiro, 11-Feb-2015.)
Assertion
Ref Expression
cpnres  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  -> 
( F  |`  S )  e.  ( ( C^n `  S ) `
 N ) )

Proof of Theorem cpnres
StepHypRef Expression
1 simpl 457 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  ->  S  e.  { RR ,  CC } )
2 simpr 461 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  ->  F  e.  ( (
C^n `  CC ) `  N )
)
3 ssid 3528 . . . . . 6  |-  CC  C_  CC
4 elfvdm 5898 . . . . . . . 8  |-  ( F  e.  ( ( C^n `  CC ) `
 N )  ->  N  e.  dom  ( C^n `  CC ) )
54adantl 466 . . . . . . 7  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  ->  N  e.  dom  ( C^n `  CC ) )
6 fncpn 22204 . . . . . . . . 9  |-  ( CC  C_  CC  ->  ( C^n `  CC )  Fn 
NN0 )
73, 6ax-mp 5 . . . . . . . 8  |-  ( C^n `  CC )  Fn  NN0
8 fndm 5686 . . . . . . . 8  |-  ( ( C^n `  CC )  Fn  NN0  ->  dom  ( C^n `  CC )  =  NN0 )
97, 8mp1i 12 . . . . . . 7  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  ->  dom  ( C^n `  CC )  =  NN0 )
105, 9eleqtrd 2557 . . . . . 6  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  ->  N  e.  NN0 )
11 elcpn 22205 . . . . . 6  |-  ( ( CC  C_  CC  /\  N  e.  NN0 )  ->  ( F  e.  ( (
C^n `  CC ) `  N )  <->  ( F  e.  ( CC 
^pm  CC )  /\  (
( CC  Dn
F ) `  N
)  e.  ( dom 
F -cn-> CC ) ) ) )
123, 10, 11sylancr 663 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  -> 
( F  e.  ( ( C^n `  CC ) `  N )  <-> 
( F  e.  ( CC  ^pm  CC )  /\  ( ( CC  Dn F ) `  N )  e.  ( dom  F -cn-> CC ) ) ) )
132, 12mpbid 210 . . . 4  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  -> 
( F  e.  ( CC  ^pm  CC )  /\  ( ( CC  Dn F ) `  N )  e.  ( dom  F -cn-> CC ) ) )
1413simpld 459 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  ->  F  e.  ( CC  ^pm 
CC ) )
15 pmresg 7458 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( F  |`  S )  e.  ( CC  ^pm  S
) )
161, 14, 15syl2anc 661 . 2  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  -> 
( F  |`  S )  e.  ( CC  ^pm  S ) )
1713simprd 463 . . . . . 6  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  -> 
( ( CC  Dn F ) `  N )  e.  ( dom  F -cn-> CC ) )
18 cncff 21265 . . . . . 6  |-  ( ( ( CC  Dn
F ) `  N
)  e.  ( dom 
F -cn-> CC )  ->  (
( CC  Dn
F ) `  N
) : dom  F --> CC )
1917, 18syl 16 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  -> 
( ( CC  Dn F ) `  N ) : dom  F --> CC )
20 fdm 5741 . . . . 5  |-  ( ( ( CC  Dn
F ) `  N
) : dom  F --> CC  ->  dom  ( ( CC  Dn F ) `
 N )  =  dom  F )
2119, 20syl 16 . . . 4  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  ->  dom  ( ( CC  Dn F ) `  N )  =  dom  F )
22 dvnres 22202 . . . 4  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC )  /\  N  e.  NN0 )  /\  dom  ( ( CC  Dn F ) `  N )  =  dom  F )  ->  ( ( S  Dn ( F  |`  S ) ) `  N )  =  ( ( ( CC  Dn F ) `  N )  |`  S ) )
231, 14, 10, 21, 22syl31anc 1231 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  -> 
( ( S  Dn ( F  |`  S ) ) `  N )  =  ( ( ( CC  Dn F ) `  N )  |`  S ) )
24 resres 5292 . . . . . . 7  |-  ( ( ( ( CC  Dn F ) `  N )  |`  S )  |`  dom  F )  =  ( ( ( CC  Dn F ) `
 N )  |`  ( S  i^i  dom  F
) )
25 rescom 5304 . . . . . . 7  |-  ( ( ( ( CC  Dn F ) `  N )  |`  S )  |`  dom  F )  =  ( ( ( ( CC  Dn F ) `  N )  |`  dom  F )  |`  S )
2624, 25eqtr3i 2498 . . . . . 6  |-  ( ( ( CC  Dn
F ) `  N
)  |`  ( S  i^i  dom 
F ) )  =  ( ( ( ( CC  Dn F ) `  N )  |`  dom  F )  |`  S )
27 ffn 5737 . . . . . . . 8  |-  ( ( ( CC  Dn
F ) `  N
) : dom  F --> CC  ->  ( ( CC  Dn F ) `
 N )  Fn 
dom  F )
28 fnresdm 5696 . . . . . . . 8  |-  ( ( ( CC  Dn
F ) `  N
)  Fn  dom  F  ->  ( ( ( CC  Dn F ) `
 N )  |`  dom  F )  =  ( ( CC  Dn
F ) `  N
) )
2919, 27, 283syl 20 . . . . . . 7  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  -> 
( ( ( CC  Dn F ) `
 N )  |`  dom  F )  =  ( ( CC  Dn
F ) `  N
) )
3029reseq1d 5278 . . . . . 6  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  -> 
( ( ( ( CC  Dn F ) `  N )  |`  dom  F )  |`  S )  =  ( ( ( CC  Dn F ) `  N )  |`  S ) )
3126, 30syl5eq 2520 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  -> 
( ( ( CC  Dn F ) `
 N )  |`  ( S  i^i  dom  F
) )  =  ( ( ( CC  Dn F ) `  N )  |`  S ) )
32 inss2 3724 . . . . . 6  |-  ( S  i^i  dom  F )  C_ 
dom  F
33 rescncf 21269 . . . . . 6  |-  ( ( S  i^i  dom  F
)  C_  dom  F  -> 
( ( ( CC  Dn F ) `
 N )  e.  ( dom  F -cn-> CC )  ->  ( (
( CC  Dn
F ) `  N
)  |`  ( S  i^i  dom 
F ) )  e.  ( ( S  i^i  dom 
F ) -cn-> CC ) ) )
3432, 17, 33mpsyl 63 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  -> 
( ( ( CC  Dn F ) `
 N )  |`  ( S  i^i  dom  F
) )  e.  ( ( S  i^i  dom  F ) -cn-> CC ) )
3531, 34eqeltrrd 2556 . . . 4  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  -> 
( ( ( CC  Dn F ) `
 N )  |`  S )  e.  ( ( S  i^i  dom  F ) -cn-> CC ) )
36 dmres 5300 . . . . 5  |-  dom  ( F  |`  S )  =  ( S  i^i  dom  F )
3736oveq1i 6305 . . . 4  |-  ( dom  ( F  |`  S )
-cn-> CC )  =  ( ( S  i^i  dom  F ) -cn-> CC )
3835, 37syl6eleqr 2566 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  -> 
( ( ( CC  Dn F ) `
 N )  |`  S )  e.  ( dom  ( F  |`  S ) -cn-> CC ) )
3923, 38eqeltrd 2555 . 2  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  -> 
( ( S  Dn ( F  |`  S ) ) `  N )  e.  ( dom  ( F  |`  S ) -cn-> CC ) )
40 recnprss 22176 . . . 4  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
4140adantr 465 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  ->  S  C_  CC )
42 elcpn 22205 . . 3  |-  ( ( S  C_  CC  /\  N  e.  NN0 )  ->  (
( F  |`  S )  e.  ( ( C^n `  S ) `
 N )  <->  ( ( F  |`  S )  e.  ( CC  ^pm  S
)  /\  ( ( S  Dn ( F  |`  S ) ) `  N )  e.  ( dom  ( F  |`  S ) -cn-> CC ) ) ) )
4341, 10, 42syl2anc 661 . 2  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  -> 
( ( F  |`  S )  e.  ( ( C^n `  S ) `  N
)  <->  ( ( F  |`  S )  e.  ( CC  ^pm  S )  /\  ( ( S  Dn ( F  |`  S ) ) `  N )  e.  ( dom  ( F  |`  S ) -cn-> CC ) ) ) )
4416, 39, 43mpbir2and 920 1  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  -> 
( F  |`  S )  e.  ( ( C^n `  S ) `
 N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    i^i cin 3480    C_ wss 3481   {cpr 4035   dom cdm 5005    |` cres 5007    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295    ^pm cpm 7433   CCcc 9502   RRcr 9503   NN0cn0 10807   -cn->ccncf 21248    Dncdvn 22136   C^nccpn 22137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fi 7883  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-icc 11548  df-fz 11685  df-seq 12088  df-exp 12147  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-plusg 14585  df-mulr 14586  df-starv 14587  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-rest 14695  df-topn 14696  df-topgen 14716  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-fbas 18286  df-fg 18287  df-cnfld 18291  df-top 19268  df-bases 19270  df-topon 19271  df-topsp 19272  df-cld 19388  df-ntr 19389  df-cls 19390  df-nei 19467  df-lp 19505  df-perf 19506  df-cnp 19597  df-haus 19684  df-fil 20215  df-fm 20307  df-flim 20308  df-flf 20309  df-xms 20691  df-ms 20692  df-cncf 21250  df-limc 22138  df-dv 22139  df-dvn 22140  df-cpn 22141
This theorem is referenced by:  aalioulem3  22597
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