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Theorem cpnres 21554
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 3486 . . . . . 6  |-  CC  C_  CC
4 elfvdm 5828 . . . . . . . 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 21550 . . . . . . . . 9  |-  ( CC  C_  CC  ->  ( C^n `  CC )  Fn 
NN0 )
73, 6ax-mp 5 . . . . . . . 8  |-  ( C^n `  CC )  Fn  NN0
8 fndm 5621 . . . . . . . 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 2544 . . . . . 6  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  ->  N  e.  NN0 )
11 elcpn 21551 . . . . . 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 7353 . . 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 20611 . . . . . 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 5674 . . . . 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 21548 . . . 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 1222 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  -> 
( ( S  Dn ( F  |`  S ) ) `  N )  =  ( ( ( CC  Dn F ) `  N )  |`  S ) )
24 resres 5234 . . . . . . 7  |-  ( ( ( ( CC  Dn F ) `  N )  |`  S )  |`  dom  F )  =  ( ( ( CC  Dn F ) `
 N )  |`  ( S  i^i  dom  F
) )
25 rescom 5246 . . . . . . 7  |-  ( ( ( ( CC  Dn F ) `  N )  |`  S )  |`  dom  F )  =  ( ( ( ( CC  Dn F ) `  N )  |`  dom  F )  |`  S )
2624, 25eqtr3i 2485 . . . . . 6  |-  ( ( ( CC  Dn
F ) `  N
)  |`  ( S  i^i  dom 
F ) )  =  ( ( ( ( CC  Dn F ) `  N )  |`  dom  F )  |`  S )
27 ffn 5670 . . . . . . . 8  |-  ( ( ( CC  Dn
F ) `  N
) : dom  F --> CC  ->  ( ( CC  Dn F ) `
 N )  Fn 
dom  F )
28 fnresdm 5631 . . . . . . . 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 5220 . . . . . 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 2507 . . . . 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 3682 . . . . . 6  |-  ( S  i^i  dom  F )  C_ 
dom  F
33 rescncf 20615 . . . . . 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 2543 . . . 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 5242 . . . . 5  |-  dom  ( F  |`  S )  =  ( S  i^i  dom  F )
3736oveq1i 6213 . . . 4  |-  ( dom  ( F  |`  S )
-cn-> CC )  =  ( ( S  i^i  dom  F ) -cn-> CC )
3835, 37syl6eleqr 2553 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  -> 
( ( ( CC  Dn F ) `
 N )  |`  S )  e.  ( dom  ( F  |`  S ) -cn-> CC ) )
3923, 38eqeltrd 2542 . 2  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  -> 
( ( S  Dn ( F  |`  S ) ) `  N )  e.  ( dom  ( F  |`  S ) -cn-> CC ) )
40 recnprss 21522 . . . 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 21551 . . 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 913 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 1370    e. wcel 1758    i^i cin 3438    C_ wss 3439   {cpr 3990   dom cdm 4951    |` cres 4953    Fn wfn 5524   -->wf 5525   ` cfv 5529  (class class class)co 6203    ^pm cpm 7328   CCcc 9395   RRcr 9396   NN0cn0 10694   -cn->ccncf 20594    Dncdvn 21482   C^nccpn 21483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7962  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-pre-sup 9475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-pm 7330  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fi 7776  df-sup 7806  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-nn 10438  df-2 10495  df-3 10496  df-4 10497  df-5 10498  df-6 10499  df-7 10500  df-8 10501  df-9 10502  df-10 10503  df-n0 10695  df-z 10762  df-dec 10871  df-uz 10977  df-q 11069  df-rp 11107  df-xneg 11204  df-xadd 11205  df-xmul 11206  df-icc 11422  df-fz 11559  df-seq 11928  df-exp 11987  df-cj 12710  df-re 12711  df-im 12712  df-sqr 12846  df-abs 12847  df-struct 14298  df-ndx 14299  df-slot 14300  df-base 14301  df-plusg 14374  df-mulr 14375  df-starv 14376  df-tset 14380  df-ple 14381  df-ds 14383  df-unif 14384  df-rest 14484  df-topn 14485  df-topgen 14505  df-psmet 17944  df-xmet 17945  df-met 17946  df-bl 17947  df-mopn 17948  df-fbas 17949  df-fg 17950  df-cnfld 17954  df-top 18645  df-bases 18647  df-topon 18648  df-topsp 18649  df-cld 18765  df-ntr 18766  df-cls 18767  df-nei 18844  df-lp 18882  df-perf 18883  df-cnp 18974  df-haus 19061  df-fil 19561  df-fm 19653  df-flim 19654  df-flf 19655  df-xms 20037  df-ms 20038  df-cncf 20596  df-limc 21484  df-dv 21485  df-dvn 21486  df-cpn 21487
This theorem is referenced by:  aalioulem3  21943
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