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Theorem cpnres 21252
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 454 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  ->  S  e.  { RR ,  CC } )
2 simpr 458 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  ->  F  e.  ( (
C^n `  CC ) `  N )
)
3 ssid 3363 . . . . . 6  |-  CC  C_  CC
4 elfvdm 5704 . . . . . . . 8  |-  ( F  e.  ( ( C^n `  CC ) `
 N )  ->  N  e.  dom  ( C^n `  CC ) )
54adantl 463 . . . . . . 7  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  ->  N  e.  dom  ( C^n `  CC ) )
6 fncpn 21248 . . . . . . . . 9  |-  ( CC  C_  CC  ->  ( C^n `  CC )  Fn 
NN0 )
73, 6ax-mp 5 . . . . . . . 8  |-  ( C^n `  CC )  Fn  NN0
8 fndm 5498 . . . . . . . 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 2509 . . . . . 6  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  ->  N  e.  NN0 )
11 elcpn 21249 . . . . . 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 656 . . . . 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 456 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  ->  F  e.  ( CC  ^pm 
CC ) )
15 pmresg 7228 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( F  |`  S )  e.  ( CC  ^pm  S
) )
161, 14, 15syl2anc 654 . 2  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  -> 
( F  |`  S )  e.  ( CC  ^pm  S ) )
1713simprd 460 . . . . . 6  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  -> 
( ( CC  Dn F ) `  N )  e.  ( dom  F -cn-> CC ) )
18 cncff 20310 . . . . . 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 5551 . . . . 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 21246 . . . 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 1214 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  -> 
( ( S  Dn ( F  |`  S ) ) `  N )  =  ( ( ( CC  Dn F ) `  N )  |`  S ) )
24 resres 5111 . . . . . . 7  |-  ( ( ( ( CC  Dn F ) `  N )  |`  S )  |`  dom  F )  =  ( ( ( CC  Dn F ) `
 N )  |`  ( S  i^i  dom  F
) )
25 rescom 5123 . . . . . . 7  |-  ( ( ( ( CC  Dn F ) `  N )  |`  S )  |`  dom  F )  =  ( ( ( ( CC  Dn F ) `  N )  |`  dom  F )  |`  S )
2624, 25eqtr3i 2455 . . . . . 6  |-  ( ( ( CC  Dn
F ) `  N
)  |`  ( S  i^i  dom 
F ) )  =  ( ( ( ( CC  Dn F ) `  N )  |`  dom  F )  |`  S )
27 ffn 5547 . . . . . . . 8  |-  ( ( ( CC  Dn
F ) `  N
) : dom  F --> CC  ->  ( ( CC  Dn F ) `
 N )  Fn 
dom  F )
28 fnresdm 5508 . . . . . . . 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 5096 . . . . . 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 2477 . . . . 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 3559 . . . . . 6  |-  ( S  i^i  dom  F )  C_ 
dom  F
33 rescncf 20314 . . . . . 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 2508 . . . 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 5119 . . . . 5  |-  dom  ( F  |`  S )  =  ( S  i^i  dom  F )
3736oveq1i 6090 . . . 4  |-  ( dom  ( F  |`  S )
-cn-> CC )  =  ( ( S  i^i  dom  F ) -cn-> CC )
3835, 37syl6eleqr 2524 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  -> 
( ( ( CC  Dn F ) `
 N )  |`  S )  e.  ( dom  ( F  |`  S ) -cn-> CC ) )
3923, 38eqeltrd 2507 . 2  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  -> 
( ( S  Dn ( F  |`  S ) ) `  N )  e.  ( dom  ( F  |`  S ) -cn-> CC ) )
40 recnprss 21220 . . . 4  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
4140adantr 462 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C^n `
 CC ) `  N ) )  ->  S  C_  CC )
42 elcpn 21249 . . 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 654 . 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 906 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 1362    e. wcel 1755    i^i cin 3315    C_ wss 3316   {cpr 3867   dom cdm 4827    |` cres 4829    Fn wfn 5401   -->wf 5402   ` cfv 5406  (class class class)co 6080    ^pm cpm 7203   CCcc 9267   RRcr 9268   NN0cn0 10566   -cn->ccncf 20293    Dncdvn 21180   C^nccpn 21181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346  ax-pre-sup 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fi 7649  df-sup 7679  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-div 9981  df-nn 10310  df-2 10367  df-3 10368  df-4 10369  df-5 10370  df-6 10371  df-7 10372  df-8 10373  df-9 10374  df-10 10375  df-n0 10567  df-z 10634  df-dec 10743  df-uz 10849  df-q 10941  df-rp 10979  df-xneg 11076  df-xadd 11077  df-xmul 11078  df-icc 11294  df-fz 11424  df-seq 11790  df-exp 11849  df-cj 12571  df-re 12572  df-im 12573  df-sqr 12707  df-abs 12708  df-struct 14158  df-ndx 14159  df-slot 14160  df-base 14161  df-plusg 14233  df-mulr 14234  df-starv 14235  df-tset 14239  df-ple 14240  df-ds 14242  df-unif 14243  df-rest 14343  df-topn 14344  df-topgen 14364  df-psmet 17652  df-xmet 17653  df-met 17654  df-bl 17655  df-mopn 17656  df-fbas 17657  df-fg 17658  df-cnfld 17662  df-top 18344  df-bases 18346  df-topon 18347  df-topsp 18348  df-cld 18464  df-ntr 18465  df-cls 18466  df-nei 18543  df-lp 18581  df-perf 18582  df-cnp 18673  df-haus 18760  df-fil 19260  df-fm 19352  df-flim 19353  df-flf 19354  df-xms 19736  df-ms 19737  df-cncf 20295  df-limc 21182  df-dv 21183  df-dvn 21184  df-cpn 21185
This theorem is referenced by:  aalioulem3  21684
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