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Theorem dvres3a 19264
Description: Restriction of a complex differentiable function to the reals. This version of dvres3 19263 assumes that  F is differentiable on its domain, but does not require  F to be differentiable on the whole real line. (Contributed by Mario Carneiro, 11-Feb-2015.)
Hypothesis
Ref Expression
dvres3a.j  |-  J  =  ( TopOpen ` fld )
Assertion
Ref Expression
dvres3a  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  -> 
( S  _D  ( F  |`  S ) )  =  ( ( CC 
_D  F )  |`  S ) )

Proof of Theorem dvres3a
StepHypRef Expression
1 reldv 19220 . . 3  |-  Rel  ( S  _D  ( F  |`  S ) )
2 recnprss 19254 . . . . . 6  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
32ad2antrr 706 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  ->  S  C_  CC )
4 simplr 731 . . . . . . 7  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  ->  F : A --> CC )
5 inss2 3390 . . . . . . 7  |-  ( S  i^i  A )  C_  A
6 fssres 5408 . . . . . . 7  |-  ( ( F : A --> CC  /\  ( S  i^i  A ) 
C_  A )  -> 
( F  |`  ( S  i^i  A ) ) : ( S  i^i  A ) --> CC )
74, 5, 6sylancl 643 . . . . . 6  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  -> 
( F  |`  ( S  i^i  A ) ) : ( S  i^i  A ) --> CC )
8 rescom 4980 . . . . . . . . 9  |-  ( ( F  |`  A )  |`  S )  =  ( ( F  |`  S )  |`  A )
9 resres 4968 . . . . . . . . 9  |-  ( ( F  |`  S )  |`  A )  =  ( F  |`  ( S  i^i  A ) )
108, 9eqtri 2303 . . . . . . . 8  |-  ( ( F  |`  A )  |`  S )  =  ( F  |`  ( S  i^i  A ) )
11 ffn 5389 . . . . . . . . . 10  |-  ( F : A --> CC  ->  F  Fn  A )
12 fnresdm 5353 . . . . . . . . . 10  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
134, 11, 123syl 18 . . . . . . . . 9  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  -> 
( F  |`  A )  =  F )
1413reseq1d 4954 . . . . . . . 8  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  -> 
( ( F  |`  A )  |`  S )  =  ( F  |`  S ) )
1510, 14syl5eqr 2329 . . . . . . 7  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  -> 
( F  |`  ( S  i^i  A ) )  =  ( F  |`  S ) )
1615feq1d 5379 . . . . . 6  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  -> 
( ( F  |`  ( S  i^i  A ) ) : ( S  i^i  A ) --> CC  <->  ( F  |`  S ) : ( S  i^i  A ) --> CC ) )
177, 16mpbid 201 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  -> 
( F  |`  S ) : ( S  i^i  A ) --> CC )
18 inss1 3389 . . . . . 6  |-  ( S  i^i  A )  C_  S
1918a1i 10 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  -> 
( S  i^i  A
)  C_  S )
203, 17, 19dvbss 19251 . . . 4  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  ->  dom  ( S  _D  ( F  |`  S ) ) 
C_  ( S  i^i  A ) )
21 dmres 4976 . . . . 5  |-  dom  (
( CC  _D  F
)  |`  S )  =  ( S  i^i  dom  ( CC  _D  F
) )
22 simprr 733 . . . . . 6  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  ->  dom  ( CC  _D  F
)  =  A )
2322ineq2d 3370 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  -> 
( S  i^i  dom  ( CC  _D  F
) )  =  ( S  i^i  A ) )
2421, 23syl5eq 2327 . . . 4  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  ->  dom  ( ( CC  _D  F )  |`  S )  =  ( S  i^i  A ) )
2520, 24sseqtr4d 3215 . . 3  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  ->  dom  ( S  _D  ( F  |`  S ) ) 
C_  dom  ( ( CC  _D  F )  |`  S ) )
26 relssres 4992 . . 3  |-  ( ( Rel  ( S  _D  ( F  |`  S ) )  /\  dom  ( S  _D  ( F  |`  S ) )  C_  dom  ( ( CC  _D  F )  |`  S ) )  ->  ( ( S  _D  ( F  |`  S ) )  |`  dom  ( ( CC  _D  F )  |`  S ) )  =  ( S  _D  ( F  |`  S ) ) )
271, 25, 26sylancr 644 . 2  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  -> 
( ( S  _D  ( F  |`  S ) )  |`  dom  ( ( CC  _D  F )  |`  S ) )  =  ( S  _D  ( F  |`  S ) ) )
28 dvfg 19256 . . . . 5  |-  ( S  e.  { RR ,  CC }  ->  ( S  _D  ( F  |`  S ) ) : dom  ( S  _D  ( F  |`  S ) ) --> CC )
2928ad2antrr 706 . . . 4  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  -> 
( S  _D  ( F  |`  S ) ) : dom  ( S  _D  ( F  |`  S ) ) --> CC )
30 ffun 5391 . . . 4  |-  ( ( S  _D  ( F  |`  S ) ) : dom  ( S  _D  ( F  |`  S ) ) --> CC  ->  Fun  ( S  _D  ( F  |`  S ) ) )
3129, 30syl 15 . . 3  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  ->  Fun  ( S  _D  ( F  |`  S ) ) )
32 ssid 3197 . . . . 5  |-  CC  C_  CC
3332a1i 10 . . . 4  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  ->  CC  C_  CC )
34 dvres3a.j . . . . . 6  |-  J  =  ( TopOpen ` fld )
3534cnfldtopon 18292 . . . . 5  |-  J  e.  (TopOn `  CC )
36 simprl 732 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  ->  A  e.  J )
37 toponss 16667 . . . . 5  |-  ( ( J  e.  (TopOn `  CC )  /\  A  e.  J )  ->  A  C_  CC )
3835, 36, 37sylancr 644 . . . 4  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  ->  A  C_  CC )
39 dvres2 19262 . . . 4  |-  ( ( ( CC  C_  CC  /\  F : A --> CC )  /\  ( A  C_  CC  /\  S  C_  CC ) )  ->  (
( CC  _D  F
)  |`  S )  C_  ( S  _D  ( F  |`  S ) ) )
4033, 4, 38, 3, 39syl22anc 1183 . . 3  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  -> 
( ( CC  _D  F )  |`  S ) 
C_  ( S  _D  ( F  |`  S ) ) )
41 funssres 5294 . . 3  |-  ( ( Fun  ( S  _D  ( F  |`  S ) )  /\  ( ( CC  _D  F )  |`  S )  C_  ( S  _D  ( F  |`  S ) ) )  ->  ( ( S  _D  ( F  |`  S ) )  |`  dom  ( ( CC  _D  F )  |`  S ) )  =  ( ( CC  _D  F )  |`  S ) )
4231, 40, 41syl2anc 642 . 2  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  -> 
( ( S  _D  ( F  |`  S ) )  |`  dom  ( ( CC  _D  F )  |`  S ) )  =  ( ( CC  _D  F )  |`  S ) )
4327, 42eqtr3d 2317 1  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  -> 
( S  _D  ( F  |`  S ) )  =  ( ( CC 
_D  F )  |`  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   {cpr 3641   dom cdm 4689    |` cres 4691   Rel wrel 4694   Fun wfun 5249    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   TopOpenctopn 13326  ℂfldccnfld 16377  TopOnctopon 16632    _D cdv 19213
This theorem is referenced by:  dvnres  19280  dvmptres3  19305
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-icc 10663  df-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-starv 13223  df-tset 13227  df-ple 13228  df-ds 13230  df-rest 13327  df-topn 13328  df-topgen 13344  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cnp 16958  df-haus 17043  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-limc 19216  df-dv 19217
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