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Theorem dvres3a 22948
Description: Restriction of a complex differentiable function to the reals. This version of dvres3 22947 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 22904 . . 3  |-  Rel  ( S  _D  ( F  |`  S ) )
2 recnprss 22938 . . . . . 6  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
32ad2antrr 740 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  ->  S  C_  CC )
4 simplr 770 . . . . . . 7  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  ->  F : A --> CC )
5 inss2 3644 . . . . . . 7  |-  ( S  i^i  A )  C_  A
6 fssres 5761 . . . . . . 7  |-  ( ( F : A --> CC  /\  ( S  i^i  A ) 
C_  A )  -> 
( F  |`  ( S  i^i  A ) ) : ( S  i^i  A ) --> CC )
74, 5, 6sylancl 675 . . . . . 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 5135 . . . . . . . . 9  |-  ( ( F  |`  A )  |`  S )  =  ( ( F  |`  S )  |`  A )
9 resres 5123 . . . . . . . . 9  |-  ( ( F  |`  S )  |`  A )  =  ( F  |`  ( S  i^i  A ) )
108, 9eqtri 2493 . . . . . . . 8  |-  ( ( F  |`  A )  |`  S )  =  ( F  |`  ( S  i^i  A ) )
11 ffn 5739 . . . . . . . . . 10  |-  ( F : A --> CC  ->  F  Fn  A )
12 fnresdm 5695 . . . . . . . . . 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 5110 . . . . . . . 8  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  -> 
( ( F  |`  A )  |`  S )  =  ( F  |`  S ) )
1510, 14syl5eqr 2519 . . . . . . 7  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  -> 
( F  |`  ( S  i^i  A ) )  =  ( F  |`  S ) )
1615feq1d 5724 . . . . . 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 215 . . . . 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 3643 . . . . . 6  |-  ( S  i^i  A )  C_  S
1918a1i 11 . . . . 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 22935 . . . 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 5131 . . . . 5  |-  dom  (
( CC  _D  F
)  |`  S )  =  ( S  i^i  dom  ( CC  _D  F
) )
22 simprr 774 . . . . . 6  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  ->  dom  ( CC  _D  F
)  =  A )
2322ineq2d 3625 . . . . 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 2517 . . . 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 3455 . . 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 5148 . . 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 676 . 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 22940 . . . . 5  |-  ( S  e.  { RR ,  CC }  ->  ( S  _D  ( F  |`  S ) ) : dom  ( S  _D  ( F  |`  S ) ) --> CC )
2928ad2antrr 740 . . . 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 5742 . . . 4  |-  ( ( S  _D  ( F  |`  S ) ) : dom  ( S  _D  ( F  |`  S ) ) --> CC  ->  Fun  ( S  _D  ( F  |`  S ) ) )
3129, 30syl 17 . . 3  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  ->  Fun  ( S  _D  ( F  |`  S ) ) )
32 ssid 3437 . . . . 5  |-  CC  C_  CC
3332a1i 11 . . . 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 21881 . . . . 5  |-  J  e.  (TopOn `  CC )
36 simprl 772 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  ->  A  e.  J )
37 toponss 20021 . . . . 5  |-  ( ( J  e.  (TopOn `  CC )  /\  A  e.  J )  ->  A  C_  CC )
3835, 36, 37sylancr 676 . . . 4  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  ->  A  C_  CC )
39 dvres2 22946 . . . 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 1293 . . 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 5629 . . 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 673 . 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 2507 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 setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904    i^i cin 3389    C_ wss 3390   {cpr 3961   dom cdm 4839    |` cres 4841   Rel wrel 4844   Fun wfun 5583    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556   TopOpenctopn 15398  ℂfldccnfld 19047  TopOnctopon 19995    _D cdv 22897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fi 7943  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-icc 11667  df-fz 11811  df-seq 12252  df-exp 12311  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-plusg 15281  df-mulr 15282  df-starv 15283  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-rest 15399  df-topn 15400  df-topgen 15420  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cnp 20321  df-haus 20408  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-limc 22900  df-dv 22901
This theorem is referenced by:  dvnres  22964  dvmptres3  22989
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