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Theorem dvres3a 22763
Description: Restriction of a complex differentiable function to the reals. This version of dvres3 22762 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 22719 . . 3  |-  Rel  ( S  _D  ( F  |`  S ) )
2 recnprss 22753 . . . . . 6  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
32ad2antrr 730 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  ->  S  C_  CC )
4 simplr 760 . . . . . . 7  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  ->  F : A --> CC )
5 inss2 3680 . . . . . . 7  |-  ( S  i^i  A )  C_  A
6 fssres 5757 . . . . . . 7  |-  ( ( F : A --> CC  /\  ( S  i^i  A ) 
C_  A )  -> 
( F  |`  ( S  i^i  A ) ) : ( S  i^i  A ) --> CC )
74, 5, 6sylancl 666 . . . . . 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 5140 . . . . . . . . 9  |-  ( ( F  |`  A )  |`  S )  =  ( ( F  |`  S )  |`  A )
9 resres 5128 . . . . . . . . 9  |-  ( ( F  |`  S )  |`  A )  =  ( F  |`  ( S  i^i  A ) )
108, 9eqtri 2449 . . . . . . . 8  |-  ( ( F  |`  A )  |`  S )  =  ( F  |`  ( S  i^i  A ) )
11 ffn 5737 . . . . . . . . . 10  |-  ( F : A --> CC  ->  F  Fn  A )
12 fnresdm 5694 . . . . . . . . . 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 5115 . . . . . . . 8  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  -> 
( ( F  |`  A )  |`  S )  =  ( F  |`  S ) )
1510, 14syl5eqr 2475 . . . . . . 7  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  -> 
( F  |`  ( S  i^i  A ) )  =  ( F  |`  S ) )
1615feq1d 5723 . . . . . 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 213 . . . . 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 3679 . . . . . 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 22750 . . . 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 5136 . . . . 5  |-  dom  (
( CC  _D  F
)  |`  S )  =  ( S  i^i  dom  ( CC  _D  F
) )
22 simprr 764 . . . . . 6  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  ->  dom  ( CC  _D  F
)  =  A )
2322ineq2d 3661 . . . . 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 2473 . . . 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 3498 . . 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 5153 . . 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 667 . 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 22755 . . . . 5  |-  ( S  e.  { RR ,  CC }  ->  ( S  _D  ( F  |`  S ) ) : dom  ( S  _D  ( F  |`  S ) ) --> CC )
2928ad2antrr 730 . . . 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 5739 . . . 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 3480 . . . . 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 21727 . . . . 5  |-  J  e.  (TopOn `  CC )
36 simprl 762 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  ->  A  e.  J )
37 toponss 19868 . . . . 5  |-  ( ( J  e.  (TopOn `  CC )  /\  A  e.  J )  ->  A  C_  CC )
3835, 36, 37sylancr 667 . . . 4  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  ->  A  C_  CC )
39 dvres2 22761 . . . 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 1265 . . 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 5632 . . 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 665 . 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 2463 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 370    = wceq 1437    e. wcel 1867    i^i cin 3432    C_ wss 3433   {cpr 3995   dom cdm 4845    |` cres 4847   Rel wrel 4850   Fun wfun 5586    Fn wfn 5587   -->wf 5588   ` cfv 5592  (class class class)co 6296   CCcc 9526   RRcr 9527   TopOpenctopn 15272  ℂfldccnfld 18898  TopOnctopon 19842    _D cdv 22712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-iin 4296  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-map 7473  df-pm 7474  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-fi 7922  df-sup 7953  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-8 10663  df-9 10664  df-10 10665  df-n0 10859  df-z 10927  df-dec 11041  df-uz 11149  df-q 11254  df-rp 11292  df-xneg 11398  df-xadd 11399  df-xmul 11400  df-icc 11631  df-fz 11772  df-seq 12200  df-exp 12259  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267  df-struct 15075  df-ndx 15076  df-slot 15077  df-base 15078  df-plusg 15155  df-mulr 15156  df-starv 15157  df-tset 15161  df-ple 15162  df-ds 15164  df-unif 15165  df-rest 15273  df-topn 15274  df-topgen 15294  df-psmet 18890  df-xmet 18891  df-met 18892  df-bl 18893  df-mopn 18894  df-fbas 18895  df-fg 18896  df-cnfld 18899  df-top 19845  df-bases 19846  df-topon 19847  df-topsp 19848  df-cld 19958  df-ntr 19959  df-cls 19960  df-nei 20038  df-lp 20076  df-perf 20077  df-cnp 20168  df-haus 20255  df-fil 20785  df-fm 20877  df-flim 20878  df-flf 20879  df-xms 21259  df-ms 21260  df-limc 22715  df-dv 22716
This theorem is referenced by:  dvnres  22779  dvmptres3  22804
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