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Theorem dvres3a 22862
Description: Restriction of a complex differentiable function to the reals. This version of dvres3 22861 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 22818 . . 3  |-  Rel  ( S  _D  ( F  |`  S ) )
2 recnprss 22852 . . . . . 6  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
32ad2antrr 731 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  ->  S  C_  CC )
4 simplr 761 . . . . . . 7  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  ->  F : A --> CC )
5 inss2 3652 . . . . . . 7  |-  ( S  i^i  A )  C_  A
6 fssres 5747 . . . . . . 7  |-  ( ( F : A --> CC  /\  ( S  i^i  A ) 
C_  A )  -> 
( F  |`  ( S  i^i  A ) ) : ( S  i^i  A ) --> CC )
74, 5, 6sylancl 667 . . . . . 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 5128 . . . . . . . . 9  |-  ( ( F  |`  A )  |`  S )  =  ( ( F  |`  S )  |`  A )
9 resres 5116 . . . . . . . . 9  |-  ( ( F  |`  S )  |`  A )  =  ( F  |`  ( S  i^i  A ) )
108, 9eqtri 2472 . . . . . . . 8  |-  ( ( F  |`  A )  |`  S )  =  ( F  |`  ( S  i^i  A ) )
11 ffn 5726 . . . . . . . . . 10  |-  ( F : A --> CC  ->  F  Fn  A )
12 fnresdm 5683 . . . . . . . . . 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 5103 . . . . . . . 8  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  -> 
( ( F  |`  A )  |`  S )  =  ( F  |`  S ) )
1510, 14syl5eqr 2498 . . . . . . 7  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  -> 
( F  |`  ( S  i^i  A ) )  =  ( F  |`  S ) )
1615feq1d 5712 . . . . . 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 214 . . . . 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 3651 . . . . . 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 22849 . . . 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 5124 . . . . 5  |-  dom  (
( CC  _D  F
)  |`  S )  =  ( S  i^i  dom  ( CC  _D  F
) )
22 simprr 765 . . . . . 6  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  ->  dom  ( CC  _D  F
)  =  A )
2322ineq2d 3633 . . . . 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 2496 . . . 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 3468 . . 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 5141 . . 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 668 . 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 22854 . . . . 5  |-  ( S  e.  { RR ,  CC }  ->  ( S  _D  ( F  |`  S ) ) : dom  ( S  _D  ( F  |`  S ) ) --> CC )
2928ad2antrr 731 . . . 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 5729 . . . 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 3450 . . . . 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 21796 . . . . 5  |-  J  e.  (TopOn `  CC )
36 simprl 763 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  ->  A  e.  J )
37 toponss 19937 . . . . 5  |-  ( ( J  e.  (TopOn `  CC )  /\  A  e.  J )  ->  A  C_  CC )
3835, 36, 37sylancr 668 . . . 4  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  ->  A  C_  CC )
39 dvres2 22860 . . . 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 1268 . . 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 5621 . . 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 666 . 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 2486 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 371    = wceq 1443    e. wcel 1886    i^i cin 3402    C_ wss 3403   {cpr 3969   dom cdm 4833    |` cres 4835   Rel wrel 4838   Fun wfun 5575    Fn wfn 5576   -->wf 5577   ` cfv 5581  (class class class)co 6288   CCcc 9534   RRcr 9535   TopOpenctopn 15313  ℂfldccnfld 18963  TopOnctopon 19911    _D cdv 22811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-map 7471  df-pm 7472  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fi 7922  df-sup 7953  df-inf 7954  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-q 11262  df-rp 11300  df-xneg 11406  df-xadd 11407  df-xmul 11408  df-icc 11639  df-fz 11782  df-seq 12211  df-exp 12270  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-plusg 15196  df-mulr 15197  df-starv 15198  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-rest 15314  df-topn 15315  df-topgen 15335  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-fbas 18960  df-fg 18961  df-cnfld 18964  df-top 19914  df-bases 19915  df-topon 19916  df-topsp 19917  df-cld 20027  df-ntr 20028  df-cls 20029  df-nei 20107  df-lp 20145  df-perf 20146  df-cnp 20237  df-haus 20324  df-fil 20854  df-fm 20946  df-flim 20947  df-flf 20948  df-xms 21328  df-ms 21329  df-limc 22814  df-dv 22815
This theorem is referenced by:  dvnres  22878  dvmptres3  22903
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