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Theorem dvres3a 21411
Description: Restriction of a complex differentiable function to the reals. This version of dvres3 21410 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 21367 . . 3  |-  Rel  ( S  _D  ( F  |`  S ) )
2 recnprss 21401 . . . . . 6  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
32ad2antrr 725 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  ->  S  C_  CC )
4 simplr 754 . . . . . . 7  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  ->  F : A --> CC )
5 inss2 3592 . . . . . . 7  |-  ( S  i^i  A )  C_  A
6 fssres 5599 . . . . . . 7  |-  ( ( F : A --> CC  /\  ( S  i^i  A ) 
C_  A )  -> 
( F  |`  ( S  i^i  A ) ) : ( S  i^i  A ) --> CC )
74, 5, 6sylancl 662 . . . . . 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 5156 . . . . . . . . 9  |-  ( ( F  |`  A )  |`  S )  =  ( ( F  |`  S )  |`  A )
9 resres 5144 . . . . . . . . 9  |-  ( ( F  |`  S )  |`  A )  =  ( F  |`  ( S  i^i  A ) )
108, 9eqtri 2463 . . . . . . . 8  |-  ( ( F  |`  A )  |`  S )  =  ( F  |`  ( S  i^i  A ) )
11 ffn 5580 . . . . . . . . . 10  |-  ( F : A --> CC  ->  F  Fn  A )
12 fnresdm 5541 . . . . . . . . . 10  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
134, 11, 123syl 20 . . . . . . . . 9  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  -> 
( F  |`  A )  =  F )
1413reseq1d 5130 . . . . . . . 8  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  -> 
( ( F  |`  A )  |`  S )  =  ( F  |`  S ) )
1510, 14syl5eqr 2489 . . . . . . 7  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  -> 
( F  |`  ( S  i^i  A ) )  =  ( F  |`  S ) )
1615feq1d 5567 . . . . . 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 210 . . . . 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 3591 . . . . . 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 21398 . . . 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 5152 . . . . 5  |-  dom  (
( CC  _D  F
)  |`  S )  =  ( S  i^i  dom  ( CC  _D  F
) )
22 simprr 756 . . . . . 6  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  ->  dom  ( CC  _D  F
)  =  A )
2322ineq2d 3573 . . . . 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 2487 . . . 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 3414 . . 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 5168 . . 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 663 . 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 21403 . . . . 5  |-  ( S  e.  { RR ,  CC }  ->  ( S  _D  ( F  |`  S ) ) : dom  ( S  _D  ( F  |`  S ) ) --> CC )
2928ad2antrr 725 . . . 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 5582 . . . 4  |-  ( ( S  _D  ( F  |`  S ) ) : dom  ( S  _D  ( F  |`  S ) ) --> CC  ->  Fun  ( S  _D  ( F  |`  S ) ) )
3129, 30syl 16 . . 3  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  ->  Fun  ( S  _D  ( F  |`  S ) ) )
32 ssid 3396 . . . . 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 20384 . . . . 5  |-  J  e.  (TopOn `  CC )
36 simprl 755 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  ->  A  e.  J )
37 toponss 18556 . . . . 5  |-  ( ( J  e.  (TopOn `  CC )  /\  A  e.  J )  ->  A  C_  CC )
3835, 36, 37sylancr 663 . . . 4  |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  ->  A  C_  CC )
39 dvres2 21409 . . . 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 1219 . . 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 5479 . . 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 661 . 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 2477 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 369    = wceq 1369    e. wcel 1756    i^i cin 3348    C_ wss 3349   {cpr 3900   dom cdm 4861    |` cres 4863   Rel wrel 4866   Fun wfun 5433    Fn wfn 5434   -->wf 5435   ` cfv 5439  (class class class)co 6112   CCcc 9301   RRcr 9302   TopOpenctopn 14381  ℂfldccnfld 17840  TopOnctopon 18521    _D cdv 21360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-iin 4195  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-map 7237  df-pm 7238  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-fi 7682  df-sup 7712  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-7 10406  df-8 10407  df-9 10408  df-10 10409  df-n0 10601  df-z 10668  df-dec 10777  df-uz 10883  df-q 10975  df-rp 11013  df-xneg 11110  df-xadd 11111  df-xmul 11112  df-icc 11328  df-fz 11459  df-seq 11828  df-exp 11887  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-struct 14197  df-ndx 14198  df-slot 14199  df-base 14200  df-plusg 14272  df-mulr 14273  df-starv 14274  df-tset 14278  df-ple 14279  df-ds 14281  df-unif 14282  df-rest 14382  df-topn 14383  df-topgen 14403  df-psmet 17831  df-xmet 17832  df-met 17833  df-bl 17834  df-mopn 17835  df-fbas 17836  df-fg 17837  df-cnfld 17841  df-top 18525  df-bases 18527  df-topon 18528  df-topsp 18529  df-cld 18645  df-ntr 18646  df-cls 18647  df-nei 18724  df-lp 18762  df-perf 18763  df-cnp 18854  df-haus 18941  df-fil 19441  df-fm 19533  df-flim 19534  df-flf 19535  df-xms 19917  df-ms 19918  df-limc 21363  df-dv 21364
This theorem is referenced by:  dvnres  21427  dvmptres3  21452
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