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Theorem dvres2 21362
Description: Restriction of the base set of a derivative. The primary application of this theorem says that if a function is complex differentiable then it is also real differentiable. Unlike dvres 21361, there is no simple reverse relation relating real differentiable functions to complex differentiability, and indeed there are functions like  Re ( x ) which are everywhere real-differentiable but nowhere complex-differentiable.) (Contributed by Mario Carneiro, 9-Feb-2015.)
Assertion
Ref Expression
dvres2  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  (
( S  _D  F
)  |`  B )  C_  ( B  _D  ( F  |`  B ) ) )

Proof of Theorem dvres2
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relres 5133 . . 3  |-  Rel  (
( S  _D  F
)  |`  B )
21a1i 11 . 2  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  Rel  ( ( S  _D  F )  |`  B ) )
3 eqid 2438 . . . . 5  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
4 eqid 2438 . . . . 5  |-  ( (
TopOpen ` fld )t  S )  =  ( ( TopOpen ` fld )t  S )
5 eqid 2438 . . . . 5  |-  ( z  e.  ( A  \  { x } ) 
|->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) )  =  ( z  e.  ( A  \  { x } ) 
|->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) )
6 simp1l 1012 . . . . 5  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
)  /\  ( x
( S  _D  F
) y  /\  x  e.  B ) )  ->  S  C_  CC )
7 simp1r 1013 . . . . 5  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
)  /\  ( x
( S  _D  F
) y  /\  x  e.  B ) )  ->  F : A --> CC )
8 simp2l 1014 . . . . 5  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
)  /\  ( x
( S  _D  F
) y  /\  x  e.  B ) )  ->  A  C_  S )
9 simp2r 1015 . . . . 5  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
)  /\  ( x
( S  _D  F
) y  /\  x  e.  B ) )  ->  B  C_  S )
10 simp3l 1016 . . . . . 6  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
)  /\  ( x
( S  _D  F
) y  /\  x  e.  B ) )  ->  x ( S  _D  F ) y )
116, 7, 8dvcl 21349 . . . . . 6  |-  ( ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S )  /\  (
x ( S  _D  F ) y  /\  x  e.  B )
)  /\  x ( S  _D  F ) y )  ->  y  e.  CC )
1210, 11mpdan 668 . . . . 5  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
)  /\  ( x
( S  _D  F
) y  /\  x  e.  B ) )  -> 
y  e.  CC )
13 simp3r 1017 . . . . 5  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
)  /\  ( x
( S  _D  F
) y  /\  x  e.  B ) )  ->  x  e.  B )
143, 4, 5, 6, 7, 8, 9, 12, 10, 13dvres2lem 21360 . . . 4  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
)  /\  ( x
( S  _D  F
) y  /\  x  e.  B ) )  ->  x ( B  _D  ( F  |`  B ) ) y )
15143expia 1189 . . 3  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  (
( x ( S  _D  F ) y  /\  x  e.  B
)  ->  x ( B  _D  ( F  |`  B ) ) y ) )
16 vex 2970 . . . . 5  |-  y  e. 
_V
1716brres 5112 . . . 4  |-  ( x ( ( S  _D  F )  |`  B ) y  <->  ( x ( S  _D  F ) y  /\  x  e.  B ) )
18 df-br 4288 . . . 4  |-  ( x ( ( S  _D  F )  |`  B ) y  <->  <. x ,  y
>.  e.  ( ( S  _D  F )  |`  B ) )
1917, 18bitr3i 251 . . 3  |-  ( ( x ( S  _D  F ) y  /\  x  e.  B )  <->  <.
x ,  y >.  e.  ( ( S  _D  F )  |`  B ) )
20 df-br 4288 . . 3  |-  ( x ( B  _D  ( F  |`  B ) ) y  <->  <. x ,  y
>.  e.  ( B  _D  ( F  |`  B ) ) )
2115, 19, 203imtr3g 269 . 2  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  ( <. x ,  y >.  e.  ( ( S  _D  F )  |`  B )  ->  <. x ,  y
>.  e.  ( B  _D  ( F  |`  B ) ) ) )
222, 21relssdv 4927 1  |-  ( ( ( S  C_  CC  /\  F : A --> CC )  /\  ( A  C_  S  /\  B  C_  S
) )  ->  (
( S  _D  F
)  |`  B )  C_  ( B  _D  ( F  |`  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    e. wcel 1756    \ cdif 3320    C_ wss 3323   {csn 3872   <.cop 3878   class class class wbr 4287    e. cmpt 4345    |` cres 4837   Rel wrel 4840   -->wf 5409   ` cfv 5413  (class class class)co 6086   CCcc 9272    - cmin 9587    / cdiv 9985   ↾t crest 14351   TopOpenctopn 14352  ℂfldccnfld 17793    _D cdv 21313
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fi 7653  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-fz 11430  df-seq 11799  df-exp 11858  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-plusg 14243  df-mulr 14244  df-starv 14245  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-rest 14353  df-topn 14354  df-topgen 14374  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-cnfld 17794  df-top 18478  df-bases 18480  df-topon 18481  df-topsp 18482  df-cld 18598  df-ntr 18599  df-cls 18600  df-cnp 18807  df-xms 19870  df-ms 19871  df-limc 21316  df-dv 21317
This theorem is referenced by:  dvres3  21363  dvres3a  21364
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