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Theorem dvres2 22879
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 22878, 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 5135 . . 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 2453 . . . . 5  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
4 eqid 2453 . . . . 5  |-  ( (
TopOpen ` fld )t  S )  =  ( ( TopOpen ` fld )t  S )
5 eqid 2453 . . . . 5  |-  ( z  e.  ( A  \  { x } ) 
|->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) )  =  ( z  e.  ( A  \  { x } ) 
|->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) )
6 simp1l 1033 . . . . 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 1034 . . . . 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 1035 . . . . 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 1036 . . . . 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 1037 . . . . . 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 22866 . . . . . 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 675 . . . . 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 1038 . . . . 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 22877 . . . 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 1211 . . 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 3050 . . . . 5  |-  y  e. 
_V
1716brres 5114 . . . 4  |-  ( x ( ( S  _D  F )  |`  B ) y  <->  ( x ( S  _D  F ) y  /\  x  e.  B ) )
18 df-br 4406 . . . 4  |-  ( x ( ( S  _D  F )  |`  B ) y  <->  <. x ,  y
>.  e.  ( ( S  _D  F )  |`  B ) )
1917, 18bitr3i 255 . . 3  |-  ( ( x ( S  _D  F ) y  /\  x  e.  B )  <->  <.
x ,  y >.  e.  ( ( S  _D  F )  |`  B ) )
20 df-br 4406 . . 3  |-  ( x ( B  _D  ( F  |`  B ) ) y  <->  <. x ,  y
>.  e.  ( B  _D  ( F  |`  B ) ) )
2115, 19, 203imtr3g 273 . 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 4930 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 371    /\ w3a 986    e. wcel 1889    \ cdif 3403    C_ wss 3406   {csn 3970   <.cop 3976   class class class wbr 4405    |-> cmpt 4464    |` cres 4839   Rel wrel 4842   -->wf 5581   ` cfv 5585  (class class class)co 6295   CCcc 9542    - cmin 9865    / cdiv 10276   ↾t crest 15331   TopOpenctopn 15332  ℂfldccnfld 18982    _D cdv 22830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-iin 4284  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fi 7930  df-sup 7961  df-inf 7962  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-7 10680  df-8 10681  df-9 10682  df-10 10683  df-n0 10877  df-z 10945  df-dec 11059  df-uz 11167  df-q 11272  df-rp 11310  df-xneg 11416  df-xadd 11417  df-xmul 11418  df-fz 11792  df-seq 12221  df-exp 12280  df-cj 13174  df-re 13175  df-im 13176  df-sqrt 13310  df-abs 13311  df-struct 15135  df-ndx 15136  df-slot 15137  df-base 15138  df-plusg 15215  df-mulr 15216  df-starv 15217  df-tset 15221  df-ple 15222  df-ds 15224  df-unif 15225  df-rest 15333  df-topn 15334  df-topgen 15354  df-psmet 18974  df-xmet 18975  df-met 18976  df-bl 18977  df-mopn 18978  df-cnfld 18983  df-top 19933  df-bases 19934  df-topon 19935  df-topsp 19936  df-cld 20046  df-ntr 20047  df-cls 20048  df-cnp 20256  df-xms 21347  df-ms 21348  df-limc 22833  df-dv 22834
This theorem is referenced by:  dvres3  22880  dvres3a  22881
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