Theorem dvres2 19752

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 19751, 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.

Step	Hyp	Ref	Expression
1		relres 5133	. . 3 |- Rel ( ( S _D F ) |` B )
2	1	a1i 11	. 2 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) -> Rel ( ( S _D F ) |` B ) )
3		eqid 2404	. . . . 5 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
4		eqid 2404	. . . . 5 |- ( ( TopOpen ` ℂfld ) ↾t S ) = ( ( TopOpen ` ℂfld ) ↾t S )
5		eqid 2404	. . . . 5 |- ( z e. ( A \ { x } ) |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = ( z e. ( A \ { x } ) |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) )
6		simp1l 981	. . . . 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 982	. . . . 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 983	. . . . 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 984	. . . . 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 985	. . . . . 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 )
11	6, 7, 8	dvcl 19739	. . . . . 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 )
12	10, 11	mpdan 650	. . . . 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 986	. . . . 5 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) /\ ( x ( S _D F ) y /\ x e. B ) ) -> x e. B )
14	3, 4, 5, 6, 7, 8, 9, 12, 10, 13	dvres2lem 19750	. . . 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 )
15	14	3expia 1155	. . 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 2919	. . . . 5 |- y e. _V
17	16	brres 5111	. . . 4 |- ( x ( ( S _D F ) |` B ) y <-> ( x ( S _D F ) y /\ x e. B ) )
18		df-br 4173	. . . 4 |- ( x ( ( S _D F ) |` B ) y <-> <. x , y >. e. ( ( S _D F ) |` B ) )
19	17, 18	bitr3i 243	. . 3 |- ( ( x ( S _D F ) y /\ x e. B ) <-> <. x , y >. e. ( ( S _D F ) |` B ) )
20		df-br 4173	. . 3 |- ( x ( B _D ( F |` B ) ) y <-> <. x , y >. e. ( B _D ( F |` B ) ) )
21	15, 19, 20	3imtr3g 261	. 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 ) ) ) )
22	2, 21	relssdv 4927	1 |- ( ( ( S C_ CC /\ F : A --> CC ) /\ ( A C_ S /\ B C_ S ) ) -> ( ( S _D F ) |` B ) C_ ( B _D ( F |` B ) ) )

Syntax hints:   -> wi 4   /\ wa 359   /\ w3a 936   e. wcel 1721   \ cdif 3277   C_ wss 3280   {csn 3774   <.cop 3777   class class class wbr 4172   e. cmpt 4226   |` cres 4839   Rel wrel 4842   -->wf 5409   ` cfv 5413  (class class class)co 6040   CCcc 8944   - cmin 9247   / cdiv 9633   ↾_t crest 13603   TopOpenctopn 13604  ℂ_fldccnfld 16658   _D cdv 19703

This theorem is referenced by:  dvres3  19753  dvres3a  19754

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  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-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-fz 11000  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-plusg 13497  df-mulr 13498  df-starv 13499  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-rest 13605  df-topn 13606  df-topgen 13622  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-cnp 17246  df-xms 18303  df-ms 18304  df-limc 19706  df-dv 19707