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Theorem eldv 22053
Description: The differentiable predicate. A function  F is differentiable at  B with derivative  C iff  F is defined in a neighborhood of  B and the difference quotient has limit  C at  B. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Mario Carneiro, 25-Dec-2016.)
Hypotheses
Ref Expression
dvval.t  |-  T  =  ( Kt  S )
dvval.k  |-  K  =  ( TopOpen ` fld )
eldv.g  |-  G  =  ( z  e.  ( A  \  { B } )  |->  ( ( ( F `  z
)  -  ( F `
 B ) )  /  ( z  -  B ) ) )
eldv.s  |-  ( ph  ->  S  C_  CC )
eldv.f  |-  ( ph  ->  F : A --> CC )
eldv.a  |-  ( ph  ->  A  C_  S )
Assertion
Ref Expression
eldv  |-  ( ph  ->  ( B ( S  _D  F ) C  <-> 
( B  e.  ( ( int `  T
) `  A )  /\  C  e.  ( G lim CC  B ) ) ) )
Distinct variable groups:    z, A    z, B    z, F    z, C    z, K    z, S
Allowed substitution hints:    ph( z)    T( z)    G( z)

Proof of Theorem eldv
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eldv.s . . . . 5  |-  ( ph  ->  S  C_  CC )
2 eldv.f . . . . 5  |-  ( ph  ->  F : A --> CC )
3 eldv.a . . . . 5  |-  ( ph  ->  A  C_  S )
4 dvval.t . . . . . 6  |-  T  =  ( Kt  S )
5 dvval.k . . . . . 6  |-  K  =  ( TopOpen ` fld )
64, 5dvfval 22052 . . . . 5  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  (
( S  _D  F
)  =  U_ x  e.  ( ( int `  T
) `  A )
( { x }  X.  ( ( z  e.  ( A  \  {
x } )  |->  ( ( ( F `  z )  -  ( F `  x )
)  /  ( z  -  x ) ) ) lim CC  x ) )  /\  ( S  _D  F )  C_  ( ( ( int `  T ) `  A
)  X.  CC ) ) )
71, 2, 3, 6syl3anc 1228 . . . 4  |-  ( ph  ->  ( ( S  _D  F )  =  U_ x  e.  ( ( int `  T ) `  A ) ( { x }  X.  (
( z  e.  ( A  \  { x } )  |->  ( ( ( F `  z
)  -  ( F `
 x ) )  /  ( z  -  x ) ) ) lim
CC  x ) )  /\  ( S  _D  F )  C_  (
( ( int `  T
) `  A )  X.  CC ) ) )
87simpld 459 . . 3  |-  ( ph  ->  ( S  _D  F
)  =  U_ x  e.  ( ( int `  T
) `  A )
( { x }  X.  ( ( z  e.  ( A  \  {
x } )  |->  ( ( ( F `  z )  -  ( F `  x )
)  /  ( z  -  x ) ) ) lim CC  x ) ) )
98eleq2d 2537 . 2  |-  ( ph  ->  ( <. B ,  C >.  e.  ( S  _D  F )  <->  <. B ,  C >.  e.  U_ x  e.  ( ( int `  T
) `  A )
( { x }  X.  ( ( z  e.  ( A  \  {
x } )  |->  ( ( ( F `  z )  -  ( F `  x )
)  /  ( z  -  x ) ) ) lim CC  x ) ) ) )
10 df-br 4448 . . 3  |-  ( B ( S  _D  F
) C  <->  <. B ,  C >.  e.  ( S  _D  F ) )
1110bicomi 202 . 2  |-  ( <. B ,  C >.  e.  ( S  _D  F
)  <->  B ( S  _D  F ) C )
12 sneq 4037 . . . . . . 7  |-  ( x  =  B  ->  { x }  =  { B } )
1312difeq2d 3622 . . . . . 6  |-  ( x  =  B  ->  ( A  \  { x }
)  =  ( A 
\  { B }
) )
14 fveq2 5865 . . . . . . . 8  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
1514oveq2d 6299 . . . . . . 7  |-  ( x  =  B  ->  (
( F `  z
)  -  ( F `
 x ) )  =  ( ( F `
 z )  -  ( F `  B ) ) )
16 oveq2 6291 . . . . . . 7  |-  ( x  =  B  ->  (
z  -  x )  =  ( z  -  B ) )
1715, 16oveq12d 6301 . . . . . 6  |-  ( x  =  B  ->  (
( ( F `  z )  -  ( F `  x )
)  /  ( z  -  x ) )  =  ( ( ( F `  z )  -  ( F `  B ) )  / 
( z  -  B
) ) )
1813, 17mpteq12dv 4525 . . . . 5  |-  ( x  =  B  ->  (
z  e.  ( A 
\  { x }
)  |->  ( ( ( F `  z )  -  ( F `  x ) )  / 
( z  -  x
) ) )  =  ( z  e.  ( A  \  { B } )  |->  ( ( ( F `  z
)  -  ( F `
 B ) )  /  ( z  -  B ) ) ) )
19 eldv.g . . . . 5  |-  G  =  ( z  e.  ( A  \  { B } )  |->  ( ( ( F `  z
)  -  ( F `
 B ) )  /  ( z  -  B ) ) )
2018, 19syl6eqr 2526 . . . 4  |-  ( x  =  B  ->  (
z  e.  ( A 
\  { x }
)  |->  ( ( ( F `  z )  -  ( F `  x ) )  / 
( z  -  x
) ) )  =  G )
21 id 22 . . . 4  |-  ( x  =  B  ->  x  =  B )
2220, 21oveq12d 6301 . . 3  |-  ( x  =  B  ->  (
( z  e.  ( A  \  { x } )  |->  ( ( ( F `  z
)  -  ( F `
 x ) )  /  ( z  -  x ) ) ) lim
CC  x )  =  ( G lim CC  B
) )
2322opeliunxp2 5140 . 2  |-  ( <. B ,  C >.  e. 
U_ x  e.  ( ( int `  T
) `  A )
( { x }  X.  ( ( z  e.  ( A  \  {
x } )  |->  ( ( ( F `  z )  -  ( F `  x )
)  /  ( z  -  x ) ) ) lim CC  x ) )  <->  ( B  e.  ( ( int `  T
) `  A )  /\  C  e.  ( G lim CC  B ) ) )
249, 11, 233bitr3g 287 1  |-  ( ph  ->  ( B ( S  _D  F ) C  <-> 
( B  e.  ( ( int `  T
) `  A )  /\  C  e.  ( G lim CC  B ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    \ cdif 3473    C_ wss 3476   {csn 4027   <.cop 4033   U_ciun 4325   class class class wbr 4447    |-> cmpt 4505    X. cxp 4997   -->wf 5583   ` cfv 5587  (class class class)co 6283   CCcc 9489    - cmin 9804    / cdiv 10205   ↾t crest 14675   TopOpenctopn 14676  ℂfldccnfld 18207   intcnt 19300   lim CC climc 22017    _D cdv 22018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fi 7870  df-sup 7900  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-3 10594  df-4 10595  df-5 10596  df-6 10597  df-7 10598  df-8 10599  df-9 10600  df-10 10601  df-n0 10795  df-z 10864  df-dec 10976  df-uz 11082  df-q 11182  df-rp 11220  df-xneg 11317  df-xadd 11318  df-xmul 11319  df-fz 11672  df-seq 12075  df-exp 12134  df-cj 12894  df-re 12895  df-im 12896  df-sqrt 13030  df-abs 13031  df-struct 14491  df-ndx 14492  df-slot 14493  df-base 14494  df-plusg 14567  df-mulr 14568  df-starv 14569  df-tset 14573  df-ple 14574  df-ds 14576  df-unif 14577  df-rest 14677  df-topn 14678  df-topgen 14698  df-psmet 18198  df-xmet 18199  df-met 18200  df-bl 18201  df-mopn 18202  df-cnfld 18208  df-top 19182  df-bases 19184  df-topon 19185  df-topsp 19186  df-cnp 19511  df-xms 20574  df-ms 20575  df-limc 22021  df-dv 22022
This theorem is referenced by:  dvcl  22054  perfdvf  22058  dvreslem  22064  dvres2lem  22065  dvidlem  22070  dvcnp  22073  dvcnp2  22074  dvaddbr  22092  dvmulbr  22093  dvcobr  22100  dvcjbr  22103  dvrec  22109  dvcnvlem  22128  dveflem  22131  dvferm1  22137  dvferm2  22139  ftc1  22194  taylthlem1  22518  ulmdvlem3  22547  ftc1cnnc  29682  fperdvper  31264
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