Theorem eldvap 15396

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 Jim Kingdon, 27-Jun-2023.)

Hypotheses

Ref	Expression
dvval.t	|- T = ( K ↾t S )
dvval.k	|- K = ( MetOpen ` ( abs o. - ) )
eldvap.g	|- G = ( z e. { w e. A | w # 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
eldvap	|- ( ph -> ( B ( S _D F ) C <-> ( B e. ( ( int ` T ) ` A ) /\ C e. ( G lim CC B ) ) ) )

Distinct variable groups:   w, A, z   w, F, z   w, S, z   z, B, w

Allowed substitution hints:   ph( z, w)   C( z, w)   T( z, w)   G( z, w)   K( z, w)

Proof of Theorem eldvap

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	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 = ( K ↾t S )
5		dvval.k	. . . . . 6 |- K = ( MetOpen ` ( abs o. - ) )
6	4, 5	dvfvalap 15395	. . . . 5 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> ( ( S _D F ) = U_ x e. ( ( int ` T ) ` A ) ( { x } X. ( ( z e. { w e. A | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) ) /\ ( S _D F ) C_ ( ( ( int ` T ) ` A ) X. CC ) ) )
7	1, 2, 3, 6	syl3anc 1271	. . . 4 |- ( ph -> ( ( S _D F ) = U_ x e. ( ( int ` T ) ` A ) ( { x } X. ( ( z e. { w e. A | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) ) /\ ( S _D F ) C_ ( ( ( int ` T ) ` A ) X. CC ) ) )
8	7	simpld 112	. . 3 |- ( ph -> ( S _D F ) = U_ x e. ( ( int ` T ) ` A ) ( { x } X. ( ( z e. { w e. A | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) ) )
9	8	eleq2d 2299	. 2 |- ( ph -> ( <. B , C >. e. ( S _D F ) <-> <. B , C >. e. U_ x e. ( ( int ` T ) ` A ) ( { x } X. ( ( z e. { w e. A | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) ) ) )
10		df-br 4087	. . 3 |- ( B ( S _D F ) C <-> <. B , C >. e. ( S _D F ) )
11	10	bicomi 132	. 2 |- ( <. B , C >. e. ( S _D F ) <-> B ( S _D F ) C )
12		breq2 4090	. . . . . . 7 |- ( x = B -> ( w # x <-> w # B ) )
13	12	rabbidv 2789	. . . . . 6 |- ( x = B -> { w e. A | w # x } = { w e. A | w # B } )
14		fveq2 5635	. . . . . . . 8 |- ( x = B -> ( F ` x ) = ( F ` B ) )
15	14	oveq2d 6029	. . . . . . 7 |- ( x = B -> ( ( F ` z ) - ( F ` x ) ) = ( ( F ` z ) - ( F ` B ) ) )
16		oveq2 6021	. . . . . . 7 |- ( x = B -> ( z - x ) = ( z - B ) )
17	15, 16	oveq12d 6031	. . . . . 6 |- ( x = B -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = ( ( ( F ` z ) - ( F ` B ) ) / ( z - B ) ) )
18	13, 17	mpteq12dv 4169	. . . . 5 |- ( x = B -> ( z e. { w e. A | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = ( z e. { w e. A | w # B } |-> ( ( ( F ` z ) - ( F ` B ) ) / ( z - B ) ) ) )
19		eldvap.g	. . . . 5 |- G = ( z e. { w e. A | w # B } |-> ( ( ( F ` z ) - ( F ` B ) ) / ( z - B ) ) )
20	18, 19	eqtr4di 2280	. . . 4 |- ( x = B -> ( z e. { w e. A | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = G )
21		id 19	. . . 4 |- ( x = B -> x = B )
22	20, 21	oveq12d 6031	. . 3 |- ( x = B -> ( ( z e. { w e. A | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) = ( G lim CC B ) )
23	22	opeliunxp2 4868	. 2 |- ( <. B , C >. e. U_ x e. ( ( int ` T ) ` A ) ( { x } X. ( ( z e. { w e. A | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) ) <-> ( B e. ( ( int ` T ) ` A ) /\ C e. ( G lim CC B ) ) )
24	9, 11, 23	3bitr3g 222	1 |- ( ph -> ( B ( S _D F ) C <-> ( B e. ( ( int ` T ) ` A ) /\ C e. ( G lim CC B ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   {crab 2512   C_ wss 3198   {csn 3667   <.cop 3670   U_ciun 3968   class class class wbr 4086   |-> cmpt 4148   X. cxp 4721   o. ccom 4727   -->wf 5320   ` cfv 5324  (class class class)co 6013   CCcc 8020   - cmin 8340   # cap 8751   / cdiv 8842   abscabs 11548   ↾_t crest 13312   MetOpencmopn 14545   intcnt 14807   lim CC climc 15368   _D cdv 15369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141  ax-caucvg 8142

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-isom 5333  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-map 6814  df-pm 6815  df-sup 7174  df-inf 7175  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-n0 9393  df-z 9470  df-uz 9746  df-q 9844  df-rp 9879  df-xneg 9997  df-xadd 9998  df-seqfrec 10700  df-exp 10791  df-cj 11393  df-re 11394  df-im 11395  df-rsqrt 11549  df-abs 11550  df-rest 13314  df-topgen 13333  df-psmet 14547  df-xmet 14548  df-met 14549  df-bl 14550  df-mopn 14551  df-top 14712  df-topon 14725  df-bases 14757  df-ntr 14810  df-limced 15370  df-dvap 15371

This theorem is referenced by:  dvcl  15397  dvfgg  15402  dvidlemap  15405  dvidrelem  15406  dvidsslem  15407  dvcnp2cntop  15413  dvaddxxbr  15415  dvmulxxbr  15416  dvcoapbr  15421  dvcjbr  15422  dvrecap  15427  dveflem  15440