Theorem eldvap 15547

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 15546	. . . . 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 1274	. . . 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 2302	. 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 4110	. . 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 4113	. . . . . . 7 |- ( x = B -> ( w # x <-> w # B ) )
13	12	rabbidv 2802	. . . . . 6 |- ( x = B -> { w e. A | w # x } = { w e. A | w # B } )
14		fveq2 5670	. . . . . . . 8 |- ( x = B -> ( F ` x ) = ( F ` B ) )
15	14	oveq2d 6066	. . . . . . 7 |- ( x = B -> ( ( F ` z ) - ( F ` x ) ) = ( ( F ` z ) - ( F ` B ) ) )
16		oveq2 6058	. . . . . . 7 |- ( x = B -> ( z - x ) = ( z - B ) )
17	15, 16	oveq12d 6068	. . . . . 6 |- ( x = B -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = ( ( ( F ` z ) - ( F ` B ) ) / ( z - B ) ) )
18	13, 17	mpteq12dv 4192	. . . . 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 2283	. . . 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 6068	. . 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 4895	. 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 1398   e. wcel 2203   {crab 2524   C_ wss 3211   {csn 3689   <.cop 3692   U_ciun 3991   class class class wbr 4109   |-> cmpt 4171   X. cxp 4747   o. ccom 4753   -->wf 5348   ` cfv 5352  (class class class)co 6050   CCcc 8125   - cmin 8444   # cap 8855   / cdiv 8946   abscabs 11682   ↾_t crest 13452   MetOpencmopn 14689   intcnt 14958   lim CC climc 15519   _D cdv 15520

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-isom 5361  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-map 6884  df-pm 6885  df-sup 7275  df-inf 7276  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-xneg 10105  df-xadd 10106  df-seqfrec 10810  df-exp 10901  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-rest 13454  df-topgen 13473  df-psmet 14691  df-xmet 14692  df-met 14693  df-bl 14694  df-mopn 14695  df-top 14863  df-topon 14876  df-bases 14908  df-ntr 14961  df-limced 15521  df-dvap 15522

This theorem is referenced by:  dvcl  15548  dvfgg  15553  dvidlemap  15556  dvidrelem  15557  dvidsslem  15558  dvcnp2cntop  15564  dvaddxxbr  15566  dvmulxxbr  15567  dvcoapbr  15572  dvcjbr  15573  dvrecap  15578  dveflem  15591