Theorem eldvap 15372

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 15371	. . . . 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 4084	. . 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 4087	. . . . . . 7 |- ( x = B -> ( w # x <-> w # B ) )
13	12	rabbidv 2788	. . . . . 6 |- ( x = B -> { w e. A | w # x } = { w e. A | w # B } )
14		fveq2 5629	. . . . . . . 8 |- ( x = B -> ( F ` x ) = ( F ` B ) )
15	14	oveq2d 6023	. . . . . . 7 |- ( x = B -> ( ( F ` z ) - ( F ` x ) ) = ( ( F ` z ) - ( F ` B ) ) )
16		oveq2 6015	. . . . . . 7 |- ( x = B -> ( z - x ) = ( z - B ) )
17	15, 16	oveq12d 6025	. . . . . 6 |- ( x = B -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = ( ( ( F ` z ) - ( F ` B ) ) / ( z - B ) ) )
18	13, 17	mpteq12dv 4166	. . . . 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 6025	. . 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 4862	. 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 3197   {csn 3666   <.cop 3669   U_ciun 3965   class class class wbr 4083   |-> cmpt 4145   X. cxp 4717   o. ccom 4723   -->wf 5314   ` cfv 5318  (class class class)co 6007   CCcc 8008   - cmin 8328   # cap 8739   / cdiv 8830   abscabs 11524   ↾_t crest 13288   MetOpencmopn 14521   intcnt 14783   lim CC climc 15344   _D cdv 15345

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 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129  ax-caucvg 8130

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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-map 6805  df-pm 6806  df-sup 7162  df-inf 7163  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-n0 9381  df-z 9458  df-uz 9734  df-q 9827  df-rp 9862  df-xneg 9980  df-xadd 9981  df-seqfrec 10682  df-exp 10773  df-cj 11369  df-re 11370  df-im 11371  df-rsqrt 11525  df-abs 11526  df-rest 13290  df-topgen 13309  df-psmet 14523  df-xmet 14524  df-met 14525  df-bl 14526  df-mopn 14527  df-top 14688  df-topon 14701  df-bases 14733  df-ntr 14786  df-limced 15346  df-dvap 15347

This theorem is referenced by:  dvcl  15373  dvfgg  15378  dvidlemap  15381  dvidrelem  15382  dvidsslem  15383  dvcnp2cntop  15389  dvaddxxbr  15391  dvmulxxbr  15392  dvcoapbr  15397  dvcjbr  15398  dvrecap  15403  dveflem  15416