Theorem dvfvalap 13365

Description: Value and set bounds on the derivative operator. (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. - ) )

Assertion

Ref	Expression
dvfvalap	|- ( ( 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 ) ) )

Distinct variable groups:   w, A, x, z   w, F, x, z   w, S, x, z   x, T

Allowed substitution hints:   T( z, w)   K( x, z, w)

Proof of Theorem dvfvalap

Dummy variables f s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-dvap 13341	. . . 4 |- _D = ( s e. ~P CC , f e. ( CC ^pm s ) |-> U_ x e. ( ( int ` ( ( MetOpen ` ( abs o. - ) ) ↾t s ) ) ` dom f ) ( { x } X. ( ( z e. { w e. dom f | w # x } |-> ( ( ( f ` z ) - ( f ` x ) ) / ( z - x ) ) ) lim CC x ) ) )
2	1	a1i 9	. . 3 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> _D = ( s e. ~P CC , f e. ( CC ^pm s ) |-> U_ x e. ( ( int ` ( ( MetOpen ` ( abs o. - ) ) ↾t s ) ) ` dom f ) ( { x } X. ( ( z e. { w e. dom f | w # x } |-> ( ( ( f ` z ) - ( f ` x ) ) / ( z - x ) ) ) lim CC x ) ) ) )
3		dvval.k	. . . . . . . 8 |- K = ( MetOpen ` ( abs o. - ) )
4	3	oveq1i 5860	. . . . . . 7 |- ( K ↾t s ) = ( ( MetOpen ` ( abs o. - ) ) ↾t s )
5		simprl 526	. . . . . . . . 9 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> s = S )
6	5	oveq2d 5866	. . . . . . . 8 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> ( K ↾t s ) = ( K ↾t S ) )
7		dvval.t	. . . . . . . 8 |- T = ( K ↾t S )
8	6, 7	eqtr4di 2221	. . . . . . 7 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> ( K ↾t s ) = T )
9	4, 8	eqtr3id 2217	. . . . . 6 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> ( ( MetOpen ` ( abs o. - ) ) ↾t s ) = T )
10	9	fveq2d 5498	. . . . 5 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> ( int ` ( ( MetOpen ` ( abs o. - ) ) ↾t s ) ) = ( int ` T ) )
11		simprr 527	. . . . . . 7 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> f = F )
12	11	dmeqd 4811	. . . . . 6 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> dom f = dom F )
13		simpl2 996	. . . . . . 7 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> F : A --> CC )
14	13	fdmd 5352	. . . . . 6 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> dom F = A )
15	12, 14	eqtrd 2203	. . . . 5 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> dom f = A )
16	10, 15	fveq12d 5501	. . . 4 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> ( ( int ` ( ( MetOpen ` ( abs o. - ) ) ↾t s ) ) ` dom f ) = ( ( int ` T ) ` A ) )
17	15	rabeqdv 2724	. . . . . . 7 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> { w e. dom f | w # x } = { w e. A | w # x } )
18	11	fveq1d 5496	. . . . . . . . 9 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> ( f ` z ) = ( F ` z ) )
19	11	fveq1d 5496	. . . . . . . . 9 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> ( f ` x ) = ( F ` x ) )
20	18, 19	oveq12d 5868	. . . . . . . 8 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> ( ( f ` z ) - ( f ` x ) ) = ( ( F ` z ) - ( F ` x ) ) )
21	20	oveq1d 5865	. . . . . . 7 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> ( ( ( f ` z ) - ( f ` x ) ) / ( z - x ) ) = ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) )
22	17, 21	mpteq12dv 4069	. . . . . 6 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> ( z e. { w e. dom f | w # x } |-> ( ( ( f ` z ) - ( f ` x ) ) / ( z - x ) ) ) = ( z e. { w e. A | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) )
23	22	oveq1d 5865	. . . . 5 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> ( ( z e. { w e. dom f | w # x } |-> ( ( ( f ` z ) - ( f ` x ) ) / ( z - x ) ) ) lim CC x ) = ( ( z e. { w e. A | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) )
24	23	xpeq2d 4633	. . . 4 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> ( { x } X. ( ( z e. { w e. dom f | w # x } |-> ( ( ( f ` z ) - ( f ` x ) ) / ( z - x ) ) ) lim CC x ) ) = ( { x } X. ( ( z e. { w e. A | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) ) )
25	16, 24	iuneq12d 3895	. . 3 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> U_ x e. ( ( int ` ( ( MetOpen ` ( abs o. - ) ) ↾t s ) ) ` dom f ) ( { x } X. ( ( z e. { w e. dom f | w # x } |-> ( ( ( f ` z ) - ( f ` x ) ) / ( z - x ) ) ) lim CC x ) ) = U_ x e. ( ( int ` T ) ` A ) ( { x } X. ( ( z e. { w e. A | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) ) )
26		simpr 109	. . . 4 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ s = S ) -> s = S )
27	26	oveq2d 5866	. . 3 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ s = S ) -> ( CC ^pm s ) = ( CC ^pm S ) )
28		simp1 992	. . . 4 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> S C_ CC )
29		cnex 7885	. . . . 5 |- CC e. _V
30	29	elpw2 4141	. . . 4 |- ( S e. ~P CC <-> S C_ CC )
31	28, 30	sylibr 133	. . 3 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> S e. ~P CC )
32	29	a1i 9	. . . 4 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> CC e. _V )
33		simp2 993	. . . 4 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> F : A --> CC )
34		simp3 994	. . . 4 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> A C_ S )
35		elpm2r 6640	. . . 4 |- ( ( ( CC e. _V /\ S e. ~P CC ) /\ ( F : A --> CC /\ A C_ S ) ) -> F e. ( CC ^pm S ) )
36	32, 31, 33, 34, 35	syl22anc 1234	. . 3 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> F e. ( CC ^pm S ) )
37	3	cntoptopon 13247	. . . . . . . . 9 |- K e. (TopOn ` CC )
38		resttopon 12886	. . . . . . . . 9 |- ( ( K e. (TopOn ` CC ) /\ S C_ CC ) -> ( K ↾t S ) e. (TopOn ` S ) )
39	37, 28, 38	sylancr 412	. . . . . . . 8 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> ( K ↾t S ) e. (TopOn ` S ) )
40	7, 39	eqeltrid 2257	. . . . . . 7 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> T e. (TopOn ` S ) )
41		topontop 12727	. . . . . . 7 |- ( T e. (TopOn ` S ) -> T e. Top )
42	40, 41	syl 14	. . . . . 6 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> T e. Top )
43		toponuni 12728	. . . . . . . 8 |- ( T e. (TopOn ` S ) -> S = U. T )
44	40, 43	syl 14	. . . . . . 7 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> S = U. T )
45	34, 44	sseqtrd 3185	. . . . . 6 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> A C_ U. T )
46		eqid 2170	. . . . . . 7 |- U. T = U. T
47	46	ntropn 12832	. . . . . 6 |- ( ( T e. Top /\ A C_ U. T ) -> ( ( int ` T ) ` A ) e. T )
48	42, 45, 47	syl2anc 409	. . . . 5 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> ( ( int ` T ) ` A ) e. T )
49		xpexg 4723	. . . . 5 |- ( ( ( ( int ` T ) ` A ) e. T /\ CC e. _V ) -> ( ( ( int ` T ) ` A ) X. CC ) e. _V )
50	48, 32, 49	syl2anc 409	. . . 4 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> ( ( ( int ` T ) ` A ) X. CC ) e. _V )
51		limccl 13343	. . . . . . . . 9 |- ( ( z e. { w e. A | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) C_ CC
52		xpss2 4720	. . . . . . . . 9 |- ( ( ( z e. { w e. A | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) C_ CC -> ( { x } X. ( ( z e. { w e. A | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) ) C_ ( { x } X. CC ) )
53	51, 52	ax-mp 5	. . . . . . . 8 |- ( { x } X. ( ( z e. { w e. A | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) ) C_ ( { x } X. CC )
54	53	rgenw 2525	. . . . . . 7 |- A. x e. ( ( int ` T ) ` A ) ( { x } X. ( ( z e. { w e. A | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) ) C_ ( { x } X. CC )
55		ss2iun 3886	. . . . . . 7 |- ( A. x e. ( ( int ` T ) ` A ) ( { x } X. ( ( z e. { w e. A | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) ) C_ ( { x } X. CC ) -> U_ x e. ( ( int ` T ) ` A ) ( { x } X. ( ( z e. { w e. A | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) ) C_ U_ x e. ( ( int ` T ) ` A ) ( { x } X. CC ) )
56	54, 55	ax-mp 5	. . . . . 6 |- U_ x e. ( ( int ` T ) ` A ) ( { x } X. ( ( z e. { w e. A | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) ) C_ U_ x e. ( ( int ` T ) ` A ) ( { x } X. CC )
57		iunxpconst 4669	. . . . . 6 |- U_ x e. ( ( int ` T ) ` A ) ( { x } X. CC ) = ( ( ( int ` T ) ` A ) X. CC )
58	56, 57	sseqtri 3181	. . . . 5 |- U_ x e. ( ( int ` T ) ` A ) ( { x } X. ( ( z e. { w e. A | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) ) C_ ( ( ( int ` T ) ` A ) X. CC )
59	58	a1i 9	. . . 4 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> U_ x e. ( ( int ` T ) ` A ) ( { x } X. ( ( z e. { w e. A | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) ) C_ ( ( ( int ` T ) ` A ) X. CC ) )
60	50, 59	ssexd 4127	. . 3 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> U_ x e. ( ( int ` T ) ` A ) ( { x } X. ( ( z e. { w e. A | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) ) e. _V )
61	2, 25, 27, 31, 36, 60	ovmpodx 5976	. 2 |- ( ( 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 ) ) )
62	61, 59	eqsstrd 3183	. 2 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> ( S _D F ) C_ ( ( ( int ` T ) ` A ) X. CC ) )
63	61, 62	jca 304	1 |- ( ( 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 ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 973   = wceq 1348   e. wcel 2141   A.wral 2448   {crab 2452   _Vcvv 2730   C_ wss 3121   ~Pcpw 3564   {csn 3581   U.cuni 3794   U_ciun 3871   class class class wbr 3987   |-> cmpt 4048   X. cxp 4607   dom cdm 4609   o. ccom 4613   -->wf 5192   ` cfv 5196  (class class class)co 5850   e. cmpo 5852   ^pm cpm 6623   CCcc 7759   - cmin 8077   # cap 8487   / cdiv 8576   abscabs 10948   ↾_t crest 12565   MetOpencmopn 12700   Topctop 12710  TopOnctopon 12723   intcnt 12808   lim CC climc 13338   _D cdv 13339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-mulrcl 7860  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-mulass 7864  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-1rid 7868  ax-0id 7869  ax-rnegex 7870  ax-precex 7871  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-apti 7876  ax-pre-ltadd 7877  ax-pre-mulgt0 7878  ax-pre-mulext 7879  ax-arch 7880  ax-caucvg 7881

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-isom 5205  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-frec 6367  df-map 6624  df-pm 6625  df-sup 6957  df-inf 6958  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-reap 8481  df-ap 8488  df-div 8577  df-inn 8866  df-2 8924  df-3 8925  df-4 8926  df-n0 9123  df-z 9200  df-uz 9475  df-q 9566  df-rp 9598  df-xneg 9716  df-xadd 9717  df-seqfrec 10389  df-exp 10463  df-cj 10793  df-re 10794  df-im 10795  df-rsqrt 10949  df-abs 10950  df-rest 12567  df-topgen 12586  df-psmet 12702  df-xmet 12703  df-met 12704  df-bl 12705  df-mopn 12706  df-top 12711  df-topon 12724  df-bases 12756  df-ntr 12811  df-limced 13340  df-dvap 13341

This theorem is referenced by:  eldvap  13366  dvbssntrcntop  13368