Theorem dvfvalap 13021

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 12997	. . . 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 5831	. . . . . . 7 |- ( K ↾t s ) = ( ( MetOpen ` ( abs o. - ) ) ↾t s )
5		simprl 521	. . . . . . . . 9 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> s = S )
6	5	oveq2d 5837	. . . . . . . 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 2208	. . . . . . 7 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> ( K ↾t s ) = T )
9	4, 8	eqtr3id 2204	. . . . . 6 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> ( ( MetOpen ` ( abs o. - ) ) ↾t s ) = T )
10	9	fveq2d 5471	. . . . 5 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> ( int ` ( ( MetOpen ` ( abs o. - ) ) ↾t s ) ) = ( int ` T ) )
11		simprr 522	. . . . . . 7 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> f = F )
12	11	dmeqd 4787	. . . . . 6 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> dom f = dom F )
13		simpl2 986	. . . . . . 7 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> F : A --> CC )
14	13	fdmd 5325	. . . . . 6 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> dom F = A )
15	12, 14	eqtrd 2190	. . . . 5 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> dom f = A )
16	10, 15	fveq12d 5474	. . . 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 2706	. . . . . . 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 5469	. . . . . . . . 9 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> ( f ` z ) = ( F ` z ) )
19	11	fveq1d 5469	. . . . . . . . 9 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> ( f ` x ) = ( F ` x ) )
20	18, 19	oveq12d 5839	. . . . . . . 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 5836	. . . . . . 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 4046	. . . . . 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 5836	. . . . 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 4609	. . . 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 3873	. . 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 5837	. . 3 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ s = S ) -> ( CC ^pm s ) = ( CC ^pm S ) )
28		simp1 982	. . . 4 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> S C_ CC )
29		cnex 7850	. . . . 5 |- CC e. _V
30	29	elpw2 4118	. . . 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 983	. . . 4 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> F : A --> CC )
34		simp3 984	. . . 4 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> A C_ S )
35		elpm2r 6608	. . . 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 1221	. . 3 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> F e. ( CC ^pm S ) )
37	3	cntoptopon 12903	. . . . . . . . 9 |- K e. (TopOn ` CC )
38		resttopon 12542	. . . . . . . . 9 |- ( ( K e. (TopOn ` CC ) /\ S C_ CC ) -> ( K ↾t S ) e. (TopOn ` S ) )
39	37, 28, 38	sylancr 411	. . . . . . . 8 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> ( K ↾t S ) e. (TopOn ` S ) )
40	7, 39	eqeltrid 2244	. . . . . . 7 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> T e. (TopOn ` S ) )
41		topontop 12383	. . . . . . 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 12384	. . . . . . . 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 3166	. . . . . 6 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> A C_ U. T )
46		eqid 2157	. . . . . . 7 |- U. T = U. T
47	46	ntropn 12488	. . . . . 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 4699	. . . . 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 12999	. . . . . . . . 9 |- ( ( z e. { w e. A | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) C_ CC
52		xpss2 4696	. . . . . . . . 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 2512	. . . . . . 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 3864	. . . . . . 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 4645	. . . . . 6 |- U_ x e. ( ( int ` T ) ` A ) ( { x } X. CC ) = ( ( ( int ` T ) ` A ) X. CC )
58	56, 57	sseqtri 3162	. . . . 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 4104	. . 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 5944	. 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 3164	. 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 963   = wceq 1335   e. wcel 2128   A.wral 2435   {crab 2439   _Vcvv 2712   C_ wss 3102   ~Pcpw 3543   {csn 3560   U.cuni 3772   U_ciun 3849   class class class wbr 3965   |-> cmpt 4025   X. cxp 4583   dom cdm 4585   o. ccom 4589   -->wf 5165   ` cfv 5169  (class class class)co 5821   e. cmpo 5823   ^pm cpm 6591   CCcc 7724   - cmin 8040   # cap 8450   / cdiv 8539   abscabs 10890   ↾_t crest 12322   MetOpencmopn 12356   Topctop 12366  TopOnctopon 12379   intcnt 12464   lim CC climc 12994   _D cdv 12995

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-iinf 4546  ax-cnex 7817  ax-resscn 7818  ax-1cn 7819  ax-1re 7820  ax-icn 7821  ax-addcl 7822  ax-addrcl 7823  ax-mulcl 7824  ax-mulrcl 7825  ax-addcom 7826  ax-mulcom 7827  ax-addass 7828  ax-mulass 7829  ax-distr 7830  ax-i2m1 7831  ax-0lt1 7832  ax-1rid 7833  ax-0id 7834  ax-rnegex 7835  ax-precex 7836  ax-cnre 7837  ax-pre-ltirr 7838  ax-pre-ltwlin 7839  ax-pre-lttrn 7840  ax-pre-apti 7841  ax-pre-ltadd 7842  ax-pre-mulgt0 7843  ax-pre-mulext 7844  ax-arch 7845  ax-caucvg 7846

This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4549  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-isom 5178  df-riota 5777  df-ov 5824  df-oprab 5825  df-mpo 5826  df-1st 6085  df-2nd 6086  df-recs 6249  df-frec 6335  df-map 6592  df-pm 6593  df-sup 6924  df-inf 6925  df-pnf 7908  df-mnf 7909  df-xr 7910  df-ltxr 7911  df-le 7912  df-sub 8042  df-neg 8043  df-reap 8444  df-ap 8451  df-div 8540  df-inn 8828  df-2 8886  df-3 8887  df-4 8888  df-n0 9085  df-z 9162  df-uz 9434  df-q 9522  df-rp 9554  df-xneg 9672  df-xadd 9673  df-seqfrec 10338  df-exp 10412  df-cj 10735  df-re 10736  df-im 10737  df-rsqrt 10891  df-abs 10892  df-rest 12324  df-topgen 12343  df-psmet 12358  df-xmet 12359  df-met 12360  df-bl 12361  df-mopn 12362  df-top 12367  df-topon 12380  df-bases 12412  df-ntr 12467  df-limced 12996  df-dvap 12997

This theorem is referenced by:  eldvap  13022  dvbssntrcntop  13024