Theorem dvfvalap 15404

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 15380	. . . 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 6027	. . . . . . 7 |- ( K ↾t s ) = ( ( MetOpen ` ( abs o. - ) ) ↾t s )
5		simprl 531	. . . . . . . . 9 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> s = S )
6	5	oveq2d 6033	. . . . . . . 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 2282	. . . . . . 7 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> ( K ↾t s ) = T )
9	4, 8	eqtr3id 2278	. . . . . 6 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> ( ( MetOpen ` ( abs o. - ) ) ↾t s ) = T )
10	9	fveq2d 5643	. . . . 5 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> ( int ` ( ( MetOpen ` ( abs o. - ) ) ↾t s ) ) = ( int ` T ) )
11		simprr 533	. . . . . . 7 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> f = F )
12	11	dmeqd 4933	. . . . . 6 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> dom f = dom F )
13		simpl2 1027	. . . . . . 7 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> F : A --> CC )
14	13	fdmd 5489	. . . . . 6 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> dom F = A )
15	12, 14	eqtrd 2264	. . . . 5 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> dom f = A )
16	10, 15	fveq12d 5646	. . . 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 2796	. . . . . . 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 5641	. . . . . . . . 9 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> ( f ` z ) = ( F ` z ) )
19	11	fveq1d 5641	. . . . . . . . 9 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> ( f ` x ) = ( F ` x ) )
20	18, 19	oveq12d 6035	. . . . . . . 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 6032	. . . . . . 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 4171	. . . . . 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 6032	. . . . 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 4749	. . . 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 3994	. . 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 110	. . . 4 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ s = S ) -> s = S )
27	26	oveq2d 6033	. . 3 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ s = S ) -> ( CC ^pm s ) = ( CC ^pm S ) )
28		simp1 1023	. . . 4 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> S C_ CC )
29		cnex 8155	. . . . 5 |- CC e. _V
30	29	elpw2 4247	. . . 4 |- ( S e. ~P CC <-> S C_ CC )
31	28, 30	sylibr 134	. . 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 1024	. . . 4 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> F : A --> CC )
34		simp3 1025	. . . 4 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> A C_ S )
35		elpm2r 6834	. . . 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 1274	. . 3 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> F e. ( CC ^pm S ) )
37	3	cntoptopon 15255	. . . . . . . . 9 |- K e. (TopOn ` CC )
38		resttopon 14894	. . . . . . . . 9 |- ( ( K e. (TopOn ` CC ) /\ S C_ CC ) -> ( K ↾t S ) e. (TopOn ` S ) )
39	37, 28, 38	sylancr 414	. . . . . . . 8 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> ( K ↾t S ) e. (TopOn ` S ) )
40	7, 39	eqeltrid 2318	. . . . . . 7 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> T e. (TopOn ` S ) )
41		topontop 14737	. . . . . . 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 14738	. . . . . . . 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 3265	. . . . . 6 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> A C_ U. T )
46		eqid 2231	. . . . . . 7 |- U. T = U. T
47	46	ntropn 14840	. . . . . 6 |- ( ( T e. Top /\ A C_ U. T ) -> ( ( int ` T ) ` A ) e. T )
48	42, 45, 47	syl2anc 411	. . . . 5 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> ( ( int ` T ) ` A ) e. T )
49		xpexg 4840	. . . . 5 |- ( ( ( ( int ` T ) ` A ) e. T /\ CC e. _V ) -> ( ( ( int ` T ) ` A ) X. CC ) e. _V )
50	48, 32, 49	syl2anc 411	. . . 4 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> ( ( ( int ` T ) ` A ) X. CC ) e. _V )
51		limccl 15382	. . . . . . . . 9 |- ( ( z e. { w e. A | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) C_ CC
52		xpss2 4837	. . . . . . . . 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 2587	. . . . . . 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 3985	. . . . . . 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 4786	. . . . . 6 |- U_ x e. ( ( int ` T ) ` A ) ( { x } X. CC ) = ( ( ( int ` T ) ` A ) X. CC )
58	56, 57	sseqtri 3261	. . . . 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 4229	. . 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 6147	. 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 3263	. 2 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> ( S _D F ) C_ ( ( ( int ` T ) ` A ) X. CC ) )
63	61, 62	jca 306	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 104   /\ w3a 1004   = wceq 1397   e. wcel 2202   A.wral 2510   {crab 2514   _Vcvv 2802   C_ wss 3200   ~Pcpw 3652   {csn 3669   U.cuni 3893   U_ciun 3970   class class class wbr 4088   |-> cmpt 4150   X. cxp 4723   dom cdm 4725   o. ccom 4729   -->wf 5322   ` cfv 5326  (class class class)co 6017   e. cmpo 6019   ^pm cpm 6817   CCcc 8029   - cmin 8349   # cap 8760   / cdiv 8851   abscabs 11557   ↾_t crest 13321   MetOpencmopn 14554   Topctop 14720  TopOnctopon 14733   intcnt 14816   lim CC climc 15377   _D cdv 15378

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-map 6818  df-pm 6819  df-sup 7182  df-inf 7183  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-xneg 10006  df-xadd 10007  df-seqfrec 10709  df-exp 10800  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-rest 13323  df-topgen 13342  df-psmet 14556  df-xmet 14557  df-met 14558  df-bl 14559  df-mopn 14560  df-top 14721  df-topon 14734  df-bases 14766  df-ntr 14819  df-limced 15379  df-dvap 15380

This theorem is referenced by:  eldvap  15405  dvbssntrcntop  15407