Theorem dvfvalap 15672

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 15648	. . . 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 6068	. . . . . . 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 6074	. . . . . . . 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 2285	. . . . . . 7 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> ( K ↾t s ) = T )
9	4, 8	eqtr3id 2281	. . . . . 6 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> ( ( MetOpen ` ( abs o. - ) ) ↾t s ) = T )
10	9	fveq2d 5679	. . . . 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 4963	. . . . . 6 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> dom f = dom F )
13		simpl2 1028	. . . . . . 7 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> F : A --> CC )
14	13	fdmd 5520	. . . . . 6 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> dom F = A )
15	12, 14	eqtrd 2267	. . . . 5 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> dom f = A )
16	10, 15	fveq12d 5682	. . . 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 2809	. . . . . . 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 5677	. . . . . . . . 9 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> ( f ` z ) = ( F ` z ) )
19	11	fveq1d 5677	. . . . . . . . 9 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> ( f ` x ) = ( F ` x ) )
20	18, 19	oveq12d 6076	. . . . . . . 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 6073	. . . . . . 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 4197	. . . . . 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 6073	. . . . 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 4778	. . . 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 4020	. . 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 6074	. . 3 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ s = S ) -> ( CC ^pm s ) = ( CC ^pm S ) )
28		simp1 1024	. . . 4 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> S C_ CC )
29		cnex 8267	. . . . 5 |- CC e. _V
30	29	elpw2 4274	. . . 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 1025	. . . 4 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> F : A --> CC )
34		simp3 1026	. . . 4 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> A C_ S )
35		elpm2r 6913	. . . 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 1275	. . 3 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> F e. ( CC ^pm S ) )
37	3	cntoptopon 15523	. . . . . . . . 9 |- K e. (TopOn ` CC )
38		resttopon 15162	. . . . . . . . 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 2321	. . . . . . 7 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> T e. (TopOn ` S ) )
41		topontop 15005	. . . . . . 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 15006	. . . . . . . 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 3280	. . . . . 6 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> A C_ U. T )
46		eqid 2234	. . . . . . 7 |- U. T = U. T
47	46	ntropn 15108	. . . . . 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 4869	. . . . 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 15650	. . . . . . . . 9 |- ( ( z e. { w e. A | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) C_ CC
52		xpss2 4866	. . . . . . . . 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 2599	. . . . . . 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 4011	. . . . . . 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 4815	. . . . . 6 |- U_ x e. ( ( int ` T ) ` A ) ( { x } X. CC ) = ( ( ( int ` T ) ` A ) X. CC )
58	56, 57	sseqtri 3276	. . . . 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 4255	. . 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 6188	. 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 3278	. 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 1005   = wceq 1398   e. wcel 2205   A.wral 2522   {crab 2526   _Vcvv 2815   C_ wss 3214   ~Pcpw 3674   {csn 3694   U.cuni 3919   U_ciun 3996   class class class wbr 4114   |-> cmpt 4176   X. cxp 4752   dom cdm 4754   o. ccom 4758   -->wf 5353   ` cfv 5357  (class class class)co 6058   e. cmpo 6060   ^pm cpm 6896   CCcc 8141   - cmin 8460   # cap 8872   / cdiv 8963   abscabs 11707   ↾_t crest 13536   MetOpencmopn 14815   Topctop 14988  TopOnctopon 15001   intcnt 15084   lim CC climc 15645   _D cdv 15646

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-map 6897  df-pm 6898  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-xneg 10124  df-xadd 10125  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-rest 13538  df-topgen 13557  df-psmet 14817  df-xmet 14818  df-met 14819  df-bl 14820  df-mopn 14821  df-top 14989  df-topon 15002  df-bases 15034  df-ntr 15087  df-limced 15647  df-dvap 15648

This theorem is referenced by:  eldvap  15673  dvbssntrcntop  15675