Theorem dvfvalap 14860

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 14836	. . . 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 5929	. . . . . . 7 |- ( K ↾t s ) = ( ( MetOpen ` ( abs o. - ) ) ↾t s )
5		simprl 529	. . . . . . . . 9 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> s = S )
6	5	oveq2d 5935	. . . . . . . 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 2244	. . . . . . 7 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> ( K ↾t s ) = T )
9	4, 8	eqtr3id 2240	. . . . . 6 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> ( ( MetOpen ` ( abs o. - ) ) ↾t s ) = T )
10	9	fveq2d 5559	. . . . 5 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> ( int ` ( ( MetOpen ` ( abs o. - ) ) ↾t s ) ) = ( int ` T ) )
11		simprr 531	. . . . . . 7 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> f = F )
12	11	dmeqd 4865	. . . . . 6 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> dom f = dom F )
13		simpl2 1003	. . . . . . 7 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> F : A --> CC )
14	13	fdmd 5411	. . . . . 6 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> dom F = A )
15	12, 14	eqtrd 2226	. . . . 5 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> dom f = A )
16	10, 15	fveq12d 5562	. . . 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 2754	. . . . . . 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 5557	. . . . . . . . 9 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> ( f ` z ) = ( F ` z ) )
19	11	fveq1d 5557	. . . . . . . . 9 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ ( s = S /\ f = F ) ) -> ( f ` x ) = ( F ` x ) )
20	18, 19	oveq12d 5937	. . . . . . . 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 5934	. . . . . . 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 4112	. . . . . 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 5934	. . . . 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 4684	. . . 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 3937	. . 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 5935	. . 3 |- ( ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) /\ s = S ) -> ( CC ^pm s ) = ( CC ^pm S ) )
28		simp1 999	. . . 4 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> S C_ CC )
29		cnex 7998	. . . . 5 |- CC e. _V
30	29	elpw2 4187	. . . 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 1000	. . . 4 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> F : A --> CC )
34		simp3 1001	. . . 4 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> A C_ S )
35		elpm2r 6722	. . . 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 1250	. . 3 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> F e. ( CC ^pm S ) )
37	3	cntoptopon 14711	. . . . . . . . 9 |- K e. (TopOn ` CC )
38		resttopon 14350	. . . . . . . . 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 2280	. . . . . . 7 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> T e. (TopOn ` S ) )
41		topontop 14193	. . . . . . 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 14194	. . . . . . . 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 3218	. . . . . 6 |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> A C_ U. T )
46		eqid 2193	. . . . . . 7 |- U. T = U. T
47	46	ntropn 14296	. . . . . 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 4774	. . . . 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 14838	. . . . . . . . 9 |- ( ( z e. { w e. A | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) C_ CC
52		xpss2 4771	. . . . . . . . 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 2549	. . . . . . 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 3928	. . . . . . 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 4720	. . . . . 6 |- U_ x e. ( ( int ` T ) ` A ) ( { x } X. CC ) = ( ( ( int ` T ) ` A ) X. CC )
58	56, 57	sseqtri 3214	. . . . 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 4170	. . 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 6046	. 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 3216	. 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 980   = wceq 1364   e. wcel 2164   A.wral 2472   {crab 2476   _Vcvv 2760   C_ wss 3154   ~Pcpw 3602   {csn 3619   U.cuni 3836   U_ciun 3913   class class class wbr 4030   |-> cmpt 4091   X. cxp 4658   dom cdm 4660   o. ccom 4664   -->wf 5251   ` cfv 5255  (class class class)co 5919   e. cmpo 5921   ^pm cpm 6705   CCcc 7872   - cmin 8192   # cap 8602   / cdiv 8693   abscabs 11144   ↾_t crest 12853   MetOpencmopn 14040   Topctop 14176  TopOnctopon 14189   intcnt 14272   lim CC climc 14833   _D cdv 14834

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993  ax-caucvg 7994

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-map 6706  df-pm 6707  df-sup 7045  df-inf 7046  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-n0 9244  df-z 9321  df-uz 9596  df-q 9688  df-rp 9723  df-xneg 9841  df-xadd 9842  df-seqfrec 10522  df-exp 10613  df-cj 10989  df-re 10990  df-im 10991  df-rsqrt 11145  df-abs 11146  df-rest 12855  df-topgen 12874  df-psmet 14042  df-xmet 14043  df-met 14044  df-bl 14045  df-mopn 14046  df-top 14177  df-topon 14190  df-bases 14222  df-ntr 14275  df-limced 14835  df-dvap 14836

This theorem is referenced by:  eldvap  14861  dvbssntrcntop  14863