Theorem dvferm2 22807

Description: One-sided version of dvferm 22808. A point U which is the local maximum of its left neighborhood has derivative at least zero. (Contributed by Mario Carneiro, 24-Feb-2015.) (Proof shortened by Mario Carneiro, 28-Dec-2016.)

Hypotheses

Ref	Expression
dvferm.a	|- ( ph -> F : X --> RR )
dvferm.b	|- ( ph -> X C_ RR )
dvferm.u	|- ( ph -> U e. ( A (,) B ) )
dvferm.s	|- ( ph -> ( A (,) B ) C_ X )
dvferm.d	|- ( ph -> U e. dom ( RR _D F ) )
dvferm2.r	|- ( ph -> A. y e. ( A (,) U ) ( F ` y ) <_ ( F ` U ) )

Assertion

Ref	Expression
dvferm2	|- ( ph -> 0 <_ ( ( RR _D F ) ` U ) )

Distinct variable groups:   y, A   y, B   y, F   y, U   y, X   ph, y

Proof of Theorem dvferm2

Dummy variables z x u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvferm.a	. . . . . . . . 9 |- ( ph -> F : X --> RR )
2		dvferm.b	. . . . . . . . 9 |- ( ph -> X C_ RR )
3		dvfre 22773	. . . . . . . . 9 |- ( ( F : X --> RR /\ X C_ RR ) -> ( RR _D F ) : dom ( RR _D F ) --> RR )
4	1, 2, 3	syl2anc 665	. . . . . . . 8 |- ( ph -> ( RR _D F ) : dom ( RR _D F ) --> RR )
5		dvferm.d	. . . . . . . 8 |- ( ph -> U e. dom ( RR _D F ) )
6	4, 5	ffvelrnd 6038	. . . . . . 7 |- ( ph -> ( ( RR _D F ) ` U ) e. RR )
7	6	adantr 466	. . . . . 6 |- ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> ( ( RR _D F ) ` U ) e. RR )
8	7	renegcld 10045	. . . . 5 |- ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> -u ( ( RR _D F ) ` U ) e. RR )
9	6	lt0neg1d 10182	. . . . . 6 |- ( ph -> ( ( ( RR _D F ) ` U ) < 0 <-> 0 < -u ( ( RR _D F ) ` U ) ) )
10	9	biimpa 486	. . . . 5 |- ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> 0 < -u ( ( RR _D F ) ` U ) )
11	8, 10	elrpd 11338	. . . 4 |- ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> -u ( ( RR _D F ) ` U ) e. RR+ )
12		dvf 22730	. . . . . . . . . . 11 |- ( RR _D F ) : dom ( RR _D F ) --> CC
13		ffun 5748	. . . . . . . . . . 11 |- ( ( RR _D F ) : dom ( RR _D F ) --> CC -> Fun ( RR _D F ) )
14		funfvbrb 6010	. . . . . . . . . . 11 |- ( Fun ( RR _D F ) -> ( U e. dom ( RR _D F ) <-> U ( RR _D F ) ( ( RR _D F ) ` U ) ) )
15	12, 13, 14	mp2b 10	. . . . . . . . . 10 |- ( U e. dom ( RR _D F ) <-> U ( RR _D F ) ( ( RR _D F ) ` U ) )
16	5, 15	sylib 199	. . . . . . . . 9 |- ( ph -> U ( RR _D F ) ( ( RR _D F ) ` U ) )
17		eqid 2429	. . . . . . . . . 10 |- ( ( TopOpen ` ℂfld ) ↾t RR ) = ( ( TopOpen ` ℂfld ) ↾t RR )
18		eqid 2429	. . . . . . . . . 10 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
19		eqid 2429	. . . . . . . . . 10 |- ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) = ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) )
20		ax-resscn 9595	. . . . . . . . . . 11 |- RR C_ CC
21	20	a1i 11	. . . . . . . . . 10 |- ( ph -> RR C_ CC )
22		fss 5754	. . . . . . . . . . 11 |- ( ( F : X --> RR /\ RR C_ CC ) -> F : X --> CC )
23	1, 20, 22	sylancl 666	. . . . . . . . . 10 |- ( ph -> F : X --> CC )
24	17, 18, 19, 21, 23, 2	eldv 22721	. . . . . . . . 9 |- ( ph -> ( U ( RR _D F ) ( ( RR _D F ) ` U ) <-> ( U e. ( ( int ` ( ( TopOpen ` ℂfld ) ↾t RR ) ) ` X ) /\ ( ( RR _D F ) ` U ) e. ( ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) lim CC U ) ) ) )
25	16, 24	mpbid 213	. . . . . . . 8 |- ( ph -> ( U e. ( ( int ` ( ( TopOpen ` ℂfld ) ↾t RR ) ) ` X ) /\ ( ( RR _D F ) ` U ) e. ( ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) lim CC U ) ) )
26	25	simprd 464	. . . . . . 7 |- ( ph -> ( ( RR _D F ) ` U ) e. ( ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) lim CC U ) )
27	26	adantr 466	. . . . . 6 |- ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> ( ( RR _D F ) ` U ) e. ( ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) lim CC U ) )
28	2, 20	syl6ss 3482	. . . . . . . . . 10 |- ( ph -> X C_ CC )
29		dvferm.s	. . . . . . . . . . 11 |- ( ph -> ( A (,) B ) C_ X )
30		dvferm.u	. . . . . . . . . . 11 |- ( ph -> U e. ( A (,) B ) )
31	29, 30	sseldd 3471	. . . . . . . . . 10 |- ( ph -> U e. X )
32	23, 28, 31	dvlem 22719	. . . . . . . . 9 |- ( ( ph /\ x e. ( X \ { U } ) ) -> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) e. CC )
33	32, 19	fmptd 6061	. . . . . . . 8 |- ( ph -> ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) : ( X \ { U } ) --> CC )
34	33	adantr 466	. . . . . . 7 |- ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) : ( X \ { U } ) --> CC )
35	28	adantr 466	. . . . . . . 8 |- ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> X C_ CC )
36	35	ssdifssd 3609	. . . . . . 7 |- ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> ( X \ { U } ) C_ CC )
37	28, 31	sseldd 3471	. . . . . . . 8 |- ( ph -> U e. CC )
38	37	adantr 466	. . . . . . 7 |- ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> U e. CC )
39	34, 36, 38	ellimc3 22702	. . . . . 6 |- ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> ( ( ( RR _D F ) ` U ) e. ( ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) lim CC U ) <-> ( ( ( RR _D F ) ` U ) e. CC /\ A. y e. RR+ E. u e. RR+ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) ` z ) - ( ( RR _D F ) ` U ) ) ) < y ) ) ) )
40	27, 39	mpbid 213	. . . . 5 |- ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> ( ( ( RR _D F ) ` U ) e. CC /\ A. y e. RR+ E. u e. RR+ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) ` z ) - ( ( RR _D F ) ` U ) ) ) < y ) ) )
41	40	simprd 464	. . . 4 |- ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> A. y e. RR+ E. u e. RR+ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) ` z ) - ( ( RR _D F ) ` U ) ) ) < y ) )
42		fveq2 5881	. . . . . . . . . . . . . 14 |- ( x = z -> ( F ` x ) = ( F ` z ) )
43	42	oveq1d 6320	. . . . . . . . . . . . 13 |- ( x = z -> ( ( F ` x ) - ( F ` U ) ) = ( ( F ` z ) - ( F ` U ) ) )
44		oveq1 6312	. . . . . . . . . . . . 13 |- ( x = z -> ( x - U ) = ( z - U ) )
45	43, 44	oveq12d 6323	. . . . . . . . . . . 12 |- ( x = z -> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) = ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) )
46		ovex 6333	. . . . . . . . . . . 12 |- ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) e. _V
47	45, 19, 46	fvmpt 5964	. . . . . . . . . . 11 |- ( z e. ( X \ { U } ) -> ( ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) ` z ) = ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) )
48	47	oveq1d 6320	. . . . . . . . . 10 |- ( z e. ( X \ { U } ) -> ( ( ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) ` z ) - ( ( RR _D F ) ` U ) ) = ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) )
49	48	fveq2d 5885	. . . . . . . . 9 |- ( z e. ( X \ { U } ) -> ( abs ` ( ( ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) ` z ) - ( ( RR _D F ) ` U ) ) ) = ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) )
50		id 23	. . . . . . . . 9 |- ( y = -u ( ( RR _D F ) ` U ) -> y = -u ( ( RR _D F ) ` U ) )
51	49, 50	breqan12rd 4442	. . . . . . . 8 |- ( ( y = -u ( ( RR _D F ) ` U ) /\ z e. ( X \ { U } ) ) -> ( ( abs ` ( ( ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) ` z ) - ( ( RR _D F ) ` U ) ) ) < y <-> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) )
52	51	imbi2d 317	. . . . . . 7 |- ( ( y = -u ( ( RR _D F ) ` U ) /\ z e. ( X \ { U } ) ) -> ( ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) ` z ) - ( ( RR _D F ) ` U ) ) ) < y ) <-> ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) ) )
53	52	ralbidva 2868	. . . . . 6 |- ( y = -u ( ( RR _D F ) ` U ) -> ( A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) ` z ) - ( ( RR _D F ) ` U ) ) ) < y ) <-> A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) ) )
54	53	rexbidv 2946	. . . . 5 |- ( y = -u ( ( RR _D F ) ` U ) -> ( E. u e. RR+ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) ` z ) - ( ( RR _D F ) ` U ) ) ) < y ) <-> E. u e. RR+ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) ) )
55	54	rspcv 3184	. . . 4 |- ( -u ( ( RR _D F ) ` U ) e. RR+ -> ( A. y e. RR+ E. u e. RR+ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) ` z ) - ( ( RR _D F ) ` U ) ) ) < y ) -> E. u e. RR+ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) ) )
56	11, 41, 55	sylc 62	. . 3 |- ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> E. u e. RR+ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) )
57	1	ad3antrrr 734	. . . . . 6 |- ( ( ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) /\ u e. RR+ ) /\ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) ) -> F : X --> RR )
58	2	ad3antrrr 734	. . . . . 6 |- ( ( ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) /\ u e. RR+ ) /\ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) ) -> X C_ RR )
59	30	ad3antrrr 734	. . . . . 6 |- ( ( ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) /\ u e. RR+ ) /\ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) ) -> U e. ( A (,) B ) )
60	29	ad3antrrr 734	. . . . . 6 |- ( ( ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) /\ u e. RR+ ) /\ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) ) -> ( A (,) B ) C_ X )
61	5	ad3antrrr 734	. . . . . 6 |- ( ( ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) /\ u e. RR+ ) /\ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) ) -> U e. dom ( RR _D F ) )
62		dvferm2.r	. . . . . . 7 |- ( ph -> A. y e. ( A (,) U ) ( F ` y ) <_ ( F ` U ) )
63	62	ad3antrrr 734	. . . . . 6 |- ( ( ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) /\ u e. RR+ ) /\ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) ) -> A. y e. ( A (,) U ) ( F ` y ) <_ ( F ` U ) )
64		simpllr 767	. . . . . 6 |- ( ( ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) /\ u e. RR+ ) /\ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) ) -> ( ( RR _D F ) ` U ) < 0 )
65		simplr 760	. . . . . 6 |- ( ( ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) /\ u e. RR+ ) /\ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) ) -> u e. RR+ )
66		simpr 462	. . . . . 6 |- ( ( ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) /\ u e. RR+ ) /\ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) ) -> A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) )
67		eqid 2429	. . . . . 6 |- ( ( if ( A <_ ( U - u ) , ( U - u ) , A ) + U ) / 2 ) = ( ( if ( A <_ ( U - u ) , ( U - u ) , A ) + U ) / 2 )
68	57, 58, 59, 60, 61, 63, 64, 65, 66, 67	dvferm2lem 22806	. . . . 5 |- -. ( ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) /\ u e. RR+ ) /\ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) )
69	68	imnani 424	. . . 4 |- ( ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) /\ u e. RR+ ) -> -. A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) )
70	69	nrexdv 2888	. . 3 |- ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> -. E. u e. RR+ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) )
71	56, 70	pm2.65da 578	. 2 |- ( ph -> -. ( ( RR _D F ) ` U ) < 0 )
72		0re 9642	. . 3 |- 0 e. RR
73		lenlt 9711	. . 3 |- ( ( 0 e. RR /\ ( ( RR _D F ) ` U ) e. RR ) -> ( 0 <_ ( ( RR _D F ) ` U ) <-> -. ( ( RR _D F ) ` U ) < 0 ) )
74	72, 6, 73	sylancr 667	. 2 |- ( ph -> ( 0 <_ ( ( RR _D F ) ` U ) <-> -. ( ( RR _D F ) ` U ) < 0 ) )
75	71, 74	mpbird 235	1 |- ( ph -> 0 <_ ( ( RR _D F ) ` U ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 187   /\ wa 370   = wceq 1437   e. wcel 1870   =/= wne 2625   A.wral 2782   E.wrex 2783   \ cdif 3439   C_ wss 3442   ifcif 3915   {csn 4002   class class class wbr 4426   |-> cmpt 4484   dom cdm 4854   Fun wfun 5595   -->wf 5597   ` cfv 5601  (class class class)co 6305   CCcc 9536   RRcr 9537   0cc0 9538   + caddc 9541   < clt 9674   <_ cle 9675   - cmin 9859   -ucneg 9860   / cdiv 10268   2c2 10659   RR+crp 11302   (,)cioo 11635   abscabs 13276   ↾_t crest 15269   TopOpenctopn 15270  ℂ_fldccnfld 18896   intcnt 19954   lim CC climc 22685   _D cdv 22686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fi 7931  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-icc 11642  df-fz 11783  df-seq 12211  df-exp 12270  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-struct 15077  df-ndx 15078  df-slot 15079  df-base 15080  df-plusg 15156  df-mulr 15157  df-starv 15158  df-tset 15162  df-ple 15163  df-ds 15165  df-unif 15166  df-rest 15271  df-topn 15272  df-topgen 15292  df-psmet 18888  df-xmet 18889  df-met 18890  df-bl 18891  df-mopn 18892  df-fbas 18893  df-fg 18894  df-cnfld 18897  df-top 19843  df-bases 19844  df-topon 19845  df-topsp 19846  df-cld 19956  df-ntr 19957  df-cls 19958  df-nei 20036  df-lp 20074  df-perf 20075  df-cn 20165  df-cnp 20166  df-haus 20253  df-fil 20783  df-fm 20875  df-flim 20876  df-flf 20877  df-xms 21257  df-ms 21258  df-cncf 21797  df-limc 22689  df-dv 22690

This theorem is referenced by:  dvferm  22808  dvivthlem1  22828