Theorem dvferm2 21439

Description: One-sided version of dvferm 21440. 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 21405	. . . . . . . . 9 |- ( ( F : X --> RR /\ X C_ RR ) -> ( RR _D F ) : dom ( RR _D F ) --> RR )
4	1, 2, 3	syl2anc 661	. . . . . . . 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 5839	. . . . . . 7 |- ( ph -> ( ( RR _D F ) ` U ) e. RR )
7	6	adantr 465	. . . . . 6 |- ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> ( ( RR _D F ) ` U ) e. RR )
8	7	renegcld 9767	. . . . 5 |- ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> -u ( ( RR _D F ) ` U ) e. RR )
9	6	lt0neg1d 9901	. . . . . 6 |- ( ph -> ( ( ( RR _D F ) ` U ) < 0 <-> 0 < -u ( ( RR _D F ) ` U ) ) )
10	9	biimpa 484	. . . . 5 |- ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> 0 < -u ( ( RR _D F ) ` U ) )
11	8, 10	elrpd 11017	. . . 4 |- ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> -u ( ( RR _D F ) ` U ) e. RR+ )
12		dvf 21362	. . . . . . . . . . 11 |- ( RR _D F ) : dom ( RR _D F ) --> CC
13		ffun 5556	. . . . . . . . . . 11 |- ( ( RR _D F ) : dom ( RR _D F ) --> CC -> Fun ( RR _D F ) )
14		funfvbrb 5811	. . . . . . . . . . 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 196	. . . . . . . . 9 |- ( ph -> U ( RR _D F ) ( ( RR _D F ) ` U ) )
17		eqid 2438	. . . . . . . . . 10 |- ( ( TopOpen ` ℂfld ) ↾t RR ) = ( ( TopOpen ` ℂfld ) ↾t RR )
18		eqid 2438	. . . . . . . . . 10 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
19		eqid 2438	. . . . . . . . . 10 |- ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) = ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) )
20		ax-resscn 9331	. . . . . . . . . . 11 |- RR C_ CC
21	20	a1i 11	. . . . . . . . . 10 |- ( ph -> RR C_ CC )
22		fss 5562	. . . . . . . . . . 11 |- ( ( F : X --> RR /\ RR C_ CC ) -> F : X --> CC )
23	1, 20, 22	sylancl 662	. . . . . . . . . 10 |- ( ph -> F : X --> CC )
24	17, 18, 19, 21, 23, 2	eldv 21353	. . . . . . . . 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 210	. . . . . . . 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 463	. . . . . . 7 |- ( ph -> ( ( RR _D F ) ` U ) e. ( ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) lim CC U ) )
27	26	adantr 465	. . . . . 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 3363	. . . . . . . . . 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 3352	. . . . . . . . . 10 |- ( ph -> U e. X )
32	23, 28, 31	dvlem 21351	. . . . . . . . 9 |- ( ( ph /\ x e. ( X \ { U } ) ) -> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) e. CC )
33	32, 19	fmptd 5862	. . . . . . . 8 |- ( ph -> ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) : ( X \ { U } ) --> CC )
34	33	adantr 465	. . . . . . 7 |- ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) : ( X \ { U } ) --> CC )
35	28	adantr 465	. . . . . . . 8 |- ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> X C_ CC )
36	35	ssdifssd 3489	. . . . . . 7 |- ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> ( X \ { U } ) C_ CC )
37	28, 31	sseldd 3352	. . . . . . . 8 |- ( ph -> U e. CC )
38	37	adantr 465	. . . . . . 7 |- ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> U e. CC )
39	34, 36, 38	ellimc3 21334	. . . . . 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 210	. . . . 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 463	. . . 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 5686	. . . . . . . . . . . . . 14 |- ( x = z -> ( F ` x ) = ( F ` z ) )
43	42	oveq1d 6101	. . . . . . . . . . . . 13 |- ( x = z -> ( ( F ` x ) - ( F ` U ) ) = ( ( F ` z ) - ( F ` U ) ) )
44		oveq1 6093	. . . . . . . . . . . . 13 |- ( x = z -> ( x - U ) = ( z - U ) )
45	43, 44	oveq12d 6104	. . . . . . . . . . . 12 |- ( x = z -> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) = ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) )
46		ovex 6111	. . . . . . . . . . . 12 |- ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) e. _V
47	45, 19, 46	fvmpt 5769	. . . . . . . . . . 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 6101	. . . . . . . . . 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 5690	. . . . . . . . 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 22	. . . . . . . . 9 |- ( y = -u ( ( RR _D F ) ` U ) -> y = -u ( ( RR _D F ) ` U ) )
51	49, 50	breqan12rd 4303	. . . . . . . 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 316	. . . . . . 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 2726	. . . . . 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 2731	. . . . 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 3064	. . . 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 60	. . 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 729	. . . . . 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 729	. . . . . 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 729	. . . . . 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 729	. . . . . 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 729	. . . . . 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 729	. . . . . 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 758	. . . . . 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 754	. . . . . 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 461	. . . . . 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 2438	. . . . . 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 21438	. . . . 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 423	. . . 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 2814	. . 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 576	. 2 |- ( ph -> -. ( ( RR _D F ) ` U ) < 0 )
72		0re 9378	. . 3 |- 0 e. RR
73		lenlt 9445	. . 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 663	. 2 |- ( ph -> ( 0 <_ ( ( RR _D F ) ` U ) <-> -. ( ( RR _D F ) ` U ) < 0 ) )
75	71, 74	mpbird 232	1 |- ( ph -> 0 <_ ( ( RR _D F ) ` U ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   =/= wne 2601   A.wral 2710   E.wrex 2711   \ cdif 3320   C_ wss 3323   ifcif 3786   {csn 3872   class class class wbr 4287   e. cmpt 4345   dom cdm 4835   Fun wfun 5407   -->wf 5409   ` cfv 5413  (class class class)co 6086   CCcc 9272   RRcr 9273   0cc0 9274   + caddc 9277   < clt 9410   <_ cle 9411   - cmin 9587   -ucneg 9588   / cdiv 9985   2c2 10363   RR+crp 10983   (,)cioo 11292   abscabs 12715   ↾_t crest 14351   TopOpenctopn 14352  ℂ_fldccnfld 17798   intcnt 18601   lim CC climc 21317   _D cdv 21318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fi 7653  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-icc 11299  df-fz 11430  df-seq 11799  df-exp 11858  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-plusg 14243  df-mulr 14244  df-starv 14245  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-rest 14353  df-topn 14354  df-topgen 14374  df-psmet 17789  df-xmet 17790  df-met 17791  df-bl 17792  df-mopn 17793  df-fbas 17794  df-fg 17795  df-cnfld 17799  df-top 18483  df-bases 18485  df-topon 18486  df-topsp 18487  df-cld 18603  df-ntr 18604  df-cls 18605  df-nei 18682  df-lp 18720  df-perf 18721  df-cn 18811  df-cnp 18812  df-haus 18899  df-fil 19399  df-fm 19491  df-flim 19492  df-flf 19493  df-xms 19875  df-ms 19876  df-cncf 20434  df-limc 21321  df-dv 21322

This theorem is referenced by:  dvferm  21440  dvivthlem1  21460