Theorem dvferm2 22123

Description: One-sided version of dvferm 22124. 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 22089	. . . . . . . . 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 6020	. . . . . . 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 9982	. . . . 5 |- ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> -u ( ( RR _D F ) ` U ) e. RR )
9	6	lt0neg1d 10118	. . . . . 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 11250	. . . 4 |- ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> -u ( ( RR _D F ) ` U ) e. RR+ )
12		dvf 22046	. . . . . . . . . . 11 |- ( RR _D F ) : dom ( RR _D F ) --> CC
13		ffun 5731	. . . . . . . . . . 11 |- ( ( RR _D F ) : dom ( RR _D F ) --> CC -> Fun ( RR _D F ) )
14		funfvbrb 5992	. . . . . . . . . . 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 2467	. . . . . . . . . 10 |- ( ( TopOpen ` ℂfld ) ↾t RR ) = ( ( TopOpen ` ℂfld ) ↾t RR )
18		eqid 2467	. . . . . . . . . 10 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
19		eqid 2467	. . . . . . . . . 10 |- ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) = ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) )
20		ax-resscn 9545	. . . . . . . . . . 11 |- RR C_ CC
21	20	a1i 11	. . . . . . . . . 10 |- ( ph -> RR C_ CC )
22		fss 5737	. . . . . . . . . . 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 22037	. . . . . . . . 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 3516	. . . . . . . . . 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 3505	. . . . . . . . . 10 |- ( ph -> U e. X )
32	23, 28, 31	dvlem 22035	. . . . . . . . 9 |- ( ( ph /\ x e. ( X \ { U } ) ) -> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) e. CC )
33	32, 19	fmptd 6043	. . . . . . . 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 3642	. . . . . . 7 |- ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> ( X \ { U } ) C_ CC )
37	28, 31	sseldd 3505	. . . . . . . 8 |- ( ph -> U e. CC )
38	37	adantr 465	. . . . . . 7 |- ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> U e. CC )
39	34, 36, 38	ellimc3 22018	. . . . . 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 5864	. . . . . . . . . . . . . 14 |- ( x = z -> ( F ` x ) = ( F ` z ) )
43	42	oveq1d 6297	. . . . . . . . . . . . 13 |- ( x = z -> ( ( F ` x ) - ( F ` U ) ) = ( ( F ` z ) - ( F ` U ) ) )
44		oveq1 6289	. . . . . . . . . . . . 13 |- ( x = z -> ( x - U ) = ( z - U ) )
45	43, 44	oveq12d 6300	. . . . . . . . . . . 12 |- ( x = z -> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) = ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) )
46		ovex 6307	. . . . . . . . . . . 12 |- ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) e. _V
47	45, 19, 46	fvmpt 5948	. . . . . . . . . . 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 6297	. . . . . . . . . 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 5868	. . . . . . . . 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 4463	. . . . . . . 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 2900	. . . . . 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 2973	. . . . 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 3210	. . . 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 2467	. . . . . 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 22122	. . . . 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 2920	. . 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 9592	. . 3 |- 0 e. RR
73		lenlt 9659	. . 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 1379   e. wcel 1767   =/= wne 2662   A.wral 2814   E.wrex 2815   \ cdif 3473   C_ wss 3476   ifcif 3939   {csn 4027   class class class wbr 4447   |-> cmpt 4505   dom cdm 4999   Fun wfun 5580   -->wf 5582   ` cfv 5586  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488   + caddc 9491   < clt 9624   <_ cle 9625   - cmin 9801   -ucneg 9802   / cdiv 10202   2c2 10581   RR+crp 11216   (,)cioo 11525   abscabs 13026   ↾_t crest 14672   TopOpenctopn 14673  ℂ_fldccnfld 18191   intcnt 19284   lim CC climc 22001   _D cdv 22002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fi 7867  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-icc 11532  df-fz 11669  df-seq 12072  df-exp 12131  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-plusg 14564  df-mulr 14565  df-starv 14566  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-rest 14674  df-topn 14675  df-topgen 14695  df-psmet 18182  df-xmet 18183  df-met 18184  df-bl 18185  df-mopn 18186  df-fbas 18187  df-fg 18188  df-cnfld 18192  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-cld 19286  df-ntr 19287  df-cls 19288  df-nei 19365  df-lp 19403  df-perf 19404  df-cn 19494  df-cnp 19495  df-haus 19582  df-fil 20082  df-fm 20174  df-flim 20175  df-flf 20176  df-xms 20558  df-ms 20559  df-cncf 21117  df-limc 22005  df-dv 22006

This theorem is referenced by:  dvferm  22124  dvivthlem1  22144