Theorem dvferm2 21577

Description: One-sided version of dvferm 21578. 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 21543	. . . . . . . . 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 5945	. . . . . . 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 9878	. . . . 5 |- ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> -u ( ( RR _D F ) ` U ) e. RR )
9	6	lt0neg1d 10012	. . . . . 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 11128	. . . 4 |- ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> -u ( ( RR _D F ) ` U ) e. RR+ )
12		dvf 21500	. . . . . . . . . . 11 |- ( RR _D F ) : dom ( RR _D F ) --> CC
13		ffun 5661	. . . . . . . . . . 11 |- ( ( RR _D F ) : dom ( RR _D F ) --> CC -> Fun ( RR _D F ) )
14		funfvbrb 5917	. . . . . . . . . . 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 2451	. . . . . . . . . 10 |- ( ( TopOpen ` ℂfld ) ↾t RR ) = ( ( TopOpen ` ℂfld ) ↾t RR )
18		eqid 2451	. . . . . . . . . 10 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
19		eqid 2451	. . . . . . . . . 10 |- ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) = ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) )
20		ax-resscn 9442	. . . . . . . . . . 11 |- RR C_ CC
21	20	a1i 11	. . . . . . . . . 10 |- ( ph -> RR C_ CC )
22		fss 5667	. . . . . . . . . . 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 21491	. . . . . . . . 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 3468	. . . . . . . . . 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 3457	. . . . . . . . . 10 |- ( ph -> U e. X )
32	23, 28, 31	dvlem 21489	. . . . . . . . 9 |- ( ( ph /\ x e. ( X \ { U } ) ) -> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) e. CC )
33	32, 19	fmptd 5968	. . . . . . . 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 3594	. . . . . . 7 |- ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> ( X \ { U } ) C_ CC )
37	28, 31	sseldd 3457	. . . . . . . 8 |- ( ph -> U e. CC )
38	37	adantr 465	. . . . . . 7 |- ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> U e. CC )
39	34, 36, 38	ellimc3 21472	. . . . . 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 5791	. . . . . . . . . . . . . 14 |- ( x = z -> ( F ` x ) = ( F ` z ) )
43	42	oveq1d 6207	. . . . . . . . . . . . 13 |- ( x = z -> ( ( F ` x ) - ( F ` U ) ) = ( ( F ` z ) - ( F ` U ) ) )
44		oveq1 6199	. . . . . . . . . . . . 13 |- ( x = z -> ( x - U ) = ( z - U ) )
45	43, 44	oveq12d 6210	. . . . . . . . . . . 12 |- ( x = z -> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) = ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) )
46		ovex 6217	. . . . . . . . . . . 12 |- ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) e. _V
47	45, 19, 46	fvmpt 5875	. . . . . . . . . . 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 6207	. . . . . . . . . 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 5795	. . . . . . . . 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 4408	. . . . . . . 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 2836	. . . . . 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 2848	. . . . 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 3167	. . . 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 2451	. . . . . 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 21576	. . . . 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 2917	. . 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 9489	. . 3 |- 0 e. RR
73		lenlt 9556	. . 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 1370   e. wcel 1758   =/= wne 2644   A.wral 2795   E.wrex 2796   \ cdif 3425   C_ wss 3428   ifcif 3891   {csn 3977   class class class wbr 4392   |-> cmpt 4450   dom cdm 4940   Fun wfun 5512   -->wf 5514   ` cfv 5518  (class class class)co 6192   CCcc 9383   RRcr 9384   0cc0 9385   + caddc 9388   < clt 9521   <_ cle 9522   - cmin 9698   -ucneg 9699   / cdiv 10096   2c2 10474   RR+crp 11094   (,)cioo 11403   abscabs 12827   ↾_t crest 14463   TopOpenctopn 14464  ℂ_fldccnfld 17929   intcnt 18739   lim CC climc 21455   _D cdv 21456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462  ax-pre-sup 9463

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-iin 4274  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-map 7318  df-pm 7319  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-fi 7764  df-sup 7794  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-div 10097  df-nn 10426  df-2 10483  df-3 10484  df-4 10485  df-5 10486  df-6 10487  df-7 10488  df-8 10489  df-9 10490  df-10 10491  df-n0 10683  df-z 10750  df-dec 10859  df-uz 10965  df-q 11057  df-rp 11095  df-xneg 11192  df-xadd 11193  df-xmul 11194  df-ioo 11407  df-icc 11410  df-fz 11541  df-seq 11910  df-exp 11969  df-cj 12692  df-re 12693  df-im 12694  df-sqr 12828  df-abs 12829  df-struct 14280  df-ndx 14281  df-slot 14282  df-base 14283  df-plusg 14355  df-mulr 14356  df-starv 14357  df-tset 14361  df-ple 14362  df-ds 14364  df-unif 14365  df-rest 14465  df-topn 14466  df-topgen 14486  df-psmet 17920  df-xmet 17921  df-met 17922  df-bl 17923  df-mopn 17924  df-fbas 17925  df-fg 17926  df-cnfld 17930  df-top 18621  df-bases 18623  df-topon 18624  df-topsp 18625  df-cld 18741  df-ntr 18742  df-cls 18743  df-nei 18820  df-lp 18858  df-perf 18859  df-cn 18949  df-cnp 18950  df-haus 19037  df-fil 19537  df-fm 19629  df-flim 19630  df-flf 19631  df-xms 20013  df-ms 20014  df-cncf 20572  df-limc 21459  df-dv 21460

This theorem is referenced by:  dvferm  21578  dvivthlem1  21598