Theorem dvle 22836

Description: If A ( x ) , C ( x ) are differentiable functions and A ` <_ C ` , then for x <_ y, A ( y ) - A ( x ) <_ C ( y ) - C ( x ). (Contributed by Mario Carneiro, 16-May-2016.)

Hypotheses

Ref	Expression
dvle.m	|- ( ph -> M e. RR )
dvle.n	|- ( ph -> N e. RR )
dvle.a	|- ( ph -> ( x e. ( M [,] N ) |-> A ) e. ( ( M [,] N ) -cn-> RR ) )
dvle.b	|- ( ph -> ( RR _D ( x e. ( M (,) N ) |-> A ) ) = ( x e. ( M (,) N ) |-> B ) )
dvle.c	|- ( ph -> ( x e. ( M [,] N ) |-> C ) e. ( ( M [,] N ) -cn-> RR ) )
dvle.d	|- ( ph -> ( RR _D ( x e. ( M (,) N ) |-> C ) ) = ( x e. ( M (,) N ) |-> D ) )
dvle.f	|- ( ( ph /\ x e. ( M (,) N ) ) -> B <_ D )
dvle.x	|- ( ph -> X e. ( M [,] N ) )
dvle.y	|- ( ph -> Y e. ( M [,] N ) )
dvle.l	|- ( ph -> X <_ Y )
dvle.p	|- ( x = X -> A = P )
dvle.q	|- ( x = X -> C = Q )
dvle.r	|- ( x = Y -> A = R )
dvle.s	|- ( x = Y -> C = S )

Assertion

Ref	Expression
dvle	|- ( ph -> ( R - P ) <_ ( S - Q ) )

Distinct variable groups:   x, M   x, N   x, P   x, Q   x, R   x, S   x, X   ph, x   x, Y

Allowed substitution hints:   A( x)   B( x)   C( x)   D( x)

Proof of Theorem dvle

Step	Hyp	Ref	Expression
1		dvle.y	. . 3 |- ( ph -> Y e. ( M [,] N ) )
2		dvle.a	. . . . 5 |- ( ph -> ( x e. ( M [,] N ) |-> A ) e. ( ( M [,] N ) -cn-> RR ) )
3		cncff 21821	. . . . 5 |- ( ( x e. ( M [,] N ) |-> A ) e. ( ( M [,] N ) -cn-> RR ) -> ( x e. ( M [,] N ) |-> A ) : ( M [,] N ) --> RR )
4	2, 3	syl 17	. . . 4 |- ( ph -> ( x e. ( M [,] N ) |-> A ) : ( M [,] N ) --> RR )
5		eqid 2429	. . . . 5 |- ( x e. ( M [,] N ) |-> A ) = ( x e. ( M [,] N ) |-> A )
6	5	fmpt 6058	. . . 4 |- ( A. x e. ( M [,] N ) A e. RR <-> ( x e. ( M [,] N ) |-> A ) : ( M [,] N ) --> RR )
7	4, 6	sylibr 215	. . 3 |- ( ph -> A. x e. ( M [,] N ) A e. RR )
8		dvle.r	. . . . 5 |- ( x = Y -> A = R )
9	8	eleq1d 2498	. . . 4 |- ( x = Y -> ( A e. RR <-> R e. RR ) )
10	9	rspcv 3184	. . 3 |- ( Y e. ( M [,] N ) -> ( A. x e. ( M [,] N ) A e. RR -> R e. RR ) )
11	1, 7, 10	sylc 62	. 2 |- ( ph -> R e. RR )
12		dvle.c	. . . . . 6 |- ( ph -> ( x e. ( M [,] N ) |-> C ) e. ( ( M [,] N ) -cn-> RR ) )
13		cncff 21821	. . . . . 6 |- ( ( x e. ( M [,] N ) |-> C ) e. ( ( M [,] N ) -cn-> RR ) -> ( x e. ( M [,] N ) |-> C ) : ( M [,] N ) --> RR )
14	12, 13	syl 17	. . . . 5 |- ( ph -> ( x e. ( M [,] N ) |-> C ) : ( M [,] N ) --> RR )
15		eqid 2429	. . . . . 6 |- ( x e. ( M [,] N ) |-> C ) = ( x e. ( M [,] N ) |-> C )
16	15	fmpt 6058	. . . . 5 |- ( A. x e. ( M [,] N ) C e. RR <-> ( x e. ( M [,] N ) |-> C ) : ( M [,] N ) --> RR )
17	14, 16	sylibr 215	. . . 4 |- ( ph -> A. x e. ( M [,] N ) C e. RR )
18		dvle.s	. . . . . 6 |- ( x = Y -> C = S )
19	18	eleq1d 2498	. . . . 5 |- ( x = Y -> ( C e. RR <-> S e. RR ) )
20	19	rspcv 3184	. . . 4 |- ( Y e. ( M [,] N ) -> ( A. x e. ( M [,] N ) C e. RR -> S e. RR ) )
21	1, 17, 20	sylc 62	. . 3 |- ( ph -> S e. RR )
22		dvle.x	. . . 4 |- ( ph -> X e. ( M [,] N ) )
23		dvle.q	. . . . . 6 |- ( x = X -> C = Q )
24	23	eleq1d 2498	. . . . 5 |- ( x = X -> ( C e. RR <-> Q e. RR ) )
25	24	rspcv 3184	. . . 4 |- ( X e. ( M [,] N ) -> ( A. x e. ( M [,] N ) C e. RR -> Q e. RR ) )
26	22, 17, 25	sylc 62	. . 3 |- ( ph -> Q e. RR )
27	21, 26	resubcld 10046	. 2 |- ( ph -> ( S - Q ) e. RR )
28		dvle.p	. . . . 5 |- ( x = X -> A = P )
29	28	eleq1d 2498	. . . 4 |- ( x = X -> ( A e. RR <-> P e. RR ) )
30	29	rspcv 3184	. . 3 |- ( X e. ( M [,] N ) -> ( A. x e. ( M [,] N ) A e. RR -> P e. RR ) )
31	22, 7, 30	sylc 62	. 2 |- ( ph -> P e. RR )
32	11	recnd 9668	. . . . 5 |- ( ph -> R e. CC )
33	26	recnd 9668	. . . . . 6 |- ( ph -> Q e. CC )
34	21	recnd 9668	. . . . . 6 |- ( ph -> S e. CC )
35	33, 34	subcld 9985	. . . . 5 |- ( ph -> ( Q - S ) e. CC )
36	32, 35	addcomd 9834	. . . 4 |- ( ph -> ( R + ( Q - S ) ) = ( ( Q - S ) + R ) )
37	32, 34, 33	subsub2d 10014	. . . 4 |- ( ph -> ( R - ( S - Q ) ) = ( R + ( Q - S ) ) )
38	33, 34, 32	subsubd 10013	. . . 4 |- ( ph -> ( Q - ( S - R ) ) = ( ( Q - S ) + R ) )
39	36, 37, 38	3eqtr4d 2480	. . 3 |- ( ph -> ( R - ( S - Q ) ) = ( Q - ( S - R ) ) )
40	21, 11	resubcld 10046	. . . 4 |- ( ph -> ( S - R ) e. RR )
41		dvle.m	. . . . . 6 |- ( ph -> M e. RR )
42		dvle.n	. . . . . 6 |- ( ph -> N e. RR )
43		eqid 2429	. . . . . . 7 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
44	43	subcn 21794	. . . . . . 7 |- - e. ( ( ( TopOpen ` ℂfld ) tX ( TopOpen ` ℂfld ) ) Cn ( TopOpen ` ℂfld ) )
45		ax-resscn 9595	. . . . . . 7 |- RR C_ CC
46		resubcl 9937	. . . . . . 7 |- ( ( C e. RR /\ A e. RR ) -> ( C - A ) e. RR )
47	43, 44, 12, 2, 45, 46	cncfmpt2ss 21843	. . . . . 6 |- ( ph -> ( x e. ( M [,] N ) |-> ( C - A ) ) e. ( ( M [,] N ) -cn-> RR ) )
48		ioossicc 11720	. . . . . . . . . . . . . . . . 17 |- ( M (,) N ) C_ ( M [,] N )
49	48	sseli 3466	. . . . . . . . . . . . . . . 16 |- ( x e. ( M (,) N ) -> x e. ( M [,] N ) )
50	17	r19.21bi 2801	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( M [,] N ) ) -> C e. RR )
51	49, 50	sylan2 476	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( M (,) N ) ) -> C e. RR )
52		eqid 2429	. . . . . . . . . . . . . . 15 |- ( x e. ( M (,) N ) |-> C ) = ( x e. ( M (,) N ) |-> C )
53	51, 52	fmptd 6061	. . . . . . . . . . . . . 14 |- ( ph -> ( x e. ( M (,) N ) |-> C ) : ( M (,) N ) --> RR )
54		ioossre 11696	. . . . . . . . . . . . . 14 |- ( M (,) N ) C_ RR
55		dvfre 22782	. . . . . . . . . . . . . 14 |- ( ( ( x e. ( M (,) N ) |-> C ) : ( M (,) N ) --> RR /\ ( M (,) N ) C_ RR ) -> ( RR _D ( x e. ( M (,) N ) |-> C ) ) : dom ( RR _D ( x e. ( M (,) N ) |-> C ) ) --> RR )
56	53, 54, 55	sylancl 666	. . . . . . . . . . . . 13 |- ( ph -> ( RR _D ( x e. ( M (,) N ) |-> C ) ) : dom ( RR _D ( x e. ( M (,) N ) |-> C ) ) --> RR )
57		dvle.d	. . . . . . . . . . . . . 14 |- ( ph -> ( RR _D ( x e. ( M (,) N ) |-> C ) ) = ( x e. ( M (,) N ) |-> D ) )
58	57	dmeqd 5057	. . . . . . . . . . . . . . 15 |- ( ph -> dom ( RR _D ( x e. ( M (,) N ) |-> C ) ) = dom ( x e. ( M (,) N ) |-> D ) )
59		dvle.f	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ x e. ( M (,) N ) ) -> B <_ D )
60		lerel 9697	. . . . . . . . . . . . . . . . . . 19 |- Rel <_
61	60	brrelex2i 4896	. . . . . . . . . . . . . . . . . 18 |- ( B <_ D -> D e. _V )
62	59, 61	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. ( M (,) N ) ) -> D e. _V )
63	62	ralrimiva 2846	. . . . . . . . . . . . . . . 16 |- ( ph -> A. x e. ( M (,) N ) D e. _V )
64		dmmptg 5352	. . . . . . . . . . . . . . . 16 |- ( A. x e. ( M (,) N ) D e. _V -> dom ( x e. ( M (,) N ) |-> D ) = ( M (,) N ) )
65	63, 64	syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> dom ( x e. ( M (,) N ) |-> D ) = ( M (,) N ) )
66	58, 65	eqtrd 2470	. . . . . . . . . . . . . 14 |- ( ph -> dom ( RR _D ( x e. ( M (,) N ) |-> C ) ) = ( M (,) N ) )
67	57, 66	feq12d 5735	. . . . . . . . . . . . 13 |- ( ph -> ( ( RR _D ( x e. ( M (,) N ) |-> C ) ) : dom ( RR _D ( x e. ( M (,) N ) |-> C ) ) --> RR <-> ( x e. ( M (,) N ) |-> D ) : ( M (,) N ) --> RR ) )
68	56, 67	mpbid 213	. . . . . . . . . . . 12 |- ( ph -> ( x e. ( M (,) N ) |-> D ) : ( M (,) N ) --> RR )
69		eqid 2429	. . . . . . . . . . . . 13 |- ( x e. ( M (,) N ) |-> D ) = ( x e. ( M (,) N ) |-> D )
70	69	fmpt 6058	. . . . . . . . . . . 12 |- ( A. x e. ( M (,) N ) D e. RR <-> ( x e. ( M (,) N ) |-> D ) : ( M (,) N ) --> RR )
71	68, 70	sylibr 215	. . . . . . . . . . 11 |- ( ph -> A. x e. ( M (,) N ) D e. RR )
72	71	r19.21bi 2801	. . . . . . . . . 10 |- ( ( ph /\ x e. ( M (,) N ) ) -> D e. RR )
73	7	r19.21bi 2801	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( M [,] N ) ) -> A e. RR )
74	49, 73	sylan2 476	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( M (,) N ) ) -> A e. RR )
75		eqid 2429	. . . . . . . . . . . . . . 15 |- ( x e. ( M (,) N ) |-> A ) = ( x e. ( M (,) N ) |-> A )
76	74, 75	fmptd 6061	. . . . . . . . . . . . . 14 |- ( ph -> ( x e. ( M (,) N ) |-> A ) : ( M (,) N ) --> RR )
77		dvfre 22782	. . . . . . . . . . . . . 14 |- ( ( ( x e. ( M (,) N ) |-> A ) : ( M (,) N ) --> RR /\ ( M (,) N ) C_ RR ) -> ( RR _D ( x e. ( M (,) N ) |-> A ) ) : dom ( RR _D ( x e. ( M (,) N ) |-> A ) ) --> RR )
78	76, 54, 77	sylancl 666	. . . . . . . . . . . . 13 |- ( ph -> ( RR _D ( x e. ( M (,) N ) |-> A ) ) : dom ( RR _D ( x e. ( M (,) N ) |-> A ) ) --> RR )
79		dvle.b	. . . . . . . . . . . . . 14 |- ( ph -> ( RR _D ( x e. ( M (,) N ) |-> A ) ) = ( x e. ( M (,) N ) |-> B ) )
80	79	dmeqd 5057	. . . . . . . . . . . . . . 15 |- ( ph -> dom ( RR _D ( x e. ( M (,) N ) |-> A ) ) = dom ( x e. ( M (,) N ) |-> B ) )
81	60	brrelexi 4895	. . . . . . . . . . . . . . . . . 18 |- ( B <_ D -> B e. _V )
82	59, 81	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. ( M (,) N ) ) -> B e. _V )
83	82	ralrimiva 2846	. . . . . . . . . . . . . . . 16 |- ( ph -> A. x e. ( M (,) N ) B e. _V )
84		dmmptg 5352	. . . . . . . . . . . . . . . 16 |- ( A. x e. ( M (,) N ) B e. _V -> dom ( x e. ( M (,) N ) |-> B ) = ( M (,) N ) )
85	83, 84	syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> dom ( x e. ( M (,) N ) |-> B ) = ( M (,) N ) )
86	80, 85	eqtrd 2470	. . . . . . . . . . . . . 14 |- ( ph -> dom ( RR _D ( x e. ( M (,) N ) |-> A ) ) = ( M (,) N ) )
87	79, 86	feq12d 5735	. . . . . . . . . . . . 13 |- ( ph -> ( ( RR _D ( x e. ( M (,) N ) |-> A ) ) : dom ( RR _D ( x e. ( M (,) N ) |-> A ) ) --> RR <-> ( x e. ( M (,) N ) |-> B ) : ( M (,) N ) --> RR ) )
88	78, 87	mpbid 213	. . . . . . . . . . . 12 |- ( ph -> ( x e. ( M (,) N ) |-> B ) : ( M (,) N ) --> RR )
89		eqid 2429	. . . . . . . . . . . . 13 |- ( x e. ( M (,) N ) |-> B ) = ( x e. ( M (,) N ) |-> B )
90	89	fmpt 6058	. . . . . . . . . . . 12 |- ( A. x e. ( M (,) N ) B e. RR <-> ( x e. ( M (,) N ) |-> B ) : ( M (,) N ) --> RR )
91	88, 90	sylibr 215	. . . . . . . . . . 11 |- ( ph -> A. x e. ( M (,) N ) B e. RR )
92	91	r19.21bi 2801	. . . . . . . . . 10 |- ( ( ph /\ x e. ( M (,) N ) ) -> B e. RR )
93	72, 92	resubcld 10046	. . . . . . . . 9 |- ( ( ph /\ x e. ( M (,) N ) ) -> ( D - B ) e. RR )
94	72, 92	subge0d 10202	. . . . . . . . . 10 |- ( ( ph /\ x e. ( M (,) N ) ) -> ( 0 <_ ( D - B ) <-> B <_ D ) )
95	59, 94	mpbird 235	. . . . . . . . 9 |- ( ( ph /\ x e. ( M (,) N ) ) -> 0 <_ ( D - B ) )
96		elrege0 11737	. . . . . . . . 9 |- ( ( D - B ) e. ( 0 [,) +oo ) <-> ( ( D - B ) e. RR /\ 0 <_ ( D - B ) ) )
97	93, 95, 96	sylanbrc 668	. . . . . . . 8 |- ( ( ph /\ x e. ( M (,) N ) ) -> ( D - B ) e. ( 0 [,) +oo ) )
98		eqid 2429	. . . . . . . 8 |- ( x e. ( M (,) N ) |-> ( D - B ) ) = ( x e. ( M (,) N ) |-> ( D - B ) )
99	97, 98	fmptd 6061	. . . . . . 7 |- ( ph -> ( x e. ( M (,) N ) |-> ( D - B ) ) : ( M (,) N ) --> ( 0 [,) +oo ) )
100	45	a1i 11	. . . . . . . . . 10 |- ( ph -> RR C_ CC )
101		iccssre 11716	. . . . . . . . . . 11 |- ( ( M e. RR /\ N e. RR ) -> ( M [,] N ) C_ RR )
102	41, 42, 101	syl2anc 665	. . . . . . . . . 10 |- ( ph -> ( M [,] N ) C_ RR )
103	50, 73	resubcld 10046	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( M [,] N ) ) -> ( C - A ) e. RR )
104	103	recnd 9668	. . . . . . . . . 10 |- ( ( ph /\ x e. ( M [,] N ) ) -> ( C - A ) e. CC )
105	43	tgioo2 21732	. . . . . . . . . 10 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
106		iccntr 21750	. . . . . . . . . . 11 |- ( ( M e. RR /\ N e. RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( M [,] N ) ) = ( M (,) N ) )
107	41, 42, 106	syl2anc 665	. . . . . . . . . 10 |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( M [,] N ) ) = ( M (,) N ) )
108	100, 102, 104, 105, 43, 107	dvmptntr 22802	. . . . . . . . 9 |- ( ph -> ( RR _D ( x e. ( M [,] N ) |-> ( C - A ) ) ) = ( RR _D ( x e. ( M (,) N ) |-> ( C - A ) ) ) )
109		reelprrecn 9630	. . . . . . . . . . 11 |- RR e. { RR , CC }
110	109	a1i 11	. . . . . . . . . 10 |- ( ph -> RR e. { RR , CC } )
111	50	recnd 9668	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( M [,] N ) ) -> C e. CC )
112	49, 111	sylan2 476	. . . . . . . . . 10 |- ( ( ph /\ x e. ( M (,) N ) ) -> C e. CC )
113	73	recnd 9668	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( M [,] N ) ) -> A e. CC )
114	49, 113	sylan2 476	. . . . . . . . . 10 |- ( ( ph /\ x e. ( M (,) N ) ) -> A e. CC )
115	110, 112, 62, 57, 114, 82, 79	dvmptsub 22798	. . . . . . . . 9 |- ( ph -> ( RR _D ( x e. ( M (,) N ) |-> ( C - A ) ) ) = ( x e. ( M (,) N ) |-> ( D - B ) ) )
116	108, 115	eqtrd 2470	. . . . . . . 8 |- ( ph -> ( RR _D ( x e. ( M [,] N ) |-> ( C - A ) ) ) = ( x e. ( M (,) N ) |-> ( D - B ) ) )
117	116	feq1d 5732	. . . . . . 7 |- ( ph -> ( ( RR _D ( x e. ( M [,] N ) |-> ( C - A ) ) ) : ( M (,) N ) --> ( 0 [,) +oo ) <-> ( x e. ( M (,) N ) |-> ( D - B ) ) : ( M (,) N ) --> ( 0 [,) +oo ) ) )
118	99, 117	mpbird 235	. . . . . 6 |- ( ph -> ( RR _D ( x e. ( M [,] N ) |-> ( C - A ) ) ) : ( M (,) N ) --> ( 0 [,) +oo ) )
119		dvle.l	. . . . . 6 |- ( ph -> X <_ Y )
120	41, 42, 47, 118, 22, 1, 119	dvge0 22835	. . . . 5 |- ( ph -> ( ( x e. ( M [,] N ) |-> ( C - A ) ) ` X ) <_ ( ( x e. ( M [,] N ) |-> ( C - A ) ) ` Y ) )
121	23, 28	oveq12d 6323	. . . . . . 7 |- ( x = X -> ( C - A ) = ( Q - P ) )
122		eqid 2429	. . . . . . 7 |- ( x e. ( M [,] N ) |-> ( C - A ) ) = ( x e. ( M [,] N ) |-> ( C - A ) )
123		ovex 6333	. . . . . . 7 |- ( C - A ) e. _V
124	121, 122, 123	fvmpt3i 5969	. . . . . 6 |- ( X e. ( M [,] N ) -> ( ( x e. ( M [,] N ) |-> ( C - A ) ) ` X ) = ( Q - P ) )
125	22, 124	syl 17	. . . . 5 |- ( ph -> ( ( x e. ( M [,] N ) |-> ( C - A ) ) ` X ) = ( Q - P ) )
126	18, 8	oveq12d 6323	. . . . . . 7 |- ( x = Y -> ( C - A ) = ( S - R ) )
127	126, 122, 123	fvmpt3i 5969	. . . . . 6 |- ( Y e. ( M [,] N ) -> ( ( x e. ( M [,] N ) |-> ( C - A ) ) ` Y ) = ( S - R ) )
128	1, 127	syl 17	. . . . 5 |- ( ph -> ( ( x e. ( M [,] N ) |-> ( C - A ) ) ` Y ) = ( S - R ) )
129	120, 125, 128	3brtr3d 4455	. . . 4 |- ( ph -> ( Q - P ) <_ ( S - R ) )
130	26, 31, 40, 129	subled 10215	. . 3 |- ( ph -> ( Q - ( S - R ) ) <_ P )
131	39, 130	eqbrtrd 4446	. 2 |- ( ph -> ( R - ( S - Q ) ) <_ P )
132	11, 27, 31, 131	subled 10215	1 |- ( ph -> ( R - P ) <_ ( S - Q ) )

Syntax hints:   -> wi 4   /\ wa 370   = wceq 1437   e. wcel 1870   A.wral 2782   _Vcvv 3087   C_ wss 3442   {cpr 4004   class class class wbr 4426   |-> cmpt 4484   dom cdm 4854   ran crn 4855   -->wf 5597   ` cfv 5601  (class class class)co 6305   CCcc 9536   RRcr 9537   0cc0 9538   + caddc 9541   +oocpnf 9671   <_ cle 9675   - cmin 9859   (,)cioo 11635   [,)cico 11637   [,]cicc 11638   TopOpenctopn 15279   topGenctg 15295  ℂ_fldccnfld 18905   intcnt 19963   -cn->ccncf 21804   _D cdv 22695

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-inf2 8146  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  ax-addf 9617  ax-mulf 9618

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-se 4814  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-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-fi 7931  df-sup 7962  df-oi 8025  df-card 8372  df-cda 8596  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-ico 11641  df-icc 11642  df-fz 11783  df-fzo 11914  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-starv 15167  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-unif 15175  df-hom 15176  df-cco 15177  df-rest 15280  df-topn 15281  df-0g 15299  df-gsum 15300  df-topgen 15301  df-pt 15302  df-prds 15305  df-xrs 15359  df-qtop 15364  df-imas 15365  df-xps 15367  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-mulg 16627  df-cntz 16922  df-cmn 17367  df-psmet 18897  df-xmet 18898  df-met 18899  df-bl 18900  df-mopn 18901  df-fbas 18902  df-fg 18903  df-cnfld 18906  df-top 19852  df-bases 19853  df-topon 19854  df-topsp 19855  df-cld 19965  df-ntr 19966  df-cls 19967  df-nei 20045  df-lp 20083  df-perf 20084  df-cn 20174  df-cnp 20175  df-haus 20262  df-cmp 20333  df-tx 20508  df-hmeo 20701  df-fil 20792  df-fm 20884  df-flim 20885  df-flf 20886  df-xms 21266  df-ms 21267  df-tms 21268  df-cncf 21806  df-limc 22698  df-dv 22699

This theorem is referenced by:  dvfsumle  22850  dvfsumlem2  22856  loglesqrt  23563