Theorem dvle 23770

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 22696	. . . . 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 2622	. . . . 5 |- ( x e. ( M [,] N ) |-> A ) = ( x e. ( M [,] N ) |-> A )
6	5	fmpt 6381	. . . 4 |- ( A. x e. ( M [,] N ) A e. RR <-> ( x e. ( M [,] N ) |-> A ) : ( M [,] N ) --> RR )
7	4, 6	sylibr 224	. . 3 |- ( ph -> A. x e. ( M [,] N ) A e. RR )
8		dvle.r	. . . . 5 |- ( x = Y -> A = R )
9	8	eleq1d 2686	. . . 4 |- ( x = Y -> ( A e. RR <-> R e. RR ) )
10	9	rspcv 3305	. . 3 |- ( Y e. ( M [,] N ) -> ( A. x e. ( M [,] N ) A e. RR -> R e. RR ) )
11	1, 7, 10	sylc 65	. 2 |- ( ph -> R e. RR )
12		dvle.c	. . . . . 6 |- ( ph -> ( x e. ( M [,] N ) |-> C ) e. ( ( M [,] N ) -cn-> RR ) )
13		cncff 22696	. . . . . 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 2622	. . . . . 6 |- ( x e. ( M [,] N ) |-> C ) = ( x e. ( M [,] N ) |-> C )
16	15	fmpt 6381	. . . . 5 |- ( A. x e. ( M [,] N ) C e. RR <-> ( x e. ( M [,] N ) |-> C ) : ( M [,] N ) --> RR )
17	14, 16	sylibr 224	. . . 4 |- ( ph -> A. x e. ( M [,] N ) C e. RR )
18		dvle.s	. . . . . 6 |- ( x = Y -> C = S )
19	18	eleq1d 2686	. . . . 5 |- ( x = Y -> ( C e. RR <-> S e. RR ) )
20	19	rspcv 3305	. . . 4 |- ( Y e. ( M [,] N ) -> ( A. x e. ( M [,] N ) C e. RR -> S e. RR ) )
21	1, 17, 20	sylc 65	. . 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 2686	. . . . 5 |- ( x = X -> ( C e. RR <-> Q e. RR ) )
25	24	rspcv 3305	. . . 4 |- ( X e. ( M [,] N ) -> ( A. x e. ( M [,] N ) C e. RR -> Q e. RR ) )
26	22, 17, 25	sylc 65	. . 3 |- ( ph -> Q e. RR )
27	21, 26	resubcld 10458	. 2 |- ( ph -> ( S - Q ) e. RR )
28		dvle.p	. . . . 5 |- ( x = X -> A = P )
29	28	eleq1d 2686	. . . 4 |- ( x = X -> ( A e. RR <-> P e. RR ) )
30	29	rspcv 3305	. . 3 |- ( X e. ( M [,] N ) -> ( A. x e. ( M [,] N ) A e. RR -> P e. RR ) )
31	22, 7, 30	sylc 65	. 2 |- ( ph -> P e. RR )
32	11	recnd 10068	. . . . 5 |- ( ph -> R e. CC )
33	26	recnd 10068	. . . . . 6 |- ( ph -> Q e. CC )
34	21	recnd 10068	. . . . . 6 |- ( ph -> S e. CC )
35	33, 34	subcld 10392	. . . . 5 |- ( ph -> ( Q - S ) e. CC )
36	32, 35	addcomd 10238	. . . 4 |- ( ph -> ( R + ( Q - S ) ) = ( ( Q - S ) + R ) )
37	32, 34, 33	subsub2d 10421	. . . 4 |- ( ph -> ( R - ( S - Q ) ) = ( R + ( Q - S ) ) )
38	33, 34, 32	subsubd 10420	. . . 4 |- ( ph -> ( Q - ( S - R ) ) = ( ( Q - S ) + R ) )
39	36, 37, 38	3eqtr4d 2666	. . 3 |- ( ph -> ( R - ( S - Q ) ) = ( Q - ( S - R ) ) )
40	21, 11	resubcld 10458	. . . 4 |- ( ph -> ( S - R ) e. RR )
41		dvle.m	. . . . . 6 |- ( ph -> M e. RR )
42		dvle.n	. . . . . 6 |- ( ph -> N e. RR )
43		eqid 2622	. . . . . . 7 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
44	43	subcn 22669	. . . . . . 7 |- - e. ( ( ( TopOpen ` ℂfld ) tX ( TopOpen ` ℂfld ) ) Cn ( TopOpen ` ℂfld ) )
45		ax-resscn 9993	. . . . . . 7 |- RR C_ CC
46		resubcl 10345	. . . . . . 7 |- ( ( C e. RR /\ A e. RR ) -> ( C - A ) e. RR )
47	43, 44, 12, 2, 45, 46	cncfmpt2ss 22718	. . . . . 6 |- ( ph -> ( x e. ( M [,] N ) |-> ( C - A ) ) e. ( ( M [,] N ) -cn-> RR ) )
48		ioossicc 12259	. . . . . . . . . . . . . . . . 17 |- ( M (,) N ) C_ ( M [,] N )
49	48	sseli 3599	. . . . . . . . . . . . . . . 16 |- ( x e. ( M (,) N ) -> x e. ( M [,] N ) )
50	17	r19.21bi 2932	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( M [,] N ) ) -> C e. RR )
51	49, 50	sylan2 491	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( M (,) N ) ) -> C e. RR )
52		eqid 2622	. . . . . . . . . . . . . . 15 |- ( x e. ( M (,) N ) |-> C ) = ( x e. ( M (,) N ) |-> C )
53	51, 52	fmptd 6385	. . . . . . . . . . . . . 14 |- ( ph -> ( x e. ( M (,) N ) |-> C ) : ( M (,) N ) --> RR )
54		ioossre 12235	. . . . . . . . . . . . . 14 |- ( M (,) N ) C_ RR
55		dvfre 23714	. . . . . . . . . . . . . 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 694	. . . . . . . . . . . . 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 5326	. . . . . . . . . . . . . . 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 10102	. . . . . . . . . . . . . . . . . . 19 |- Rel <_
61	60	brrelex2i 5159	. . . . . . . . . . . . . . . . . 18 |- ( B <_ D -> D e. _V )
62	59, 61	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. ( M (,) N ) ) -> D e. _V )
63	62	ralrimiva 2966	. . . . . . . . . . . . . . . 16 |- ( ph -> A. x e. ( M (,) N ) D e. _V )
64		dmmptg 5632	. . . . . . . . . . . . . . . 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 2656	. . . . . . . . . . . . . 14 |- ( ph -> dom ( RR _D ( x e. ( M (,) N ) |-> C ) ) = ( M (,) N ) )
67	57, 66	feq12d 6033	. . . . . . . . . . . . 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 222	. . . . . . . . . . . 12 |- ( ph -> ( x e. ( M (,) N ) |-> D ) : ( M (,) N ) --> RR )
69		eqid 2622	. . . . . . . . . . . . 13 |- ( x e. ( M (,) N ) |-> D ) = ( x e. ( M (,) N ) |-> D )
70	69	fmpt 6381	. . . . . . . . . . . 12 |- ( A. x e. ( M (,) N ) D e. RR <-> ( x e. ( M (,) N ) |-> D ) : ( M (,) N ) --> RR )
71	68, 70	sylibr 224	. . . . . . . . . . 11 |- ( ph -> A. x e. ( M (,) N ) D e. RR )
72	71	r19.21bi 2932	. . . . . . . . . 10 |- ( ( ph /\ x e. ( M (,) N ) ) -> D e. RR )
73	7	r19.21bi 2932	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( M [,] N ) ) -> A e. RR )
74	49, 73	sylan2 491	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( M (,) N ) ) -> A e. RR )
75		eqid 2622	. . . . . . . . . . . . . . 15 |- ( x e. ( M (,) N ) |-> A ) = ( x e. ( M (,) N ) |-> A )
76	74, 75	fmptd 6385	. . . . . . . . . . . . . 14 |- ( ph -> ( x e. ( M (,) N ) |-> A ) : ( M (,) N ) --> RR )
77		dvfre 23714	. . . . . . . . . . . . . 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 694	. . . . . . . . . . . . 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 5326	. . . . . . . . . . . . . . 15 |- ( ph -> dom ( RR _D ( x e. ( M (,) N ) |-> A ) ) = dom ( x e. ( M (,) N ) |-> B ) )
81	60	brrelexi 5158	. . . . . . . . . . . . . . . . . 18 |- ( B <_ D -> B e. _V )
82	59, 81	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. ( M (,) N ) ) -> B e. _V )
83	82	ralrimiva 2966	. . . . . . . . . . . . . . . 16 |- ( ph -> A. x e. ( M (,) N ) B e. _V )
84		dmmptg 5632	. . . . . . . . . . . . . . . 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 2656	. . . . . . . . . . . . . 14 |- ( ph -> dom ( RR _D ( x e. ( M (,) N ) |-> A ) ) = ( M (,) N ) )
87	79, 86	feq12d 6033	. . . . . . . . . . . . 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 222	. . . . . . . . . . . 12 |- ( ph -> ( x e. ( M (,) N ) |-> B ) : ( M (,) N ) --> RR )
89		eqid 2622	. . . . . . . . . . . . 13 |- ( x e. ( M (,) N ) |-> B ) = ( x e. ( M (,) N ) |-> B )
90	89	fmpt 6381	. . . . . . . . . . . 12 |- ( A. x e. ( M (,) N ) B e. RR <-> ( x e. ( M (,) N ) |-> B ) : ( M (,) N ) --> RR )
91	88, 90	sylibr 224	. . . . . . . . . . 11 |- ( ph -> A. x e. ( M (,) N ) B e. RR )
92	91	r19.21bi 2932	. . . . . . . . . 10 |- ( ( ph /\ x e. ( M (,) N ) ) -> B e. RR )
93	72, 92	resubcld 10458	. . . . . . . . 9 |- ( ( ph /\ x e. ( M (,) N ) ) -> ( D - B ) e. RR )
94	72, 92	subge0d 10617	. . . . . . . . . 10 |- ( ( ph /\ x e. ( M (,) N ) ) -> ( 0 <_ ( D - B ) <-> B <_ D ) )
95	59, 94	mpbird 247	. . . . . . . . 9 |- ( ( ph /\ x e. ( M (,) N ) ) -> 0 <_ ( D - B ) )
96		elrege0 12278	. . . . . . . . 9 |- ( ( D - B ) e. ( 0 [,) +oo ) <-> ( ( D - B ) e. RR /\ 0 <_ ( D - B ) ) )
97	93, 95, 96	sylanbrc 698	. . . . . . . 8 |- ( ( ph /\ x e. ( M (,) N ) ) -> ( D - B ) e. ( 0 [,) +oo ) )
98		eqid 2622	. . . . . . . 8 |- ( x e. ( M (,) N ) |-> ( D - B ) ) = ( x e. ( M (,) N ) |-> ( D - B ) )
99	97, 98	fmptd 6385	. . . . . . 7 |- ( ph -> ( x e. ( M (,) N ) |-> ( D - B ) ) : ( M (,) N ) --> ( 0 [,) +oo ) )
100	45	a1i 11	. . . . . . . . . 10 |- ( ph -> RR C_ CC )
101		iccssre 12255	. . . . . . . . . . 11 |- ( ( M e. RR /\ N e. RR ) -> ( M [,] N ) C_ RR )
102	41, 42, 101	syl2anc 693	. . . . . . . . . 10 |- ( ph -> ( M [,] N ) C_ RR )
103	50, 73	resubcld 10458	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( M [,] N ) ) -> ( C - A ) e. RR )
104	103	recnd 10068	. . . . . . . . . 10 |- ( ( ph /\ x e. ( M [,] N ) ) -> ( C - A ) e. CC )
105	43	tgioo2 22606	. . . . . . . . . 10 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
106		iccntr 22624	. . . . . . . . . . 11 |- ( ( M e. RR /\ N e. RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( M [,] N ) ) = ( M (,) N ) )
107	41, 42, 106	syl2anc 693	. . . . . . . . . 10 |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( M [,] N ) ) = ( M (,) N ) )
108	100, 102, 104, 105, 43, 107	dvmptntr 23734	. . . . . . . . 9 |- ( ph -> ( RR _D ( x e. ( M [,] N ) |-> ( C - A ) ) ) = ( RR _D ( x e. ( M (,) N ) |-> ( C - A ) ) ) )
109		reelprrecn 10028	. . . . . . . . . . 11 |- RR e. { RR , CC }
110	109	a1i 11	. . . . . . . . . 10 |- ( ph -> RR e. { RR , CC } )
111	50	recnd 10068	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( M [,] N ) ) -> C e. CC )
112	49, 111	sylan2 491	. . . . . . . . . 10 |- ( ( ph /\ x e. ( M (,) N ) ) -> C e. CC )
113	73	recnd 10068	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( M [,] N ) ) -> A e. CC )
114	49, 113	sylan2 491	. . . . . . . . . 10 |- ( ( ph /\ x e. ( M (,) N ) ) -> A e. CC )
115	110, 112, 62, 57, 114, 82, 79	dvmptsub 23730	. . . . . . . . 9 |- ( ph -> ( RR _D ( x e. ( M (,) N ) |-> ( C - A ) ) ) = ( x e. ( M (,) N ) |-> ( D - B ) ) )
116	108, 115	eqtrd 2656	. . . . . . . 8 |- ( ph -> ( RR _D ( x e. ( M [,] N ) |-> ( C - A ) ) ) = ( x e. ( M (,) N ) |-> ( D - B ) ) )
117	116	feq1d 6030	. . . . . . 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 247	. . . . . 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 23769	. . . . 5 |- ( ph -> ( ( x e. ( M [,] N ) |-> ( C - A ) ) ` X ) <_ ( ( x e. ( M [,] N ) |-> ( C - A ) ) ` Y ) )
121	23, 28	oveq12d 6668	. . . . . . 7 |- ( x = X -> ( C - A ) = ( Q - P ) )
122		eqid 2622	. . . . . . 7 |- ( x e. ( M [,] N ) |-> ( C - A ) ) = ( x e. ( M [,] N ) |-> ( C - A ) )
123		ovex 6678	. . . . . . 7 |- ( C - A ) e. _V
124	121, 122, 123	fvmpt3i 6287	. . . . . 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 6668	. . . . . . 7 |- ( x = Y -> ( C - A ) = ( S - R ) )
127	126, 122, 123	fvmpt3i 6287	. . . . . 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 4684	. . . 4 |- ( ph -> ( Q - P ) <_ ( S - R ) )
130	26, 31, 40, 129	subled 10630	. . 3 |- ( ph -> ( Q - ( S - R ) ) <_ P )
131	39, 130	eqbrtrd 4675	. 2 |- ( ph -> ( R - ( S - Q ) ) <_ P )
132	11, 27, 31, 131	subled 10630	1 |- ( ph -> ( R - P ) <_ ( S - Q ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   {cpr 4179   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   +oocpnf 10071   <_ cle 10075   - cmin 10266   (,)cioo 12175   [,)cico 12177   [,]cicc 12178   TopOpenctopn 16082   topGenctg 16098  ℂ_fldccnfld 19746   intcnt 20821   -cn->ccncf 22679   _D cdv 23627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631

This theorem is referenced by:  dvfsumle  23784  dvfsumlem2  23790  loglesqrt  24499