Theorem dvle 22577

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 21566	. . . . 5 |- ( ( x e. ( M [,] N ) |-> A ) e. ( ( M [,] N ) -cn-> RR ) -> ( x e. ( M [,] N ) |-> A ) : ( M [,] N ) --> RR )
4	2, 3	syl 16	. . . 4 |- ( ph -> ( x e. ( M [,] N ) |-> A ) : ( M [,] N ) --> RR )
5		eqid 2454	. . . . 5 |- ( x e. ( M [,] N ) |-> A ) = ( x e. ( M [,] N ) |-> A )
6	5	fmpt 6028	. . . 4 |- ( A. x e. ( M [,] N ) A e. RR <-> ( x e. ( M [,] N ) |-> A ) : ( M [,] N ) --> RR )
7	4, 6	sylibr 212	. . 3 |- ( ph -> A. x e. ( M [,] N ) A e. RR )
8		dvle.r	. . . . 5 |- ( x = Y -> A = R )
9	8	eleq1d 2523	. . . 4 |- ( x = Y -> ( A e. RR <-> R e. RR ) )
10	9	rspcv 3203	. . 3 |- ( Y e. ( M [,] N ) -> ( A. x e. ( M [,] N ) A e. RR -> R e. RR ) )
11	1, 7, 10	sylc 60	. 2 |- ( ph -> R e. RR )
12		dvle.c	. . . . . 6 |- ( ph -> ( x e. ( M [,] N ) |-> C ) e. ( ( M [,] N ) -cn-> RR ) )
13		cncff 21566	. . . . . 6 |- ( ( x e. ( M [,] N ) |-> C ) e. ( ( M [,] N ) -cn-> RR ) -> ( x e. ( M [,] N ) |-> C ) : ( M [,] N ) --> RR )
14	12, 13	syl 16	. . . . 5 |- ( ph -> ( x e. ( M [,] N ) |-> C ) : ( M [,] N ) --> RR )
15		eqid 2454	. . . . . 6 |- ( x e. ( M [,] N ) |-> C ) = ( x e. ( M [,] N ) |-> C )
16	15	fmpt 6028	. . . . 5 |- ( A. x e. ( M [,] N ) C e. RR <-> ( x e. ( M [,] N ) |-> C ) : ( M [,] N ) --> RR )
17	14, 16	sylibr 212	. . . 4 |- ( ph -> A. x e. ( M [,] N ) C e. RR )
18		dvle.s	. . . . . 6 |- ( x = Y -> C = S )
19	18	eleq1d 2523	. . . . 5 |- ( x = Y -> ( C e. RR <-> S e. RR ) )
20	19	rspcv 3203	. . . 4 |- ( Y e. ( M [,] N ) -> ( A. x e. ( M [,] N ) C e. RR -> S e. RR ) )
21	1, 17, 20	sylc 60	. . 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 2523	. . . . 5 |- ( x = X -> ( C e. RR <-> Q e. RR ) )
25	24	rspcv 3203	. . . 4 |- ( X e. ( M [,] N ) -> ( A. x e. ( M [,] N ) C e. RR -> Q e. RR ) )
26	22, 17, 25	sylc 60	. . 3 |- ( ph -> Q e. RR )
27	21, 26	resubcld 9983	. 2 |- ( ph -> ( S - Q ) e. RR )
28		dvle.p	. . . . 5 |- ( x = X -> A = P )
29	28	eleq1d 2523	. . . 4 |- ( x = X -> ( A e. RR <-> P e. RR ) )
30	29	rspcv 3203	. . 3 |- ( X e. ( M [,] N ) -> ( A. x e. ( M [,] N ) A e. RR -> P e. RR ) )
31	22, 7, 30	sylc 60	. 2 |- ( ph -> P e. RR )
32	11	recnd 9611	. . . . 5 |- ( ph -> R e. CC )
33	26	recnd 9611	. . . . . 6 |- ( ph -> Q e. CC )
34	21	recnd 9611	. . . . . 6 |- ( ph -> S e. CC )
35	33, 34	subcld 9922	. . . . 5 |- ( ph -> ( Q - S ) e. CC )
36	32, 35	addcomd 9771	. . . 4 |- ( ph -> ( R + ( Q - S ) ) = ( ( Q - S ) + R ) )
37	32, 34, 33	subsub2d 9951	. . . 4 |- ( ph -> ( R - ( S - Q ) ) = ( R + ( Q - S ) ) )
38	33, 34, 32	subsubd 9950	. . . 4 |- ( ph -> ( Q - ( S - R ) ) = ( ( Q - S ) + R ) )
39	36, 37, 38	3eqtr4d 2505	. . 3 |- ( ph -> ( R - ( S - Q ) ) = ( Q - ( S - R ) ) )
40	21, 11	resubcld 9983	. . . 4 |- ( ph -> ( S - R ) e. RR )
41		dvle.m	. . . . . 6 |- ( ph -> M e. RR )
42		dvle.n	. . . . . 6 |- ( ph -> N e. RR )
43		eqid 2454	. . . . . . 7 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
44	43	subcn 21539	. . . . . . 7 |- - e. ( ( ( TopOpen ` ℂfld ) tX ( TopOpen ` ℂfld ) ) Cn ( TopOpen ` ℂfld ) )
45		ax-resscn 9538	. . . . . . 7 |- RR C_ CC
46		resubcl 9874	. . . . . . 7 |- ( ( C e. RR /\ A e. RR ) -> ( C - A ) e. RR )
47	43, 44, 12, 2, 45, 46	cncfmpt2ss 21588	. . . . . 6 |- ( ph -> ( x e. ( M [,] N ) |-> ( C - A ) ) e. ( ( M [,] N ) -cn-> RR ) )
48		ioossicc 11613	. . . . . . . . . . . . . . . . 17 |- ( M (,) N ) C_ ( M [,] N )
49	48	sseli 3485	. . . . . . . . . . . . . . . 16 |- ( x e. ( M (,) N ) -> x e. ( M [,] N ) )
50	17	r19.21bi 2823	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( M [,] N ) ) -> C e. RR )
51	49, 50	sylan2 472	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( M (,) N ) ) -> C e. RR )
52		eqid 2454	. . . . . . . . . . . . . . 15 |- ( x e. ( M (,) N ) |-> C ) = ( x e. ( M (,) N ) |-> C )
53	51, 52	fmptd 6031	. . . . . . . . . . . . . 14 |- ( ph -> ( x e. ( M (,) N ) |-> C ) : ( M (,) N ) --> RR )
54		ioossre 11589	. . . . . . . . . . . . . 14 |- ( M (,) N ) C_ RR
55		dvfre 22523	. . . . . . . . . . . . . 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 660	. . . . . . . . . . . . 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 5194	. . . . . . . . . . . . . . 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 9640	. . . . . . . . . . . . . . . . . . 19 |- Rel <_
61	60	brrelex2i 5030	. . . . . . . . . . . . . . . . . 18 |- ( B <_ D -> D e. _V )
62	59, 61	syl 16	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. ( M (,) N ) ) -> D e. _V )
63	62	ralrimiva 2868	. . . . . . . . . . . . . . . 16 |- ( ph -> A. x e. ( M (,) N ) D e. _V )
64		dmmptg 5487	. . . . . . . . . . . . . . . 16 |- ( A. x e. ( M (,) N ) D e. _V -> dom ( x e. ( M (,) N ) |-> D ) = ( M (,) N ) )
65	63, 64	syl 16	. . . . . . . . . . . . . . 15 |- ( ph -> dom ( x e. ( M (,) N ) |-> D ) = ( M (,) N ) )
66	58, 65	eqtrd 2495	. . . . . . . . . . . . . 14 |- ( ph -> dom ( RR _D ( x e. ( M (,) N ) |-> C ) ) = ( M (,) N ) )
67	57, 66	feq12d 5702	. . . . . . . . . . . . 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 210	. . . . . . . . . . . 12 |- ( ph -> ( x e. ( M (,) N ) |-> D ) : ( M (,) N ) --> RR )
69		eqid 2454	. . . . . . . . . . . . 13 |- ( x e. ( M (,) N ) |-> D ) = ( x e. ( M (,) N ) |-> D )
70	69	fmpt 6028	. . . . . . . . . . . 12 |- ( A. x e. ( M (,) N ) D e. RR <-> ( x e. ( M (,) N ) |-> D ) : ( M (,) N ) --> RR )
71	68, 70	sylibr 212	. . . . . . . . . . 11 |- ( ph -> A. x e. ( M (,) N ) D e. RR )
72	71	r19.21bi 2823	. . . . . . . . . 10 |- ( ( ph /\ x e. ( M (,) N ) ) -> D e. RR )
73	7	r19.21bi 2823	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( M [,] N ) ) -> A e. RR )
74	49, 73	sylan2 472	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( M (,) N ) ) -> A e. RR )
75		eqid 2454	. . . . . . . . . . . . . . 15 |- ( x e. ( M (,) N ) |-> A ) = ( x e. ( M (,) N ) |-> A )
76	74, 75	fmptd 6031	. . . . . . . . . . . . . 14 |- ( ph -> ( x e. ( M (,) N ) |-> A ) : ( M (,) N ) --> RR )
77		dvfre 22523	. . . . . . . . . . . . . 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 660	. . . . . . . . . . . . 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 5194	. . . . . . . . . . . . . . 15 |- ( ph -> dom ( RR _D ( x e. ( M (,) N ) |-> A ) ) = dom ( x e. ( M (,) N ) |-> B ) )
81	60	brrelexi 5029	. . . . . . . . . . . . . . . . . 18 |- ( B <_ D -> B e. _V )
82	59, 81	syl 16	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. ( M (,) N ) ) -> B e. _V )
83	82	ralrimiva 2868	. . . . . . . . . . . . . . . 16 |- ( ph -> A. x e. ( M (,) N ) B e. _V )
84		dmmptg 5487	. . . . . . . . . . . . . . . 16 |- ( A. x e. ( M (,) N ) B e. _V -> dom ( x e. ( M (,) N ) |-> B ) = ( M (,) N ) )
85	83, 84	syl 16	. . . . . . . . . . . . . . 15 |- ( ph -> dom ( x e. ( M (,) N ) |-> B ) = ( M (,) N ) )
86	80, 85	eqtrd 2495	. . . . . . . . . . . . . 14 |- ( ph -> dom ( RR _D ( x e. ( M (,) N ) |-> A ) ) = ( M (,) N ) )
87	79, 86	feq12d 5702	. . . . . . . . . . . . 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 210	. . . . . . . . . . . 12 |- ( ph -> ( x e. ( M (,) N ) |-> B ) : ( M (,) N ) --> RR )
89		eqid 2454	. . . . . . . . . . . . 13 |- ( x e. ( M (,) N ) |-> B ) = ( x e. ( M (,) N ) |-> B )
90	89	fmpt 6028	. . . . . . . . . . . 12 |- ( A. x e. ( M (,) N ) B e. RR <-> ( x e. ( M (,) N ) |-> B ) : ( M (,) N ) --> RR )
91	88, 90	sylibr 212	. . . . . . . . . . 11 |- ( ph -> A. x e. ( M (,) N ) B e. RR )
92	91	r19.21bi 2823	. . . . . . . . . 10 |- ( ( ph /\ x e. ( M (,) N ) ) -> B e. RR )
93	72, 92	resubcld 9983	. . . . . . . . 9 |- ( ( ph /\ x e. ( M (,) N ) ) -> ( D - B ) e. RR )
94	72, 92	subge0d 10138	. . . . . . . . . 10 |- ( ( ph /\ x e. ( M (,) N ) ) -> ( 0 <_ ( D - B ) <-> B <_ D ) )
95	59, 94	mpbird 232	. . . . . . . . 9 |- ( ( ph /\ x e. ( M (,) N ) ) -> 0 <_ ( D - B ) )
96		elrege0 11630	. . . . . . . . 9 |- ( ( D - B ) e. ( 0 [,) +oo ) <-> ( ( D - B ) e. RR /\ 0 <_ ( D - B ) ) )
97	93, 95, 96	sylanbrc 662	. . . . . . . 8 |- ( ( ph /\ x e. ( M (,) N ) ) -> ( D - B ) e. ( 0 [,) +oo ) )
98		eqid 2454	. . . . . . . 8 |- ( x e. ( M (,) N ) |-> ( D - B ) ) = ( x e. ( M (,) N ) |-> ( D - B ) )
99	97, 98	fmptd 6031	. . . . . . 7 |- ( ph -> ( x e. ( M (,) N ) |-> ( D - B ) ) : ( M (,) N ) --> ( 0 [,) +oo ) )
100	45	a1i 11	. . . . . . . . . 10 |- ( ph -> RR C_ CC )
101		iccssre 11609	. . . . . . . . . . 11 |- ( ( M e. RR /\ N e. RR ) -> ( M [,] N ) C_ RR )
102	41, 42, 101	syl2anc 659	. . . . . . . . . 10 |- ( ph -> ( M [,] N ) C_ RR )
103	50, 73	resubcld 9983	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( M [,] N ) ) -> ( C - A ) e. RR )
104	103	recnd 9611	. . . . . . . . . 10 |- ( ( ph /\ x e. ( M [,] N ) ) -> ( C - A ) e. CC )
105	43	tgioo2 21477	. . . . . . . . . 10 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
106		iccntr 21495	. . . . . . . . . . 11 |- ( ( M e. RR /\ N e. RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( M [,] N ) ) = ( M (,) N ) )
107	41, 42, 106	syl2anc 659	. . . . . . . . . 10 |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( M [,] N ) ) = ( M (,) N ) )
108	100, 102, 104, 105, 43, 107	dvmptntr 22543	. . . . . . . . 9 |- ( ph -> ( RR _D ( x e. ( M [,] N ) |-> ( C - A ) ) ) = ( RR _D ( x e. ( M (,) N ) |-> ( C - A ) ) ) )
109		reelprrecn 9573	. . . . . . . . . . 11 |- RR e. { RR , CC }
110	109	a1i 11	. . . . . . . . . 10 |- ( ph -> RR e. { RR , CC } )
111	50	recnd 9611	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( M [,] N ) ) -> C e. CC )
112	49, 111	sylan2 472	. . . . . . . . . 10 |- ( ( ph /\ x e. ( M (,) N ) ) -> C e. CC )
113	73	recnd 9611	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( M [,] N ) ) -> A e. CC )
114	49, 113	sylan2 472	. . . . . . . . . 10 |- ( ( ph /\ x e. ( M (,) N ) ) -> A e. CC )
115	110, 112, 62, 57, 114, 82, 79	dvmptsub 22539	. . . . . . . . 9 |- ( ph -> ( RR _D ( x e. ( M (,) N ) |-> ( C - A ) ) ) = ( x e. ( M (,) N ) |-> ( D - B ) ) )
116	108, 115	eqtrd 2495	. . . . . . . 8 |- ( ph -> ( RR _D ( x e. ( M [,] N ) |-> ( C - A ) ) ) = ( x e. ( M (,) N ) |-> ( D - B ) ) )
117	116	feq1d 5699	. . . . . . 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 232	. . . . . 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 22576	. . . . 5 |- ( ph -> ( ( x e. ( M [,] N ) |-> ( C - A ) ) ` X ) <_ ( ( x e. ( M [,] N ) |-> ( C - A ) ) ` Y ) )
121	23, 28	oveq12d 6288	. . . . . . 7 |- ( x = X -> ( C - A ) = ( Q - P ) )
122		eqid 2454	. . . . . . 7 |- ( x e. ( M [,] N ) |-> ( C - A ) ) = ( x e. ( M [,] N ) |-> ( C - A ) )
123		ovex 6298	. . . . . . 7 |- ( C - A ) e. _V
124	121, 122, 123	fvmpt3i 5935	. . . . . 6 |- ( X e. ( M [,] N ) -> ( ( x e. ( M [,] N ) |-> ( C - A ) ) ` X ) = ( Q - P ) )
125	22, 124	syl 16	. . . . 5 |- ( ph -> ( ( x e. ( M [,] N ) |-> ( C - A ) ) ` X ) = ( Q - P ) )
126	18, 8	oveq12d 6288	. . . . . . 7 |- ( x = Y -> ( C - A ) = ( S - R ) )
127	126, 122, 123	fvmpt3i 5935	. . . . . 6 |- ( Y e. ( M [,] N ) -> ( ( x e. ( M [,] N ) |-> ( C - A ) ) ` Y ) = ( S - R ) )
128	1, 127	syl 16	. . . . 5 |- ( ph -> ( ( x e. ( M [,] N ) |-> ( C - A ) ) ` Y ) = ( S - R ) )
129	120, 125, 128	3brtr3d 4468	. . . 4 |- ( ph -> ( Q - P ) <_ ( S - R ) )
130	26, 31, 40, 129	subled 10151	. . 3 |- ( ph -> ( Q - ( S - R ) ) <_ P )
131	39, 130	eqbrtrd 4459	. 2 |- ( ph -> ( R - ( S - Q ) ) <_ P )
132	11, 27, 31, 131	subled 10151	1 |- ( ph -> ( R - P ) <_ ( S - Q ) )

Syntax hints:   -> wi 4   /\ wa 367   = wceq 1398   e. wcel 1823   A.wral 2804   _Vcvv 3106   C_ wss 3461   {cpr 4018   class class class wbr 4439   |-> cmpt 4497   dom cdm 4988   ran crn 4989   -->wf 5566   ` cfv 5570  (class class class)co 6270   CCcc 9479   RRcr 9480   0cc0 9481   + caddc 9484   +oocpnf 9614   <_ cle 9618   - cmin 9796   (,)cioo 11532   [,)cico 11534   [,]cicc 11535   TopOpenctopn 14914   topGenctg 14930  ℂ_fldccnfld 18618   intcnt 19688   -cn->ccncf 21549   _D cdv 22436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  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 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-seq 12093  df-exp 12152  df-hash 12391  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-mulr 14801  df-starv 14802  df-sca 14803  df-vsca 14804  df-ip 14805  df-tset 14806  df-ple 14807  df-ds 14809  df-unif 14810  df-hom 14811  df-cco 14812  df-rest 14915  df-topn 14916  df-0g 14934  df-gsum 14935  df-topgen 14936  df-pt 14937  df-prds 14940  df-xrs 14994  df-qtop 14999  df-imas 15000  df-xps 15002  df-mre 15078  df-mrc 15079  df-acs 15081  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-submnd 16169  df-mulg 16262  df-cntz 16557  df-cmn 17002  df-psmet 18609  df-xmet 18610  df-met 18611  df-bl 18612  df-mopn 18613  df-fbas 18614  df-fg 18615  df-cnfld 18619  df-top 19569  df-bases 19571  df-topon 19572  df-topsp 19573  df-cld 19690  df-ntr 19691  df-cls 19692  df-nei 19769  df-lp 19807  df-perf 19808  df-cn 19898  df-cnp 19899  df-haus 19986  df-cmp 20057  df-tx 20232  df-hmeo 20425  df-fil 20516  df-fm 20608  df-flim 20609  df-flf 20610  df-xms 20992  df-ms 20993  df-tms 20994  df-cncf 21551  df-limc 22439  df-dv 22440

This theorem is referenced by:  dvfsumle  22591  dvfsumlem2  22597  loglesqrt  23303