Theorem logccv 24409

Description: The natural logarithm function on the reals is a strictly concave function. (Contributed by Mario Carneiro, 20-Jun-2015.)

Assertion

Ref	Expression
logccv	|- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( T x. ( log ` A ) ) + ( ( 1 - T ) x. ( log ` B ) ) ) < ( log ` ( ( T x. A ) + ( ( 1 - T ) x. B ) ) ) )

Proof of Theorem logccv

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1064	. . . . 5 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> A e. RR+ )
2	1	rpred 11872	. . . 4 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> A e. RR )
3		simpl2 1065	. . . . 5 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> B e. RR+ )
4	3	rpred 11872	. . . 4 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> B e. RR )
5		simpl3 1066	. . . 4 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> A < B )
6	1	rpgt0d 11875	. . . . . . . . . . 11 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> 0 < A )
7		ltpnf 11954	. . . . . . . . . . . 12 |- ( B e. RR -> B < +oo )
8	4, 7	syl 17	. . . . . . . . . . 11 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> B < +oo )
9		0xr 10086	. . . . . . . . . . . 12 |- 0 e. RR*
10		pnfxr 10092	. . . . . . . . . . . 12 |- +oo e. RR*
11		iccssioo 12242	. . . . . . . . . . . 12 |- ( ( ( 0 e. RR* /\ +oo e. RR* ) /\ ( 0 < A /\ B < +oo ) ) -> ( A [,] B ) C_ ( 0 (,) +oo ) )
12	9, 10, 11	mpanl12 718	. . . . . . . . . . 11 |- ( ( 0 < A /\ B < +oo ) -> ( A [,] B ) C_ ( 0 (,) +oo ) )
13	6, 8, 12	syl2anc 693	. . . . . . . . . 10 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( A [,] B ) C_ ( 0 (,) +oo ) )
14		ioorp 12251	. . . . . . . . . 10 |- ( 0 (,) +oo ) = RR+
15	13, 14	syl6sseq 3651	. . . . . . . . 9 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( A [,] B ) C_ RR+ )
16	15	sselda 3603	. . . . . . . 8 |- ( ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) /\ x e. ( A [,] B ) ) -> x e. RR+ )
17	16	relogcld 24369	. . . . . . 7 |- ( ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) /\ x e. ( A [,] B ) ) -> ( log ` x ) e. RR )
18	17	renegcld 10457	. . . . . 6 |- ( ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) /\ x e. ( A [,] B ) ) -> -u ( log ` x ) e. RR )
19		eqid 2622	. . . . . 6 |- ( x e. ( A [,] B ) |-> -u ( log ` x ) ) = ( x e. ( A [,] B ) |-> -u ( log ` x ) )
20	18, 19	fmptd 6385	. . . . 5 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( x e. ( A [,] B ) |-> -u ( log ` x ) ) : ( A [,] B ) --> RR )
21		ax-resscn 9993	. . . . . 6 |- RR C_ CC
22	15	resabs1d 5428	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( log |` RR+ ) |` ( A [,] B ) ) = ( log |` ( A [,] B ) ) )
23		ssid 3624	. . . . . . . . . . 11 |- CC C_ CC
24		cncfss 22702	. . . . . . . . . . 11 |- ( ( RR C_ CC /\ CC C_ CC ) -> ( RR+ -cn-> RR ) C_ ( RR+ -cn-> CC ) )
25	21, 23, 24	mp2an 708	. . . . . . . . . 10 |- ( RR+ -cn-> RR ) C_ ( RR+ -cn-> CC )
26		relogcn 24384	. . . . . . . . . 10 |- ( log |` RR+ ) e. ( RR+ -cn-> RR )
27	25, 26	sselii 3600	. . . . . . . . 9 |- ( log |` RR+ ) e. ( RR+ -cn-> CC )
28		rescncf 22700	. . . . . . . . 9 |- ( ( A [,] B ) C_ RR+ -> ( ( log |` RR+ ) e. ( RR+ -cn-> CC ) -> ( ( log |` RR+ ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) ) )
29	15, 27, 28	mpisyl 21	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( log |` RR+ ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) )
30	22, 29	eqeltrrd 2702	. . . . . . 7 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( log |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) )
31		fvres 6207	. . . . . . . . . . 11 |- ( x e. ( A [,] B ) -> ( ( log |` ( A [,] B ) ) ` x ) = ( log ` x ) )
32	31	negeqd 10275	. . . . . . . . . 10 |- ( x e. ( A [,] B ) -> -u ( ( log |` ( A [,] B ) ) ` x ) = -u ( log ` x ) )
33	32	mpteq2ia 4740	. . . . . . . . 9 |- ( x e. ( A [,] B ) |-> -u ( ( log |` ( A [,] B ) ) ` x ) ) = ( x e. ( A [,] B ) |-> -u ( log ` x ) )
34	33	eqcomi 2631	. . . . . . . 8 |- ( x e. ( A [,] B ) |-> -u ( log ` x ) ) = ( x e. ( A [,] B ) |-> -u ( ( log |` ( A [,] B ) ) ` x ) )
35	34	negfcncf 22722	. . . . . . 7 |- ( ( log |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) -> ( x e. ( A [,] B ) |-> -u ( log ` x ) ) e. ( ( A [,] B ) -cn-> CC ) )
36	30, 35	syl 17	. . . . . 6 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( x e. ( A [,] B ) |-> -u ( log ` x ) ) e. ( ( A [,] B ) -cn-> CC ) )
37		cncffvrn 22701	. . . . . 6 |- ( ( RR C_ CC /\ ( x e. ( A [,] B ) |-> -u ( log ` x ) ) e. ( ( A [,] B ) -cn-> CC ) ) -> ( ( x e. ( A [,] B ) |-> -u ( log ` x ) ) e. ( ( A [,] B ) -cn-> RR ) <-> ( x e. ( A [,] B ) |-> -u ( log ` x ) ) : ( A [,] B ) --> RR ) )
38	21, 36, 37	sylancr 695	. . . . 5 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( x e. ( A [,] B ) |-> -u ( log ` x ) ) e. ( ( A [,] B ) -cn-> RR ) <-> ( x e. ( A [,] B ) |-> -u ( log ` x ) ) : ( A [,] B ) --> RR ) )
39	20, 38	mpbird 247	. . . 4 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( x e. ( A [,] B ) |-> -u ( log ` x ) ) e. ( ( A [,] B ) -cn-> RR ) )
40		ioossre 12235	. . . . . . . 8 |- ( A (,) B ) C_ RR
41		ltso 10118	. . . . . . . 8 |- < Or RR
42		soss 5053	. . . . . . . 8 |- ( ( A (,) B ) C_ RR -> ( < Or RR -> < Or ( A (,) B ) ) )
43	40, 41, 42	mp2 9	. . . . . . 7 |- < Or ( A (,) B )
44	43	a1i 11	. . . . . 6 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> < Or ( A (,) B ) )
45		ioossicc 12259	. . . . . . . . . . . . . 14 |- ( A (,) B ) C_ ( A [,] B )
46	45, 15	syl5ss 3614	. . . . . . . . . . . . 13 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( A (,) B ) C_ RR+ )
47	46	sselda 3603	. . . . . . . . . . . 12 |- ( ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) /\ x e. ( A (,) B ) ) -> x e. RR+ )
48	47	rprecred 11883	. . . . . . . . . . 11 |- ( ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) /\ x e. ( A (,) B ) ) -> ( 1 / x ) e. RR )
49	48	renegcld 10457	. . . . . . . . . 10 |- ( ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) /\ x e. ( A (,) B ) ) -> -u ( 1 / x ) e. RR )
50		eqid 2622	. . . . . . . . . 10 |- ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) = ( x e. ( A (,) B ) |-> -u ( 1 / x ) )
51	49, 50	fmptd 6385	. . . . . . . . 9 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) : ( A (,) B ) --> RR )
52		frn 6053	. . . . . . . . 9 |- ( ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) : ( A (,) B ) --> RR -> ran ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) C_ RR )
53	51, 52	syl 17	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ran ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) C_ RR )
54		soss 5053	. . . . . . . 8 |- ( ran ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) C_ RR -> ( < Or RR -> < Or ran ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) ) )
55	53, 41, 54	mpisyl 21	. . . . . . 7 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> < Or ran ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) )
56		sopo 5052	. . . . . . 7 |- ( < Or ran ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) -> < Po ran ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) )
57	55, 56	syl 17	. . . . . 6 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> < Po ran ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) )
58		negex 10279	. . . . . . . . 9 |- -u ( 1 / x ) e. _V
59	58, 50	fnmpti 6022	. . . . . . . 8 |- ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) Fn ( A (,) B )
60		dffn4 6121	. . . . . . . 8 |- ( ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) Fn ( A (,) B ) <-> ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) : ( A (,) B ) -onto-> ran ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) )
61	59, 60	mpbi 220	. . . . . . 7 |- ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) : ( A (,) B ) -onto-> ran ( x e. ( A (,) B ) |-> -u ( 1 / x ) )
62	61	a1i 11	. . . . . 6 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) : ( A (,) B ) -onto-> ran ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) )
63	46	sselda 3603	. . . . . . . . . . . 12 |- ( ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) /\ z e. ( A (,) B ) ) -> z e. RR+ )
64	63	adantrl 752	. . . . . . . . . . 11 |- ( ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) /\ ( y e. ( A (,) B ) /\ z e. ( A (,) B ) ) ) -> z e. RR+ )
65	64	rprecred 11883	. . . . . . . . . 10 |- ( ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) /\ ( y e. ( A (,) B ) /\ z e. ( A (,) B ) ) ) -> ( 1 / z ) e. RR )
66	46	sselda 3603	. . . . . . . . . . . 12 |- ( ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) /\ y e. ( A (,) B ) ) -> y e. RR+ )
67	66	adantrr 753	. . . . . . . . . . 11 |- ( ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) /\ ( y e. ( A (,) B ) /\ z e. ( A (,) B ) ) ) -> y e. RR+ )
68	67	rprecred 11883	. . . . . . . . . 10 |- ( ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) /\ ( y e. ( A (,) B ) /\ z e. ( A (,) B ) ) ) -> ( 1 / y ) e. RR )
69	65, 68	ltnegd 10605	. . . . . . . . 9 |- ( ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) /\ ( y e. ( A (,) B ) /\ z e. ( A (,) B ) ) ) -> ( ( 1 / z ) < ( 1 / y ) <-> -u ( 1 / y ) < -u ( 1 / z ) ) )
70	67, 64	ltrecd 11890	. . . . . . . . 9 |- ( ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) /\ ( y e. ( A (,) B ) /\ z e. ( A (,) B ) ) ) -> ( y < z <-> ( 1 / z ) < ( 1 / y ) ) )
71		oveq2 6658	. . . . . . . . . . . . 13 |- ( x = y -> ( 1 / x ) = ( 1 / y ) )
72	71	negeqd 10275	. . . . . . . . . . . 12 |- ( x = y -> -u ( 1 / x ) = -u ( 1 / y ) )
73		negex 10279	. . . . . . . . . . . 12 |- -u ( 1 / y ) e. _V
74	72, 50, 73	fvmpt 6282	. . . . . . . . . . 11 |- ( y e. ( A (,) B ) -> ( ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) ` y ) = -u ( 1 / y ) )
75		oveq2 6658	. . . . . . . . . . . . 13 |- ( x = z -> ( 1 / x ) = ( 1 / z ) )
76	75	negeqd 10275	. . . . . . . . . . . 12 |- ( x = z -> -u ( 1 / x ) = -u ( 1 / z ) )
77		negex 10279	. . . . . . . . . . . 12 |- -u ( 1 / z ) e. _V
78	76, 50, 77	fvmpt 6282	. . . . . . . . . . 11 |- ( z e. ( A (,) B ) -> ( ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) ` z ) = -u ( 1 / z ) )
79	74, 78	breqan12d 4669	. . . . . . . . . 10 |- ( ( y e. ( A (,) B ) /\ z e. ( A (,) B ) ) -> ( ( ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) ` y ) < ( ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) ` z ) <-> -u ( 1 / y ) < -u ( 1 / z ) ) )
80	79	adantl 482	. . . . . . . . 9 |- ( ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) /\ ( y e. ( A (,) B ) /\ z e. ( A (,) B ) ) ) -> ( ( ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) ` y ) < ( ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) ` z ) <-> -u ( 1 / y ) < -u ( 1 / z ) ) )
81	69, 70, 80	3bitr4d 300	. . . . . . . 8 |- ( ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) /\ ( y e. ( A (,) B ) /\ z e. ( A (,) B ) ) ) -> ( y < z <-> ( ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) ` y ) < ( ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) ` z ) ) )
82	81	biimpd 219	. . . . . . 7 |- ( ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) /\ ( y e. ( A (,) B ) /\ z e. ( A (,) B ) ) ) -> ( y < z -> ( ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) ` y ) < ( ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) ` z ) ) )
83	82	ralrimivva 2971	. . . . . 6 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> A. y e. ( A (,) B ) A. z e. ( A (,) B ) ( y < z -> ( ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) ` y ) < ( ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) ` z ) ) )
84		soisoi 6578	. . . . . 6 |- ( ( ( < Or ( A (,) B ) /\ < Po ran ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) ) /\ ( ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) : ( A (,) B ) -onto-> ran ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) /\ A. y e. ( A (,) B ) A. z e. ( A (,) B ) ( y < z -> ( ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) ` y ) < ( ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) ` z ) ) ) ) -> ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) Isom < , < ( ( A (,) B ) , ran ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) ) )
85	44, 57, 62, 83, 84	syl22anc 1327	. . . . 5 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) Isom < , < ( ( A (,) B ) , ran ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) ) )
86		reelprrecn 10028	. . . . . . . 8 |- RR e. { RR , CC }
87	86	a1i 11	. . . . . . 7 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> RR e. { RR , CC } )
88		relogcl 24322	. . . . . . . . . 10 |- ( x e. RR+ -> ( log ` x ) e. RR )
89	88	adantl 482	. . . . . . . . 9 |- ( ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) /\ x e. RR+ ) -> ( log ` x ) e. RR )
90	89	recnd 10068	. . . . . . . 8 |- ( ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) /\ x e. RR+ ) -> ( log ` x ) e. CC )
91	90	negcld 10379	. . . . . . 7 |- ( ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) /\ x e. RR+ ) -> -u ( log ` x ) e. CC )
92	58	a1i 11	. . . . . . 7 |- ( ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) /\ x e. RR+ ) -> -u ( 1 / x ) e. _V )
93		ovexd 6680	. . . . . . . 8 |- ( ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) /\ x e. RR+ ) -> ( 1 / x ) e. _V )
94		dvrelog 24383	. . . . . . . . 9 |- ( RR _D ( log |` RR+ ) ) = ( x e. RR+ |-> ( 1 / x ) )
95		relogf1o 24313	. . . . . . . . . . . . 13 |- ( log |` RR+ ) : RR+ -1-1-onto-> RR
96		f1of 6137	. . . . . . . . . . . . 13 |- ( ( log |` RR+ ) : RR+ -1-1-onto-> RR -> ( log |` RR+ ) : RR+ --> RR )
97	95, 96	mp1i 13	. . . . . . . . . . . 12 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( log |` RR+ ) : RR+ --> RR )
98	97	feqmptd 6249	. . . . . . . . . . 11 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( log |` RR+ ) = ( x e. RR+ |-> ( ( log |` RR+ ) ` x ) ) )
99		fvres 6207	. . . . . . . . . . . 12 |- ( x e. RR+ -> ( ( log |` RR+ ) ` x ) = ( log ` x ) )
100	99	mpteq2ia 4740	. . . . . . . . . . 11 |- ( x e. RR+ |-> ( ( log |` RR+ ) ` x ) ) = ( x e. RR+ |-> ( log ` x ) )
101	98, 100	syl6eq 2672	. . . . . . . . . 10 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( log |` RR+ ) = ( x e. RR+ |-> ( log ` x ) ) )
102	101	oveq2d 6666	. . . . . . . . 9 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( RR _D ( log |` RR+ ) ) = ( RR _D ( x e. RR+ |-> ( log ` x ) ) ) )
103	94, 102	syl5reqr 2671	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( RR _D ( x e. RR+ |-> ( log ` x ) ) ) = ( x e. RR+ |-> ( 1 / x ) ) )
104	87, 90, 93, 103	dvmptneg 23729	. . . . . . 7 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( RR _D ( x e. RR+ |-> -u ( log ` x ) ) ) = ( x e. RR+ |-> -u ( 1 / x ) ) )
105		eqid 2622	. . . . . . . 8 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
106	105	tgioo2 22606	. . . . . . 7 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
107		iccntr 22624	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
108	2, 4, 107	syl2anc 693	. . . . . . 7 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
109	87, 91, 92, 104, 15, 106, 105, 108	dvmptres2 23725	. . . . . 6 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( RR _D ( x e. ( A [,] B ) |-> -u ( log ` x ) ) ) = ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) )
110		isoeq1 6567	. . . . . 6 |- ( ( RR _D ( x e. ( A [,] B ) |-> -u ( log ` x ) ) ) = ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) -> ( ( RR _D ( x e. ( A [,] B ) |-> -u ( log ` x ) ) ) Isom < , < ( ( A (,) B ) , ran ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) ) <-> ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) Isom < , < ( ( A (,) B ) , ran ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) ) ) )
111	109, 110	syl 17	. . . . 5 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( RR _D ( x e. ( A [,] B ) |-> -u ( log ` x ) ) ) Isom < , < ( ( A (,) B ) , ran ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) ) <-> ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) Isom < , < ( ( A (,) B ) , ran ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) ) ) )
112	85, 111	mpbird 247	. . . 4 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( RR _D ( x e. ( A [,] B ) |-> -u ( log ` x ) ) ) Isom < , < ( ( A (,) B ) , ran ( x e. ( A (,) B ) |-> -u ( 1 / x ) ) ) )
113		simpr 477	. . . 4 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> T e. ( 0 (,) 1 ) )
114		eqid 2622	. . . 4 |- ( ( T x. A ) + ( ( 1 - T ) x. B ) ) = ( ( T x. A ) + ( ( 1 - T ) x. B ) )
115	2, 4, 5, 39, 112, 113, 114	dvcvx 23783	. . 3 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( x e. ( A [,] B ) |-> -u ( log ` x ) ) ` ( ( T x. A ) + ( ( 1 - T ) x. B ) ) ) < ( ( T x. ( ( x e. ( A [,] B ) |-> -u ( log ` x ) ) ` A ) ) + ( ( 1 - T ) x. ( ( x e. ( A [,] B ) |-> -u ( log ` x ) ) ` B ) ) ) )
116		ax-1cn 9994	. . . . . . . 8 |- 1 e. CC
117		elioore 12205	. . . . . . . . . 10 |- ( T e. ( 0 (,) 1 ) -> T e. RR )
118	117	adantl 482	. . . . . . . . 9 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> T e. RR )
119	118	recnd 10068	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> T e. CC )
120		nncan 10310	. . . . . . . 8 |- ( ( 1 e. CC /\ T e. CC ) -> ( 1 - ( 1 - T ) ) = T )
121	116, 119, 120	sylancr 695	. . . . . . 7 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( 1 - ( 1 - T ) ) = T )
122	121	oveq1d 6665	. . . . . 6 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( 1 - ( 1 - T ) ) x. A ) = ( T x. A ) )
123	122	oveq1d 6665	. . . . 5 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( ( 1 - ( 1 - T ) ) x. A ) + ( ( 1 - T ) x. B ) ) = ( ( T x. A ) + ( ( 1 - T ) x. B ) ) )
124		ioossicc 12259	. . . . . . . 8 |- ( 0 (,) 1 ) C_ ( 0 [,] 1 )
125	124, 113	sseldi 3601	. . . . . . 7 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> T e. ( 0 [,] 1 ) )
126		iirev 22728	. . . . . . 7 |- ( T e. ( 0 [,] 1 ) -> ( 1 - T ) e. ( 0 [,] 1 ) )
127	125, 126	syl 17	. . . . . 6 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( 1 - T ) e. ( 0 [,] 1 ) )
128		lincmb01cmp 12315	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( 1 - T ) e. ( 0 [,] 1 ) ) -> ( ( ( 1 - ( 1 - T ) ) x. A ) + ( ( 1 - T ) x. B ) ) e. ( A [,] B ) )
129	2, 4, 5, 127, 128	syl31anc 1329	. . . . 5 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( ( 1 - ( 1 - T ) ) x. A ) + ( ( 1 - T ) x. B ) ) e. ( A [,] B ) )
130	123, 129	eqeltrrd 2702	. . . 4 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( T x. A ) + ( ( 1 - T ) x. B ) ) e. ( A [,] B ) )
131		fveq2 6191	. . . . . 6 |- ( x = ( ( T x. A ) + ( ( 1 - T ) x. B ) ) -> ( log ` x ) = ( log ` ( ( T x. A ) + ( ( 1 - T ) x. B ) ) ) )
132	131	negeqd 10275	. . . . 5 |- ( x = ( ( T x. A ) + ( ( 1 - T ) x. B ) ) -> -u ( log ` x ) = -u ( log ` ( ( T x. A ) + ( ( 1 - T ) x. B ) ) ) )
133		negex 10279	. . . . 5 |- -u ( log ` ( ( T x. A ) + ( ( 1 - T ) x. B ) ) ) e. _V
134	132, 19, 133	fvmpt 6282	. . . 4 |- ( ( ( T x. A ) + ( ( 1 - T ) x. B ) ) e. ( A [,] B ) -> ( ( x e. ( A [,] B ) |-> -u ( log ` x ) ) ` ( ( T x. A ) + ( ( 1 - T ) x. B ) ) ) = -u ( log ` ( ( T x. A ) + ( ( 1 - T ) x. B ) ) ) )
135	130, 134	syl 17	. . 3 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( x e. ( A [,] B ) |-> -u ( log ` x ) ) ` ( ( T x. A ) + ( ( 1 - T ) x. B ) ) ) = -u ( log ` ( ( T x. A ) + ( ( 1 - T ) x. B ) ) ) )
136	1	rpxrd 11873	. . . . . . . . 9 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> A e. RR* )
137	3	rpxrd 11873	. . . . . . . . 9 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> B e. RR* )
138	2, 4, 5	ltled 10185	. . . . . . . . 9 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> A <_ B )
139		lbicc2 12288	. . . . . . . . 9 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
140	136, 137, 138, 139	syl3anc 1326	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> A e. ( A [,] B ) )
141		fveq2 6191	. . . . . . . . . 10 |- ( x = A -> ( log ` x ) = ( log ` A ) )
142	141	negeqd 10275	. . . . . . . . 9 |- ( x = A -> -u ( log ` x ) = -u ( log ` A ) )
143		negex 10279	. . . . . . . . 9 |- -u ( log ` A ) e. _V
144	142, 19, 143	fvmpt 6282	. . . . . . . 8 |- ( A e. ( A [,] B ) -> ( ( x e. ( A [,] B ) |-> -u ( log ` x ) ) ` A ) = -u ( log ` A ) )
145	140, 144	syl 17	. . . . . . 7 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( x e. ( A [,] B ) |-> -u ( log ` x ) ) ` A ) = -u ( log ` A ) )
146	145	oveq2d 6666	. . . . . 6 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( T x. ( ( x e. ( A [,] B ) |-> -u ( log ` x ) ) ` A ) ) = ( T x. -u ( log ` A ) ) )
147	1	relogcld 24369	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( log ` A ) e. RR )
148	147	recnd 10068	. . . . . . 7 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( log ` A ) e. CC )
149	119, 148	mulneg2d 10484	. . . . . 6 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( T x. -u ( log ` A ) ) = -u ( T x. ( log ` A ) ) )
150	146, 149	eqtrd 2656	. . . . 5 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( T x. ( ( x e. ( A [,] B ) |-> -u ( log ` x ) ) ` A ) ) = -u ( T x. ( log ` A ) ) )
151		ubicc2 12289	. . . . . . . . 9 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
152	136, 137, 138, 151	syl3anc 1326	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> B e. ( A [,] B ) )
153		fveq2 6191	. . . . . . . . . 10 |- ( x = B -> ( log ` x ) = ( log ` B ) )
154	153	negeqd 10275	. . . . . . . . 9 |- ( x = B -> -u ( log ` x ) = -u ( log ` B ) )
155		negex 10279	. . . . . . . . 9 |- -u ( log ` B ) e. _V
156	154, 19, 155	fvmpt 6282	. . . . . . . 8 |- ( B e. ( A [,] B ) -> ( ( x e. ( A [,] B ) |-> -u ( log ` x ) ) ` B ) = -u ( log ` B ) )
157	152, 156	syl 17	. . . . . . 7 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( x e. ( A [,] B ) |-> -u ( log ` x ) ) ` B ) = -u ( log ` B ) )
158	157	oveq2d 6666	. . . . . 6 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( 1 - T ) x. ( ( x e. ( A [,] B ) |-> -u ( log ` x ) ) ` B ) ) = ( ( 1 - T ) x. -u ( log ` B ) ) )
159		1re 10039	. . . . . . . . 9 |- 1 e. RR
160		resubcl 10345	. . . . . . . . 9 |- ( ( 1 e. RR /\ T e. RR ) -> ( 1 - T ) e. RR )
161	159, 118, 160	sylancr 695	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( 1 - T ) e. RR )
162	161	recnd 10068	. . . . . . 7 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( 1 - T ) e. CC )
163	3	relogcld 24369	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( log ` B ) e. RR )
164	163	recnd 10068	. . . . . . 7 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( log ` B ) e. CC )
165	162, 164	mulneg2d 10484	. . . . . 6 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( 1 - T ) x. -u ( log ` B ) ) = -u ( ( 1 - T ) x. ( log ` B ) ) )
166	158, 165	eqtrd 2656	. . . . 5 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( 1 - T ) x. ( ( x e. ( A [,] B ) |-> -u ( log ` x ) ) ` B ) ) = -u ( ( 1 - T ) x. ( log ` B ) ) )
167	150, 166	oveq12d 6668	. . . 4 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( T x. ( ( x e. ( A [,] B ) |-> -u ( log ` x ) ) ` A ) ) + ( ( 1 - T ) x. ( ( x e. ( A [,] B ) |-> -u ( log ` x ) ) ` B ) ) ) = ( -u ( T x. ( log ` A ) ) + -u ( ( 1 - T ) x. ( log ` B ) ) ) )
168	118, 147	remulcld 10070	. . . . . 6 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( T x. ( log ` A ) ) e. RR )
169	168	recnd 10068	. . . . 5 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( T x. ( log ` A ) ) e. CC )
170	161, 163	remulcld 10070	. . . . . 6 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( 1 - T ) x. ( log ` B ) ) e. RR )
171	170	recnd 10068	. . . . 5 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( 1 - T ) x. ( log ` B ) ) e. CC )
172	169, 171	negdid 10405	. . . 4 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> -u ( ( T x. ( log ` A ) ) + ( ( 1 - T ) x. ( log ` B ) ) ) = ( -u ( T x. ( log ` A ) ) + -u ( ( 1 - T ) x. ( log ` B ) ) ) )
173	167, 172	eqtr4d 2659	. . 3 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( T x. ( ( x e. ( A [,] B ) |-> -u ( log ` x ) ) ` A ) ) + ( ( 1 - T ) x. ( ( x e. ( A [,] B ) |-> -u ( log ` x ) ) ` B ) ) ) = -u ( ( T x. ( log ` A ) ) + ( ( 1 - T ) x. ( log ` B ) ) ) )
174	115, 135, 173	3brtr3d 4684	. 2 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> -u ( log ` ( ( T x. A ) + ( ( 1 - T ) x. B ) ) ) < -u ( ( T x. ( log ` A ) ) + ( ( 1 - T ) x. ( log ` B ) ) ) )
175	168, 170	readdcld 10069	. . 3 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( T x. ( log ` A ) ) + ( ( 1 - T ) x. ( log ` B ) ) ) e. RR )
176	15, 130	sseldd 3604	. . . 4 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( T x. A ) + ( ( 1 - T ) x. B ) ) e. RR+ )
177	176	relogcld 24369	. . 3 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( log ` ( ( T x. A ) + ( ( 1 - T ) x. B ) ) ) e. RR )
178	175, 177	ltnegd 10605	. 2 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( ( T x. ( log ` A ) ) + ( ( 1 - T ) x. ( log ` B ) ) ) < ( log ` ( ( T x. A ) + ( ( 1 - T ) x. B ) ) ) <-> -u ( log ` ( ( T x. A ) + ( ( 1 - T ) x. B ) ) ) < -u ( ( T x. ( log ` A ) ) + ( ( 1 - T ) x. ( log ` B ) ) ) ) )
179	174, 178	mpbird 247	1 |- ( ( ( A e. RR+ /\ B e. RR+ /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( T x. ( log ` A ) ) + ( ( 1 - T ) x. ( log ` B ) ) ) < ( log ` ( ( T x. A ) + ( ( 1 - T ) x. B ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   {cpr 4179   class class class wbr 4653   |-> cmpt 4729   Po wpo 5033   Or wor 5034   ran crn 5115   |` cres 5116   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888   Isom wiso 5889  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   -ucneg 10267   / cdiv 10684   RR+crp 11832   (,)cioo 12175   [,]cicc 12178   TopOpenctopn 16082   topGenctg 16098  ℂ_fldccnfld 19746   intcnt 20821   -cn->ccncf 22679   _D cdv 23627   logclog 24301

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-fal 1489  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-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  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  df-log 24303

This theorem is referenced by:  amgmlem  24716  amgmwlem  42548