Theorem efcvx 24203

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

Assertion

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

Proof of Theorem efcvx

Step	Hyp	Ref	Expression
1		simpl1 1064	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> A e. RR )
2		simpl2 1065	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> B e. RR )
3		simpl3 1066	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> A < B )
4		reeff1o 24201	. . . . . . 7 |- ( exp |` RR ) : RR -1-1-onto-> RR+
5		f1of 6137	. . . . . . 7 |- ( ( exp |` RR ) : RR -1-1-onto-> RR+ -> ( exp |` RR ) : RR --> RR+ )
6	4, 5	ax-mp 5	. . . . . 6 |- ( exp |` RR ) : RR --> RR+
7		rpssre 11843	. . . . . 6 |- RR+ C_ RR
8		fss 6056	. . . . . 6 |- ( ( ( exp |` RR ) : RR --> RR+ /\ RR+ C_ RR ) -> ( exp |` RR ) : RR --> RR )
9	6, 7, 8	mp2an 708	. . . . 5 |- ( exp |` RR ) : RR --> RR
10		iccssre 12255	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
11	1, 2, 10	syl2anc 693	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( A [,] B ) C_ RR )
12		fssres2 6072	. . . . 5 |- ( ( ( exp |` RR ) : RR --> RR /\ ( A [,] B ) C_ RR ) -> ( exp |` ( A [,] B ) ) : ( A [,] B ) --> RR )
13	9, 11, 12	sylancr 695	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( exp |` ( A [,] B ) ) : ( A [,] B ) --> RR )
14		ax-resscn 9993	. . . . 5 |- RR C_ CC
15	11, 14	syl6ss 3615	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( A [,] B ) C_ CC )
16		efcn 24197	. . . . . 6 |- exp e. ( CC -cn-> CC )
17		rescncf 22700	. . . . . 6 |- ( ( A [,] B ) C_ CC -> ( exp e. ( CC -cn-> CC ) -> ( exp |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) ) )
18	15, 16, 17	mpisyl 21	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( exp |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) )
19		cncffvrn 22701	. . . . 5 |- ( ( RR C_ CC /\ ( exp |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) ) -> ( ( exp |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) <-> ( exp |` ( A [,] B ) ) : ( A [,] B ) --> RR ) )
20	14, 18, 19	sylancr 695	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( exp |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) <-> ( exp |` ( A [,] B ) ) : ( A [,] B ) --> RR ) )
21	13, 20	mpbird 247	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( exp |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> RR ) )
22		reefiso 24202	. . . . . 6 |- ( exp |` RR ) Isom < , < ( RR , RR+ )
23	22	a1i 11	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( exp |` RR ) Isom < , < ( RR , RR+ ) )
24		ioossre 12235	. . . . . 6 |- ( A (,) B ) C_ RR
25	24	a1i 11	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( A (,) B ) C_ RR )
26		eqidd 2623	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( exp |` RR ) " ( A (,) B ) ) = ( ( exp |` RR ) " ( A (,) B ) ) )
27		isores3 6585	. . . . 5 |- ( ( ( exp |` RR ) Isom < , < ( RR , RR+ ) /\ ( A (,) B ) C_ RR /\ ( ( exp |` RR ) " ( A (,) B ) ) = ( ( exp |` RR ) " ( A (,) B ) ) ) -> ( ( exp |` RR ) |` ( A (,) B ) ) Isom < , < ( ( A (,) B ) , ( ( exp |` RR ) " ( A (,) B ) ) ) )
28	23, 25, 26, 27	syl3anc 1326	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( exp |` RR ) |` ( A (,) B ) ) Isom < , < ( ( A (,) B ) , ( ( exp |` RR ) " ( A (,) B ) ) ) )
29		ssid 3624	. . . . . . 7 |- RR C_ RR
30		fss 6056	. . . . . . . . 9 |- ( ( ( exp |` RR ) : RR --> RR /\ RR C_ CC ) -> ( exp |` RR ) : RR --> CC )
31	9, 14, 30	mp2an 708	. . . . . . . 8 |- ( exp |` RR ) : RR --> CC
32		eqid 2622	. . . . . . . . 9 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
33	32	tgioo2 22606	. . . . . . . . 9 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
34	32, 33	dvres 23675	. . . . . . . 8 |- ( ( ( RR C_ CC /\ ( exp |` RR ) : RR --> CC ) /\ ( RR C_ RR /\ ( A [,] B ) C_ RR ) ) -> ( RR _D ( ( exp |` RR ) |` ( A [,] B ) ) ) = ( ( RR _D ( exp |` RR ) ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) ) )
35	14, 31, 34	mpanl12 718	. . . . . . 7 |- ( ( RR C_ RR /\ ( A [,] B ) C_ RR ) -> ( RR _D ( ( exp |` RR ) |` ( A [,] B ) ) ) = ( ( RR _D ( exp |` RR ) ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) ) )
36	29, 11, 35	sylancr 695	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( RR _D ( ( exp |` RR ) |` ( A [,] B ) ) ) = ( ( RR _D ( exp |` RR ) ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) ) )
37	11	resabs1d 5428	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( exp |` RR ) |` ( A [,] B ) ) = ( exp |` ( A [,] B ) ) )
38	37	oveq2d 6666	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( RR _D ( ( exp |` RR ) |` ( A [,] B ) ) ) = ( RR _D ( exp |` ( A [,] B ) ) ) )
39		reelprrecn 10028	. . . . . . . . . 10 |- RR e. { RR , CC }
40		eff 14812	. . . . . . . . . 10 |- exp : CC --> CC
41		ssid 3624	. . . . . . . . . 10 |- CC C_ CC
42		dvef 23743	. . . . . . . . . . . . 13 |- ( CC _D exp ) = exp
43	42	dmeqi 5325	. . . . . . . . . . . 12 |- dom ( CC _D exp ) = dom exp
44	40	fdmi 6052	. . . . . . . . . . . 12 |- dom exp = CC
45	43, 44	eqtri 2644	. . . . . . . . . . 11 |- dom ( CC _D exp ) = CC
46	14, 45	sseqtr4i 3638	. . . . . . . . . 10 |- RR C_ dom ( CC _D exp )
47		dvres3 23677	. . . . . . . . . 10 |- ( ( ( RR e. { RR , CC } /\ exp : CC --> CC ) /\ ( CC C_ CC /\ RR C_ dom ( CC _D exp ) ) ) -> ( RR _D ( exp |` RR ) ) = ( ( CC _D exp ) |` RR ) )
48	39, 40, 41, 46, 47	mp4an 709	. . . . . . . . 9 |- ( RR _D ( exp |` RR ) ) = ( ( CC _D exp ) |` RR )
49	42	reseq1i 5392	. . . . . . . . 9 |- ( ( CC _D exp ) |` RR ) = ( exp |` RR )
50	48, 49	eqtri 2644	. . . . . . . 8 |- ( RR _D ( exp |` RR ) ) = ( exp |` RR )
51	50	a1i 11	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( RR _D ( exp |` RR ) ) = ( exp |` RR ) )
52		iccntr 22624	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
53	1, 2, 52	syl2anc 693	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
54	51, 53	reseq12d 5397	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( RR _D ( exp |` RR ) ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) ) = ( ( exp |` RR ) |` ( A (,) B ) ) )
55	36, 38, 54	3eqtr3d 2664	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( RR _D ( exp |` ( A [,] B ) ) ) = ( ( exp |` RR ) |` ( A (,) B ) ) )
56		isoeq1 6567	. . . . 5 |- ( ( RR _D ( exp |` ( A [,] B ) ) ) = ( ( exp |` RR ) |` ( A (,) B ) ) -> ( ( RR _D ( exp |` ( A [,] B ) ) ) Isom < , < ( ( A (,) B ) , ( ( exp |` RR ) " ( A (,) B ) ) ) <-> ( ( exp |` RR ) |` ( A (,) B ) ) Isom < , < ( ( A (,) B ) , ( ( exp |` RR ) " ( A (,) B ) ) ) ) )
57	55, 56	syl 17	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( RR _D ( exp |` ( A [,] B ) ) ) Isom < , < ( ( A (,) B ) , ( ( exp |` RR ) " ( A (,) B ) ) ) <-> ( ( exp |` RR ) |` ( A (,) B ) ) Isom < , < ( ( A (,) B ) , ( ( exp |` RR ) " ( A (,) B ) ) ) ) )
58	28, 57	mpbird 247	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( RR _D ( exp |` ( A [,] B ) ) ) Isom < , < ( ( A (,) B ) , ( ( exp |` RR ) " ( A (,) B ) ) ) )
59		simpr 477	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> T e. ( 0 (,) 1 ) )
60		eqid 2622	. . 3 |- ( ( T x. A ) + ( ( 1 - T ) x. B ) ) = ( ( T x. A ) + ( ( 1 - T ) x. B ) )
61	1, 2, 3, 21, 58, 59, 60	dvcvx 23783	. 2 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( exp |` ( A [,] B ) ) ` ( ( T x. A ) + ( ( 1 - T ) x. B ) ) ) < ( ( T x. ( ( exp |` ( A [,] B ) ) ` A ) ) + ( ( 1 - T ) x. ( ( exp |` ( A [,] B ) ) ` B ) ) ) )
62		ax-1cn 9994	. . . . . . 7 |- 1 e. CC
63		ioossre 12235	. . . . . . . . 9 |- ( 0 (,) 1 ) C_ RR
64	63, 59	sseldi 3601	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> T e. RR )
65	64	recnd 10068	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> T e. CC )
66		nncan 10310	. . . . . . 7 |- ( ( 1 e. CC /\ T e. CC ) -> ( 1 - ( 1 - T ) ) = T )
67	62, 65, 66	sylancr 695	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( 1 - ( 1 - T ) ) = T )
68	67	oveq1d 6665	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( 1 - ( 1 - T ) ) x. A ) = ( T x. A ) )
69	68	oveq1d 6665	. . . 4 |- ( ( ( 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 ) ) )
70		ioossicc 12259	. . . . . . 7 |- ( 0 (,) 1 ) C_ ( 0 [,] 1 )
71	70, 59	sseldi 3601	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> T e. ( 0 [,] 1 ) )
72		iirev 22728	. . . . . 6 |- ( T e. ( 0 [,] 1 ) -> ( 1 - T ) e. ( 0 [,] 1 ) )
73	71, 72	syl 17	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( 1 - T ) e. ( 0 [,] 1 ) )
74		lincmb01cmp 12315	. . . . 5 |- ( ( ( 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 ) )
75	73, 74	syldan 487	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( ( 1 - ( 1 - T ) ) x. A ) + ( ( 1 - T ) x. B ) ) e. ( A [,] B ) )
76	69, 75	eqeltrrd 2702	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( T x. A ) + ( ( 1 - T ) x. B ) ) e. ( A [,] B ) )
77		fvres 6207	. . 3 |- ( ( ( T x. A ) + ( ( 1 - T ) x. B ) ) e. ( A [,] B ) -> ( ( exp |` ( A [,] B ) ) ` ( ( T x. A ) + ( ( 1 - T ) x. B ) ) ) = ( exp ` ( ( T x. A ) + ( ( 1 - T ) x. B ) ) ) )
78	76, 77	syl 17	. 2 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( exp |` ( A [,] B ) ) ` ( ( T x. A ) + ( ( 1 - T ) x. B ) ) ) = ( exp ` ( ( T x. A ) + ( ( 1 - T ) x. B ) ) ) )
79	1	rexrd 10089	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> A e. RR* )
80	2	rexrd 10089	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> B e. RR* )
81	1, 2, 3	ltled 10185	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> A <_ B )
82		lbicc2 12288	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
83	79, 80, 81, 82	syl3anc 1326	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> A e. ( A [,] B ) )
84		fvres 6207	. . . . 5 |- ( A e. ( A [,] B ) -> ( ( exp |` ( A [,] B ) ) ` A ) = ( exp ` A ) )
85	83, 84	syl 17	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( exp |` ( A [,] B ) ) ` A ) = ( exp ` A ) )
86	85	oveq2d 6666	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( T x. ( ( exp |` ( A [,] B ) ) ` A ) ) = ( T x. ( exp ` A ) ) )
87		ubicc2 12289	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
88	79, 80, 81, 87	syl3anc 1326	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> B e. ( A [,] B ) )
89		fvres 6207	. . . . 5 |- ( B e. ( A [,] B ) -> ( ( exp |` ( A [,] B ) ) ` B ) = ( exp ` B ) )
90	88, 89	syl 17	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( exp |` ( A [,] B ) ) ` B ) = ( exp ` B ) )
91	90	oveq2d 6666	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( 1 - T ) x. ( ( exp |` ( A [,] B ) ) ` B ) ) = ( ( 1 - T ) x. ( exp ` B ) ) )
92	86, 91	oveq12d 6668	. 2 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( ( T x. ( ( exp |` ( A [,] B ) ) ` A ) ) + ( ( 1 - T ) x. ( ( exp |` ( A [,] B ) ) ` B ) ) ) = ( ( T x. ( exp ` A ) ) + ( ( 1 - T ) x. ( exp ` B ) ) ) )
93	61, 78, 92	3brtr3d 4684	1 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 (,) 1 ) ) -> ( exp ` ( ( T x. A ) + ( ( 1 - T ) x. B ) ) ) < ( ( T x. ( exp ` A ) ) + ( ( 1 - T ) x. ( exp ` B ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   {cpr 4179   class class class wbr 4653   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   -->wf 5884   -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   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   RR+crp 11832   (,)cioo 12175   [,]cicc 12178   expce 14792   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-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-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  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-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: (None)