Theorem icccvx 22749

Description: A linear combination of two reals lies in the interval between them. Equivalently, a closed interval is a convex set. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
icccvx	|- ( ( A e. RR /\ B e. RR ) -> ( ( C e. ( A [,] B ) /\ D e. ( A [,] B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) x. C ) + ( T x. D ) ) e. ( A [,] B ) ) )

Proof of Theorem icccvx

Step	Hyp	Ref	Expression
1		iccss2 12244	. . . . . . 7 |- ( ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) -> ( C [,] D ) C_ ( A [,] B ) )
2	1	adantl 482	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) ) -> ( C [,] D ) C_ ( A [,] B ) )
3	2	3adantr3 1222	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. ( A [,] B ) /\ D e. ( A [,] B ) /\ T e. ( 0 [,] 1 ) ) ) -> ( C [,] D ) C_ ( A [,] B ) )
4	3	adantr 481	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. ( A [,] B ) /\ D e. ( A [,] B ) /\ T e. ( 0 [,] 1 ) ) ) /\ C < D ) -> ( C [,] D ) C_ ( A [,] B ) )
5		iccssre 12255	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
6	5	sselda 3603	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A [,] B ) ) -> C e. RR )
7	6	adantrr 753	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) ) -> C e. RR )
8	5	sselda 3603	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR ) /\ D e. ( A [,] B ) ) -> D e. RR )
9	8	adantrl 752	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) ) -> D e. RR )
10	7, 9	jca 554	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) ) -> ( C e. RR /\ D e. RR ) )
11	10	3adantr3 1222	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. ( A [,] B ) /\ D e. ( A [,] B ) /\ T e. ( 0 [,] 1 ) ) ) -> ( C e. RR /\ D e. RR ) )
12		simpr3 1069	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. ( A [,] B ) /\ D e. ( A [,] B ) /\ T e. ( 0 [,] 1 ) ) ) -> T e. ( 0 [,] 1 ) )
13	11, 12	jca 554	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. ( A [,] B ) /\ D e. ( A [,] B ) /\ T e. ( 0 [,] 1 ) ) ) -> ( ( C e. RR /\ D e. RR ) /\ T e. ( 0 [,] 1 ) ) )
14		lincmb01cmp 12315	. . . . . . . . 9 |- ( ( ( C e. RR /\ D e. RR /\ C < D ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) x. C ) + ( T x. D ) ) e. ( C [,] D ) )
15	14	ex 450	. . . . . . . 8 |- ( ( C e. RR /\ D e. RR /\ C < D ) -> ( T e. ( 0 [,] 1 ) -> ( ( ( 1 - T ) x. C ) + ( T x. D ) ) e. ( C [,] D ) ) )
16	15	3expa 1265	. . . . . . 7 |- ( ( ( C e. RR /\ D e. RR ) /\ C < D ) -> ( T e. ( 0 [,] 1 ) -> ( ( ( 1 - T ) x. C ) + ( T x. D ) ) e. ( C [,] D ) ) )
17	16	imp 445	. . . . . 6 |- ( ( ( ( C e. RR /\ D e. RR ) /\ C < D ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) x. C ) + ( T x. D ) ) e. ( C [,] D ) )
18	17	an32s 846	. . . . 5 |- ( ( ( ( C e. RR /\ D e. RR ) /\ T e. ( 0 [,] 1 ) ) /\ C < D ) -> ( ( ( 1 - T ) x. C ) + ( T x. D ) ) e. ( C [,] D ) )
19	13, 18	sylan 488	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. ( A [,] B ) /\ D e. ( A [,] B ) /\ T e. ( 0 [,] 1 ) ) ) /\ C < D ) -> ( ( ( 1 - T ) x. C ) + ( T x. D ) ) e. ( C [,] D ) )
20	4, 19	sseldd 3604	. . 3 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. ( A [,] B ) /\ D e. ( A [,] B ) /\ T e. ( 0 [,] 1 ) ) ) /\ C < D ) -> ( ( ( 1 - T ) x. C ) + ( T x. D ) ) e. ( A [,] B ) )
21		oveq2 6658	. . . . . 6 |- ( C = D -> ( ( 1 - T ) x. C ) = ( ( 1 - T ) x. D ) )
22	21	oveq1d 6665	. . . . 5 |- ( C = D -> ( ( ( 1 - T ) x. C ) + ( T x. D ) ) = ( ( ( 1 - T ) x. D ) + ( T x. D ) ) )
23		unitssre 12319	. . . . . . . . . 10 |- ( 0 [,] 1 ) C_ RR
24	23	sseli 3599	. . . . . . . . 9 |- ( T e. ( 0 [,] 1 ) -> T e. RR )
25	24	recnd 10068	. . . . . . . 8 |- ( T e. ( 0 [,] 1 ) -> T e. CC )
26	25	ad2antll 765	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( D e. ( A [,] B ) /\ T e. ( 0 [,] 1 ) ) ) -> T e. CC )
27	8	recnd 10068	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ D e. ( A [,] B ) ) -> D e. CC )
28	27	adantrr 753	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( D e. ( A [,] B ) /\ T e. ( 0 [,] 1 ) ) ) -> D e. CC )
29		ax-1cn 9994	. . . . . . . . . . 11 |- 1 e. CC
30		npcan 10290	. . . . . . . . . . 11 |- ( ( 1 e. CC /\ T e. CC ) -> ( ( 1 - T ) + T ) = 1 )
31	29, 30	mpan 706	. . . . . . . . . 10 |- ( T e. CC -> ( ( 1 - T ) + T ) = 1 )
32	31	adantr 481	. . . . . . . . 9 |- ( ( T e. CC /\ D e. CC ) -> ( ( 1 - T ) + T ) = 1 )
33	32	oveq1d 6665	. . . . . . . 8 |- ( ( T e. CC /\ D e. CC ) -> ( ( ( 1 - T ) + T ) x. D ) = ( 1 x. D ) )
34		subcl 10280	. . . . . . . . . . 11 |- ( ( 1 e. CC /\ T e. CC ) -> ( 1 - T ) e. CC )
35	29, 34	mpan 706	. . . . . . . . . 10 |- ( T e. CC -> ( 1 - T ) e. CC )
36	35	ancri 575	. . . . . . . . 9 |- ( T e. CC -> ( ( 1 - T ) e. CC /\ T e. CC ) )
37		adddir 10031	. . . . . . . . . 10 |- ( ( ( 1 - T ) e. CC /\ T e. CC /\ D e. CC ) -> ( ( ( 1 - T ) + T ) x. D ) = ( ( ( 1 - T ) x. D ) + ( T x. D ) ) )
38	37	3expa 1265	. . . . . . . . 9 |- ( ( ( ( 1 - T ) e. CC /\ T e. CC ) /\ D e. CC ) -> ( ( ( 1 - T ) + T ) x. D ) = ( ( ( 1 - T ) x. D ) + ( T x. D ) ) )
39	36, 38	sylan 488	. . . . . . . 8 |- ( ( T e. CC /\ D e. CC ) -> ( ( ( 1 - T ) + T ) x. D ) = ( ( ( 1 - T ) x. D ) + ( T x. D ) ) )
40		mulid2 10038	. . . . . . . . 9 |- ( D e. CC -> ( 1 x. D ) = D )
41	40	adantl 482	. . . . . . . 8 |- ( ( T e. CC /\ D e. CC ) -> ( 1 x. D ) = D )
42	33, 39, 41	3eqtr3d 2664	. . . . . . 7 |- ( ( T e. CC /\ D e. CC ) -> ( ( ( 1 - T ) x. D ) + ( T x. D ) ) = D )
43	26, 28, 42	syl2anc 693	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( D e. ( A [,] B ) /\ T e. ( 0 [,] 1 ) ) ) -> ( ( ( 1 - T ) x. D ) + ( T x. D ) ) = D )
44	43	3adantr1 1220	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. ( A [,] B ) /\ D e. ( A [,] B ) /\ T e. ( 0 [,] 1 ) ) ) -> ( ( ( 1 - T ) x. D ) + ( T x. D ) ) = D )
45	22, 44	sylan9eqr 2678	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. ( A [,] B ) /\ D e. ( A [,] B ) /\ T e. ( 0 [,] 1 ) ) ) /\ C = D ) -> ( ( ( 1 - T ) x. C ) + ( T x. D ) ) = D )
46		simplr2 1104	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. ( A [,] B ) /\ D e. ( A [,] B ) /\ T e. ( 0 [,] 1 ) ) ) /\ C = D ) -> D e. ( A [,] B ) )
47	45, 46	eqeltrd 2701	. . 3 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. ( A [,] B ) /\ D e. ( A [,] B ) /\ T e. ( 0 [,] 1 ) ) ) /\ C = D ) -> ( ( ( 1 - T ) x. C ) + ( T x. D ) ) e. ( A [,] B ) )
48		iccss2 12244	. . . . . . . 8 |- ( ( D e. ( A [,] B ) /\ C e. ( A [,] B ) ) -> ( D [,] C ) C_ ( A [,] B ) )
49	48	adantl 482	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( D e. ( A [,] B ) /\ C e. ( A [,] B ) ) ) -> ( D [,] C ) C_ ( A [,] B ) )
50	49	ancom2s 844	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) ) -> ( D [,] C ) C_ ( A [,] B ) )
51	50	3adantr3 1222	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. ( A [,] B ) /\ D e. ( A [,] B ) /\ T e. ( 0 [,] 1 ) ) ) -> ( D [,] C ) C_ ( A [,] B ) )
52	51	adantr 481	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. ( A [,] B ) /\ D e. ( A [,] B ) /\ T e. ( 0 [,] 1 ) ) ) /\ D < C ) -> ( D [,] C ) C_ ( A [,] B ) )
53	9, 7	jca 554	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) ) -> ( D e. RR /\ C e. RR ) )
54	53	3adantr3 1222	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. ( A [,] B ) /\ D e. ( A [,] B ) /\ T e. ( 0 [,] 1 ) ) ) -> ( D e. RR /\ C e. RR ) )
55	54, 12	jca 554	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. ( A [,] B ) /\ D e. ( A [,] B ) /\ T e. ( 0 [,] 1 ) ) ) -> ( ( D e. RR /\ C e. RR ) /\ T e. ( 0 [,] 1 ) ) )
56		iirev 22728	. . . . . . . . . . . . . . . . 17 |- ( T e. ( 0 [,] 1 ) -> ( 1 - T ) e. ( 0 [,] 1 ) )
57	23, 56	sseldi 3601	. . . . . . . . . . . . . . . 16 |- ( T e. ( 0 [,] 1 ) -> ( 1 - T ) e. RR )
58	57	recnd 10068	. . . . . . . . . . . . . . 15 |- ( T e. ( 0 [,] 1 ) -> ( 1 - T ) e. CC )
59		recn 10026	. . . . . . . . . . . . . . 15 |- ( C e. RR -> C e. CC )
60		mulcl 10020	. . . . . . . . . . . . . . 15 |- ( ( ( 1 - T ) e. CC /\ C e. CC ) -> ( ( 1 - T ) x. C ) e. CC )
61	58, 59, 60	syl2anr 495	. . . . . . . . . . . . . 14 |- ( ( C e. RR /\ T e. ( 0 [,] 1 ) ) -> ( ( 1 - T ) x. C ) e. CC )
62	61	adantll 750	. . . . . . . . . . . . 13 |- ( ( ( D e. RR /\ C e. RR ) /\ T e. ( 0 [,] 1 ) ) -> ( ( 1 - T ) x. C ) e. CC )
63		recn 10026	. . . . . . . . . . . . . . 15 |- ( D e. RR -> D e. CC )
64		mulcl 10020	. . . . . . . . . . . . . . 15 |- ( ( T e. CC /\ D e. CC ) -> ( T x. D ) e. CC )
65	25, 63, 64	syl2anr 495	. . . . . . . . . . . . . 14 |- ( ( D e. RR /\ T e. ( 0 [,] 1 ) ) -> ( T x. D ) e. CC )
66	65	adantlr 751	. . . . . . . . . . . . 13 |- ( ( ( D e. RR /\ C e. RR ) /\ T e. ( 0 [,] 1 ) ) -> ( T x. D ) e. CC )
67	62, 66	addcomd 10238	. . . . . . . . . . . 12 |- ( ( ( D e. RR /\ C e. RR ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) x. C ) + ( T x. D ) ) = ( ( T x. D ) + ( ( 1 - T ) x. C ) ) )
68	67	3adantl3 1219	. . . . . . . . . . 11 |- ( ( ( D e. RR /\ C e. RR /\ D < C ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) x. C ) + ( T x. D ) ) = ( ( T x. D ) + ( ( 1 - T ) x. C ) ) )
69		nncan 10310	. . . . . . . . . . . . . . . . 17 |- ( ( 1 e. CC /\ T e. CC ) -> ( 1 - ( 1 - T ) ) = T )
70	29, 69	mpan 706	. . . . . . . . . . . . . . . 16 |- ( T e. CC -> ( 1 - ( 1 - T ) ) = T )
71	70	eqcomd 2628	. . . . . . . . . . . . . . 15 |- ( T e. CC -> T = ( 1 - ( 1 - T ) ) )
72	71	oveq1d 6665	. . . . . . . . . . . . . 14 |- ( T e. CC -> ( T x. D ) = ( ( 1 - ( 1 - T ) ) x. D ) )
73	72	oveq1d 6665	. . . . . . . . . . . . 13 |- ( T e. CC -> ( ( T x. D ) + ( ( 1 - T ) x. C ) ) = ( ( ( 1 - ( 1 - T ) ) x. D ) + ( ( 1 - T ) x. C ) ) )
74	25, 73	syl 17	. . . . . . . . . . . 12 |- ( T e. ( 0 [,] 1 ) -> ( ( T x. D ) + ( ( 1 - T ) x. C ) ) = ( ( ( 1 - ( 1 - T ) ) x. D ) + ( ( 1 - T ) x. C ) ) )
75	74	adantl 482	. . . . . . . . . . 11 |- ( ( ( D e. RR /\ C e. RR /\ D < C ) /\ T e. ( 0 [,] 1 ) ) -> ( ( T x. D ) + ( ( 1 - T ) x. C ) ) = ( ( ( 1 - ( 1 - T ) ) x. D ) + ( ( 1 - T ) x. C ) ) )
76	68, 75	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( D e. RR /\ C e. RR /\ D < C ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) x. C ) + ( T x. D ) ) = ( ( ( 1 - ( 1 - T ) ) x. D ) + ( ( 1 - T ) x. C ) ) )
77		lincmb01cmp 12315	. . . . . . . . . . 11 |- ( ( ( D e. RR /\ C e. RR /\ D < C ) /\ ( 1 - T ) e. ( 0 [,] 1 ) ) -> ( ( ( 1 - ( 1 - T ) ) x. D ) + ( ( 1 - T ) x. C ) ) e. ( D [,] C ) )
78	56, 77	sylan2 491	. . . . . . . . . 10 |- ( ( ( D e. RR /\ C e. RR /\ D < C ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - ( 1 - T ) ) x. D ) + ( ( 1 - T ) x. C ) ) e. ( D [,] C ) )
79	76, 78	eqeltrd 2701	. . . . . . . . 9 |- ( ( ( D e. RR /\ C e. RR /\ D < C ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) x. C ) + ( T x. D ) ) e. ( D [,] C ) )
80	79	ex 450	. . . . . . . 8 |- ( ( D e. RR /\ C e. RR /\ D < C ) -> ( T e. ( 0 [,] 1 ) -> ( ( ( 1 - T ) x. C ) + ( T x. D ) ) e. ( D [,] C ) ) )
81	80	3expa 1265	. . . . . . 7 |- ( ( ( D e. RR /\ C e. RR ) /\ D < C ) -> ( T e. ( 0 [,] 1 ) -> ( ( ( 1 - T ) x. C ) + ( T x. D ) ) e. ( D [,] C ) ) )
82	81	imp 445	. . . . . 6 |- ( ( ( ( D e. RR /\ C e. RR ) /\ D < C ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) x. C ) + ( T x. D ) ) e. ( D [,] C ) )
83	82	an32s 846	. . . . 5 |- ( ( ( ( D e. RR /\ C e. RR ) /\ T e. ( 0 [,] 1 ) ) /\ D < C ) -> ( ( ( 1 - T ) x. C ) + ( T x. D ) ) e. ( D [,] C ) )
84	55, 83	sylan 488	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. ( A [,] B ) /\ D e. ( A [,] B ) /\ T e. ( 0 [,] 1 ) ) ) /\ D < C ) -> ( ( ( 1 - T ) x. C ) + ( T x. D ) ) e. ( D [,] C ) )
85	52, 84	sseldd 3604	. . 3 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( C e. ( A [,] B ) /\ D e. ( A [,] B ) /\ T e. ( 0 [,] 1 ) ) ) /\ D < C ) -> ( ( ( 1 - T ) x. C ) + ( T x. D ) ) e. ( A [,] B ) )
86	7, 9	lttri4d 10178	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. ( A [,] B ) /\ D e. ( A [,] B ) ) ) -> ( C < D \/ C = D \/ D < C ) )
87	86	3adantr3 1222	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. ( A [,] B ) /\ D e. ( A [,] B ) /\ T e. ( 0 [,] 1 ) ) ) -> ( C < D \/ C = D \/ D < C ) )
88	20, 47, 85, 87	mpjao3dan 1395	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. ( A [,] B ) /\ D e. ( A [,] B ) /\ T e. ( 0 [,] 1 ) ) ) -> ( ( ( 1 - T ) x. C ) + ( T x. D ) ) e. ( A [,] B ) )
89	88	ex 450	1 |- ( ( A e. RR /\ B e. RR ) -> ( ( C e. ( A [,] B ) /\ D e. ( A [,] B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) x. C ) + ( T x. D ) ) e. ( A [,] B ) ) )

Syntax hints:   -> wi 4   /\ wa 384   \/ w3o 1036   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   class class class wbr 4653  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   - cmin 10266   [,]cicc 12178

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  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

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-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-rp 11833  df-icc 12182

This theorem is referenced by:  reparphti  22797