Theorem recreclt 7978

Description: Given a positive number A, construct a new positive number less than both A and 1. (Contributed by NM, 28-Dec-2005.)

Assertion

Ref	Expression
recreclt	|- ( ( A e. RR /\ 0 < A ) -> ( ( 1 / ( 1 + ( 1 / A ) ) ) < 1 /\ ( 1 / ( 1 + ( 1 / A ) ) ) < A ) )

Proof of Theorem recreclt

Step	Hyp	Ref	Expression
1		recgt0 7928	. . . 4 |- ( ( A e. RR /\ 0 < A ) -> 0 < ( 1 / A ) )
2		simpl 107	. . . . . 6 |- ( ( A e. RR /\ 0 < A ) -> A e. RR )
3		gt0ap0 7725	. . . . . 6 |- ( ( A e. RR /\ 0 < A ) -> A # 0 )
4	2, 3	rerecclapd 7919	. . . . 5 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) e. RR )
5		1re 7118	. . . . 5 |- 1 e. RR
6		ltaddpos 7556	. . . . 5 |- ( ( ( 1 / A ) e. RR /\ 1 e. RR ) -> ( 0 < ( 1 / A ) <-> 1 < ( 1 + ( 1 / A ) ) ) )
7	4, 5, 6	sylancl 404	. . . 4 |- ( ( A e. RR /\ 0 < A ) -> ( 0 < ( 1 / A ) <-> 1 < ( 1 + ( 1 / A ) ) ) )
8	1, 7	mpbid 145	. . 3 |- ( ( A e. RR /\ 0 < A ) -> 1 < ( 1 + ( 1 / A ) ) )
9		readdcl 7099	. . . . 5 |- ( ( 1 e. RR /\ ( 1 / A ) e. RR ) -> ( 1 + ( 1 / A ) ) e. RR )
10	5, 4, 9	sylancr 405	. . . 4 |- ( ( A e. RR /\ 0 < A ) -> ( 1 + ( 1 / A ) ) e. RR )
11		0lt1 7236	. . . . . 6 |- 0 < 1
12		0re 7119	. . . . . . . 8 |- 0 e. RR
13		lttr 7185	. . . . . . . 8 |- ( ( 0 e. RR /\ 1 e. RR /\ ( 1 + ( 1 / A ) ) e. RR ) -> ( ( 0 < 1 /\ 1 < ( 1 + ( 1 / A ) ) ) -> 0 < ( 1 + ( 1 / A ) ) ) )
14	12, 5, 13	mp3an12 1258	. . . . . . 7 |- ( ( 1 + ( 1 / A ) ) e. RR -> ( ( 0 < 1 /\ 1 < ( 1 + ( 1 / A ) ) ) -> 0 < ( 1 + ( 1 / A ) ) ) )
15	10, 14	syl 14	. . . . . 6 |- ( ( A e. RR /\ 0 < A ) -> ( ( 0 < 1 /\ 1 < ( 1 + ( 1 / A ) ) ) -> 0 < ( 1 + ( 1 / A ) ) ) )
16	11, 15	mpani 420	. . . . 5 |- ( ( A e. RR /\ 0 < A ) -> ( 1 < ( 1 + ( 1 / A ) ) -> 0 < ( 1 + ( 1 / A ) ) ) )
17	8, 16	mpd 13	. . . 4 |- ( ( A e. RR /\ 0 < A ) -> 0 < ( 1 + ( 1 / A ) ) )
18		recgt1 7975	. . . 4 |- ( ( ( 1 + ( 1 / A ) ) e. RR /\ 0 < ( 1 + ( 1 / A ) ) ) -> ( 1 < ( 1 + ( 1 / A ) ) <-> ( 1 / ( 1 + ( 1 / A ) ) ) < 1 ) )
19	10, 17, 18	syl2anc 403	. . 3 |- ( ( A e. RR /\ 0 < A ) -> ( 1 < ( 1 + ( 1 / A ) ) <-> ( 1 / ( 1 + ( 1 / A ) ) ) < 1 ) )
20	8, 19	mpbid 145	. 2 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / ( 1 + ( 1 / A ) ) ) < 1 )
21		ltaddpos 7556	. . . . . 6 |- ( ( 1 e. RR /\ ( 1 / A ) e. RR ) -> ( 0 < 1 <-> ( 1 / A ) < ( ( 1 / A ) + 1 ) ) )
22	5, 4, 21	sylancr 405	. . . . 5 |- ( ( A e. RR /\ 0 < A ) -> ( 0 < 1 <-> ( 1 / A ) < ( ( 1 / A ) + 1 ) ) )
23	11, 22	mpbii 146	. . . 4 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) < ( ( 1 / A ) + 1 ) )
24	4	recnd 7147	. . . . 5 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) e. CC )
25		ax-1cn 7069	. . . . 5 |- 1 e. CC
26		addcom 7245	. . . . 5 |- ( ( ( 1 / A ) e. CC /\ 1 e. CC ) -> ( ( 1 / A ) + 1 ) = ( 1 + ( 1 / A ) ) )
27	24, 25, 26	sylancl 404	. . . 4 |- ( ( A e. RR /\ 0 < A ) -> ( ( 1 / A ) + 1 ) = ( 1 + ( 1 / A ) ) )
28	23, 27	breqtrd 3809	. . 3 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) < ( 1 + ( 1 / A ) ) )
29		simpr 108	. . . 4 |- ( ( A e. RR /\ 0 < A ) -> 0 < A )
30		ltrec1 7966	. . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( ( 1 + ( 1 / A ) ) e. RR /\ 0 < ( 1 + ( 1 / A ) ) ) ) -> ( ( 1 / A ) < ( 1 + ( 1 / A ) ) <-> ( 1 / ( 1 + ( 1 / A ) ) ) < A ) )
31	2, 29, 10, 17, 30	syl22anc 1170	. . 3 |- ( ( A e. RR /\ 0 < A ) -> ( ( 1 / A ) < ( 1 + ( 1 / A ) ) <-> ( 1 / ( 1 + ( 1 / A ) ) ) < A ) )
32	28, 31	mpbid 145	. 2 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / ( 1 + ( 1 / A ) ) ) < A )
33	20, 32	jca 300	1 |- ( ( A e. RR /\ 0 < A ) -> ( ( 1 / ( 1 + ( 1 / A ) ) ) < 1 /\ ( 1 / ( 1 + ( 1 / A ) ) ) < A ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   class class class wbr 3785  (class class class)co 5532   CCcc 6979   RRcr 6980   0cc0 6981   1c1 6982   + caddc 6984   < clt 7153   / cdiv 7760

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-po 4051  df-iso 4052  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761

This theorem is referenced by: (None)