Theorem recreclt 8921

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 8871	. . . 4 |- ( ( A e. RR /\ 0 < A ) -> 0 < ( 1 / A ) )
2		simpl 109	. . . . . 6 |- ( ( A e. RR /\ 0 < A ) -> A e. RR )
3		gt0ap0 8647	. . . . . 6 |- ( ( A e. RR /\ 0 < A ) -> A # 0 )
4	2, 3	rerecclapd 8855	. . . . 5 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) e. RR )
5		1re 8020	. . . . 5 |- 1 e. RR
6		ltaddpos 8473	. . . . 5 |- ( ( ( 1 / A ) e. RR /\ 1 e. RR ) -> ( 0 < ( 1 / A ) <-> 1 < ( 1 + ( 1 / A ) ) ) )
7	4, 5, 6	sylancl 413	. . . 4 |- ( ( A e. RR /\ 0 < A ) -> ( 0 < ( 1 / A ) <-> 1 < ( 1 + ( 1 / A ) ) ) )
8	1, 7	mpbid 147	. . 3 |- ( ( A e. RR /\ 0 < A ) -> 1 < ( 1 + ( 1 / A ) ) )
9		readdcl 8000	. . . . 5 |- ( ( 1 e. RR /\ ( 1 / A ) e. RR ) -> ( 1 + ( 1 / A ) ) e. RR )
10	5, 4, 9	sylancr 414	. . . 4 |- ( ( A e. RR /\ 0 < A ) -> ( 1 + ( 1 / A ) ) e. RR )
11		0lt1 8148	. . . . . 6 |- 0 < 1
12		0re 8021	. . . . . . . 8 |- 0 e. RR
13		lttr 8095	. . . . . . . 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 1338	. . . . . . 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 430	. . . . 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 8918	. . . 4 |- ( ( ( 1 + ( 1 / A ) ) e. RR /\ 0 < ( 1 + ( 1 / A ) ) ) -> ( 1 < ( 1 + ( 1 / A ) ) <-> ( 1 / ( 1 + ( 1 / A ) ) ) < 1 ) )
19	10, 17, 18	syl2anc 411	. . 3 |- ( ( A e. RR /\ 0 < A ) -> ( 1 < ( 1 + ( 1 / A ) ) <-> ( 1 / ( 1 + ( 1 / A ) ) ) < 1 ) )
20	8, 19	mpbid 147	. 2 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / ( 1 + ( 1 / A ) ) ) < 1 )
21		ltaddpos 8473	. . . . . 6 |- ( ( 1 e. RR /\ ( 1 / A ) e. RR ) -> ( 0 < 1 <-> ( 1 / A ) < ( ( 1 / A ) + 1 ) ) )
22	5, 4, 21	sylancr 414	. . . . 5 |- ( ( A e. RR /\ 0 < A ) -> ( 0 < 1 <-> ( 1 / A ) < ( ( 1 / A ) + 1 ) ) )
23	11, 22	mpbii 148	. . . 4 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) < ( ( 1 / A ) + 1 ) )
24	4	recnd 8050	. . . . 5 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) e. CC )
25		ax-1cn 7967	. . . . 5 |- 1 e. CC
26		addcom 8158	. . . . 5 |- ( ( ( 1 / A ) e. CC /\ 1 e. CC ) -> ( ( 1 / A ) + 1 ) = ( 1 + ( 1 / A ) ) )
27	24, 25, 26	sylancl 413	. . . 4 |- ( ( A e. RR /\ 0 < A ) -> ( ( 1 / A ) + 1 ) = ( 1 + ( 1 / A ) ) )
28	23, 27	breqtrd 4056	. . 3 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) < ( 1 + ( 1 / A ) ) )
29		simpr 110	. . . 4 |- ( ( A e. RR /\ 0 < A ) -> 0 < A )
30		ltrec1 8909	. . . 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 1250	. . 3 |- ( ( A e. RR /\ 0 < A ) -> ( ( 1 / A ) < ( 1 + ( 1 / A ) ) <-> ( 1 / ( 1 + ( 1 / A ) ) ) < A ) )
32	28, 31	mpbid 147	. 2 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / ( 1 + ( 1 / A ) ) ) < A )
33	20, 32	jca 306	1 |- ( ( A e. RR /\ 0 < A ) -> ( ( 1 / ( 1 + ( 1 / A ) ) ) < 1 /\ ( 1 / ( 1 + ( 1 / A ) ) ) < A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   class class class wbr 4030  (class class class)co 5919   CCcc 7872   RRcr 7873   0cc0 7874   1c1 7875   + caddc 7877   < clt 8056   / cdiv 8693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-id 4325  df-po 4328  df-iso 4329  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694

This theorem is referenced by: (None)