Theorem recreclt 8975

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 8925	. . . 4 |- ( ( A e. RR /\ 0 < A ) -> 0 < ( 1 / A ) )
2		simpl 109	. . . . . 6 |- ( ( A e. RR /\ 0 < A ) -> A e. RR )
3		gt0ap0 8701	. . . . . 6 |- ( ( A e. RR /\ 0 < A ) -> A # 0 )
4	2, 3	rerecclapd 8909	. . . . 5 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) e. RR )
5		1re 8073	. . . . 5 |- 1 e. RR
6		ltaddpos 8527	. . . . 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 8053	. . . . 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 8201	. . . . . 6 |- 0 < 1
12		0re 8074	. . . . . . . 8 |- 0 e. RR
13		lttr 8148	. . . . . . . 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 1340	. . . . . . 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 8972	. . . 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 8527	. . . . . 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 8103	. . . . 5 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) e. CC )
25		ax-1cn 8020	. . . . 5 |- 1 e. CC
26		addcom 8211	. . . . 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 4071	. . 3 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) < ( 1 + ( 1 / A ) ) )
29		simpr 110	. . . 4 |- ( ( A e. RR /\ 0 < A ) -> 0 < A )
30		ltrec1 8963	. . . 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 1251	. . 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 1373   e. wcel 2176   class class class wbr 4045  (class class class)co 5946   CCcc 7925   RRcr 7926   0cc0 7927   1c1 7928   + caddc 7930   < clt 8109   / cdiv 8747

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-mulrcl 8026  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-precex 8037  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043  ax-pre-mulgt0 8044  ax-pre-mulext 8045

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-id 4341  df-po 4344  df-iso 4345  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-iota 5233  df-fun 5274  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-reap 8650  df-ap 8657  df-div 8748

This theorem is referenced by: (None)