Theorem recreclt 9008

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 8958	. . . 4 |- ( ( A e. RR /\ 0 < A ) -> 0 < ( 1 / A ) )
2		simpl 109	. . . . . 6 |- ( ( A e. RR /\ 0 < A ) -> A e. RR )
3		gt0ap0 8734	. . . . . 6 |- ( ( A e. RR /\ 0 < A ) -> A # 0 )
4	2, 3	rerecclapd 8942	. . . . 5 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) e. RR )
5		1re 8106	. . . . 5 |- 1 e. RR
6		ltaddpos 8560	. . . . 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 8086	. . . . 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 8234	. . . . . 6 |- 0 < 1
12		0re 8107	. . . . . . . 8 |- 0 e. RR
13		lttr 8181	. . . . . . . 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 9005	. . . 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 8560	. . . . . 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 8136	. . . . 5 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) e. CC )
25		ax-1cn 8053	. . . . 5 |- 1 e. CC
26		addcom 8244	. . . . 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 4085	. . 3 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) < ( 1 + ( 1 / A ) ) )
29		simpr 110	. . . 4 |- ( ( A e. RR /\ 0 < A ) -> 0 < A )
30		ltrec1 8996	. . . 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 2178   class class class wbr 4059  (class class class)co 5967   CCcc 7958   RRcr 7959   0cc0 7960   1c1 7961   + caddc 7963   < clt 8142   / cdiv 8780

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-id 4358  df-po 4361  df-iso 4362  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781

This theorem is referenced by: (None)