Theorem recp1lt1 9175

Description: Construct a number less than 1 from any nonnegative number. (Contributed by NM, 30-Dec-2005.)

Assertion

Ref	Expression
recp1lt1	|- ( ( A e. RR /\ 0 <_ A ) -> ( A / ( 1 + A ) ) < 1 )

Proof of Theorem recp1lt1

Step	Hyp	Ref	Expression
1		simpl 109	. . . . 5 |- ( ( A e. RR /\ 0 <_ A ) -> A e. RR )
2		ltp1 9120	. . . . 5 |- ( A e. RR -> A < ( A + 1 ) )
3	1, 2	syl 14	. . . 4 |- ( ( A e. RR /\ 0 <_ A ) -> A < ( A + 1 ) )
4	1	recnd 8304	. . . . 5 |- ( ( A e. RR /\ 0 <_ A ) -> A e. CC )
5		1cnd 8292	. . . . 5 |- ( ( A e. RR /\ 0 <_ A ) -> 1 e. CC )
6	4, 5	addcomd 8426	. . . 4 |- ( ( A e. RR /\ 0 <_ A ) -> ( A + 1 ) = ( 1 + A ) )
7	3, 6	breqtrd 4137	. . 3 |- ( ( A e. RR /\ 0 <_ A ) -> A < ( 1 + A ) )
8	5, 4	addcld 8295	. . . 4 |- ( ( A e. RR /\ 0 <_ A ) -> ( 1 + A ) e. CC )
9		1red 8291	. . . . . 6 |- ( ( A e. RR /\ 0 <_ A ) -> 1 e. RR )
10	9, 1	readdcld 8305	. . . . 5 |- ( ( A e. RR /\ 0 <_ A ) -> ( 1 + A ) e. RR )
11		1re 8275	. . . . . 6 |- 1 e. RR
12		0lt1 8402	. . . . . . 7 |- 0 < 1
13		addgtge0 8726	. . . . . . 7 |- ( ( ( 1 e. RR /\ A e. RR ) /\ ( 0 < 1 /\ 0 <_ A ) ) -> 0 < ( 1 + A ) )
14	12, 13	mpanr1 437	. . . . . 6 |- ( ( ( 1 e. RR /\ A e. RR ) /\ 0 <_ A ) -> 0 < ( 1 + A ) )
15	11, 14	mpanl1 434	. . . . 5 |- ( ( A e. RR /\ 0 <_ A ) -> 0 < ( 1 + A ) )
16	10, 15	gt0ap0d 8905	. . . 4 |- ( ( A e. RR /\ 0 <_ A ) -> ( 1 + A ) # 0 )
17	4, 8, 16	divcanap1d 9067	. . 3 |- ( ( A e. RR /\ 0 <_ A ) -> ( ( A / ( 1 + A ) ) x. ( 1 + A ) ) = A )
18	8	mullidd 8294	. . 3 |- ( ( A e. RR /\ 0 <_ A ) -> ( 1 x. ( 1 + A ) ) = ( 1 + A ) )
19	7, 17, 18	3brtr4d 4143	. 2 |- ( ( A e. RR /\ 0 <_ A ) -> ( ( A / ( 1 + A ) ) x. ( 1 + A ) ) < ( 1 x. ( 1 + A ) ) )
20	1, 10, 16	redivclapd 9111	. . 3 |- ( ( A e. RR /\ 0 <_ A ) -> ( A / ( 1 + A ) ) e. RR )
21		ltmul1 8868	. . 3 |- ( ( ( A / ( 1 + A ) ) e. RR /\ 1 e. RR /\ ( ( 1 + A ) e. RR /\ 0 < ( 1 + A ) ) ) -> ( ( A / ( 1 + A ) ) < 1 <-> ( ( A / ( 1 + A ) ) x. ( 1 + A ) ) < ( 1 x. ( 1 + A ) ) ) )
22	20, 9, 10, 15, 21	syl112anc 1278	. 2 |- ( ( A e. RR /\ 0 <_ A ) -> ( ( A / ( 1 + A ) ) < 1 <-> ( ( A / ( 1 + A ) ) x. ( 1 + A ) ) < ( 1 x. ( 1 + A ) ) ) )
23	19, 22	mpbird 167	1 |- ( ( A e. RR /\ 0 <_ A ) -> ( A / ( 1 + A ) ) < 1 )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2205   class class class wbr 4111  (class class class)co 6052   RRcr 8128   0cc0 8129   1c1 8130   + caddc 8132   x. cmul 8134   < clt 8310   <_ cle 8311   / cdiv 8948

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-mulrcl 8228  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-precex 8239  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245  ax-pre-mulgt0 8246  ax-pre-mulext 8247

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-id 4416  df-po 4419  df-iso 4420  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-iota 5314  df-fun 5356  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-reap 8851  df-ap 8858  df-div 8949

This theorem is referenced by: (None)