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Theorem recgt0 8031
Description: The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 25-Aug-1999.) (Revised by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
recgt0  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( 1  /  A ) )

Proof of Theorem recgt0
StepHypRef Expression
1 0lt1 7339 . . . . 5  |-  0  <  1
2 0re 7217 . . . . . 6  |-  0  e.  RR
3 1re 7216 . . . . . 6  |-  1  e.  RR
42, 3ltnsymi 7313 . . . . 5  |-  ( 0  <  1  ->  -.  1  <  0 )
51, 4ax-mp 7 . . . 4  |-  -.  1  <  0
6 simpll 496 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  A  e.  RR )
7 gt0ap0 7828 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A #  0 )
87adantr 270 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  A #  0
)
96, 8rerecclapd 8022 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( 1  /  A )  e.  RR )
109renegcld 7587 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  -u ( 1  /  A )  e.  RR )
11 simpr 108 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( 1  /  A )  <  0 )
12 simpl 107 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  e.  RR )
1312, 7rerecclapd 8022 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  e.  RR )
1413adantr 270 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( 1  /  A )  e.  RR )
1514lt0neg1d 7719 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( (
1  /  A )  <  0  <->  0  <  -u ( 1  /  A
) ) )
1611, 15mpbid 145 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  0  <  -u ( 1  /  A
) )
17 simplr 497 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  0  <  A )
1810, 6, 16, 17mulgt0d 7335 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  0  <  (
-u ( 1  /  A )  x.  A
) )
1912recnd 7245 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  e.  CC )
2019adantr 270 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  A  e.  CC )
21 recclap 7870 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A #  0 )  ->  (
1  /  A )  e.  CC )
2220, 8, 21syl2anc 403 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( 1  /  A )  e.  CC )
2322, 20mulneg1d 7618 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( -u (
1  /  A )  x.  A )  = 
-u ( ( 1  /  A )  x.  A ) )
24 recidap2 7878 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A #  0 )  ->  (
( 1  /  A
)  x.  A )  =  1 )
2520, 8, 24syl2anc 403 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( (
1  /  A )  x.  A )  =  1 )
2625negeqd 7406 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  -u ( ( 1  /  A )  x.  A )  = 
-u 1 )
2723, 26eqtrd 2115 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( -u (
1  /  A )  x.  A )  = 
-u 1 )
2818, 27breqtrd 3830 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  0  <  -u 1 )
29 1red 7232 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  1  e.  RR )
3029lt0neg1d 7719 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( 1  <  0  <->  0  <  -u 1 ) )
3128, 30mpbird 165 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  1  <  0 )
3231ex 113 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( ( 1  /  A )  <  0  ->  1  <  0 ) )
335, 32mtoi 623 . . 3  |-  ( ( A  e.  RR  /\  0  <  A )  ->  -.  ( 1  /  A
)  <  0 )
34 lenlt 7290 . . . 4  |-  ( ( 0  e.  RR  /\  ( 1  /  A
)  e.  RR )  ->  ( 0  <_ 
( 1  /  A
)  <->  -.  ( 1  /  A )  <  0 ) )
352, 13, 34sylancr 405 . . 3  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 0  <_  (
1  /  A )  <->  -.  ( 1  /  A
)  <  0 ) )
3633, 35mpbird 165 . 2  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <_  ( 1  /  A ) )
37 recap0 7876 . . . 4  |-  ( ( A  e.  CC  /\  A #  0 )  ->  (
1  /  A ) #  0 )
3819, 7, 37syl2anc 403 . . 3  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
) #  0 )
3919, 7, 21syl2anc 403 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  e.  CC )
40 0cn 7209 . . . 4  |-  0  e.  CC
41 apsym 7809 . . . 4  |-  ( ( ( 1  /  A
)  e.  CC  /\  0  e.  CC )  ->  ( ( 1  /  A ) #  0  <->  0 #  (
1  /  A ) ) )
4239, 40, 41sylancl 404 . . 3  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( ( 1  /  A ) #  0  <->  0 #  (
1  /  A ) ) )
4338, 42mpbid 145 . 2  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0 #  ( 1  /  A ) )
44 ltleap 7833 . . 3  |-  ( ( 0  e.  RR  /\  ( 1  /  A
)  e.  RR )  ->  ( 0  < 
( 1  /  A
)  <->  ( 0  <_ 
( 1  /  A
)  /\  0 #  (
1  /  A ) ) ) )
452, 13, 44sylancr 405 . 2  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 0  <  (
1  /  A )  <-> 
( 0  <_  (
1  /  A )  /\  0 #  ( 1  /  A ) ) ) )
4636, 43, 45mpbir2and 886 1  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( 1  /  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   class class class wbr 3806  (class class class)co 5564   CCcc 7077   RRcr 7078   0cc0 7079   1c1 7080    x. cmul 7084    < clt 7251    <_ cle 7252   -ucneg 7383   # cap 7784    / cdiv 7863
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-cnex 7165  ax-resscn 7166  ax-1cn 7167  ax-1re 7168  ax-icn 7169  ax-addcl 7170  ax-addrcl 7171  ax-mulcl 7172  ax-mulrcl 7173  ax-addcom 7174  ax-mulcom 7175  ax-addass 7176  ax-mulass 7177  ax-distr 7178  ax-i2m1 7179  ax-0lt1 7180  ax-1rid 7181  ax-0id 7182  ax-rnegex 7183  ax-precex 7184  ax-cnre 7185  ax-pre-ltirr 7186  ax-pre-ltwlin 7187  ax-pre-lttrn 7188  ax-pre-apti 7189  ax-pre-ltadd 7190  ax-pre-mulgt0 7191  ax-pre-mulext 7192
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2826  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-br 3807  df-opab 3861  df-id 4077  df-po 4080  df-iso 4081  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-iota 4918  df-fun 4955  df-fv 4961  df-riota 5520  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-pnf 7253  df-mnf 7254  df-xr 7255  df-ltxr 7256  df-le 7257  df-sub 7384  df-neg 7385  df-reap 7778  df-ap 7785  df-div 7864
This theorem is referenced by:  prodgt0gt0  8032  ltdiv1  8049  ltrec1  8069  lerec2  8070  lediv12a  8075  recgt1i  8079  recreclt  8081  recgt0i  8087  recgt0ii  8088  recgt0d  8115  nnrecgt0  8179  nnrecl  8389
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