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Theorem recgt0 8627
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 7908 . . . . 5  |-  0  <  1
2 0re 7785 . . . . . 6  |-  0  e.  RR
3 1re 7784 . . . . . 6  |-  1  e.  RR
42, 3ltnsymi 7882 . . . . 5  |-  ( 0  <  1  ->  -.  1  <  0 )
51, 4ax-mp 5 . . . 4  |-  -.  1  <  0
6 simpll 518 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  A  e.  RR )
7 gt0ap0 8407 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A #  0 )
87adantr 274 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  A #  0
)
96, 8rerecclapd 8612 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( 1  /  A )  e.  RR )
109renegcld 8161 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  -u ( 1  /  A )  e.  RR )
11 simpr 109 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( 1  /  A )  <  0 )
12 simpl 108 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  e.  RR )
1312, 7rerecclapd 8612 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  e.  RR )
1413adantr 274 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( 1  /  A )  e.  RR )
1514lt0neg1d 8296 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( (
1  /  A )  <  0  <->  0  <  -u ( 1  /  A
) ) )
1611, 15mpbid 146 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  0  <  -u ( 1  /  A
) )
17 simplr 519 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  0  <  A )
1810, 6, 16, 17mulgt0d 7904 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  0  <  (
-u ( 1  /  A )  x.  A
) )
1912recnd 7813 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  e.  CC )
2019adantr 274 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  A  e.  CC )
21 recclap 8458 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A #  0 )  ->  (
1  /  A )  e.  CC )
2220, 8, 21syl2anc 408 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( 1  /  A )  e.  CC )
2322, 20mulneg1d 8192 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( -u (
1  /  A )  x.  A )  = 
-u ( ( 1  /  A )  x.  A ) )
24 recidap2 8466 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A #  0 )  ->  (
( 1  /  A
)  x.  A )  =  1 )
2520, 8, 24syl2anc 408 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( (
1  /  A )  x.  A )  =  1 )
2625negeqd 7976 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  -u ( ( 1  /  A )  x.  A )  = 
-u 1 )
2723, 26eqtrd 2172 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( -u (
1  /  A )  x.  A )  = 
-u 1 )
2818, 27breqtrd 3957 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  0  <  -u 1 )
29 1red 7800 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  1  e.  RR )
3029lt0neg1d 8296 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  ( 1  <  0  <->  0  <  -u 1 ) )
3128, 30mpbird 166 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( 1  /  A )  <  0
)  ->  1  <  0 )
3231ex 114 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( ( 1  /  A )  <  0  ->  1  <  0 ) )
335, 32mtoi 653 . . 3  |-  ( ( A  e.  RR  /\  0  <  A )  ->  -.  ( 1  /  A
)  <  0 )
34 lenlt 7859 . . . 4  |-  ( ( 0  e.  RR  /\  ( 1  /  A
)  e.  RR )  ->  ( 0  <_ 
( 1  /  A
)  <->  -.  ( 1  /  A )  <  0 ) )
352, 13, 34sylancr 410 . . 3  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 0  <_  (
1  /  A )  <->  -.  ( 1  /  A
)  <  0 ) )
3633, 35mpbird 166 . 2  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <_  ( 1  /  A ) )
37 recap0 8464 . . . 4  |-  ( ( A  e.  CC  /\  A #  0 )  ->  (
1  /  A ) #  0 )
3819, 7, 37syl2anc 408 . . 3  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
) #  0 )
3919, 7, 21syl2anc 408 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  e.  CC )
40 0cn 7777 . . . 4  |-  0  e.  CC
41 apsym 8387 . . . 4  |-  ( ( ( 1  /  A
)  e.  CC  /\  0  e.  CC )  ->  ( ( 1  /  A ) #  0  <->  0 #  (
1  /  A ) ) )
4239, 40, 41sylancl 409 . . 3  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( ( 1  /  A ) #  0  <->  0 #  (
1  /  A ) ) )
4338, 42mpbid 146 . 2  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0 #  ( 1  /  A ) )
44 ltleap 8413 . . 3  |-  ( ( 0  e.  RR  /\  ( 1  /  A
)  e.  RR )  ->  ( 0  < 
( 1  /  A
)  <->  ( 0  <_ 
( 1  /  A
)  /\  0 #  (
1  /  A ) ) ) )
452, 13, 44sylancr 410 . 2  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 0  <  (
1  /  A )  <-> 
( 0  <_  (
1  /  A )  /\  0 #  ( 1  /  A ) ) ) )
4636, 43, 45mpbir2and 928 1  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( 1  /  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   class class class wbr 3932  (class class class)co 5777   CCcc 7637   RRcr 7638   0cc0 7639   1c1 7640    x. cmul 7644    < clt 7819    <_ cle 7820   -ucneg 7953   # cap 8362    / cdiv 8451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4049  ax-pow 4101  ax-pr 4134  ax-un 4358  ax-setind 4455  ax-cnex 7730  ax-resscn 7731  ax-1cn 7732  ax-1re 7733  ax-icn 7734  ax-addcl 7735  ax-addrcl 7736  ax-mulcl 7737  ax-mulrcl 7738  ax-addcom 7739  ax-mulcom 7740  ax-addass 7741  ax-mulass 7742  ax-distr 7743  ax-i2m1 7744  ax-0lt1 7745  ax-1rid 7746  ax-0id 7747  ax-rnegex 7748  ax-precex 7749  ax-cnre 7750  ax-pre-ltirr 7751  ax-pre-ltwlin 7752  ax-pre-lttrn 7753  ax-pre-apti 7754  ax-pre-ltadd 7755  ax-pre-mulgt0 7756  ax-pre-mulext 7757
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-br 3933  df-opab 3993  df-id 4218  df-po 4221  df-iso 4222  df-xp 4548  df-rel 4549  df-cnv 4550  df-co 4551  df-dm 4552  df-iota 5091  df-fun 5128  df-fv 5134  df-riota 5733  df-ov 5780  df-oprab 5781  df-mpo 5782  df-pnf 7821  df-mnf 7822  df-xr 7823  df-ltxr 7824  df-le 7825  df-sub 7954  df-neg 7955  df-reap 8356  df-ap 8363  df-div 8452
This theorem is referenced by:  prodgt0gt0  8628  ltdiv1  8645  ltrec1  8665  lerec2  8666  lediv12a  8671  recgt1i  8675  recreclt  8677  recgt0i  8683  recgt0ii  8684  recgt0d  8711  nnrecgt0  8777  nnrecl  8994
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