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Theorem qbtwnrelemcalc 10281
Description: Lemma for qbtwnre 10282. Calculations involved in showing the constructed rational number is less than 
B. (Contributed by Jim Kingdon, 14-Oct-2021.)
Hypotheses
Ref Expression
qbtwnrelemcalc.m  |-  ( ph  ->  M  e.  ZZ )
qbtwnrelemcalc.n  |-  ( ph  ->  N  e.  NN )
qbtwnrelemcalc.a  |-  ( ph  ->  A  e.  RR )
qbtwnrelemcalc.b  |-  ( ph  ->  B  e.  RR )
qbtwnrelemcalc.lt  |-  ( ph  ->  M  <  ( A  x.  ( 2  x.  N ) ) )
qbtwnrelemcalc.1n  |-  ( ph  ->  ( 1  /  N
)  <  ( B  -  A ) )
Assertion
Ref Expression
qbtwnrelemcalc  |-  ( ph  ->  ( ( M  + 
2 )  /  (
2  x.  N ) )  <  B )

Proof of Theorem qbtwnrelemcalc
StepHypRef Expression
1 2re 9014 . . . . 5  |-  2  e.  RR
21a1i 9 . . . 4  |-  ( ph  ->  2  e.  RR )
3 qbtwnrelemcalc.b . . . . . 6  |-  ( ph  ->  B  e.  RR )
4 qbtwnrelemcalc.n . . . . . . . 8  |-  ( ph  ->  N  e.  NN )
54nnred 8957 . . . . . . 7  |-  ( ph  ->  N  e.  RR )
62, 5remulcld 8013 . . . . . 6  |-  ( ph  ->  ( 2  x.  N
)  e.  RR )
73, 6remulcld 8013 . . . . 5  |-  ( ph  ->  ( B  x.  (
2  x.  N ) )  e.  RR )
8 qbtwnrelemcalc.a . . . . . 6  |-  ( ph  ->  A  e.  RR )
98, 6remulcld 8013 . . . . 5  |-  ( ph  ->  ( A  x.  (
2  x.  N ) )  e.  RR )
107, 9resubcld 8363 . . . 4  |-  ( ph  ->  ( ( B  x.  ( 2  x.  N
) )  -  ( A  x.  ( 2  x.  N ) ) )  e.  RR )
11 qbtwnrelemcalc.m . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
1211zred 9400 . . . . 5  |-  ( ph  ->  M  e.  RR )
137, 12resubcld 8363 . . . 4  |-  ( ph  ->  ( ( B  x.  ( 2  x.  N
) )  -  M
)  e.  RR )
14 2t1e2 9097 . . . . . . . . 9  |-  ( 2  x.  1 )  =  2
1514oveq1i 5902 . . . . . . . 8  |-  ( ( 2  x.  1 )  /  ( 2  x.  N ) )  =  ( 2  /  (
2  x.  N ) )
16 1cnd 7998 . . . . . . . . 9  |-  ( ph  ->  1  e.  CC )
175recnd 8011 . . . . . . . . 9  |-  ( ph  ->  N  e.  CC )
182recnd 8011 . . . . . . . . 9  |-  ( ph  ->  2  e.  CC )
194nnap0d 8990 . . . . . . . . 9  |-  ( ph  ->  N #  0 )
20 2ap0 9037 . . . . . . . . . 10  |-  2 #  0
2120a1i 9 . . . . . . . . 9  |-  ( ph  ->  2 #  0 )
2216, 17, 18, 19, 21divcanap5d 8799 . . . . . . . 8  |-  ( ph  ->  ( ( 2  x.  1 )  /  (
2  x.  N ) )  =  ( 1  /  N ) )
2315, 22eqtr3id 2236 . . . . . . 7  |-  ( ph  ->  ( 2  /  (
2  x.  N ) )  =  ( 1  /  N ) )
24 qbtwnrelemcalc.1n . . . . . . 7  |-  ( ph  ->  ( 1  /  N
)  <  ( B  -  A ) )
2523, 24eqbrtrd 4040 . . . . . 6  |-  ( ph  ->  ( 2  /  (
2  x.  N ) )  <  ( B  -  A ) )
263, 8resubcld 8363 . . . . . . 7  |-  ( ph  ->  ( B  -  A
)  e.  RR )
27 2rp 9683 . . . . . . . . 9  |-  2  e.  RR+
2827a1i 9 . . . . . . . 8  |-  ( ph  ->  2  e.  RR+ )
294nnrpd 9719 . . . . . . . 8  |-  ( ph  ->  N  e.  RR+ )
3028, 29rpmulcld 9738 . . . . . . 7  |-  ( ph  ->  ( 2  x.  N
)  e.  RR+ )
312, 26, 30ltdivmul2d 9774 . . . . . 6  |-  ( ph  ->  ( ( 2  / 
( 2  x.  N
) )  <  ( B  -  A )  <->  2  <  ( ( B  -  A )  x.  ( 2  x.  N
) ) ) )
3225, 31mpbid 147 . . . . 5  |-  ( ph  ->  2  <  ( ( B  -  A )  x.  ( 2  x.  N ) ) )
333recnd 8011 . . . . . 6  |-  ( ph  ->  B  e.  CC )
348recnd 8011 . . . . . 6  |-  ( ph  ->  A  e.  CC )
3518, 17mulcld 8003 . . . . . 6  |-  ( ph  ->  ( 2  x.  N
)  e.  CC )
3633, 34, 35subdird 8397 . . . . 5  |-  ( ph  ->  ( ( B  -  A )  x.  (
2  x.  N ) )  =  ( ( B  x.  ( 2  x.  N ) )  -  ( A  x.  ( 2  x.  N
) ) ) )
3732, 36breqtrd 4044 . . . 4  |-  ( ph  ->  2  <  ( ( B  x.  ( 2  x.  N ) )  -  ( A  x.  ( 2  x.  N
) ) ) )
38 qbtwnrelemcalc.lt . . . . 5  |-  ( ph  ->  M  <  ( A  x.  ( 2  x.  N ) ) )
3912, 9, 7, 38ltsub2dd 8540 . . . 4  |-  ( ph  ->  ( ( B  x.  ( 2  x.  N
) )  -  ( A  x.  ( 2  x.  N ) ) )  <  ( ( B  x.  ( 2  x.  N ) )  -  M ) )
402, 10, 13, 37, 39lttrd 8108 . . 3  |-  ( ph  ->  2  <  ( ( B  x.  ( 2  x.  N ) )  -  M ) )
4112, 2, 7ltaddsub2d 8528 . . 3  |-  ( ph  ->  ( ( M  + 
2 )  <  ( B  x.  ( 2  x.  N ) )  <->  2  <  ( ( B  x.  ( 2  x.  N ) )  -  M ) ) )
4240, 41mpbird 167 . 2  |-  ( ph  ->  ( M  +  2 )  <  ( B  x.  ( 2  x.  N ) ) )
4312, 2readdcld 8012 . . 3  |-  ( ph  ->  ( M  +  2 )  e.  RR )
4443, 3, 30ltdivmul2d 9774 . 2  |-  ( ph  ->  ( ( ( M  +  2 )  / 
( 2  x.  N
) )  <  B  <->  ( M  +  2 )  <  ( B  x.  ( 2  x.  N
) ) ) )
4542, 44mpbird 167 1  |-  ( ph  ->  ( ( M  + 
2 )  /  (
2  x.  N ) )  <  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2160   class class class wbr 4018  (class class class)co 5892   RRcr 7835   0cc0 7836   1c1 7837    + caddc 7839    x. cmul 7841    < clt 8017    - cmin 8153   # cap 8563    / cdiv 8654   NNcn 8944   2c2 8995   ZZcz 9278   RR+crp 9678
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7927  ax-resscn 7928  ax-1cn 7929  ax-1re 7930  ax-icn 7931  ax-addcl 7932  ax-addrcl 7933  ax-mulcl 7934  ax-mulrcl 7935  ax-addcom 7936  ax-mulcom 7937  ax-addass 7938  ax-mulass 7939  ax-distr 7940  ax-i2m1 7941  ax-0lt1 7942  ax-1rid 7943  ax-0id 7944  ax-rnegex 7945  ax-precex 7946  ax-cnre 7947  ax-pre-ltirr 7948  ax-pre-ltwlin 7949  ax-pre-lttrn 7950  ax-pre-apti 7951  ax-pre-ltadd 7952  ax-pre-mulgt0 7953  ax-pre-mulext 7954
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-id 4308  df-po 4311  df-iso 4312  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-iota 5193  df-fun 5234  df-fv 5240  df-riota 5848  df-ov 5895  df-oprab 5896  df-mpo 5897  df-pnf 8019  df-mnf 8020  df-xr 8021  df-ltxr 8022  df-le 8023  df-sub 8155  df-neg 8156  df-reap 8557  df-ap 8564  df-div 8655  df-inn 8945  df-2 9003  df-z 9279  df-rp 9679
This theorem is referenced by:  qbtwnre  10282
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