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Theorem qbtwnrelemcalc 9973
Description: Lemma for qbtwnre 9974. 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 8747 . . . . 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 8690 . . . . . . 7  |-  ( ph  ->  N  e.  RR )
62, 5remulcld 7760 . . . . . 6  |-  ( ph  ->  ( 2  x.  N
)  e.  RR )
73, 6remulcld 7760 . . . . 5  |-  ( ph  ->  ( B  x.  (
2  x.  N ) )  e.  RR )
8 qbtwnrelemcalc.a . . . . . 6  |-  ( ph  ->  A  e.  RR )
98, 6remulcld 7760 . . . . 5  |-  ( ph  ->  ( A  x.  (
2  x.  N ) )  e.  RR )
107, 9resubcld 8107 . . . 4  |-  ( ph  ->  ( ( B  x.  ( 2  x.  N
) )  -  ( A  x.  ( 2  x.  N ) ) )  e.  RR )
11 qbtwnrelemcalc.m . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
1211zred 9124 . . . . 5  |-  ( ph  ->  M  e.  RR )
137, 12resubcld 8107 . . . 4  |-  ( ph  ->  ( ( B  x.  ( 2  x.  N
) )  -  M
)  e.  RR )
14 2t1e2 8824 . . . . . . . . 9  |-  ( 2  x.  1 )  =  2
1514oveq1i 5750 . . . . . . . 8  |-  ( ( 2  x.  1 )  /  ( 2  x.  N ) )  =  ( 2  /  (
2  x.  N ) )
16 1cnd 7746 . . . . . . . . 9  |-  ( ph  ->  1  e.  CC )
175recnd 7758 . . . . . . . . 9  |-  ( ph  ->  N  e.  CC )
182recnd 7758 . . . . . . . . 9  |-  ( ph  ->  2  e.  CC )
194nnap0d 8723 . . . . . . . . 9  |-  ( ph  ->  N #  0 )
20 2ap0 8770 . . . . . . . . . 10  |-  2 #  0
2120a1i 9 . . . . . . . . 9  |-  ( ph  ->  2 #  0 )
2216, 17, 18, 19, 21divcanap5d 8537 . . . . . . . 8  |-  ( ph  ->  ( ( 2  x.  1 )  /  (
2  x.  N ) )  =  ( 1  /  N ) )
2315, 22syl5eqr 2162 . . . . . . 7  |-  ( ph  ->  ( 2  /  (
2  x.  N ) )  =  ( 1  /  N ) )
24 qbtwnrelemcalc.1n . . . . . . 7  |-  ( ph  ->  ( 1  /  N
)  <  ( B  -  A ) )
2523, 24eqbrtrd 3918 . . . . . 6  |-  ( ph  ->  ( 2  /  (
2  x.  N ) )  <  ( B  -  A ) )
263, 8resubcld 8107 . . . . . . 7  |-  ( ph  ->  ( B  -  A
)  e.  RR )
27 2rp 9395 . . . . . . . . 9  |-  2  e.  RR+
2827a1i 9 . . . . . . . 8  |-  ( ph  ->  2  e.  RR+ )
294nnrpd 9428 . . . . . . . 8  |-  ( ph  ->  N  e.  RR+ )
3028, 29rpmulcld 9446 . . . . . . 7  |-  ( ph  ->  ( 2  x.  N
)  e.  RR+ )
312, 26, 30ltdivmul2d 9482 . . . . . 6  |-  ( ph  ->  ( ( 2  / 
( 2  x.  N
) )  <  ( B  -  A )  <->  2  <  ( ( B  -  A )  x.  ( 2  x.  N
) ) ) )
3225, 31mpbid 146 . . . . 5  |-  ( ph  ->  2  <  ( ( B  -  A )  x.  ( 2  x.  N ) ) )
333recnd 7758 . . . . . 6  |-  ( ph  ->  B  e.  CC )
348recnd 7758 . . . . . 6  |-  ( ph  ->  A  e.  CC )
3518, 17mulcld 7750 . . . . . 6  |-  ( ph  ->  ( 2  x.  N
)  e.  CC )
3633, 34, 35subdird 8141 . . . . 5  |-  ( ph  ->  ( ( B  -  A )  x.  (
2  x.  N ) )  =  ( ( B  x.  ( 2  x.  N ) )  -  ( A  x.  ( 2  x.  N
) ) ) )
3732, 36breqtrd 3922 . . . 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 8283 . . . 4  |-  ( ph  ->  ( ( B  x.  ( 2  x.  N
) )  -  ( A  x.  ( 2  x.  N ) ) )  <  ( ( B  x.  ( 2  x.  N ) )  -  M ) )
402, 10, 13, 37, 39lttrd 7852 . . 3  |-  ( ph  ->  2  <  ( ( B  x.  ( 2  x.  N ) )  -  M ) )
4112, 2, 7ltaddsub2d 8271 . . 3  |-  ( ph  ->  ( ( M  + 
2 )  <  ( B  x.  ( 2  x.  N ) )  <->  2  <  ( ( B  x.  ( 2  x.  N ) )  -  M ) ) )
4240, 41mpbird 166 . 2  |-  ( ph  ->  ( M  +  2 )  <  ( B  x.  ( 2  x.  N ) ) )
4312, 2readdcld 7759 . . 3  |-  ( ph  ->  ( M  +  2 )  e.  RR )
4443, 3, 30ltdivmul2d 9482 . 2  |-  ( ph  ->  ( ( ( M  +  2 )  / 
( 2  x.  N
) )  <  B  <->  ( M  +  2 )  <  ( B  x.  ( 2  x.  N
) ) ) )
4542, 44mpbird 166 1  |-  ( ph  ->  ( ( M  + 
2 )  /  (
2  x.  N ) )  <  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1463   class class class wbr 3897  (class class class)co 5740   RRcr 7583   0cc0 7584   1c1 7585    + caddc 7587    x. cmul 7589    < clt 7764    - cmin 7897   # cap 8306    / cdiv 8392   NNcn 8677   2c2 8728   ZZcz 9005   RR+crp 9390
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-opab 3958  df-id 4183  df-po 4186  df-iso 4187  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8393  df-inn 8678  df-2 8736  df-z 9006  df-rp 9391
This theorem is referenced by:  qbtwnre  9974
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