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Theorem qbtwnrelemcalc 9632
Description: Lemma for qbtwnre 9633. 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 8463 . . . . 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 8407 . . . . . . 7  |-  ( ph  ->  N  e.  RR )
62, 5remulcld 7497 . . . . . 6  |-  ( ph  ->  ( 2  x.  N
)  e.  RR )
73, 6remulcld 7497 . . . . 5  |-  ( ph  ->  ( B  x.  (
2  x.  N ) )  e.  RR )
8 qbtwnrelemcalc.a . . . . . 6  |-  ( ph  ->  A  e.  RR )
98, 6remulcld 7497 . . . . 5  |-  ( ph  ->  ( A  x.  (
2  x.  N ) )  e.  RR )
107, 9resubcld 7838 . . . 4  |-  ( ph  ->  ( ( B  x.  ( 2  x.  N
) )  -  ( A  x.  ( 2  x.  N ) ) )  e.  RR )
11 qbtwnrelemcalc.m . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
1211zred 8838 . . . . 5  |-  ( ph  ->  M  e.  RR )
137, 12resubcld 7838 . . . 4  |-  ( ph  ->  ( ( B  x.  ( 2  x.  N
) )  -  M
)  e.  RR )
14 2t1e2 8539 . . . . . . . . 9  |-  ( 2  x.  1 )  =  2
1514oveq1i 5644 . . . . . . . 8  |-  ( ( 2  x.  1 )  /  ( 2  x.  N ) )  =  ( 2  /  (
2  x.  N ) )
16 1cnd 7483 . . . . . . . . 9  |-  ( ph  ->  1  e.  CC )
175recnd 7495 . . . . . . . . 9  |-  ( ph  ->  N  e.  CC )
182recnd 7495 . . . . . . . . 9  |-  ( ph  ->  2  e.  CC )
194nnap0d 8439 . . . . . . . . 9  |-  ( ph  ->  N #  0 )
20 2ap0 8486 . . . . . . . . . 10  |-  2 #  0
2120a1i 9 . . . . . . . . 9  |-  ( ph  ->  2 #  0 )
2216, 17, 18, 19, 21divcanap5d 8256 . . . . . . . 8  |-  ( ph  ->  ( ( 2  x.  1 )  /  (
2  x.  N ) )  =  ( 1  /  N ) )
2315, 22syl5eqr 2134 . . . . . . 7  |-  ( ph  ->  ( 2  /  (
2  x.  N ) )  =  ( 1  /  N ) )
24 qbtwnrelemcalc.1n . . . . . . 7  |-  ( ph  ->  ( 1  /  N
)  <  ( B  -  A ) )
2523, 24eqbrtrd 3857 . . . . . 6  |-  ( ph  ->  ( 2  /  (
2  x.  N ) )  <  ( B  -  A ) )
263, 8resubcld 7838 . . . . . . 7  |-  ( ph  ->  ( B  -  A
)  e.  RR )
27 2rp 9108 . . . . . . . . 9  |-  2  e.  RR+
2827a1i 9 . . . . . . . 8  |-  ( ph  ->  2  e.  RR+ )
294nnrpd 9141 . . . . . . . 8  |-  ( ph  ->  N  e.  RR+ )
3028, 29rpmulcld 9159 . . . . . . 7  |-  ( ph  ->  ( 2  x.  N
)  e.  RR+ )
312, 26, 30ltdivmul2d 9195 . . . . . 6  |-  ( ph  ->  ( ( 2  / 
( 2  x.  N
) )  <  ( B  -  A )  <->  2  <  ( ( B  -  A )  x.  ( 2  x.  N
) ) ) )
3225, 31mpbid 145 . . . . 5  |-  ( ph  ->  2  <  ( ( B  -  A )  x.  ( 2  x.  N ) ) )
333recnd 7495 . . . . . 6  |-  ( ph  ->  B  e.  CC )
348recnd 7495 . . . . . 6  |-  ( ph  ->  A  e.  CC )
3518, 17mulcld 7487 . . . . . 6  |-  ( ph  ->  ( 2  x.  N
)  e.  CC )
3633, 34, 35subdird 7872 . . . . 5  |-  ( ph  ->  ( ( B  -  A )  x.  (
2  x.  N ) )  =  ( ( B  x.  ( 2  x.  N ) )  -  ( A  x.  ( 2  x.  N
) ) ) )
3732, 36breqtrd 3861 . . . 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 8011 . . . 4  |-  ( ph  ->  ( ( B  x.  ( 2  x.  N
) )  -  ( A  x.  ( 2  x.  N ) ) )  <  ( ( B  x.  ( 2  x.  N ) )  -  M ) )
402, 10, 13, 37, 39lttrd 7588 . . 3  |-  ( ph  ->  2  <  ( ( B  x.  ( 2  x.  N ) )  -  M ) )
4112, 2, 7ltaddsub2d 7999 . . 3  |-  ( ph  ->  ( ( M  + 
2 )  <  ( B  x.  ( 2  x.  N ) )  <->  2  <  ( ( B  x.  ( 2  x.  N ) )  -  M ) ) )
4240, 41mpbird 165 . 2  |-  ( ph  ->  ( M  +  2 )  <  ( B  x.  ( 2  x.  N ) ) )
4312, 2readdcld 7496 . . 3  |-  ( ph  ->  ( M  +  2 )  e.  RR )
4443, 3, 30ltdivmul2d 9195 . 2  |-  ( ph  ->  ( ( ( M  +  2 )  / 
( 2  x.  N
) )  <  B  <->  ( M  +  2 )  <  ( B  x.  ( 2  x.  N
) ) ) )
4542, 44mpbird 165 1  |-  ( ph  ->  ( ( M  + 
2 )  /  (
2  x.  N ) )  <  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1438   class class class wbr 3837  (class class class)co 5634   RRcr 7328   0cc0 7329   1c1 7330    + caddc 7332    x. cmul 7334    < clt 7501    - cmin 7632   # cap 8034    / cdiv 8113   NNcn 8394   2c2 8444   ZZcz 8720   RR+crp 9103
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-id 4111  df-po 4114  df-iso 4115  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-2 8452  df-z 8721  df-rp 9104
This theorem is referenced by:  qbtwnre  9633
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