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Theorem qbtwnrelemcalc 9342
Description: Lemma for qbtwnre 9343. 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 8176 . . . . 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 8119 . . . . . . 7  |-  ( ph  ->  N  e.  RR )
62, 5remulcld 7211 . . . . . 6  |-  ( ph  ->  ( 2  x.  N
)  e.  RR )
73, 6remulcld 7211 . . . . 5  |-  ( ph  ->  ( B  x.  (
2  x.  N ) )  e.  RR )
8 qbtwnrelemcalc.a . . . . . 6  |-  ( ph  ->  A  e.  RR )
98, 6remulcld 7211 . . . . 5  |-  ( ph  ->  ( A  x.  (
2  x.  N ) )  e.  RR )
107, 9resubcld 7552 . . . 4  |-  ( ph  ->  ( ( B  x.  ( 2  x.  N
) )  -  ( A  x.  ( 2  x.  N ) ) )  e.  RR )
11 qbtwnrelemcalc.m . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
1211zred 8550 . . . . 5  |-  ( ph  ->  M  e.  RR )
137, 12resubcld 7552 . . . 4  |-  ( ph  ->  ( ( B  x.  ( 2  x.  N
) )  -  M
)  e.  RR )
14 2t1e2 8252 . . . . . . . . 9  |-  ( 2  x.  1 )  =  2
1514oveq1i 5553 . . . . . . . 8  |-  ( ( 2  x.  1 )  /  ( 2  x.  N ) )  =  ( 2  /  (
2  x.  N ) )
16 1cnd 7197 . . . . . . . . 9  |-  ( ph  ->  1  e.  CC )
175recnd 7209 . . . . . . . . 9  |-  ( ph  ->  N  e.  CC )
182recnd 7209 . . . . . . . . 9  |-  ( ph  ->  2  e.  CC )
194nnap0d 8151 . . . . . . . . 9  |-  ( ph  ->  N #  0 )
20 2ap0 8199 . . . . . . . . . 10  |-  2 #  0
2120a1i 9 . . . . . . . . 9  |-  ( ph  ->  2 #  0 )
2216, 17, 18, 19, 21divcanap5d 7970 . . . . . . . 8  |-  ( ph  ->  ( ( 2  x.  1 )  /  (
2  x.  N ) )  =  ( 1  /  N ) )
2315, 22syl5eqr 2128 . . . . . . 7  |-  ( ph  ->  ( 2  /  (
2  x.  N ) )  =  ( 1  /  N ) )
24 qbtwnrelemcalc.1n . . . . . . 7  |-  ( ph  ->  ( 1  /  N
)  <  ( B  -  A ) )
2523, 24eqbrtrd 3813 . . . . . 6  |-  ( ph  ->  ( 2  /  (
2  x.  N ) )  <  ( B  -  A ) )
263, 8resubcld 7552 . . . . . . 7  |-  ( ph  ->  ( B  -  A
)  e.  RR )
27 2rp 8820 . . . . . . . . 9  |-  2  e.  RR+
2827a1i 9 . . . . . . . 8  |-  ( ph  ->  2  e.  RR+ )
294nnrpd 8853 . . . . . . . 8  |-  ( ph  ->  N  e.  RR+ )
3028, 29rpmulcld 8871 . . . . . . 7  |-  ( ph  ->  ( 2  x.  N
)  e.  RR+ )
312, 26, 30ltdivmul2d 8907 . . . . . 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 7209 . . . . . 6  |-  ( ph  ->  B  e.  CC )
348recnd 7209 . . . . . 6  |-  ( ph  ->  A  e.  CC )
3518, 17mulcld 7201 . . . . . 6  |-  ( ph  ->  ( 2  x.  N
)  e.  CC )
3633, 34, 35subdird 7586 . . . . 5  |-  ( ph  ->  ( ( B  -  A )  x.  (
2  x.  N ) )  =  ( ( B  x.  ( 2  x.  N ) )  -  ( A  x.  ( 2  x.  N
) ) ) )
3732, 36breqtrd 3817 . . . 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 7725 . . . 4  |-  ( ph  ->  ( ( B  x.  ( 2  x.  N
) )  -  ( A  x.  ( 2  x.  N ) ) )  <  ( ( B  x.  ( 2  x.  N ) )  -  M ) )
402, 10, 13, 37, 39lttrd 7302 . . 3  |-  ( ph  ->  2  <  ( ( B  x.  ( 2  x.  N ) )  -  M ) )
4112, 2, 7ltaddsub2d 7713 . . 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 7210 . . 3  |-  ( ph  ->  ( M  +  2 )  e.  RR )
4443, 3, 30ltdivmul2d 8907 . 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 1434   class class class wbr 3793  (class class class)co 5543   RRcr 7042   0cc0 7043   1c1 7044    + caddc 7046    x. cmul 7048    < clt 7215    - cmin 7346   # cap 7748    / cdiv 7827   NNcn 8106   2c2 8156   ZZcz 8432   RR+crp 8815
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 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-precex 7148  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-apti 7153  ax-pre-ltadd 7154  ax-pre-mulgt0 7155  ax-pre-mulext 7156
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-br 3794  df-opab 3848  df-id 4056  df-po 4059  df-iso 4060  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-reap 7742  df-ap 7749  df-div 7828  df-inn 8107  df-2 8165  df-z 8433  df-rp 8816
This theorem is referenced by:  qbtwnre  9343
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