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Theorem qbtwnrelemcalc 10229
Description: Lemma for qbtwnre 10230. 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 8965 . . . . 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 8908 . . . . . . 7  |-  ( ph  ->  N  e.  RR )
62, 5remulcld 7965 . . . . . 6  |-  ( ph  ->  ( 2  x.  N
)  e.  RR )
73, 6remulcld 7965 . . . . 5  |-  ( ph  ->  ( B  x.  (
2  x.  N ) )  e.  RR )
8 qbtwnrelemcalc.a . . . . . 6  |-  ( ph  ->  A  e.  RR )
98, 6remulcld 7965 . . . . 5  |-  ( ph  ->  ( A  x.  (
2  x.  N ) )  e.  RR )
107, 9resubcld 8315 . . . 4  |-  ( ph  ->  ( ( B  x.  ( 2  x.  N
) )  -  ( A  x.  ( 2  x.  N ) ) )  e.  RR )
11 qbtwnrelemcalc.m . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
1211zred 9351 . . . . 5  |-  ( ph  ->  M  e.  RR )
137, 12resubcld 8315 . . . 4  |-  ( ph  ->  ( ( B  x.  ( 2  x.  N
) )  -  M
)  e.  RR )
14 2t1e2 9048 . . . . . . . . 9  |-  ( 2  x.  1 )  =  2
1514oveq1i 5878 . . . . . . . 8  |-  ( ( 2  x.  1 )  /  ( 2  x.  N ) )  =  ( 2  /  (
2  x.  N ) )
16 1cnd 7951 . . . . . . . . 9  |-  ( ph  ->  1  e.  CC )
175recnd 7963 . . . . . . . . 9  |-  ( ph  ->  N  e.  CC )
182recnd 7963 . . . . . . . . 9  |-  ( ph  ->  2  e.  CC )
194nnap0d 8941 . . . . . . . . 9  |-  ( ph  ->  N #  0 )
20 2ap0 8988 . . . . . . . . . 10  |-  2 #  0
2120a1i 9 . . . . . . . . 9  |-  ( ph  ->  2 #  0 )
2216, 17, 18, 19, 21divcanap5d 8750 . . . . . . . 8  |-  ( ph  ->  ( ( 2  x.  1 )  /  (
2  x.  N ) )  =  ( 1  /  N ) )
2315, 22eqtr3id 2224 . . . . . . 7  |-  ( ph  ->  ( 2  /  (
2  x.  N ) )  =  ( 1  /  N ) )
24 qbtwnrelemcalc.1n . . . . . . 7  |-  ( ph  ->  ( 1  /  N
)  <  ( B  -  A ) )
2523, 24eqbrtrd 4022 . . . . . 6  |-  ( ph  ->  ( 2  /  (
2  x.  N ) )  <  ( B  -  A ) )
263, 8resubcld 8315 . . . . . . 7  |-  ( ph  ->  ( B  -  A
)  e.  RR )
27 2rp 9632 . . . . . . . . 9  |-  2  e.  RR+
2827a1i 9 . . . . . . . 8  |-  ( ph  ->  2  e.  RR+ )
294nnrpd 9668 . . . . . . . 8  |-  ( ph  ->  N  e.  RR+ )
3028, 29rpmulcld 9687 . . . . . . 7  |-  ( ph  ->  ( 2  x.  N
)  e.  RR+ )
312, 26, 30ltdivmul2d 9723 . . . . . 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 7963 . . . . . 6  |-  ( ph  ->  B  e.  CC )
348recnd 7963 . . . . . 6  |-  ( ph  ->  A  e.  CC )
3518, 17mulcld 7955 . . . . . 6  |-  ( ph  ->  ( 2  x.  N
)  e.  CC )
3633, 34, 35subdird 8349 . . . . 5  |-  ( ph  ->  ( ( B  -  A )  x.  (
2  x.  N ) )  =  ( ( B  x.  ( 2  x.  N ) )  -  ( A  x.  ( 2  x.  N
) ) ) )
3732, 36breqtrd 4026 . . . 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 8492 . . . 4  |-  ( ph  ->  ( ( B  x.  ( 2  x.  N
) )  -  ( A  x.  ( 2  x.  N ) ) )  <  ( ( B  x.  ( 2  x.  N ) )  -  M ) )
402, 10, 13, 37, 39lttrd 8060 . . 3  |-  ( ph  ->  2  <  ( ( B  x.  ( 2  x.  N ) )  -  M ) )
4112, 2, 7ltaddsub2d 8480 . . 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 7964 . . 3  |-  ( ph  ->  ( M  +  2 )  e.  RR )
4443, 3, 30ltdivmul2d 9723 . 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 2148   class class class wbr 4000  (class class class)co 5868   RRcr 7788   0cc0 7789   1c1 7790    + caddc 7792    x. cmul 7794    < clt 7969    - cmin 8105   # cap 8515    / cdiv 8605   NNcn 8895   2c2 8946   ZZcz 9229   RR+crp 9627
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-mulrcl 7888  ax-addcom 7889  ax-mulcom 7890  ax-addass 7891  ax-mulass 7892  ax-distr 7893  ax-i2m1 7894  ax-0lt1 7895  ax-1rid 7896  ax-0id 7897  ax-rnegex 7898  ax-precex 7899  ax-cnre 7900  ax-pre-ltirr 7901  ax-pre-ltwlin 7902  ax-pre-lttrn 7903  ax-pre-apti 7904  ax-pre-ltadd 7905  ax-pre-mulgt0 7906  ax-pre-mulext 7907
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-id 4289  df-po 4292  df-iso 4293  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-iota 5173  df-fun 5213  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975  df-sub 8107  df-neg 8108  df-reap 8509  df-ap 8516  df-div 8606  df-inn 8896  df-2 8954  df-z 9230  df-rp 9628
This theorem is referenced by:  qbtwnre  10230
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