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Theorem cvg1nlemcxze 10722
Description: Lemma for cvg1n 10726. Rearranging an expression related to the rate of convergence. (Contributed by Jim Kingdon, 6-Aug-2021.)
Hypotheses
Ref Expression
cvg1nlemcxze.c  |-  ( ph  ->  C  e.  RR+ )
cvg1nlemcxze.x  |-  ( ph  ->  X  e.  RR+ )
cvg1nlemcxze.z  |-  ( ph  ->  Z  e.  NN )
cvg1nlemcxze.e  |-  ( ph  ->  E  e.  NN )
cvg1nlemcxze.a  |-  ( ph  ->  A  e.  NN )
cvg1nlemcxze.1  |-  ( ph  ->  ( ( ( ( C  x.  2 )  /  X )  /  Z )  +  A
)  <  E )
Assertion
Ref Expression
cvg1nlemcxze  |-  ( ph  ->  ( C  /  ( E  x.  Z )
)  <  ( X  /  2 ) )

Proof of Theorem cvg1nlemcxze
StepHypRef Expression
1 cvg1nlemcxze.c . . . . . . . 8  |-  ( ph  ->  C  e.  RR+ )
21rpcnd 9453 . . . . . . 7  |-  ( ph  ->  C  e.  CC )
3 2cnd 8761 . . . . . . 7  |-  ( ph  ->  2  e.  CC )
4 cvg1nlemcxze.x . . . . . . . 8  |-  ( ph  ->  X  e.  RR+ )
54rpcnd 9453 . . . . . . 7  |-  ( ph  ->  X  e.  CC )
64rpap0d 9457 . . . . . . 7  |-  ( ph  ->  X #  0 )
72, 3, 5, 6div23apd 8556 . . . . . 6  |-  ( ph  ->  ( ( C  x.  2 )  /  X
)  =  ( ( C  /  X )  x.  2 ) )
8 2rp 9414 . . . . . . . . . . . . 13  |-  2  e.  RR+
98a1i 9 . . . . . . . . . . . 12  |-  ( ph  ->  2  e.  RR+ )
101, 9rpmulcld 9468 . . . . . . . . . . 11  |-  ( ph  ->  ( C  x.  2 )  e.  RR+ )
1110, 4rpdivcld 9469 . . . . . . . . . 10  |-  ( ph  ->  ( ( C  x.  2 )  /  X
)  e.  RR+ )
12 cvg1nlemcxze.z . . . . . . . . . . 11  |-  ( ph  ->  Z  e.  NN )
1312nnrpd 9450 . . . . . . . . . 10  |-  ( ph  ->  Z  e.  RR+ )
1411, 13rpdivcld 9469 . . . . . . . . 9  |-  ( ph  ->  ( ( ( C  x.  2 )  /  X )  /  Z
)  e.  RR+ )
1514rpred 9451 . . . . . . . 8  |-  ( ph  ->  ( ( ( C  x.  2 )  /  X )  /  Z
)  e.  RR )
16 cvg1nlemcxze.a . . . . . . . . . 10  |-  ( ph  ->  A  e.  NN )
1716nnred 8701 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR )
1815, 17readdcld 7763 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( C  x.  2 )  /  X )  /  Z )  +  A
)  e.  RR )
19 cvg1nlemcxze.e . . . . . . . . 9  |-  ( ph  ->  E  e.  NN )
2019nnred 8701 . . . . . . . 8  |-  ( ph  ->  E  e.  RR )
2116nnrpd 9450 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR+ )
2215, 21ltaddrpd 9485 . . . . . . . 8  |-  ( ph  ->  ( ( ( C  x.  2 )  /  X )  /  Z
)  <  ( (
( ( C  x.  2 )  /  X
)  /  Z )  +  A ) )
23 cvg1nlemcxze.1 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( C  x.  2 )  /  X )  /  Z )  +  A
)  <  E )
2415, 18, 20, 22, 23lttrd 7856 . . . . . . 7  |-  ( ph  ->  ( ( ( C  x.  2 )  /  X )  /  Z
)  <  E )
2511rpred 9451 . . . . . . . 8  |-  ( ph  ->  ( ( C  x.  2 )  /  X
)  e.  RR )
2625, 20, 13ltdivmul2d 9504 . . . . . . 7  |-  ( ph  ->  ( ( ( ( C  x.  2 )  /  X )  /  Z )  <  E  <->  ( ( C  x.  2 )  /  X )  <  ( E  x.  Z ) ) )
2724, 26mpbid 146 . . . . . 6  |-  ( ph  ->  ( ( C  x.  2 )  /  X
)  <  ( E  x.  Z ) )
287, 27eqbrtrrd 3922 . . . . 5  |-  ( ph  ->  ( ( C  /  X )  x.  2 )  <  ( E  x.  Z ) )
291rpred 9451 . . . . . . 7  |-  ( ph  ->  C  e.  RR )
3029, 4rerpdivcld 9483 . . . . . 6  |-  ( ph  ->  ( C  /  X
)  e.  RR )
3119, 12nnmulcld 8737 . . . . . . 7  |-  ( ph  ->  ( E  x.  Z
)  e.  NN )
3231nnred 8701 . . . . . 6  |-  ( ph  ->  ( E  x.  Z
)  e.  RR )
3330, 32, 9ltmuldivd 9499 . . . . 5  |-  ( ph  ->  ( ( ( C  /  X )  x.  2 )  <  ( E  x.  Z )  <->  ( C  /  X )  <  ( ( E  x.  Z )  / 
2 ) ) )
3428, 33mpbid 146 . . . 4  |-  ( ph  ->  ( C  /  X
)  <  ( ( E  x.  Z )  /  2 ) )
3529, 9, 32, 4lt2mul2divd 9520 . . . 4  |-  ( ph  ->  ( ( C  x.  2 )  <  (
( E  x.  Z
)  x.  X )  <-> 
( C  /  X
)  <  ( ( E  x.  Z )  /  2 ) ) )
3634, 35mpbird 166 . . 3  |-  ( ph  ->  ( C  x.  2 )  <  ( ( E  x.  Z )  x.  X ) )
3731nncnd 8702 . . . 4  |-  ( ph  ->  ( E  x.  Z
)  e.  CC )
3837, 5mulcomd 7755 . . 3  |-  ( ph  ->  ( ( E  x.  Z )  x.  X
)  =  ( X  x.  ( E  x.  Z ) ) )
3936, 38breqtrd 3924 . 2  |-  ( ph  ->  ( C  x.  2 )  <  ( X  x.  ( E  x.  Z ) ) )
404rpred 9451 . . 3  |-  ( ph  ->  X  e.  RR )
4131nnrpd 9450 . . 3  |-  ( ph  ->  ( E  x.  Z
)  e.  RR+ )
4229, 9, 40, 41lt2mul2divd 9520 . 2  |-  ( ph  ->  ( ( C  x.  2 )  <  ( X  x.  ( E  x.  Z ) )  <->  ( C  /  ( E  x.  Z ) )  < 
( X  /  2
) ) )
4339, 42mpbid 146 1  |-  ( ph  ->  ( C  /  ( E  x.  Z )
)  <  ( X  /  2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1465   class class class wbr 3899  (class class class)co 5742    + caddc 7591    x. cmul 7593    < clt 7768    / cdiv 8400   NNcn 8688   2c2 8739   RR+crp 9409
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-id 4185  df-po 4188  df-iso 4189  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-div 8401  df-inn 8689  df-2 8747  df-rp 9410
This theorem is referenced by:  cvg1nlemres  10725
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