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Theorem cvg1nlemcxze 10761
Description: Lemma for cvg1n 10765. 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 9492 . . . . . . 7  |-  ( ph  ->  C  e.  CC )
3 2cnd 8800 . . . . . . 7  |-  ( ph  ->  2  e.  CC )
4 cvg1nlemcxze.x . . . . . . . 8  |-  ( ph  ->  X  e.  RR+ )
54rpcnd 9492 . . . . . . 7  |-  ( ph  ->  X  e.  CC )
64rpap0d 9496 . . . . . . 7  |-  ( ph  ->  X #  0 )
72, 3, 5, 6div23apd 8595 . . . . . 6  |-  ( ph  ->  ( ( C  x.  2 )  /  X
)  =  ( ( C  /  X )  x.  2 ) )
8 2rp 9453 . . . . . . . . . . . . 13  |-  2  e.  RR+
98a1i 9 . . . . . . . . . . . 12  |-  ( ph  ->  2  e.  RR+ )
101, 9rpmulcld 9507 . . . . . . . . . . 11  |-  ( ph  ->  ( C  x.  2 )  e.  RR+ )
1110, 4rpdivcld 9508 . . . . . . . . . 10  |-  ( ph  ->  ( ( C  x.  2 )  /  X
)  e.  RR+ )
12 cvg1nlemcxze.z . . . . . . . . . . 11  |-  ( ph  ->  Z  e.  NN )
1312nnrpd 9489 . . . . . . . . . 10  |-  ( ph  ->  Z  e.  RR+ )
1411, 13rpdivcld 9508 . . . . . . . . 9  |-  ( ph  ->  ( ( ( C  x.  2 )  /  X )  /  Z
)  e.  RR+ )
1514rpred 9490 . . . . . . . 8  |-  ( ph  ->  ( ( ( C  x.  2 )  /  X )  /  Z
)  e.  RR )
16 cvg1nlemcxze.a . . . . . . . . . 10  |-  ( ph  ->  A  e.  NN )
1716nnred 8740 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR )
1815, 17readdcld 7802 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( C  x.  2 )  /  X )  /  Z )  +  A
)  e.  RR )
19 cvg1nlemcxze.e . . . . . . . . 9  |-  ( ph  ->  E  e.  NN )
2019nnred 8740 . . . . . . . 8  |-  ( ph  ->  E  e.  RR )
2116nnrpd 9489 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR+ )
2215, 21ltaddrpd 9524 . . . . . . . 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 7895 . . . . . . 7  |-  ( ph  ->  ( ( ( C  x.  2 )  /  X )  /  Z
)  <  E )
2511rpred 9490 . . . . . . . 8  |-  ( ph  ->  ( ( C  x.  2 )  /  X
)  e.  RR )
2625, 20, 13ltdivmul2d 9543 . . . . . . 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 3952 . . . . 5  |-  ( ph  ->  ( ( C  /  X )  x.  2 )  <  ( E  x.  Z ) )
291rpred 9490 . . . . . . 7  |-  ( ph  ->  C  e.  RR )
3029, 4rerpdivcld 9522 . . . . . 6  |-  ( ph  ->  ( C  /  X
)  e.  RR )
3119, 12nnmulcld 8776 . . . . . . 7  |-  ( ph  ->  ( E  x.  Z
)  e.  NN )
3231nnred 8740 . . . . . 6  |-  ( ph  ->  ( E  x.  Z
)  e.  RR )
3330, 32, 9ltmuldivd 9538 . . . . 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 9559 . . . 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 8741 . . . 4  |-  ( ph  ->  ( E  x.  Z
)  e.  CC )
3837, 5mulcomd 7794 . . 3  |-  ( ph  ->  ( ( E  x.  Z )  x.  X
)  =  ( X  x.  ( E  x.  Z ) ) )
3936, 38breqtrd 3954 . 2  |-  ( ph  ->  ( C  x.  2 )  <  ( X  x.  ( E  x.  Z ) ) )
404rpred 9490 . . 3  |-  ( ph  ->  X  e.  RR )
4131nnrpd 9489 . . 3  |-  ( ph  ->  ( E  x.  Z
)  e.  RR+ )
4229, 9, 40, 41lt2mul2divd 9559 . 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 1480   class class class wbr 3929  (class class class)co 5774    + caddc 7630    x. cmul 7632    < clt 7807    / cdiv 8439   NNcn 8727   2c2 8778   RR+crp 9448
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-mulrcl 7726  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-1rid 7734  ax-0id 7735  ax-rnegex 7736  ax-precex 7737  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743  ax-pre-mulgt0 7744  ax-pre-mulext 7745
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-reap 8344  df-ap 8351  df-div 8440  df-inn 8728  df-2 8786  df-rp 9449
This theorem is referenced by:  cvg1nlemres  10764
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