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Theorem dchrisum0lem1a 20635
Description: Lemma for dchrisum0lem1 20665. (Contributed by Mario Carneiro, 7-Jun-2016.)
Assertion
Ref Expression
dchrisum0lem1a  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( X  <_  ( ( X ^
2 )  /  D
)  /\  ( |_ `  ( ( X ^
2 )  /  D
) )  e.  (
ZZ>= `  ( |_ `  X ) ) ) )

Proof of Theorem dchrisum0lem1a
StepHypRef Expression
1 elfznn 10819 . . . . . . 7  |-  ( D  e.  ( 1 ... ( |_ `  X
) )  ->  D  e.  NN )
21adantl 452 . . . . . 6  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  D  e.  NN )
32nnred 9761 . . . . 5  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  D  e.  RR )
4 simpr 447 . . . . . . . 8  |-  ( (
ph  /\  X  e.  RR+ )  ->  X  e.  RR+ )
54rpregt0d 10396 . . . . . . 7  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X  e.  RR  /\  0  < 
X ) )
65adantr 451 . . . . . 6  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( X  e.  RR  /\  0  < 
X ) )
76simpld 445 . . . . 5  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  X  e.  RR )
84adantr 451 . . . . . 6  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  X  e.  RR+ )
98rpge0d 10394 . . . . 5  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  0  <_  X )
104rpred 10390 . . . . . . 7  |-  ( (
ph  /\  X  e.  RR+ )  ->  X  e.  RR )
11 fznnfl 10966 . . . . . . 7  |-  ( X  e.  RR  ->  ( D  e.  ( 1 ... ( |_ `  X ) )  <->  ( D  e.  NN  /\  D  <_  X ) ) )
1210, 11syl 15 . . . . . 6  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( D  e.  ( 1 ... ( |_ `  X ) )  <-> 
( D  e.  NN  /\  D  <_  X )
) )
1312simplbda 607 . . . . 5  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  D  <_  X )
143, 7, 7, 9, 13lemul2ad 9697 . . . 4  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( X  x.  D )  <_  ( X  x.  X )
)
15 rpcn 10362 . . . . . . 7  |-  ( X  e.  RR+  ->  X  e.  CC )
1615adantl 452 . . . . . 6  |-  ( (
ph  /\  X  e.  RR+ )  ->  X  e.  CC )
1716sqvald 11242 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X ^ 2 )  =  ( X  x.  X
) )
1817adantr 451 . . . 4  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( X ^ 2 )  =  ( X  x.  X
) )
1914, 18breqtrrd 4049 . . 3  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( X  x.  D )  <_  ( X ^ 2 ) )
20 2z 10054 . . . . . . 7  |-  2  e.  ZZ
21 rpexpcl 11122 . . . . . . 7  |-  ( ( X  e.  RR+  /\  2  e.  ZZ )  ->  ( X ^ 2 )  e.  RR+ )
224, 20, 21sylancl 643 . . . . . 6  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X ^ 2 )  e.  RR+ )
2322rpred 10390 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X ^ 2 )  e.  RR )
2423adantr 451 . . . 4  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( X ^ 2 )  e.  RR )
252nnrpd 10389 . . . 4  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  D  e.  RR+ )
267, 24, 25lemuldivd 10435 . . 3  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( ( X  x.  D )  <_  ( X ^ 2 )  <->  X  <_  ( ( X ^ 2 )  /  D ) ) )
2719, 26mpbid 201 . 2  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  X  <_  ( ( X ^ 2 )  /  D ) )
28 nndivre 9781 . . . 4  |-  ( ( ( X ^ 2 )  e.  RR  /\  D  e.  NN )  ->  ( ( X ^
2 )  /  D
)  e.  RR )
2923, 1, 28syl2an 463 . . 3  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( ( X ^ 2 )  /  D )  e.  RR )
30 flword2 10943 . . 3  |-  ( ( X  e.  RR  /\  ( ( X ^
2 )  /  D
)  e.  RR  /\  X  <_  ( ( X ^ 2 )  /  D ) )  -> 
( |_ `  (
( X ^ 2 )  /  D ) )  e.  ( ZZ>= `  ( |_ `  X ) ) )
317, 29, 27, 30syl3anc 1182 . 2  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( |_ `  ( ( X ^
2 )  /  D
) )  e.  (
ZZ>= `  ( |_ `  X ) ) )
3227, 31jca 518 1  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( X  <_  ( ( X ^
2 )  /  D
)  /\  ( |_ `  ( ( X ^
2 )  /  D
) )  e.  (
ZZ>= `  ( |_ `  X ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    < clt 8867    <_ cle 8868    / cdiv 9423   NNcn 9746   2c2 9795   ZZcz 10024   ZZ>=cuz 10230   RR+crp 10354   ...cfz 10782   |_cfl 10924   ^cexp 11104
This theorem is referenced by:  dchrisum0lem1b  20664  dchrisum0lem1  20665
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fl 10925  df-seq 11047  df-exp 11105
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