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Theorem dchrisum0lem1a 22871
Description: Lemma for dchrisum0lem1 22901. (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 11598 . . . . . . 7  |-  ( D  e.  ( 1 ... ( |_ `  X
) )  ->  D  e.  NN )
21adantl 466 . . . . . 6  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  D  e.  NN )
32nnred 10451 . . . . 5  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  D  e.  RR )
4 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  X  e.  RR+ )  ->  X  e.  RR+ )
54rpregt0d 11147 . . . . . . 7  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X  e.  RR  /\  0  < 
X ) )
65adantr 465 . . . . . 6  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( X  e.  RR  /\  0  < 
X ) )
76simpld 459 . . . . 5  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  X  e.  RR )
84adantr 465 . . . . . 6  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  X  e.  RR+ )
98rpge0d 11145 . . . . 5  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  0  <_  X )
104rpred 11141 . . . . . . 7  |-  ( (
ph  /\  X  e.  RR+ )  ->  X  e.  RR )
11 fznnfl 11821 . . . . . . 7  |-  ( X  e.  RR  ->  ( D  e.  ( 1 ... ( |_ `  X ) )  <->  ( D  e.  NN  /\  D  <_  X ) ) )
1210, 11syl 16 . . . . . 6  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( D  e.  ( 1 ... ( |_ `  X ) )  <-> 
( D  e.  NN  /\  D  <_  X )
) )
1312simplbda 624 . . . . 5  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  D  <_  X )
143, 7, 7, 9, 13lemul2ad 10387 . . . 4  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( X  x.  D )  <_  ( X  x.  X )
)
15 rpcn 11113 . . . . . . 7  |-  ( X  e.  RR+  ->  X  e.  CC )
1615adantl 466 . . . . . 6  |-  ( (
ph  /\  X  e.  RR+ )  ->  X  e.  CC )
1716sqvald 12125 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X ^ 2 )  =  ( X  x.  X
) )
1817adantr 465 . . . 4  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( X ^ 2 )  =  ( X  x.  X
) )
1914, 18breqtrrd 4429 . . 3  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( X  x.  D )  <_  ( X ^ 2 ) )
20 2z 10792 . . . . . . 7  |-  2  e.  ZZ
21 rpexpcl 12004 . . . . . . 7  |-  ( ( X  e.  RR+  /\  2  e.  ZZ )  ->  ( X ^ 2 )  e.  RR+ )
224, 20, 21sylancl 662 . . . . . 6  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X ^ 2 )  e.  RR+ )
2322rpred 11141 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X ^ 2 )  e.  RR )
2423adantr 465 . . . 4  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( X ^ 2 )  e.  RR )
252nnrpd 11140 . . . 4  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  D  e.  RR+ )
267, 24, 25lemuldivd 11186 . . 3  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( ( X  x.  D )  <_  ( X ^ 2 )  <->  X  <_  ( ( X ^ 2 )  /  D ) ) )
2719, 26mpbid 210 . 2  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  X  <_  ( ( X ^ 2 )  /  D ) )
28 nndivre 10471 . . . 4  |-  ( ( ( X ^ 2 )  e.  RR  /\  D  e.  NN )  ->  ( ( X ^
2 )  /  D
)  e.  RR )
2923, 1, 28syl2an 477 . . 3  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( ( X ^ 2 )  /  D )  e.  RR )
30 flword2 11781 . . 3  |-  ( ( X  e.  RR  /\  ( ( X ^
2 )  /  D
)  e.  RR  /\  X  <_  ( ( X ^ 2 )  /  D ) )  -> 
( |_ `  (
( X ^ 2 )  /  D ) )  e.  ( ZZ>= `  ( |_ `  X ) ) )
317, 29, 27, 30syl3anc 1219 . 2  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( |_ `  ( ( X ^
2 )  /  D
) )  e.  (
ZZ>= `  ( |_ `  X ) ) )
3227, 31jca 532 1  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( X  <_  ( ( X ^
2 )  /  D
)  /\  ( |_ `  ( ( X ^
2 )  /  D
) )  e.  (
ZZ>= `  ( |_ `  X ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   CCcc 9394   RRcr 9395   0cc0 9396   1c1 9397    x. cmul 9401    < clt 9532    <_ cle 9533    / cdiv 10107   NNcn 10436   2c2 10485   ZZcz 10760   ZZ>=cuz 10975   RR+crp 11105   ...cfz 11557   |_cfl 11760   ^cexp 11985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-sup 7805  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-n0 10694  df-z 10761  df-uz 10976  df-rp 11106  df-fz 11558  df-fl 11762  df-seq 11927  df-exp 11986
This theorem is referenced by:  dchrisum0lem1b  22900  dchrisum0lem1  22901
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