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Theorem dchrisum0lem1a 23396
Description: Lemma for dchrisum0lem1 23426. (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 11710 . . . . . . 7  |-  ( D  e.  ( 1 ... ( |_ `  X
) )  ->  D  e.  NN )
21adantl 466 . . . . . 6  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  D  e.  NN )
32nnred 10547 . . . . 5  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  D  e.  RR )
4 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  X  e.  RR+ )  ->  X  e.  RR+ )
54rpregt0d 11258 . . . . . . 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 11256 . . . . 5  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  0  <_  X )
104rpred 11252 . . . . . . 7  |-  ( (
ph  /\  X  e.  RR+ )  ->  X  e.  RR )
11 fznnfl 11952 . . . . . . 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 10482 . . . 4  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( X  x.  D )  <_  ( X  x.  X )
)
15 rpcn 11224 . . . . . . 7  |-  ( X  e.  RR+  ->  X  e.  CC )
1615adantl 466 . . . . . 6  |-  ( (
ph  /\  X  e.  RR+ )  ->  X  e.  CC )
1716sqvald 12269 . . . . 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 4473 . . 3  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( X  x.  D )  <_  ( X ^ 2 ) )
20 2z 10892 . . . . . . 7  |-  2  e.  ZZ
21 rpexpcl 12148 . . . . . . 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 11252 . . . . 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 11251 . . . 4  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  D  e.  RR+ )
267, 24, 25lemuldivd 11297 . . 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 10567 . . . 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 11912 . . 3  |-  ( ( X  e.  RR  /\  ( ( X ^
2 )  /  D
)  e.  RR  /\  X  <_  ( ( X ^ 2 )  /  D ) )  -> 
( |_ `  (
( X ^ 2 )  /  D ) )  e.  ( ZZ>= `  ( |_ `  X ) ) )
317, 29, 27, 30syl3anc 1228 . 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 1379    e. wcel 1767   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489    x. cmul 9493    < clt 9624    <_ cle 9625    / cdiv 10202   NNcn 10532   2c2 10581   ZZcz 10860   ZZ>=cuz 11078   RR+crp 11216   ...cfz 11668   |_cfl 11891   ^cexp 12129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-fz 11669  df-fl 11893  df-seq 12071  df-exp 12130
This theorem is referenced by:  dchrisum0lem1b  23425  dchrisum0lem1  23426
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