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Theorem dchrisum0lem1a 24052
Description: Lemma for dchrisum0lem1 24082. (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 11768 . . . . . . 7  |-  ( D  e.  ( 1 ... ( |_ `  X
) )  ->  D  e.  NN )
21adantl 464 . . . . . 6  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  D  e.  NN )
32nnred 10591 . . . . 5  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  D  e.  RR )
4 simpr 459 . . . . . . . 8  |-  ( (
ph  /\  X  e.  RR+ )  ->  X  e.  RR+ )
54rpregt0d 11310 . . . . . . 7  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X  e.  RR  /\  0  < 
X ) )
65adantr 463 . . . . . 6  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( X  e.  RR  /\  0  < 
X ) )
76simpld 457 . . . . 5  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  X  e.  RR )
84adantr 463 . . . . . 6  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  X  e.  RR+ )
98rpge0d 11308 . . . . 5  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  0  <_  X )
104rpred 11304 . . . . . . 7  |-  ( (
ph  /\  X  e.  RR+ )  ->  X  e.  RR )
11 fznnfl 12027 . . . . . . 7  |-  ( X  e.  RR  ->  ( D  e.  ( 1 ... ( |_ `  X ) )  <->  ( D  e.  NN  /\  D  <_  X ) ) )
1210, 11syl 17 . . . . . 6  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( D  e.  ( 1 ... ( |_ `  X ) )  <-> 
( D  e.  NN  /\  D  <_  X )
) )
1312simplbda 622 . . . . 5  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  D  <_  X )
143, 7, 7, 9, 13lemul2ad 10526 . . . 4  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( X  x.  D )  <_  ( X  x.  X )
)
15 rpcn 11273 . . . . . . 7  |-  ( X  e.  RR+  ->  X  e.  CC )
1615adantl 464 . . . . . 6  |-  ( (
ph  /\  X  e.  RR+ )  ->  X  e.  CC )
1716sqvald 12351 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X ^ 2 )  =  ( X  x.  X
) )
1817adantr 463 . . . 4  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( X ^ 2 )  =  ( X  x.  X
) )
1914, 18breqtrrd 4421 . . 3  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( X  x.  D )  <_  ( X ^ 2 ) )
20 2z 10937 . . . . . . 7  |-  2  e.  ZZ
21 rpexpcl 12229 . . . . . . 7  |-  ( ( X  e.  RR+  /\  2  e.  ZZ )  ->  ( X ^ 2 )  e.  RR+ )
224, 20, 21sylancl 660 . . . . . 6  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X ^ 2 )  e.  RR+ )
2322rpred 11304 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X ^ 2 )  e.  RR )
2423adantr 463 . . . 4  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( X ^ 2 )  e.  RR )
252nnrpd 11302 . . . 4  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  D  e.  RR+ )
267, 24, 25lemuldivd 11349 . . 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 10612 . . . 4  |-  ( ( ( X ^ 2 )  e.  RR  /\  D  e.  NN )  ->  ( ( X ^
2 )  /  D
)  e.  RR )
2923, 1, 28syl2an 475 . . 3  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( ( X ^ 2 )  /  D )  e.  RR )
30 flword2 11986 . . 3  |-  ( ( X  e.  RR  /\  ( ( X ^
2 )  /  D
)  e.  RR  /\  X  <_  ( ( X ^ 2 )  /  D ) )  -> 
( |_ `  (
( X ^ 2 )  /  D ) )  e.  ( ZZ>= `  ( |_ `  X ) ) )
317, 29, 27, 30syl3anc 1230 . 2  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( |_ `  ( ( X ^
2 )  /  D
) )  e.  (
ZZ>= `  ( |_ `  X ) ) )
3227, 31jca 530 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 367    = wceq 1405    e. wcel 1842   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   CCcc 9520   RRcr 9521   0cc0 9522   1c1 9523    x. cmul 9527    < clt 9658    <_ cle 9659    / cdiv 10247   NNcn 10576   2c2 10626   ZZcz 10905   ZZ>=cuz 11127   RR+crp 11265   ...cfz 11726   |_cfl 11964   ^cexp 12210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-sup 7935  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-n0 10837  df-z 10906  df-uz 11128  df-rp 11266  df-fz 11727  df-fl 11966  df-seq 12152  df-exp 12211
This theorem is referenced by:  dchrisum0lem1b  24081  dchrisum0lem1  24082
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