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Theorem dchrisum0lem1a 21182
Description: Lemma for dchrisum0lem1 21212. (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 11082 . . . . . . 7  |-  ( D  e.  ( 1 ... ( |_ `  X
) )  ->  D  e.  NN )
21adantl 454 . . . . . 6  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  D  e.  NN )
32nnred 10017 . . . . 5  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  D  e.  RR )
4 simpr 449 . . . . . . . 8  |-  ( (
ph  /\  X  e.  RR+ )  ->  X  e.  RR+ )
54rpregt0d 10656 . . . . . . 7  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X  e.  RR  /\  0  < 
X ) )
65adantr 453 . . . . . 6  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( X  e.  RR  /\  0  < 
X ) )
76simpld 447 . . . . 5  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  X  e.  RR )
84adantr 453 . . . . . 6  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  X  e.  RR+ )
98rpge0d 10654 . . . . 5  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  0  <_  X )
104rpred 10650 . . . . . . 7  |-  ( (
ph  /\  X  e.  RR+ )  ->  X  e.  RR )
11 fznnfl 11245 . . . . . . 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 609 . . . . 5  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  D  <_  X )
143, 7, 7, 9, 13lemul2ad 9953 . . . 4  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( X  x.  D )  <_  ( X  x.  X )
)
15 rpcn 10622 . . . . . . 7  |-  ( X  e.  RR+  ->  X  e.  CC )
1615adantl 454 . . . . . 6  |-  ( (
ph  /\  X  e.  RR+ )  ->  X  e.  CC )
1716sqvald 11522 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X ^ 2 )  =  ( X  x.  X
) )
1817adantr 453 . . . 4  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( X ^ 2 )  =  ( X  x.  X
) )
1914, 18breqtrrd 4240 . . 3  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( X  x.  D )  <_  ( X ^ 2 ) )
20 2z 10314 . . . . . . 7  |-  2  e.  ZZ
21 rpexpcl 11402 . . . . . . 7  |-  ( ( X  e.  RR+  /\  2  e.  ZZ )  ->  ( X ^ 2 )  e.  RR+ )
224, 20, 21sylancl 645 . . . . . 6  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X ^ 2 )  e.  RR+ )
2322rpred 10650 . . . . 5  |-  ( (
ph  /\  X  e.  RR+ )  ->  ( X ^ 2 )  e.  RR )
2423adantr 453 . . . 4  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( X ^ 2 )  e.  RR )
252nnrpd 10649 . . . 4  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  D  e.  RR+ )
267, 24, 25lemuldivd 10695 . . 3  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( ( X  x.  D )  <_  ( X ^ 2 )  <->  X  <_  ( ( X ^ 2 )  /  D ) ) )
2719, 26mpbid 203 . 2  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  X  <_  ( ( X ^ 2 )  /  D ) )
28 nndivre 10037 . . . 4  |-  ( ( ( X ^ 2 )  e.  RR  /\  D  e.  NN )  ->  ( ( X ^
2 )  /  D
)  e.  RR )
2923, 1, 28syl2an 465 . . 3  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( ( X ^ 2 )  /  D )  e.  RR )
30 flword2 11222 . . 3  |-  ( ( X  e.  RR  /\  ( ( X ^
2 )  /  D
)  e.  RR  /\  X  <_  ( ( X ^ 2 )  /  D ) )  -> 
( |_ `  (
( X ^ 2 )  /  D ) )  e.  ( ZZ>= `  ( |_ `  X ) ) )
317, 29, 27, 30syl3anc 1185 . 2  |-  ( ( ( ph  /\  X  e.  RR+ )  /\  D  e.  ( 1 ... ( |_ `  X ) ) )  ->  ( |_ `  ( ( X ^
2 )  /  D
) )  e.  (
ZZ>= `  ( |_ `  X ) ) )
3227, 31jca 520 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   CCcc 8990   RRcr 8991   0cc0 8992   1c1 8993    x. cmul 8997    < clt 9122    <_ cle 9123    / cdiv 9679   NNcn 10002   2c2 10051   ZZcz 10284   ZZ>=cuz 10490   RR+crp 10614   ...cfz 11045   |_cfl 11203   ^cexp 11384
This theorem is referenced by:  dchrisum0lem1b  21211  dchrisum0lem1  21212
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-fz 11046  df-fl 11204  df-seq 11326  df-exp 11385
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