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Theorem bposlem4 20526
Description: Lemma for bpos 20532. (Contributed by Mario Carneiro, 13-Mar-2014.)
Hypotheses
Ref Expression
bpos.1  |-  ( ph  ->  N  e.  ( ZZ>= ` 
5 ) )
bpos.2  |-  ( ph  ->  -.  E. p  e. 
Prime  ( N  <  p  /\  p  <_  ( 2  x.  N ) ) )
bpos.3  |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( n ^ (
n  pCnt  ( (
2  x.  N )  _C  N ) ) ) ,  1 ) )
bpos.4  |-  K  =  ( |_ `  (
( 2  x.  N
)  /  3 ) )
bpos.5  |-  M  =  ( |_ `  ( sqr `  ( 2  x.  N ) ) )
Assertion
Ref Expression
bposlem4  |-  ( ph  ->  M  e.  ( 3 ... K ) )
Distinct variable groups:    F, p    n, p, K    M, p    n, N, p    ph, n, p
Allowed substitution hints:    F( n)    M( n)

Proof of Theorem bposlem4
StepHypRef Expression
1 2nn 9877 . . . . . . . 8  |-  2  e.  NN
2 5nn 9880 . . . . . . . . 9  |-  5  e.  NN
3 bpos.1 . . . . . . . . 9  |-  ( ph  ->  N  e.  ( ZZ>= ` 
5 ) )
4 nnuz 10263 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
54uztrn2 10245 . . . . . . . . 9  |-  ( ( 5  e.  NN  /\  N  e.  ( ZZ>= ` 
5 ) )  ->  N  e.  NN )
62, 3, 5sylancr 644 . . . . . . . 8  |-  ( ph  ->  N  e.  NN )
7 nnmulcl 9769 . . . . . . . 8  |-  ( ( 2  e.  NN  /\  N  e.  NN )  ->  ( 2  x.  N
)  e.  NN )
81, 6, 7sylancr 644 . . . . . . 7  |-  ( ph  ->  ( 2  x.  N
)  e.  NN )
98nnred 9761 . . . . . 6  |-  ( ph  ->  ( 2  x.  N
)  e.  RR )
108nnrpd 10389 . . . . . . 7  |-  ( ph  ->  ( 2  x.  N
)  e.  RR+ )
1110rpge0d 10394 . . . . . 6  |-  ( ph  ->  0  <_  ( 2  x.  N ) )
129, 11resqrcld 11900 . . . . 5  |-  ( ph  ->  ( sqr `  (
2  x.  N ) )  e.  RR )
1312flcld 10930 . . . 4  |-  ( ph  ->  ( |_ `  ( sqr `  ( 2  x.  N ) ) )  e.  ZZ )
14 sqr9 11759 . . . . . 6  |-  ( sqr `  9 )  =  3
15 9re 9825 . . . . . . . . 9  |-  9  e.  RR
1615a1i 10 . . . . . . . 8  |-  ( ph  ->  9  e.  RR )
17 10re 9826 . . . . . . . . 9  |-  10  e.  RR
1817a1i 10 . . . . . . . 8  |-  ( ph  ->  10  e.  RR )
19 lep1 9595 . . . . . . . . . . 11  |-  ( 9  e.  RR  ->  9  <_  ( 9  +  1 ) )
2015, 19ax-mp 8 . . . . . . . . . 10  |-  9  <_  ( 9  +  1 )
21 df-10 9812 . . . . . . . . . 10  |-  10  =  ( 9  +  1 )
2220, 21breqtrri 4048 . . . . . . . . 9  |-  9  <_  10
2322a1i 10 . . . . . . . 8  |-  ( ph  ->  9  <_  10 )
242nncni 9756 . . . . . . . . . 10  |-  5  e.  CC
25 2cn 9816 . . . . . . . . . 10  |-  2  e.  CC
26 5t2e10 9875 . . . . . . . . . 10  |-  ( 5  x.  2 )  =  10
2724, 25, 26mulcomli 8844 . . . . . . . . 9  |-  ( 2  x.  5 )  =  10
28 eluzle 10240 . . . . . . . . . . 11  |-  ( N  e.  ( ZZ>= `  5
)  ->  5  <_  N )
293, 28syl 15 . . . . . . . . . 10  |-  ( ph  ->  5  <_  N )
306nnred 9761 . . . . . . . . . . 11  |-  ( ph  ->  N  e.  RR )
31 5re 9821 . . . . . . . . . . . 12  |-  5  e.  RR
32 2re 9815 . . . . . . . . . . . . 13  |-  2  e.  RR
33 2pos 9828 . . . . . . . . . . . . 13  |-  0  <  2
3432, 33pm3.2i 441 . . . . . . . . . . . 12  |-  ( 2  e.  RR  /\  0  <  2 )
35 lemul2 9609 . . . . . . . . . . . 12  |-  ( ( 5  e.  RR  /\  N  e.  RR  /\  (
2  e.  RR  /\  0  <  2 ) )  ->  ( 5  <_  N 
<->  ( 2  x.  5 )  <_  ( 2  x.  N ) ) )
3631, 34, 35mp3an13 1268 . . . . . . . . . . 11  |-  ( N  e.  RR  ->  (
5  <_  N  <->  ( 2  x.  5 )  <_ 
( 2  x.  N
) ) )
3730, 36syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( 5  <_  N  <->  ( 2  x.  5 )  <_  ( 2  x.  N ) ) )
3829, 37mpbid 201 . . . . . . . . 9  |-  ( ph  ->  ( 2  x.  5 )  <_  ( 2  x.  N ) )
3927, 38syl5eqbrr 4057 . . . . . . . 8  |-  ( ph  ->  10  <_  ( 2  x.  N ) )
4016, 18, 9, 23, 39letrd 8973 . . . . . . 7  |-  ( ph  ->  9  <_  ( 2  x.  N ) )
41 0re 8838 . . . . . . . . . 10  |-  0  e.  RR
42 9pos 9837 . . . . . . . . . 10  |-  0  <  9
4341, 15, 42ltleii 8941 . . . . . . . . 9  |-  0  <_  9
4415, 43pm3.2i 441 . . . . . . . 8  |-  ( 9  e.  RR  /\  0  <_  9 )
4510rprege0d 10397 . . . . . . . 8  |-  ( ph  ->  ( ( 2  x.  N )  e.  RR  /\  0  <_  ( 2  x.  N ) ) )
46 sqrle 11746 . . . . . . . 8  |-  ( ( ( 9  e.  RR  /\  0  <_  9 )  /\  ( ( 2  x.  N )  e.  RR  /\  0  <_ 
( 2  x.  N
) ) )  -> 
( 9  <_  (
2  x.  N )  <-> 
( sqr `  9
)  <_  ( sqr `  ( 2  x.  N
) ) ) )
4744, 45, 46sylancr 644 . . . . . . 7  |-  ( ph  ->  ( 9  <_  (
2  x.  N )  <-> 
( sqr `  9
)  <_  ( sqr `  ( 2  x.  N
) ) ) )
4840, 47mpbid 201 . . . . . 6  |-  ( ph  ->  ( sqr `  9
)  <_  ( sqr `  ( 2  x.  N
) ) )
4914, 48syl5eqbrr 4057 . . . . 5  |-  ( ph  ->  3  <_  ( sqr `  ( 2  x.  N
) ) )
50 3nn 9878 . . . . . . 7  |-  3  e.  NN
5150nnzi 10047 . . . . . 6  |-  3  e.  ZZ
52 flge 10937 . . . . . 6  |-  ( ( ( sqr `  (
2  x.  N ) )  e.  RR  /\  3  e.  ZZ )  ->  ( 3  <_  ( sqr `  ( 2  x.  N ) )  <->  3  <_  ( |_ `  ( sqr `  ( 2  x.  N
) ) ) ) )
5312, 51, 52sylancl 643 . . . . 5  |-  ( ph  ->  ( 3  <_  ( sqr `  ( 2  x.  N ) )  <->  3  <_  ( |_ `  ( sqr `  ( 2  x.  N
) ) ) ) )
5449, 53mpbid 201 . . . 4  |-  ( ph  ->  3  <_  ( |_ `  ( sqr `  (
2  x.  N ) ) ) )
5551eluz1i 10237 . . . 4  |-  ( ( |_ `  ( sqr `  ( 2  x.  N
) ) )  e.  ( ZZ>= `  3 )  <->  ( ( |_ `  ( sqr `  ( 2  x.  N ) ) )  e.  ZZ  /\  3  <_  ( |_ `  ( sqr `  ( 2  x.  N ) ) ) ) )
5613, 54, 55sylanbrc 645 . . 3  |-  ( ph  ->  ( |_ `  ( sqr `  ( 2  x.  N ) ) )  e.  ( ZZ>= `  3
) )
57 nndivre 9781 . . . . 5  |-  ( ( ( 2  x.  N
)  e.  RR  /\  3  e.  NN )  ->  ( ( 2  x.  N )  /  3
)  e.  RR )
589, 50, 57sylancl 643 . . . 4  |-  ( ph  ->  ( ( 2  x.  N )  /  3
)  e.  RR )
59 3re 9817 . . . . . . . . 9  |-  3  e.  RR
6059a1i 10 . . . . . . . 8  |-  ( ph  ->  3  e.  RR )
6110sqrgt0d 11895 . . . . . . . 8  |-  ( ph  ->  0  <  ( sqr `  ( 2  x.  N
) ) )
62 lemul2 9609 . . . . . . . 8  |-  ( ( 3  e.  RR  /\  ( sqr `  ( 2  x.  N ) )  e.  RR  /\  (
( sqr `  (
2  x.  N ) )  e.  RR  /\  0  <  ( sqr `  (
2  x.  N ) ) ) )  -> 
( 3  <_  ( sqr `  ( 2  x.  N ) )  <->  ( ( sqr `  ( 2  x.  N ) )  x.  3 )  <_  (
( sqr `  (
2  x.  N ) )  x.  ( sqr `  ( 2  x.  N
) ) ) ) )
6360, 12, 12, 61, 62syl112anc 1186 . . . . . . 7  |-  ( ph  ->  ( 3  <_  ( sqr `  ( 2  x.  N ) )  <->  ( ( sqr `  ( 2  x.  N ) )  x.  3 )  <_  (
( sqr `  (
2  x.  N ) )  x.  ( sqr `  ( 2  x.  N
) ) ) ) )
6449, 63mpbid 201 . . . . . 6  |-  ( ph  ->  ( ( sqr `  (
2  x.  N ) )  x.  3 )  <_  ( ( sqr `  ( 2  x.  N
) )  x.  ( sqr `  ( 2  x.  N ) ) ) )
65 remsqsqr 11742 . . . . . . 7  |-  ( ( ( 2  x.  N
)  e.  RR  /\  0  <_  ( 2  x.  N ) )  -> 
( ( sqr `  (
2  x.  N ) )  x.  ( sqr `  ( 2  x.  N
) ) )  =  ( 2  x.  N
) )
669, 11, 65syl2anc 642 . . . . . 6  |-  ( ph  ->  ( ( sqr `  (
2  x.  N ) )  x.  ( sqr `  ( 2  x.  N
) ) )  =  ( 2  x.  N
) )
6764, 66breqtrd 4047 . . . . 5  |-  ( ph  ->  ( ( sqr `  (
2  x.  N ) )  x.  3 )  <_  ( 2  x.  N ) )
68 3pos 9830 . . . . . . . 8  |-  0  <  3
6959, 68pm3.2i 441 . . . . . . 7  |-  ( 3  e.  RR  /\  0  <  3 )
7069a1i 10 . . . . . 6  |-  ( ph  ->  ( 3  e.  RR  /\  0  <  3 ) )
71 lemuldiv 9635 . . . . . 6  |-  ( ( ( sqr `  (
2  x.  N ) )  e.  RR  /\  ( 2  x.  N
)  e.  RR  /\  ( 3  e.  RR  /\  0  <  3 ) )  ->  ( (
( sqr `  (
2  x.  N ) )  x.  3 )  <_  ( 2  x.  N )  <->  ( sqr `  ( 2  x.  N
) )  <_  (
( 2  x.  N
)  /  3 ) ) )
7212, 9, 70, 71syl3anc 1182 . . . . 5  |-  ( ph  ->  ( ( ( sqr `  ( 2  x.  N
) )  x.  3 )  <_  ( 2  x.  N )  <->  ( sqr `  ( 2  x.  N
) )  <_  (
( 2  x.  N
)  /  3 ) ) )
7367, 72mpbid 201 . . . 4  |-  ( ph  ->  ( sqr `  (
2  x.  N ) )  <_  ( (
2  x.  N )  /  3 ) )
74 flword2 10943 . . . 4  |-  ( ( ( sqr `  (
2  x.  N ) )  e.  RR  /\  ( ( 2  x.  N )  /  3
)  e.  RR  /\  ( sqr `  ( 2  x.  N ) )  <_  ( ( 2  x.  N )  / 
3 ) )  -> 
( |_ `  (
( 2  x.  N
)  /  3 ) )  e.  ( ZZ>= `  ( |_ `  ( sqr `  ( 2  x.  N
) ) ) ) )
7512, 58, 73, 74syl3anc 1182 . . 3  |-  ( ph  ->  ( |_ `  (
( 2  x.  N
)  /  3 ) )  e.  ( ZZ>= `  ( |_ `  ( sqr `  ( 2  x.  N
) ) ) ) )
76 elfzuzb 10792 . . 3  |-  ( ( |_ `  ( sqr `  ( 2  x.  N
) ) )  e.  ( 3 ... ( |_ `  ( ( 2  x.  N )  / 
3 ) ) )  <-> 
( ( |_ `  ( sqr `  ( 2  x.  N ) ) )  e.  ( ZZ>= ` 
3 )  /\  ( |_ `  ( ( 2  x.  N )  / 
3 ) )  e.  ( ZZ>= `  ( |_ `  ( sqr `  (
2  x.  N ) ) ) ) ) )
7756, 75, 76sylanbrc 645 . 2  |-  ( ph  ->  ( |_ `  ( sqr `  ( 2  x.  N ) ) )  e.  ( 3 ... ( |_ `  (
( 2  x.  N
)  /  3 ) ) ) )
78 bpos.5 . . 3  |-  M  =  ( |_ `  ( sqr `  ( 2  x.  N ) ) )
79 bpos.4 . . . 4  |-  K  =  ( |_ `  (
( 2  x.  N
)  /  3 ) )
8079oveq2i 5869 . . 3  |-  ( 3 ... K )  =  ( 3 ... ( |_ `  ( ( 2  x.  N )  / 
3 ) ) )
8178, 80eleq12i 2348 . 2  |-  ( M  e.  ( 3 ... K )  <->  ( |_ `  ( sqr `  (
2  x.  N ) ) )  e.  ( 3 ... ( |_
`  ( ( 2  x.  N )  / 
3 ) ) ) )
8277, 81sylibr 203 1  |-  ( ph  ->  M  e.  ( 3 ... K ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   ifcif 3565   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868    / cdiv 9423   NNcn 9746   2c2 9795   3c3 9796   5c5 9798   9c9 9802   10c10 9803   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782   |_cfl 10924   ^cexp 11104    _C cbc 11315   sqrcsqr 11718   Primecprime 12758    pCnt cpc 12889
This theorem is referenced by:  bposlem6  20528
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-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fl 10925  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720
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