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Theorem ppiltx 20960
Description: The prime pi function is strictly less than the identity. (Contributed by Mario Carneiro, 22-Sep-2014.)
Assertion
Ref Expression
ppiltx  |-  ( A  e.  RR+  ->  (π `  A
)  <  A )

Proof of Theorem ppiltx
StepHypRef Expression
1 rpre 10618 . . . . . 6  |-  ( A  e.  RR+  ->  A  e.  RR )
2 ppicl 20914 . . . . . 6  |-  ( A  e.  RR  ->  (π `  A )  e.  NN0 )
31, 2syl 16 . . . . 5  |-  ( A  e.  RR+  ->  (π `  A
)  e.  NN0 )
43nn0red 10275 . . . 4  |-  ( A  e.  RR+  ->  (π `  A
)  e.  RR )
54adantr 452 . . 3  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (π `  A )  e.  RR )
6 reflcl 11205 . . . . 5  |-  ( A  e.  RR  ->  ( |_ `  A )  e.  RR )
71, 6syl 16 . . . 4  |-  ( A  e.  RR+  ->  ( |_
`  A )  e.  RR )
87adantr 452 . . 3  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  ( |_ `  A )  e.  RR )
91adantr 452 . . 3  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  A  e.  RR )
10 fzfi 11311 . . . . . 6  |-  ( 1 ... ( |_ `  A ) )  e. 
Fin
11 inss1 3561 . . . . . . 7  |-  ( ( 2 ... ( |_
`  A ) )  i^i  Prime )  C_  (
2 ... ( |_ `  A ) )
12 2nn 10133 . . . . . . . . . 10  |-  2  e.  NN
13 nnuz 10521 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
1412, 13eleqtri 2508 . . . . . . . . 9  |-  2  e.  ( ZZ>= `  1 )
15 fzss1 11091 . . . . . . . . 9  |-  ( 2  e.  ( ZZ>= `  1
)  ->  ( 2 ... ( |_ `  A ) )  C_  ( 1 ... ( |_ `  A ) ) )
1614, 15mp1i 12 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (
2 ... ( |_ `  A ) )  C_  ( 1 ... ( |_ `  A ) ) )
17 simpr 448 . . . . . . . . . . . 12  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  ( |_ `  A )  e.  NN )
1817, 13syl6eleq 2526 . . . . . . . . . . 11  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  ( |_ `  A )  e.  ( ZZ>= `  1 )
)
19 eluzfz1 11064 . . . . . . . . . . 11  |-  ( ( |_ `  A )  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... ( |_ `  A ) ) )
2018, 19syl 16 . . . . . . . . . 10  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  1  e.  ( 1 ... ( |_ `  A ) ) )
21 1lt2 10142 . . . . . . . . . . . 12  |-  1  <  2
22 1re 9090 . . . . . . . . . . . . 13  |-  1  e.  RR
23 2re 10069 . . . . . . . . . . . . 13  |-  2  e.  RR
2422, 23ltnlei 9194 . . . . . . . . . . . 12  |-  ( 1  <  2  <->  -.  2  <_  1 )
2521, 24mpbi 200 . . . . . . . . . . 11  |-  -.  2  <_  1
26 elfzle1 11060 . . . . . . . . . . 11  |-  ( 1  e.  ( 2 ... ( |_ `  A
) )  ->  2  <_  1 )
2725, 26mto 169 . . . . . . . . . 10  |-  -.  1  e.  ( 2 ... ( |_ `  A ) )
28 nelne1 2693 . . . . . . . . . 10  |-  ( ( 1  e.  ( 1 ... ( |_ `  A ) )  /\  -.  1  e.  (
2 ... ( |_ `  A ) ) )  ->  ( 1 ... ( |_ `  A
) )  =/=  (
2 ... ( |_ `  A ) ) )
2920, 27, 28sylancl 644 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (
1 ... ( |_ `  A ) )  =/=  ( 2 ... ( |_ `  A ) ) )
3029necomd 2687 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (
2 ... ( |_ `  A ) )  =/=  ( 1 ... ( |_ `  A ) ) )
31 df-pss 3336 . . . . . . . 8  |-  ( ( 2 ... ( |_
`  A ) ) 
C.  ( 1 ... ( |_ `  A
) )  <->  ( (
2 ... ( |_ `  A ) )  C_  ( 1 ... ( |_ `  A ) )  /\  ( 2 ... ( |_ `  A
) )  =/=  (
1 ... ( |_ `  A ) ) ) )
3216, 30, 31sylanbrc 646 . . . . . . 7  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (
2 ... ( |_ `  A ) )  C.  ( 1 ... ( |_ `  A ) ) )
33 sspsstr 3452 . . . . . . 7  |-  ( ( ( ( 2 ... ( |_ `  A
) )  i^i  Prime ) 
C_  ( 2 ... ( |_ `  A
) )  /\  (
2 ... ( |_ `  A ) )  C.  ( 1 ... ( |_ `  A ) ) )  ->  ( (
2 ... ( |_ `  A ) )  i^i 
Prime )  C.  ( 1 ... ( |_ `  A ) ) )
3411, 32, 33sylancr 645 . . . . . 6  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (
( 2 ... ( |_ `  A ) )  i^i  Prime )  C.  (
1 ... ( |_ `  A ) ) )
35 php3 7293 . . . . . 6  |-  ( ( ( 1 ... ( |_ `  A ) )  e.  Fin  /\  (
( 2 ... ( |_ `  A ) )  i^i  Prime )  C.  (
1 ... ( |_ `  A ) ) )  ->  ( ( 2 ... ( |_ `  A ) )  i^i 
Prime )  ~<  ( 1 ... ( |_ `  A ) ) )
3610, 34, 35sylancr 645 . . . . 5  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (
( 2 ... ( |_ `  A ) )  i^i  Prime )  ~<  (
1 ... ( |_ `  A ) ) )
37 fzfi 11311 . . . . . . 7  |-  ( 2 ... ( |_ `  A ) )  e. 
Fin
38 ssfi 7329 . . . . . . 7  |-  ( ( ( 2 ... ( |_ `  A ) )  e.  Fin  /\  (
( 2 ... ( |_ `  A ) )  i^i  Prime )  C_  (
2 ... ( |_ `  A ) ) )  ->  ( ( 2 ... ( |_ `  A ) )  i^i 
Prime )  e.  Fin )
3937, 11, 38mp2an 654 . . . . . 6  |-  ( ( 2 ... ( |_
`  A ) )  i^i  Prime )  e.  Fin
40 hashsdom 11655 . . . . . 6  |-  ( ( ( ( 2 ... ( |_ `  A
) )  i^i  Prime )  e.  Fin  /\  (
1 ... ( |_ `  A ) )  e. 
Fin )  ->  (
( # `  ( ( 2 ... ( |_
`  A ) )  i^i  Prime ) )  < 
( # `  ( 1 ... ( |_ `  A ) ) )  <-> 
( ( 2 ... ( |_ `  A
) )  i^i  Prime ) 
~<  ( 1 ... ( |_ `  A ) ) ) )
4139, 10, 40mp2an 654 . . . . 5  |-  ( (
# `  ( (
2 ... ( |_ `  A ) )  i^i 
Prime ) )  <  ( # `
 ( 1 ... ( |_ `  A
) ) )  <->  ( (
2 ... ( |_ `  A ) )  i^i 
Prime )  ~<  ( 1 ... ( |_ `  A ) ) )
4236, 41sylibr 204 . . . 4  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  ( # `
 ( ( 2 ... ( |_ `  A ) )  i^i 
Prime ) )  <  ( # `
 ( 1 ... ( |_ `  A
) ) ) )
431flcld 11207 . . . . . . 7  |-  ( A  e.  RR+  ->  ( |_
`  A )  e.  ZZ )
44 ppival2 20911 . . . . . . 7  |-  ( ( |_ `  A )  e.  ZZ  ->  (π `  ( |_ `  A
) )  =  (
# `  ( (
2 ... ( |_ `  A ) )  i^i 
Prime ) ) )
4543, 44syl 16 . . . . . 6  |-  ( A  e.  RR+  ->  (π `  ( |_ `  A ) )  =  ( # `  (
( 2 ... ( |_ `  A ) )  i^i  Prime ) ) )
46 ppifl 20943 . . . . . . 7  |-  ( A  e.  RR  ->  (π `  ( |_ `  A
) )  =  (π `  A ) )
471, 46syl 16 . . . . . 6  |-  ( A  e.  RR+  ->  (π `  ( |_ `  A ) )  =  (π `  A ) )
4845, 47eqtr3d 2470 . . . . 5  |-  ( A  e.  RR+  ->  ( # `  ( ( 2 ... ( |_ `  A
) )  i^i  Prime ) )  =  (π `  A
) )
4948adantr 452 . . . 4  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  ( # `
 ( ( 2 ... ( |_ `  A ) )  i^i 
Prime ) )  =  (π `  A ) )
50 rpge0 10624 . . . . . . 7  |-  ( A  e.  RR+  ->  0  <_  A )
51 flge0nn0 11225 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( |_ `  A
)  e.  NN0 )
521, 50, 51syl2anc 643 . . . . . 6  |-  ( A  e.  RR+  ->  ( |_
`  A )  e. 
NN0 )
53 hashfz1 11630 . . . . . 6  |-  ( ( |_ `  A )  e.  NN0  ->  ( # `  ( 1 ... ( |_ `  A ) ) )  =  ( |_
`  A ) )
5452, 53syl 16 . . . . 5  |-  ( A  e.  RR+  ->  ( # `  ( 1 ... ( |_ `  A ) ) )  =  ( |_
`  A ) )
5554adantr 452 . . . 4  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  ( # `
 ( 1 ... ( |_ `  A
) ) )  =  ( |_ `  A
) )
5642, 49, 553brtr3d 4241 . . 3  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (π `  A )  <  ( |_ `  A ) )
57 flle 11208 . . . 4  |-  ( A  e.  RR  ->  ( |_ `  A )  <_  A )
589, 57syl 16 . . 3  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  ( |_ `  A )  <_  A )
595, 8, 9, 56, 58ltletrd 9230 . 2  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (π `  A )  <  A
)
6047adantr 452 . . . 4  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  =  0 )  ->  (π `  ( |_ `  A
) )  =  (π `  A ) )
61 simpr 448 . . . . . 6  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  =  0 )  ->  ( |_ `  A )  =  0 )
6261fveq2d 5732 . . . . 5  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  =  0 )  ->  (π `  ( |_ `  A
) )  =  (π `  0 ) )
63 2pos 10082 . . . . . 6  |-  0  <  2
64 0re 9091 . . . . . . 7  |-  0  e.  RR
65 ppieq0 20959 . . . . . . 7  |-  ( 0  e.  RR  ->  (
(π `  0 )  =  0  <->  0  <  2
) )
6664, 65ax-mp 8 . . . . . 6  |-  ( (π `  0 )  =  0  <->  0  <  2 )
6763, 66mpbir 201 . . . . 5  |-  (π `  0
)  =  0
6862, 67syl6eq 2484 . . . 4  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  =  0 )  ->  (π `  ( |_ `  A
) )  =  0 )
6960, 68eqtr3d 2470 . . 3  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  =  0 )  ->  (π `  A )  =  0 )
70 rpgt0 10623 . . . 4  |-  ( A  e.  RR+  ->  0  < 
A )
7170adantr 452 . . 3  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  =  0 )  ->  0  <  A )
7269, 71eqbrtrd 4232 . 2  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  =  0 )  ->  (π `  A )  <  A
)
73 elnn0 10223 . . 3  |-  ( ( |_ `  A )  e.  NN0  <->  ( ( |_
`  A )  e.  NN  \/  ( |_
`  A )  =  0 ) )
7452, 73sylib 189 . 2  |-  ( A  e.  RR+  ->  ( ( |_ `  A )  e.  NN  \/  ( |_ `  A )  =  0 ) )
7559, 72, 74mpjaodan 762 1  |-  ( A  e.  RR+  ->  (π `  A
)  <  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599    i^i cin 3319    C_ wss 3320    C. wpss 3321   class class class wbr 4212   ` cfv 5454  (class class class)co 6081    ~< csdm 7108   Fincfn 7109   RRcr 8989   0cc0 8990   1c1 8991    < clt 9120    <_ cle 9121   NNcn 10000   2c2 10049   NN0cn0 10221   ZZcz 10282   ZZ>=cuz 10488   RR+crp 10612   ...cfz 11043   |_cfl 11201   #chash 11618   Primecprime 13079  πcppi 20876
This theorem is referenced by:  chtppilimlem1  21167
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-icc 10923  df-fz 11044  df-fl 11202  df-hash 11619  df-dvds 12853  df-prm 13080  df-ppi 20882
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