MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ppiltx Unicode version

Theorem ppiltx 20377
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 10327 . . . . . 6  |-  ( A  e.  RR+  ->  A  e.  RR )
2 ppicl 20331 . . . . . 6  |-  ( A  e.  RR  ->  (π `  A )  e.  NN0 )
31, 2syl 17 . . . . 5  |-  ( A  e.  RR+  ->  (π `  A
)  e.  NN0 )
43nn0red 9986 . . . 4  |-  ( A  e.  RR+  ->  (π `  A
)  e.  RR )
54adantr 453 . . 3  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (π `  A )  e.  RR )
6 reflcl 10894 . . . . 5  |-  ( A  e.  RR  ->  ( |_ `  A )  e.  RR )
71, 6syl 17 . . . 4  |-  ( A  e.  RR+  ->  ( |_
`  A )  e.  RR )
87adantr 453 . . 3  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  ( |_ `  A )  e.  RR )
91adantr 453 . . 3  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  A  e.  RR )
10 fzfi 11000 . . . . . 6  |-  ( 1 ... ( |_ `  A ) )  e. 
Fin
11 inss1 3364 . . . . . . 7  |-  ( ( 2 ... ( |_
`  A ) )  i^i  Prime )  C_  (
2 ... ( |_ `  A ) )
12 2nn 9844 . . . . . . . . . 10  |-  2  e.  NN
13 nnuz 10230 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
1412, 13eleqtri 2330 . . . . . . . . 9  |-  2  e.  ( ZZ>= `  1 )
15 fzss1 10796 . . . . . . . . 9  |-  ( 2  e.  ( ZZ>= `  1
)  ->  ( 2 ... ( |_ `  A ) )  C_  ( 1 ... ( |_ `  A ) ) )
1614, 15mp1i 13 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (
2 ... ( |_ `  A ) )  C_  ( 1 ... ( |_ `  A ) ) )
17 simpr 449 . . . . . . . . . . . 12  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  ( |_ `  A )  e.  NN )
1817, 13syl6eleq 2348 . . . . . . . . . . 11  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  ( |_ `  A )  e.  ( ZZ>= `  1 )
)
19 eluzfz1 10769 . . . . . . . . . . 11  |-  ( ( |_ `  A )  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... ( |_ `  A ) ) )
2018, 19syl 17 . . . . . . . . . 10  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  1  e.  ( 1 ... ( |_ `  A ) ) )
21 1lt2 9853 . . . . . . . . . . . 12  |-  1  <  2
22 1re 8805 . . . . . . . . . . . . 13  |-  1  e.  RR
23 2re 9783 . . . . . . . . . . . . 13  |-  2  e.  RR
2422, 23ltnlei 8907 . . . . . . . . . . . 12  |-  ( 1  <  2  <->  -.  2  <_  1 )
2521, 24mpbi 201 . . . . . . . . . . 11  |-  -.  2  <_  1
26 elfzle1 10765 . . . . . . . . . . 11  |-  ( 1  e.  ( 2 ... ( |_ `  A
) )  ->  2  <_  1 )
2725, 26mto 169 . . . . . . . . . 10  |-  -.  1  e.  ( 2 ... ( |_ `  A ) )
28 nelne1 2510 . . . . . . . . . 10  |-  ( ( 1  e.  ( 1 ... ( |_ `  A ) )  /\  -.  1  e.  (
2 ... ( |_ `  A ) ) )  ->  ( 1 ... ( |_ `  A
) )  =/=  (
2 ... ( |_ `  A ) ) )
2920, 27, 28sylancl 646 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (
1 ... ( |_ `  A ) )  =/=  ( 2 ... ( |_ `  A ) ) )
3029necomd 2504 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (
2 ... ( |_ `  A ) )  =/=  ( 1 ... ( |_ `  A ) ) )
31 df-pss 3143 . . . . . . . 8  |-  ( ( 2 ... ( |_
`  A ) ) 
C.  ( 1 ... ( |_ `  A
) )  <->  ( (
2 ... ( |_ `  A ) )  C_  ( 1 ... ( |_ `  A ) )  /\  ( 2 ... ( |_ `  A
) )  =/=  (
1 ... ( |_ `  A ) ) ) )
3216, 30, 31sylanbrc 648 . . . . . . 7  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (
2 ... ( |_ `  A ) )  C.  ( 1 ... ( |_ `  A ) ) )
33 sspsstr 3256 . . . . . . 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 647 . . . . . 6  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (
( 2 ... ( |_ `  A ) )  i^i  Prime )  C.  (
1 ... ( |_ `  A ) ) )
35 php3 7015 . . . . . 6  |-  ( ( ( 1 ... ( |_ `  A ) )  e.  Fin  /\  (
( 2 ... ( |_ `  A ) )  i^i  Prime )  C.  (
1 ... ( |_ `  A ) ) )  ->  ( ( 2 ... ( |_ `  A ) )  i^i 
Prime )  ~<  ( 1 ... ( |_ `  A ) ) )
3610, 34, 35sylancr 647 . . . . 5  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (
( 2 ... ( |_ `  A ) )  i^i  Prime )  ~<  (
1 ... ( |_ `  A ) ) )
37 fzfi 11000 . . . . . . 7  |-  ( 2 ... ( |_ `  A ) )  e. 
Fin
38 ssfi 7051 . . . . . . 7  |-  ( ( ( 2 ... ( |_ `  A ) )  e.  Fin  /\  (
( 2 ... ( |_ `  A ) )  i^i  Prime )  C_  (
2 ... ( |_ `  A ) ) )  ->  ( ( 2 ... ( |_ `  A ) )  i^i 
Prime )  e.  Fin )
3937, 11, 38mp2an 656 . . . . . 6  |-  ( ( 2 ... ( |_
`  A ) )  i^i  Prime )  e.  Fin
40 hashsdom 11329 . . . . . 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 656 . . . . 5  |-  ( (
# `  ( (
2 ... ( |_ `  A ) )  i^i 
Prime ) )  <  ( # `
 ( 1 ... ( |_ `  A
) ) )  <->  ( (
2 ... ( |_ `  A ) )  i^i 
Prime )  ~<  ( 1 ... ( |_ `  A ) ) )
4236, 41sylibr 205 . . . 4  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  ( # `
 ( ( 2 ... ( |_ `  A ) )  i^i 
Prime ) )  <  ( # `
 ( 1 ... ( |_ `  A
) ) ) )
431flcld 10896 . . . . . . 7  |-  ( A  e.  RR+  ->  ( |_
`  A )  e.  ZZ )
44 ppival2 20328 . . . . . . 7  |-  ( ( |_ `  A )  e.  ZZ  ->  (π `  ( |_ `  A
) )  =  (
# `  ( (
2 ... ( |_ `  A ) )  i^i 
Prime ) ) )
4543, 44syl 17 . . . . . 6  |-  ( A  e.  RR+  ->  (π `  ( |_ `  A ) )  =  ( # `  (
( 2 ... ( |_ `  A ) )  i^i  Prime ) ) )
46 ppifl 20360 . . . . . . 7  |-  ( A  e.  RR  ->  (π `  ( |_ `  A
) )  =  (π `  A ) )
471, 46syl 17 . . . . . 6  |-  ( A  e.  RR+  ->  (π `  ( |_ `  A ) )  =  (π `  A ) )
4845, 47eqtr3d 2292 . . . . 5  |-  ( A  e.  RR+  ->  ( # `  ( ( 2 ... ( |_ `  A
) )  i^i  Prime ) )  =  (π `  A
) )
4948adantr 453 . . . 4  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  ( # `
 ( ( 2 ... ( |_ `  A ) )  i^i 
Prime ) )  =  (π `  A ) )
50 rpge0 10333 . . . . . . 7  |-  ( A  e.  RR+  ->  0  <_  A )
51 flge0nn0 10914 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( |_ `  A
)  e.  NN0 )
521, 50, 51syl2anc 645 . . . . . 6  |-  ( A  e.  RR+  ->  ( |_
`  A )  e. 
NN0 )
53 hashfz1 11311 . . . . . 6  |-  ( ( |_ `  A )  e.  NN0  ->  ( # `  ( 1 ... ( |_ `  A ) ) )  =  ( |_
`  A ) )
5452, 53syl 17 . . . . 5  |-  ( A  e.  RR+  ->  ( # `  ( 1 ... ( |_ `  A ) ) )  =  ( |_
`  A ) )
5554adantr 453 . . . 4  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  ( # `
 ( 1 ... ( |_ `  A
) ) )  =  ( |_ `  A
) )
5642, 49, 553brtr3d 4026 . . 3  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (π `  A )  <  ( |_ `  A ) )
57 flle 10897 . . . 4  |-  ( A  e.  RR  ->  ( |_ `  A )  <_  A )
589, 57syl 17 . . 3  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  ( |_ `  A )  <_  A )
595, 8, 9, 56, 58ltletrd 8944 . 2  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (π `  A )  <  A
)
6047adantr 453 . . . 4  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  =  0 )  ->  (π `  ( |_ `  A
) )  =  (π `  A ) )
61 simpr 449 . . . . . 6  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  =  0 )  ->  ( |_ `  A )  =  0 )
6261fveq2d 5462 . . . . 5  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  =  0 )  ->  (π `  ( |_ `  A
) )  =  (π `  0 ) )
63 2pos 9796 . . . . . 6  |-  0  <  2
64 0re 8806 . . . . . . 7  |-  0  e.  RR
65 ppieq0 20376 . . . . . . 7  |-  ( 0  e.  RR  ->  (
(π `  0 )  =  0  <->  0  <  2
) )
6664, 65ax-mp 10 . . . . . 6  |-  ( (π `  0 )  =  0  <->  0  <  2 )
6763, 66mpbir 202 . . . . 5  |-  (π `  0
)  =  0
6862, 67syl6eq 2306 . . . 4  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  =  0 )  ->  (π `  ( |_ `  A
) )  =  0 )
6960, 68eqtr3d 2292 . . 3  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  =  0 )  ->  (π `  A )  =  0 )
70 rpgt0 10332 . . . 4  |-  ( A  e.  RR+  ->  0  < 
A )
7170adantr 453 . . 3  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  =  0 )  ->  0  <  A )
7269, 71eqbrtrd 4017 . 2  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  =  0 )  ->  (π `  A )  <  A
)
73 elnn0 9934 . . 3  |-  ( ( |_ `  A )  e.  NN0  <->  ( ( |_
`  A )  e.  NN  \/  ( |_
`  A )  =  0 ) )
7452, 73sylib 190 . 2  |-  ( A  e.  RR+  ->  ( ( |_ `  A )  e.  NN  \/  ( |_ `  A )  =  0 ) )
7559, 72, 74mpjaodan 764 1  |-  ( A  e.  RR+  ->  (π `  A
)  <  A )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2421    i^i cin 3126    C_ wss 3127    C. wpss 3128   class class class wbr 3997   ` cfv 4673  (class class class)co 5792    ~< csdm 6830   Fincfn 6831   RRcr 8704   0cc0 8705   1c1 8706    < clt 8835    <_ cle 8836   NNcn 9714   2c2 9763   NN0cn0 9932   ZZcz 9991   ZZ>=cuz 10197   RR+crp 10321   ...cfz 10748   |_cfl 10890   #chash 11303   Primecprime 12720  πcppi 20293
This theorem is referenced by:  chtppilimlem1  20584
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-2o 6448  df-oadd 6451  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-sup 7162  df-card 7540  df-cda 7762  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-n 9715  df-2 9772  df-n0 9933  df-z 9992  df-uz 10198  df-rp 10322  df-icc 10629  df-fz 10749  df-fl 10891  df-hash 11304  df-divides 12494  df-prime 12721  df-ppi 20299
  Copyright terms: Public domain W3C validator