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

Theorem ppiltx 22537
Description: The prime-counting function π 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 11018 . . . . . 6  |-  ( A  e.  RR+  ->  A  e.  RR )
2 ppicl 22491 . . . . . 6  |-  ( A  e.  RR  ->  (π `  A )  e.  NN0 )
31, 2syl 16 . . . . 5  |-  ( A  e.  RR+  ->  (π `  A
)  e.  NN0 )
43nn0red 10658 . . . 4  |-  ( A  e.  RR+  ->  (π `  A
)  e.  RR )
54adantr 465 . . 3  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (π `  A )  e.  RR )
6 reflcl 11667 . . . . 5  |-  ( A  e.  RR  ->  ( |_ `  A )  e.  RR )
71, 6syl 16 . . . 4  |-  ( A  e.  RR+  ->  ( |_
`  A )  e.  RR )
87adantr 465 . . 3  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  ( |_ `  A )  e.  RR )
91adantr 465 . . 3  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  A  e.  RR )
10 fzfi 11815 . . . . . 6  |-  ( 1 ... ( |_ `  A ) )  e. 
Fin
11 inss1 3591 . . . . . . 7  |-  ( ( 2 ... ( |_
`  A ) )  i^i  Prime )  C_  (
2 ... ( |_ `  A ) )
12 2nn 10500 . . . . . . . . . 10  |-  2  e.  NN
13 nnuz 10917 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
1412, 13eleqtri 2515 . . . . . . . . 9  |-  2  e.  ( ZZ>= `  1 )
15 fzss1 11518 . . . . . . . . 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 461 . . . . . . . . . . . 12  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  ( |_ `  A )  e.  NN )
1817, 13syl6eleq 2533 . . . . . . . . . . 11  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  ( |_ `  A )  e.  ( ZZ>= `  1 )
)
19 eluzfz1 11479 . . . . . . . . . . 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 10509 . . . . . . . . . . . 12  |-  1  <  2
22 1re 9406 . . . . . . . . . . . . 13  |-  1  e.  RR
23 2re 10412 . . . . . . . . . . . . 13  |-  2  e.  RR
2422, 23ltnlei 9516 . . . . . . . . . . . 12  |-  ( 1  <  2  <->  -.  2  <_  1 )
2521, 24mpbi 208 . . . . . . . . . . 11  |-  -.  2  <_  1
26 elfzle1 11475 . . . . . . . . . . 11  |-  ( 1  e.  ( 2 ... ( |_ `  A
) )  ->  2  <_  1 )
2725, 26mto 176 . . . . . . . . . 10  |-  -.  1  e.  ( 2 ... ( |_ `  A ) )
28 nelne1 2722 . . . . . . . . . 10  |-  ( ( 1  e.  ( 1 ... ( |_ `  A ) )  /\  -.  1  e.  (
2 ... ( |_ `  A ) ) )  ->  ( 1 ... ( |_ `  A
) )  =/=  (
2 ... ( |_ `  A ) ) )
2920, 27, 28sylancl 662 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (
1 ... ( |_ `  A ) )  =/=  ( 2 ... ( |_ `  A ) ) )
3029necomd 2640 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (
2 ... ( |_ `  A ) )  =/=  ( 1 ... ( |_ `  A ) ) )
31 df-pss 3365 . . . . . . . 8  |-  ( ( 2 ... ( |_
`  A ) ) 
C.  ( 1 ... ( |_ `  A
) )  <->  ( (
2 ... ( |_ `  A ) )  C_  ( 1 ... ( |_ `  A ) )  /\  ( 2 ... ( |_ `  A
) )  =/=  (
1 ... ( |_ `  A ) ) ) )
3216, 30, 31sylanbrc 664 . . . . . . 7  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (
2 ... ( |_ `  A ) )  C.  ( 1 ... ( |_ `  A ) ) )
33 sspsstr 3482 . . . . . . 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 663 . . . . . 6  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (
( 2 ... ( |_ `  A ) )  i^i  Prime )  C.  (
1 ... ( |_ `  A ) ) )
35 php3 7518 . . . . . 6  |-  ( ( ( 1 ... ( |_ `  A ) )  e.  Fin  /\  (
( 2 ... ( |_ `  A ) )  i^i  Prime )  C.  (
1 ... ( |_ `  A ) ) )  ->  ( ( 2 ... ( |_ `  A ) )  i^i 
Prime )  ~<  ( 1 ... ( |_ `  A ) ) )
3610, 34, 35sylancr 663 . . . . 5  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (
( 2 ... ( |_ `  A ) )  i^i  Prime )  ~<  (
1 ... ( |_ `  A ) ) )
37 fzfi 11815 . . . . . . 7  |-  ( 2 ... ( |_ `  A ) )  e. 
Fin
38 ssfi 7554 . . . . . . 7  |-  ( ( ( 2 ... ( |_ `  A ) )  e.  Fin  /\  (
( 2 ... ( |_ `  A ) )  i^i  Prime )  C_  (
2 ... ( |_ `  A ) ) )  ->  ( ( 2 ... ( |_ `  A ) )  i^i 
Prime )  e.  Fin )
3937, 11, 38mp2an 672 . . . . . 6  |-  ( ( 2 ... ( |_
`  A ) )  i^i  Prime )  e.  Fin
40 hashsdom 12165 . . . . . 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 672 . . . . 5  |-  ( (
# `  ( (
2 ... ( |_ `  A ) )  i^i 
Prime ) )  <  ( # `
 ( 1 ... ( |_ `  A
) ) )  <->  ( (
2 ... ( |_ `  A ) )  i^i 
Prime )  ~<  ( 1 ... ( |_ `  A ) ) )
4236, 41sylibr 212 . . . 4  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  ( # `
 ( ( 2 ... ( |_ `  A ) )  i^i 
Prime ) )  <  ( # `
 ( 1 ... ( |_ `  A
) ) ) )
431flcld 11669 . . . . . . 7  |-  ( A  e.  RR+  ->  ( |_
`  A )  e.  ZZ )
44 ppival2 22488 . . . . . . 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 22520 . . . . . . 7  |-  ( A  e.  RR  ->  (π `  ( |_ `  A
) )  =  (π `  A ) )
471, 46syl 16 . . . . . 6  |-  ( A  e.  RR+  ->  (π `  ( |_ `  A ) )  =  (π `  A ) )
4845, 47eqtr3d 2477 . . . . 5  |-  ( A  e.  RR+  ->  ( # `  ( ( 2 ... ( |_ `  A
) )  i^i  Prime ) )  =  (π `  A
) )
4948adantr 465 . . . 4  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  ( # `
 ( ( 2 ... ( |_ `  A ) )  i^i 
Prime ) )  =  (π `  A ) )
50 rpge0 11024 . . . . . . 7  |-  ( A  e.  RR+  ->  0  <_  A )
51 flge0nn0 11687 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( |_ `  A
)  e.  NN0 )
521, 50, 51syl2anc 661 . . . . . 6  |-  ( A  e.  RR+  ->  ( |_
`  A )  e. 
NN0 )
53 hashfz1 12138 . . . . . 6  |-  ( ( |_ `  A )  e.  NN0  ->  ( # `  ( 1 ... ( |_ `  A ) ) )  =  ( |_
`  A ) )
5452, 53syl 16 . . . . 5  |-  ( A  e.  RR+  ->  ( # `  ( 1 ... ( |_ `  A ) ) )  =  ( |_
`  A ) )
5554adantr 465 . . . 4  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  ( # `
 ( 1 ... ( |_ `  A
) ) )  =  ( |_ `  A
) )
5642, 49, 553brtr3d 4342 . . 3  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (π `  A )  <  ( |_ `  A ) )
57 flle 11670 . . . 4  |-  ( A  e.  RR  ->  ( |_ `  A )  <_  A )
589, 57syl 16 . . 3  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  ( |_ `  A )  <_  A )
595, 8, 9, 56, 58ltletrd 9552 . 2  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (π `  A )  <  A
)
6047adantr 465 . . . 4  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  =  0 )  ->  (π `  ( |_ `  A
) )  =  (π `  A ) )
61 simpr 461 . . . . . 6  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  =  0 )  ->  ( |_ `  A )  =  0 )
6261fveq2d 5716 . . . . 5  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  =  0 )  ->  (π `  ( |_ `  A
) )  =  (π `  0 ) )
63 2pos 10434 . . . . . 6  |-  0  <  2
64 0re 9407 . . . . . . 7  |-  0  e.  RR
65 ppieq0 22536 . . . . . . 7  |-  ( 0  e.  RR  ->  (
(π `  0 )  =  0  <->  0  <  2
) )
6664, 65ax-mp 5 . . . . . 6  |-  ( (π `  0 )  =  0  <->  0  <  2 )
6763, 66mpbir 209 . . . . 5  |-  (π `  0
)  =  0
6862, 67syl6eq 2491 . . . 4  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  =  0 )  ->  (π `  ( |_ `  A
) )  =  0 )
6960, 68eqtr3d 2477 . . 3  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  =  0 )  ->  (π `  A )  =  0 )
70 rpgt0 11023 . . . 4  |-  ( A  e.  RR+  ->  0  < 
A )
7170adantr 465 . . 3  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  =  0 )  ->  0  <  A )
7269, 71eqbrtrd 4333 . 2  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  =  0 )  ->  (π `  A )  <  A
)
73 elnn0 10602 . . 3  |-  ( ( |_ `  A )  e.  NN0  <->  ( ( |_
`  A )  e.  NN  \/  ( |_
`  A )  =  0 ) )
7452, 73sylib 196 . 2  |-  ( A  e.  RR+  ->  ( ( |_ `  A )  e.  NN  \/  ( |_ `  A )  =  0 ) )
7559, 72, 74mpjaodan 784 1  |-  ( A  e.  RR+  ->  (π `  A
)  <  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2620    i^i cin 3348    C_ wss 3349    C. wpss 3350   class class class wbr 4313   ` cfv 5439  (class class class)co 6112    ~< csdm 7330   Fincfn 7331   RRcr 9302   0cc0 9303   1c1 9304    < clt 9439    <_ cle 9440   NNcn 10343   2c2 10392   NN0cn0 10600   ZZcz 10667   ZZ>=cuz 10882   RR+crp 11012   ...cfz 11458   |_cfl 11661   #chash 12124   Primecprime 13784  πcppi 22453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-2o 6942  df-oadd 6945  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-sup 7712  df-card 8130  df-cda 8358  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-n0 10601  df-z 10668  df-uz 10883  df-rp 11013  df-icc 11328  df-fz 11459  df-fl 11663  df-hash 12125  df-dvds 13557  df-prm 13785  df-ppi 22459
This theorem is referenced by:  chtppilimlem1  22744
  Copyright terms: Public domain W3C validator