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Theorem ppiltx 20828
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 10551 . . . . . 6  |-  ( A  e.  RR+  ->  A  e.  RR )
2 ppicl 20782 . . . . . 6  |-  ( A  e.  RR  ->  (π `  A )  e.  NN0 )
31, 2syl 16 . . . . 5  |-  ( A  e.  RR+  ->  (π `  A
)  e.  NN0 )
43nn0red 10208 . . . 4  |-  ( A  e.  RR+  ->  (π `  A
)  e.  RR )
54adantr 452 . . 3  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (π `  A )  e.  RR )
6 reflcl 11133 . . . . 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 11239 . . . . . 6  |-  ( 1 ... ( |_ `  A ) )  e. 
Fin
11 inss1 3505 . . . . . . 7  |-  ( ( 2 ... ( |_
`  A ) )  i^i  Prime )  C_  (
2 ... ( |_ `  A ) )
12 2nn 10066 . . . . . . . . . 10  |-  2  e.  NN
13 nnuz 10454 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
1412, 13eleqtri 2460 . . . . . . . . 9  |-  2  e.  ( ZZ>= `  1 )
15 fzss1 11024 . . . . . . . . 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 2478 . . . . . . . . . . 11  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  ( |_ `  A )  e.  ( ZZ>= `  1 )
)
19 eluzfz1 10997 . . . . . . . . . . 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 10075 . . . . . . . . . . . 12  |-  1  <  2
22 1re 9024 . . . . . . . . . . . . 13  |-  1  e.  RR
23 2re 10002 . . . . . . . . . . . . 13  |-  2  e.  RR
2422, 23ltnlei 9126 . . . . . . . . . . . 12  |-  ( 1  <  2  <->  -.  2  <_  1 )
2521, 24mpbi 200 . . . . . . . . . . 11  |-  -.  2  <_  1
26 elfzle1 10993 . . . . . . . . . . 11  |-  ( 1  e.  ( 2 ... ( |_ `  A
) )  ->  2  <_  1 )
2725, 26mto 169 . . . . . . . . . 10  |-  -.  1  e.  ( 2 ... ( |_ `  A ) )
28 nelne1 2640 . . . . . . . . . 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 2634 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (
2 ... ( |_ `  A ) )  =/=  ( 1 ... ( |_ `  A ) ) )
31 df-pss 3280 . . . . . . . 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 3396 . . . . . . 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 7230 . . . . . 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 11239 . . . . . . 7  |-  ( 2 ... ( |_ `  A ) )  e. 
Fin
38 ssfi 7266 . . . . . . 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 11583 . . . . . 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 11135 . . . . . . 7  |-  ( A  e.  RR+  ->  ( |_
`  A )  e.  ZZ )
44 ppival2 20779 . . . . . . 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 20811 . . . . . . 7  |-  ( A  e.  RR  ->  (π `  ( |_ `  A
) )  =  (π `  A ) )
471, 46syl 16 . . . . . 6  |-  ( A  e.  RR+  ->  (π `  ( |_ `  A ) )  =  (π `  A ) )
4845, 47eqtr3d 2422 . . . . 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 10557 . . . . . . 7  |-  ( A  e.  RR+  ->  0  <_  A )
51 flge0nn0 11153 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( |_ `  A
)  e.  NN0 )
521, 50, 51syl2anc 643 . . . . . 6  |-  ( A  e.  RR+  ->  ( |_
`  A )  e. 
NN0 )
53 hashfz1 11558 . . . . . 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 4183 . . 3  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (π `  A )  <  ( |_ `  A ) )
57 flle 11136 . . . 4  |-  ( A  e.  RR  ->  ( |_ `  A )  <_  A )
589, 57syl 16 . . 3  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  ( |_ `  A )  <_  A )
595, 8, 9, 56, 58ltletrd 9163 . 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 5673 . . . . 5  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  =  0 )  ->  (π `  ( |_ `  A
) )  =  (π `  0 ) )
63 2pos 10015 . . . . . 6  |-  0  <  2
64 0re 9025 . . . . . . 7  |-  0  e.  RR
65 ppieq0 20827 . . . . . . 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 2436 . . . 4  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  =  0 )  ->  (π `  ( |_ `  A
) )  =  0 )
6960, 68eqtr3d 2422 . . 3  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  =  0 )  ->  (π `  A )  =  0 )
70 rpgt0 10556 . . . 4  |-  ( A  e.  RR+  ->  0  < 
A )
7170adantr 452 . . 3  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  =  0 )  ->  0  <  A )
7269, 71eqbrtrd 4174 . 2  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  =  0 )  ->  (π `  A )  <  A
)
73 elnn0 10156 . . 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 1649    e. wcel 1717    =/= wne 2551    i^i cin 3263    C_ wss 3264    C. wpss 3265   class class class wbr 4154   ` cfv 5395  (class class class)co 6021    ~< csdm 7045   Fincfn 7046   RRcr 8923   0cc0 8924   1c1 8925    < clt 9054    <_ cle 9055   NNcn 9933   2c2 9982   NN0cn0 10154   ZZcz 10215   ZZ>=cuz 10421   RR+crp 10545   ...cfz 10976   |_cfl 11129   #chash 11546   Primecprime 13007  πcppi 20744
This theorem is referenced by:  chtppilimlem1  21035
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-2o 6662  df-oadd 6665  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-sup 7382  df-card 7760  df-cda 7982  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-2 9991  df-n0 10155  df-z 10216  df-uz 10422  df-rp 10546  df-icc 10856  df-fz 10977  df-fl 11130  df-hash 11547  df-dvds 12781  df-prm 13008  df-ppi 20750
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