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Theorem ppiltx 20409
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 10355 . . . . . 6  |-  ( A  e.  RR+  ->  A  e.  RR )
2 ppicl 20363 . . . . . 6  |-  ( A  e.  RR  ->  (π `  A )  e.  NN0 )
31, 2syl 17 . . . . 5  |-  ( A  e.  RR+  ->  (π `  A
)  e.  NN0 )
43nn0red 10014 . . . 4  |-  ( A  e.  RR+  ->  (π `  A
)  e.  RR )
54adantr 453 . . 3  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (π `  A )  e.  RR )
6 reflcl 10922 . . . . 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 11028 . . . . . 6  |-  ( 1 ... ( |_ `  A ) )  e. 
Fin
11 inss1 3390 . . . . . . 7  |-  ( ( 2 ... ( |_
`  A ) )  i^i  Prime )  C_  (
2 ... ( |_ `  A ) )
12 2nn 9872 . . . . . . . . . 10  |-  2  e.  NN
13 nnuz 10258 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
1412, 13eleqtri 2356 . . . . . . . . 9  |-  2  e.  ( ZZ>= `  1 )
15 fzss1 10824 . . . . . . . . 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 2374 . . . . . . . . . . 11  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  ( |_ `  A )  e.  ( ZZ>= `  1 )
)
19 eluzfz1 10797 . . . . . . . . . . 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 9881 . . . . . . . . . . . 12  |-  1  <  2
22 1re 8832 . . . . . . . . . . . . 13  |-  1  e.  RR
23 2re 9810 . . . . . . . . . . . . 13  |-  2  e.  RR
2422, 23ltnlei 8934 . . . . . . . . . . . 12  |-  ( 1  <  2  <->  -.  2  <_  1 )
2521, 24mpbi 201 . . . . . . . . . . 11  |-  -.  2  <_  1
26 elfzle1 10793 . . . . . . . . . . 11  |-  ( 1  e.  ( 2 ... ( |_ `  A
) )  ->  2  <_  1 )
2725, 26mto 169 . . . . . . . . . 10  |-  -.  1  e.  ( 2 ... ( |_ `  A ) )
28 nelne1 2536 . . . . . . . . . 10  |-  ( ( 1  e.  ( 1 ... ( |_ `  A ) )  /\  -.  1  e.  (
2 ... ( |_ `  A ) ) )  ->  ( 1 ... ( |_ `  A
) )  =/=  (
2 ... ( |_ `  A ) ) )
2920, 27, 28sylancl 645 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (
1 ... ( |_ `  A ) )  =/=  ( 2 ... ( |_ `  A ) ) )
3029necomd 2530 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (
2 ... ( |_ `  A ) )  =/=  ( 1 ... ( |_ `  A ) ) )
31 df-pss 3169 . . . . . . . 8  |-  ( ( 2 ... ( |_
`  A ) ) 
C.  ( 1 ... ( |_ `  A
) )  <->  ( (
2 ... ( |_ `  A ) )  C_  ( 1 ... ( |_ `  A ) )  /\  ( 2 ... ( |_ `  A
) )  =/=  (
1 ... ( |_ `  A ) ) ) )
3216, 30, 31sylanbrc 647 . . . . . . 7  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (
2 ... ( |_ `  A ) )  C.  ( 1 ... ( |_ `  A ) ) )
33 sspsstr 3282 . . . . . . 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 646 . . . . . 6  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (
( 2 ... ( |_ `  A ) )  i^i  Prime )  C.  (
1 ... ( |_ `  A ) ) )
35 php3 7042 . . . . . 6  |-  ( ( ( 1 ... ( |_ `  A ) )  e.  Fin  /\  (
( 2 ... ( |_ `  A ) )  i^i  Prime )  C.  (
1 ... ( |_ `  A ) ) )  ->  ( ( 2 ... ( |_ `  A ) )  i^i 
Prime )  ~<  ( 1 ... ( |_ `  A ) ) )
3610, 34, 35sylancr 646 . . . . 5  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (
( 2 ... ( |_ `  A ) )  i^i  Prime )  ~<  (
1 ... ( |_ `  A ) ) )
37 fzfi 11028 . . . . . . 7  |-  ( 2 ... ( |_ `  A ) )  e. 
Fin
38 ssfi 7078 . . . . . . 7  |-  ( ( ( 2 ... ( |_ `  A ) )  e.  Fin  /\  (
( 2 ... ( |_ `  A ) )  i^i  Prime )  C_  (
2 ... ( |_ `  A ) ) )  ->  ( ( 2 ... ( |_ `  A ) )  i^i 
Prime )  e.  Fin )
3937, 11, 38mp2an 655 . . . . . 6  |-  ( ( 2 ... ( |_
`  A ) )  i^i  Prime )  e.  Fin
40 hashsdom 11357 . . . . . 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 655 . . . . 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 10924 . . . . . . 7  |-  ( A  e.  RR+  ->  ( |_
`  A )  e.  ZZ )
44 ppival2 20360 . . . . . . 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 20392 . . . . . . 7  |-  ( A  e.  RR  ->  (π `  ( |_ `  A
) )  =  (π `  A ) )
471, 46syl 17 . . . . . 6  |-  ( A  e.  RR+  ->  (π `  ( |_ `  A ) )  =  (π `  A ) )
4845, 47eqtr3d 2318 . . . . 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 10361 . . . . . . 7  |-  ( A  e.  RR+  ->  0  <_  A )
51 flge0nn0 10942 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( |_ `  A
)  e.  NN0 )
521, 50, 51syl2anc 644 . . . . . 6  |-  ( A  e.  RR+  ->  ( |_
`  A )  e. 
NN0 )
53 hashfz1 11339 . . . . . 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 4053 . . 3  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  (π `  A )  <  ( |_ `  A ) )
57 flle 10925 . . . 4  |-  ( A  e.  RR  ->  ( |_ `  A )  <_  A )
589, 57syl 17 . . 3  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  e.  NN )  ->  ( |_ `  A )  <_  A )
595, 8, 9, 56, 58ltletrd 8971 . 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 5489 . . . . 5  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  =  0 )  ->  (π `  ( |_ `  A
) )  =  (π `  0 ) )
63 2pos 9823 . . . . . 6  |-  0  <  2
64 0re 8833 . . . . . . 7  |-  0  e.  RR
65 ppieq0 20408 . . . . . . 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 2332 . . . 4  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  =  0 )  ->  (π `  ( |_ `  A
) )  =  0 )
6960, 68eqtr3d 2318 . . 3  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  =  0 )  ->  (π `  A )  =  0 )
70 rpgt0 10360 . . . 4  |-  ( A  e.  RR+  ->  0  < 
A )
7170adantr 453 . . 3  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  =  0 )  ->  0  <  A )
7269, 71eqbrtrd 4044 . 2  |-  ( ( A  e.  RR+  /\  ( |_ `  A )  =  0 )  ->  (π `  A )  <  A
)
73 elnn0 9962 . . 3  |-  ( ( |_ `  A )  e.  NN0  <->  ( ( |_
`  A )  e.  NN  \/  ( |_
`  A )  =  0 ) )
7452, 73sylib 190 . 2  |-  ( A  e.  RR+  ->  ( ( |_ `  A )  e.  NN  \/  ( |_ `  A )  =  0 ) )
7559, 72, 74mpjaodan 763 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 1624    e. wcel 1685    =/= wne 2447    i^i cin 3152    C_ wss 3153    C. wpss 3154   class class class wbr 4024   ` cfv 5221  (class class class)co 5819    ~< csdm 6857   Fincfn 6858   RRcr 8731   0cc0 8732   1c1 8733    < clt 8862    <_ cle 8863   NNcn 9741   2c2 9790   NN0cn0 9960   ZZcz 10019   ZZ>=cuz 10225   RR+crp 10349   ...cfz 10776   |_cfl 10918   #chash 11331   Primecprime 12752  πcppi 20325
This theorem is referenced by:  chtppilimlem1  20616
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-2o 6475  df-oadd 6478  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-sup 7189  df-card 7567  df-cda 7789  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-nn 9742  df-2 9799  df-n0 9961  df-z 10020  df-uz 10226  df-rp 10350  df-icc 10657  df-fz 10777  df-fl 10919  df-hash 11332  df-dvds 12526  df-prm 12753  df-ppi 20331
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