MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ppip1le Unicode version

Theorem ppip1le 20393
Description: The prime pi function cannot locally increase faster than the identity function. (Contributed by Mario Carneiro, 21-Sep-2014.)
Assertion
Ref Expression
ppip1le  |-  ( A  e.  RR  ->  (π `  ( A  +  1 ) )  <_  (
(π `  A )  +  1 ) )

Proof of Theorem ppip1le
StepHypRef Expression
1 flcl 10921 . . 3  |-  ( A  e.  RR  ->  ( |_ `  A )  e.  ZZ )
2 zre 10023 . . . . . . . . 9  |-  ( ( |_ `  A )  e.  ZZ  ->  ( |_ `  A )  e.  RR )
3 peano2re 8980 . . . . . . . . 9  |-  ( ( |_ `  A )  e.  RR  ->  (
( |_ `  A
)  +  1 )  e.  RR )
42, 3syl 17 . . . . . . . 8  |-  ( ( |_ `  A )  e.  ZZ  ->  (
( |_ `  A
)  +  1 )  e.  RR )
54adantr 453 . . . . . . 7  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  ( ( |_ `  A )  +  1 )  e.  RR )
6 ppicl 20363 . . . . . . 7  |-  ( ( ( |_ `  A
)  +  1 )  e.  RR  ->  (π `  ( ( |_ `  A )  +  1 ) )  e.  NN0 )
75, 6syl 17 . . . . . 6  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( ( |_
`  A )  +  1 ) )  e. 
NN0 )
87nn0red 10014 . . . . 5  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( ( |_
`  A )  +  1 ) )  e.  RR )
9 ppiprm 20383 . . . . 5  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( ( |_
`  A )  +  1 ) )  =  ( (π `  ( |_ `  A ) )  +  1 ) )
10 eqle 8918 . . . . 5  |-  ( ( (π `  ( ( |_
`  A )  +  1 ) )  e.  RR  /\  (π `  (
( |_ `  A
)  +  1 ) )  =  ( (π `  ( |_ `  A
) )  +  1 ) )  ->  (π `  ( ( |_ `  A )  +  1 ) )  <_  (
(π `  ( |_ `  A ) )  +  1 ) )
118, 9, 10syl2anc 644 . . . 4  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( ( |_
`  A )  +  1 ) )  <_ 
( (π `  ( |_ `  A ) )  +  1 ) )
12 ppinprm 20384 . . . . 5  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  -.  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( ( |_
`  A )  +  1 ) )  =  (π `  ( |_ `  A ) ) )
13 ppicl 20363 . . . . . . . . 9  |-  ( ( |_ `  A )  e.  RR  ->  (π `  ( |_ `  A
) )  e.  NN0 )
142, 13syl 17 . . . . . . . 8  |-  ( ( |_ `  A )  e.  ZZ  ->  (π `  ( |_ `  A
) )  e.  NN0 )
1514nn0red 10014 . . . . . . 7  |-  ( ( |_ `  A )  e.  ZZ  ->  (π `  ( |_ `  A
) )  e.  RR )
1615adantr 453 . . . . . 6  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  -.  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( |_ `  A ) )  e.  RR )
1716lep1d 9683 . . . . 5  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  -.  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( |_ `  A ) )  <_ 
( (π `  ( |_ `  A ) )  +  1 ) )
1812, 17eqbrtrd 4044 . . . 4  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  -.  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( ( |_
`  A )  +  1 ) )  <_ 
( (π `  ( |_ `  A ) )  +  1 ) )
1911, 18pm2.61dan 768 . . 3  |-  ( ( |_ `  A )  e.  ZZ  ->  (π `  ( ( |_ `  A )  +  1 ) )  <_  (
(π `  ( |_ `  A ) )  +  1 ) )
201, 19syl 17 . 2  |-  ( A  e.  RR  ->  (π `  ( ( |_ `  A )  +  1 ) )  <_  (
(π `  ( |_ `  A ) )  +  1 ) )
21 1z 10048 . . . . 5  |-  1  e.  ZZ
22 fladdz 10944 . . . . 5  |-  ( ( A  e.  RR  /\  1  e.  ZZ )  ->  ( |_ `  ( A  +  1 ) )  =  ( ( |_ `  A )  +  1 ) )
2321, 22mpan2 654 . . . 4  |-  ( A  e.  RR  ->  ( |_ `  ( A  + 
1 ) )  =  ( ( |_ `  A )  +  1 ) )
2423fveq2d 5489 . . 3  |-  ( A  e.  RR  ->  (π `  ( |_ `  ( A  +  1 ) ) )  =  (π `  ( ( |_ `  A )  +  1 ) ) )
25 peano2re 8980 . . . 4  |-  ( A  e.  RR  ->  ( A  +  1 )  e.  RR )
26 ppifl 20392 . . . 4  |-  ( ( A  +  1 )  e.  RR  ->  (π `  ( |_ `  ( A  +  1 ) ) )  =  (π `  ( A  +  1 ) ) )
2725, 26syl 17 . . 3  |-  ( A  e.  RR  ->  (π `  ( |_ `  ( A  +  1 ) ) )  =  (π `  ( A  +  1 ) ) )
2824, 27eqtr3d 2318 . 2  |-  ( A  e.  RR  ->  (π `  ( ( |_ `  A )  +  1 ) )  =  (π `  ( A  +  1 ) ) )
29 ppifl 20392 . . 3  |-  ( A  e.  RR  ->  (π `  ( |_ `  A
) )  =  (π `  A ) )
3029oveq1d 5834 . 2  |-  ( A  e.  RR  ->  (
(π `  ( |_ `  A ) )  +  1 )  =  ( (π `  A )  +  1 ) )
3120, 28, 303brtr3d 4053 1  |-  ( A  e.  RR  ->  (π `  ( A  +  1 ) )  <_  (
(π `  A )  +  1 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    = wceq 1624    e. wcel 1685   class class class wbr 4024   ` cfv 5221  (class class class)co 5819   RRcr 8731   1c1 8733    + caddc 8735    <_ cle 8863   NN0cn0 9960   ZZcz 10019   |_cfl 10918   Primecprime 12752  πcppi 20325
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-icc 10657  df-fz 10777  df-fl 10919  df-hash 11332  df-dvds 12526  df-prm 12753  df-ppi 20331
  Copyright terms: Public domain W3C validator