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Theorem ppip1le 23575
Description: The prime-counting function π 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 11854 . . 3  |-  ( A  e.  RR  ->  ( |_ `  A )  e.  ZZ )
2 zre 10807 . . . . . . . . 9  |-  ( ( |_ `  A )  e.  ZZ  ->  ( |_ `  A )  e.  RR )
3 peano2re 9686 . . . . . . . . 9  |-  ( ( |_ `  A )  e.  RR  ->  (
( |_ `  A
)  +  1 )  e.  RR )
42, 3syl 16 . . . . . . . 8  |-  ( ( |_ `  A )  e.  ZZ  ->  (
( |_ `  A
)  +  1 )  e.  RR )
54adantr 463 . . . . . . 7  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  ( ( |_ `  A )  +  1 )  e.  RR )
6 ppicl 23545 . . . . . . 7  |-  ( ( ( |_ `  A
)  +  1 )  e.  RR  ->  (π `  ( ( |_ `  A )  +  1 ) )  e.  NN0 )
75, 6syl 16 . . . . . 6  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( ( |_
`  A )  +  1 ) )  e. 
NN0 )
87nn0red 10792 . . . . 5  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( ( |_
`  A )  +  1 ) )  e.  RR )
9 ppiprm 23565 . . . . 5  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( ( |_
`  A )  +  1 ) )  =  ( (π `  ( |_ `  A ) )  +  1 ) )
10 eqle 9620 . . . . 5  |-  ( ( (π `  ( ( |_
`  A )  +  1 ) )  e.  RR  /\  (π `  (
( |_ `  A
)  +  1 ) )  =  ( (π `  ( |_ `  A
) )  +  1 ) )  ->  (π `  ( ( |_ `  A )  +  1 ) )  <_  (
(π `  ( |_ `  A ) )  +  1 ) )
118, 9, 10syl2anc 659 . . . 4  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( ( |_
`  A )  +  1 ) )  <_ 
( (π `  ( |_ `  A ) )  +  1 ) )
12 ppinprm 23566 . . . . 5  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  -.  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( ( |_
`  A )  +  1 ) )  =  (π `  ( |_ `  A ) ) )
13 ppicl 23545 . . . . . . . . 9  |-  ( ( |_ `  A )  e.  RR  ->  (π `  ( |_ `  A
) )  e.  NN0 )
142, 13syl 16 . . . . . . . 8  |-  ( ( |_ `  A )  e.  ZZ  ->  (π `  ( |_ `  A
) )  e.  NN0 )
1514nn0red 10792 . . . . . . 7  |-  ( ( |_ `  A )  e.  ZZ  ->  (π `  ( |_ `  A
) )  e.  RR )
1615adantr 463 . . . . . 6  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  -.  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( |_ `  A ) )  e.  RR )
1716lep1d 10415 . . . . 5  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  -.  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( |_ `  A ) )  <_ 
( (π `  ( |_ `  A ) )  +  1 ) )
1812, 17eqbrtrd 4404 . . . 4  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  -.  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( ( |_
`  A )  +  1 ) )  <_ 
( (π `  ( |_ `  A ) )  +  1 ) )
1911, 18pm2.61dan 789 . . 3  |-  ( ( |_ `  A )  e.  ZZ  ->  (π `  ( ( |_ `  A )  +  1 ) )  <_  (
(π `  ( |_ `  A ) )  +  1 ) )
201, 19syl 16 . 2  |-  ( A  e.  RR  ->  (π `  ( ( |_ `  A )  +  1 ) )  <_  (
(π `  ( |_ `  A ) )  +  1 ) )
21 1z 10833 . . . . 5  |-  1  e.  ZZ
22 fladdz 11881 . . . . 5  |-  ( ( A  e.  RR  /\  1  e.  ZZ )  ->  ( |_ `  ( A  +  1 ) )  =  ( ( |_ `  A )  +  1 ) )
2321, 22mpan2 669 . . . 4  |-  ( A  e.  RR  ->  ( |_ `  ( A  + 
1 ) )  =  ( ( |_ `  A )  +  1 ) )
2423fveq2d 5795 . . 3  |-  ( A  e.  RR  ->  (π `  ( |_ `  ( A  +  1 ) ) )  =  (π `  ( ( |_ `  A )  +  1 ) ) )
25 peano2re 9686 . . . 4  |-  ( A  e.  RR  ->  ( A  +  1 )  e.  RR )
26 ppifl 23574 . . . 4  |-  ( ( A  +  1 )  e.  RR  ->  (π `  ( |_ `  ( A  +  1 ) ) )  =  (π `  ( A  +  1 ) ) )
2725, 26syl 16 . . 3  |-  ( A  e.  RR  ->  (π `  ( |_ `  ( A  +  1 ) ) )  =  (π `  ( A  +  1 ) ) )
2824, 27eqtr3d 2439 . 2  |-  ( A  e.  RR  ->  (π `  ( ( |_ `  A )  +  1 ) )  =  (π `  ( A  +  1 ) ) )
29 ppifl 23574 . . 3  |-  ( A  e.  RR  ->  (π `  ( |_ `  A
) )  =  (π `  A ) )
3029oveq1d 6233 . 2  |-  ( A  e.  RR  ->  (
(π `  ( |_ `  A ) )  +  1 )  =  ( (π `  A )  +  1 ) )
3120, 28, 303brtr3d 4413 1  |-  ( A  e.  RR  ->  (π `  ( A  +  1 ) )  <_  (
(π `  A )  +  1 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1836   class class class wbr 4384   ` cfv 5513  (class class class)co 6218   RRcr 9424   1c1 9426    + caddc 9428    <_ cle 9562   NN0cn0 10734   ZZcz 10803   |_cfl 11849   Primecprime 14242  πcppi 23507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513  ax-cnex 9481  ax-resscn 9482  ax-1cn 9483  ax-icn 9484  ax-addcl 9485  ax-addrcl 9486  ax-mulcl 9487  ax-mulrcl 9488  ax-mulcom 9489  ax-addass 9490  ax-mulass 9491  ax-distr 9492  ax-i2m1 9493  ax-1ne0 9494  ax-1rid 9495  ax-rnegex 9496  ax-rrecex 9497  ax-cnre 9498  ax-pre-lttri 9499  ax-pre-lttrn 9500  ax-pre-ltadd 9501  ax-pre-mulgt0 9502  ax-pre-sup 9503
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-nel 2594  df-ral 2751  df-rex 2752  df-reu 2753  df-rmo 2754  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4181  df-int 4217  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-tr 4478  df-eprel 4722  df-id 4726  df-po 4731  df-so 4732  df-fr 4769  df-we 4771  df-ord 4812  df-on 4813  df-lim 4814  df-suc 4815  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6180  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-om 6622  df-1st 6721  df-2nd 6722  df-recs 6982  df-rdg 7016  df-1o 7070  df-2o 7071  df-oadd 7074  df-er 7251  df-en 7458  df-dom 7459  df-sdom 7460  df-fin 7461  df-sup 7838  df-card 8255  df-cda 8483  df-pnf 9563  df-mnf 9564  df-xr 9565  df-ltxr 9566  df-le 9567  df-sub 9742  df-neg 9743  df-nn 10475  df-2 10533  df-n0 10735  df-z 10804  df-uz 11024  df-icc 11479  df-fz 11616  df-fl 11851  df-hash 12331  df-dvds 14012  df-prm 14243  df-ppi 23513
This theorem is referenced by: (None)
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