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Theorem ppip1le 22633
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 11763 . . 3  |-  ( A  e.  RR  ->  ( |_ `  A )  e.  ZZ )
2 zre 10762 . . . . . . . . 9  |-  ( ( |_ `  A )  e.  ZZ  ->  ( |_ `  A )  e.  RR )
3 peano2re 9654 . . . . . . . . 9  |-  ( ( |_ `  A )  e.  RR  ->  (
( |_ `  A
)  +  1 )  e.  RR )
42, 3syl 16 . . . . . . . 8  |-  ( ( |_ `  A )  e.  ZZ  ->  (
( |_ `  A
)  +  1 )  e.  RR )
54adantr 465 . . . . . . 7  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  ( ( |_ `  A )  +  1 )  e.  RR )
6 ppicl 22603 . . . . . . 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 10749 . . . . 5  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( ( |_
`  A )  +  1 ) )  e.  RR )
9 ppiprm 22623 . . . . 5  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( ( |_
`  A )  +  1 ) )  =  ( (π `  ( |_ `  A ) )  +  1 ) )
10 eqle 9589 . . . . 5  |-  ( ( (π `  ( ( |_
`  A )  +  1 ) )  e.  RR  /\  (π `  (
( |_ `  A
)  +  1 ) )  =  ( (π `  ( |_ `  A
) )  +  1 ) )  ->  (π `  ( ( |_ `  A )  +  1 ) )  <_  (
(π `  ( |_ `  A ) )  +  1 ) )
118, 9, 10syl2anc 661 . . . 4  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( ( |_
`  A )  +  1 ) )  <_ 
( (π `  ( |_ `  A ) )  +  1 ) )
12 ppinprm 22624 . . . . 5  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  -.  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( ( |_
`  A )  +  1 ) )  =  (π `  ( |_ `  A ) ) )
13 ppicl 22603 . . . . . . . . 9  |-  ( ( |_ `  A )  e.  RR  ->  (π `  ( |_ `  A
) )  e.  NN0 )
142, 13syl 16 . . . . . . . 8  |-  ( ( |_ `  A )  e.  ZZ  ->  (π `  ( |_ `  A
) )  e.  NN0 )
1514nn0red 10749 . . . . . . 7  |-  ( ( |_ `  A )  e.  ZZ  ->  (π `  ( |_ `  A
) )  e.  RR )
1615adantr 465 . . . . . 6  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  -.  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( |_ `  A ) )  e.  RR )
1716lep1d 10376 . . . . 5  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  -.  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( |_ `  A ) )  <_ 
( (π `  ( |_ `  A ) )  +  1 ) )
1812, 17eqbrtrd 4421 . . . 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 10788 . . . . 5  |-  1  e.  ZZ
22 fladdz 11788 . . . . 5  |-  ( ( A  e.  RR  /\  1  e.  ZZ )  ->  ( |_ `  ( A  +  1 ) )  =  ( ( |_ `  A )  +  1 ) )
2321, 22mpan2 671 . . . 4  |-  ( A  e.  RR  ->  ( |_ `  ( A  + 
1 ) )  =  ( ( |_ `  A )  +  1 ) )
2423fveq2d 5804 . . 3  |-  ( A  e.  RR  ->  (π `  ( |_ `  ( A  +  1 ) ) )  =  (π `  ( ( |_ `  A )  +  1 ) ) )
25 peano2re 9654 . . . 4  |-  ( A  e.  RR  ->  ( A  +  1 )  e.  RR )
26 ppifl 22632 . . . 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 2497 . 2  |-  ( A  e.  RR  ->  (π `  ( ( |_ `  A )  +  1 ) )  =  (π `  ( A  +  1 ) ) )
29 ppifl 22632 . . 3  |-  ( A  e.  RR  ->  (π `  ( |_ `  A
) )  =  (π `  A ) )
3029oveq1d 6216 . 2  |-  ( A  e.  RR  ->  (
(π `  ( |_ `  A ) )  +  1 )  =  ( (π `  A )  +  1 ) )
3120, 28, 303brtr3d 4430 1  |-  ( A  e.  RR  ->  (π `  ( A  +  1 ) )  <_  (
(π `  A )  +  1 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   class class class wbr 4401   ` cfv 5527  (class class class)co 6201   RRcr 9393   1c1 9395    + caddc 9397    <_ cle 9531   NN0cn0 10691   ZZcz 10758   |_cfl 11758   Primecprime 13882  πcppi 22565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471  ax-pre-sup 9472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-2o 7032  df-oadd 7035  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-sup 7803  df-card 8221  df-cda 8449  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-n0 10692  df-z 10759  df-uz 10974  df-icc 11419  df-fz 11556  df-fl 11760  df-hash 12222  df-dvds 13655  df-prm 13883  df-ppi 22571
This theorem is referenced by: (None)
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