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Theorem ppip1le 20812
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 11132 . . 3  |-  ( A  e.  RR  ->  ( |_ `  A )  e.  ZZ )
2 zre 10219 . . . . . . . . 9  |-  ( ( |_ `  A )  e.  ZZ  ->  ( |_ `  A )  e.  RR )
3 peano2re 9172 . . . . . . . . 9  |-  ( ( |_ `  A )  e.  RR  ->  (
( |_ `  A
)  +  1 )  e.  RR )
42, 3syl 16 . . . . . . . 8  |-  ( ( |_ `  A )  e.  ZZ  ->  (
( |_ `  A
)  +  1 )  e.  RR )
54adantr 452 . . . . . . 7  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  ( ( |_ `  A )  +  1 )  e.  RR )
6 ppicl 20782 . . . . . . 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 10208 . . . . 5  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( ( |_
`  A )  +  1 ) )  e.  RR )
9 ppiprm 20802 . . . . 5  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( ( |_
`  A )  +  1 ) )  =  ( (π `  ( |_ `  A ) )  +  1 ) )
10 eqle 9110 . . . . 5  |-  ( ( (π `  ( ( |_
`  A )  +  1 ) )  e.  RR  /\  (π `  (
( |_ `  A
)  +  1 ) )  =  ( (π `  ( |_ `  A
) )  +  1 ) )  ->  (π `  ( ( |_ `  A )  +  1 ) )  <_  (
(π `  ( |_ `  A ) )  +  1 ) )
118, 9, 10syl2anc 643 . . . 4  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( ( |_
`  A )  +  1 ) )  <_ 
( (π `  ( |_ `  A ) )  +  1 ) )
12 ppinprm 20803 . . . . 5  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  -.  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( ( |_
`  A )  +  1 ) )  =  (π `  ( |_ `  A ) ) )
13 ppicl 20782 . . . . . . . . 9  |-  ( ( |_ `  A )  e.  RR  ->  (π `  ( |_ `  A
) )  e.  NN0 )
142, 13syl 16 . . . . . . . 8  |-  ( ( |_ `  A )  e.  ZZ  ->  (π `  ( |_ `  A
) )  e.  NN0 )
1514nn0red 10208 . . . . . . 7  |-  ( ( |_ `  A )  e.  ZZ  ->  (π `  ( |_ `  A
) )  e.  RR )
1615adantr 452 . . . . . 6  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  -.  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( |_ `  A ) )  e.  RR )
1716lep1d 9875 . . . . 5  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  -.  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( |_ `  A ) )  <_ 
( (π `  ( |_ `  A ) )  +  1 ) )
1812, 17eqbrtrd 4174 . . . 4  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  -.  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( ( |_
`  A )  +  1 ) )  <_ 
( (π `  ( |_ `  A ) )  +  1 ) )
1911, 18pm2.61dan 767 . . 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 10244 . . . . 5  |-  1  e.  ZZ
22 fladdz 11155 . . . . 5  |-  ( ( A  e.  RR  /\  1  e.  ZZ )  ->  ( |_ `  ( A  +  1 ) )  =  ( ( |_ `  A )  +  1 ) )
2321, 22mpan2 653 . . . 4  |-  ( A  e.  RR  ->  ( |_ `  ( A  + 
1 ) )  =  ( ( |_ `  A )  +  1 ) )
2423fveq2d 5673 . . 3  |-  ( A  e.  RR  ->  (π `  ( |_ `  ( A  +  1 ) ) )  =  (π `  ( ( |_ `  A )  +  1 ) ) )
25 peano2re 9172 . . . 4  |-  ( A  e.  RR  ->  ( A  +  1 )  e.  RR )
26 ppifl 20811 . . . 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 2422 . 2  |-  ( A  e.  RR  ->  (π `  ( ( |_ `  A )  +  1 ) )  =  (π `  ( A  +  1 ) ) )
29 ppifl 20811 . . 3  |-  ( A  e.  RR  ->  (π `  ( |_ `  A
) )  =  (π `  A ) )
3029oveq1d 6036 . 2  |-  ( A  e.  RR  ->  (
(π `  ( |_ `  A ) )  +  1 )  =  ( (π `  A )  +  1 ) )
3120, 28, 303brtr3d 4183 1  |-  ( A  e.  RR  ->  (π `  ( A  +  1 ) )  <_  (
(π `  A )  +  1 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   class class class wbr 4154   ` cfv 5395  (class class class)co 6021   RRcr 8923   1c1 8925    + caddc 8927    <_ cle 9055   NN0cn0 10154   ZZcz 10215   |_cfl 11129   Primecprime 13007  πcppi 20744
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-icc 10856  df-fz 10977  df-fl 11130  df-hash 11547  df-dvds 12781  df-prm 13008  df-ppi 20750
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