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Theorem ppip1le 20347
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 10879 . . 3  |-  ( A  e.  RR  ->  ( |_ `  A )  e.  ZZ )
2 zre 9981 . . . . . . . . 9  |-  ( ( |_ `  A )  e.  ZZ  ->  ( |_ `  A )  e.  RR )
3 peano2re 8939 . . . . . . . . 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 20317 . . . . . . 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 9972 . . . . 5  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( ( |_
`  A )  +  1 ) )  e.  RR )
9 ppiprm 20337 . . . . 5  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( ( |_
`  A )  +  1 ) )  =  ( (π `  ( |_ `  A ) )  +  1 ) )
10 eqle 8877 . . . . 5  |-  ( ( (π `  ( ( |_
`  A )  +  1 ) )  e.  RR  /\  (π `  (
( |_ `  A
)  +  1 ) )  =  ( (π `  ( |_ `  A
) )  +  1 ) )  ->  (π `  ( ( |_ `  A )  +  1 ) )  <_  (
(π `  ( |_ `  A ) )  +  1 ) )
118, 9, 10syl2anc 645 . . . 4  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( ( |_
`  A )  +  1 ) )  <_ 
( (π `  ( |_ `  A ) )  +  1 ) )
12 ppinprm 20338 . . . . 5  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  -.  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( ( |_
`  A )  +  1 ) )  =  (π `  ( |_ `  A ) ) )
13 ppicl 20317 . . . . . . . . 9  |-  ( ( |_ `  A )  e.  RR  ->  (π `  ( |_ `  A
) )  e.  NN0 )
142, 13syl 17 . . . . . . . 8  |-  ( ( |_ `  A )  e.  ZZ  ->  (π `  ( |_ `  A
) )  e.  NN0 )
1514nn0red 9972 . . . . . . 7  |-  ( ( |_ `  A )  e.  ZZ  ->  (π `  ( |_ `  A
) )  e.  RR )
1615adantr 453 . . . . . 6  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  -.  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( |_ `  A ) )  e.  RR )
1716lep1d 9642 . . . . 5  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  -.  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( |_ `  A ) )  <_ 
( (π `  ( |_ `  A ) )  +  1 ) )
1812, 17eqbrtrd 4003 . . . 4  |-  ( ( ( |_ `  A
)  e.  ZZ  /\  -.  ( ( |_ `  A )  +  1 )  e.  Prime )  ->  (π `  ( ( |_
`  A )  +  1 ) )  <_ 
( (π `  ( |_ `  A ) )  +  1 ) )
1911, 18pm2.61dan 769 . . 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 10006 . . . . 5  |-  1  e.  ZZ
22 fladdz 10902 . . . . 5  |-  ( ( A  e.  RR  /\  1  e.  ZZ )  ->  ( |_ `  ( A  +  1 ) )  =  ( ( |_ `  A )  +  1 ) )
2321, 22mpan2 655 . . . 4  |-  ( A  e.  RR  ->  ( |_ `  ( A  + 
1 ) )  =  ( ( |_ `  A )  +  1 ) )
2423fveq2d 5448 . . 3  |-  ( A  e.  RR  ->  (π `  ( |_ `  ( A  +  1 ) ) )  =  (π `  ( ( |_ `  A )  +  1 ) ) )
25 peano2re 8939 . . . 4  |-  ( A  e.  RR  ->  ( A  +  1 )  e.  RR )
26 ppifl 20346 . . . 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 2290 . 2  |-  ( A  e.  RR  ->  (π `  ( ( |_ `  A )  +  1 ) )  =  (π `  ( A  +  1 ) ) )
29 ppifl 20346 . . 3  |-  ( A  e.  RR  ->  (π `  ( |_ `  A
) )  =  (π `  A ) )
3029oveq1d 5793 . 2  |-  ( A  e.  RR  ->  (
(π `  ( |_ `  A ) )  +  1 )  =  ( (π `  A )  +  1 ) )
3120, 28, 303brtr3d 4012 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 1619    e. wcel 1621   class class class wbr 3983   ` cfv 4659  (class class class)co 5778   RRcr 8690   1c1 8692    + caddc 8694    <_ cle 8822   NN0cn0 9918   ZZcz 9977   |_cfl 10876   Primecprime 12706  πcppi 20279
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768  ax-pre-sup 8769
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-recs 6342  df-rdg 6377  df-1o 6433  df-2o 6434  df-oadd 6437  df-er 6614  df-en 6818  df-dom 6819  df-sdom 6820  df-fin 6821  df-sup 7148  df-card 7526  df-cda 7748  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-n 9701  df-2 9758  df-n0 9919  df-z 9978  df-uz 10184  df-icc 10615  df-fz 10735  df-fl 10877  df-hash 11290  df-divides 12480  df-prime 12707  df-ppi 20285
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