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Theorem ppiprm 22494
Description: The prime-counting function π at a prime. (Contributed by Mario Carneiro, 19-Sep-2014.)
Assertion
Ref Expression
ppiprm  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  (π `  ( A  + 
1 ) )  =  ( (π `  A )  +  1 ) )

Proof of Theorem ppiprm
StepHypRef Expression
1 fzfid 11800 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( 2 ... A
)  e.  Fin )
2 inss1 3575 . . . 4  |-  ( ( 2 ... A )  i^i  Prime )  C_  (
2 ... A )
3 ssfi 7538 . . . 4  |-  ( ( ( 2 ... A
)  e.  Fin  /\  ( ( 2 ... A )  i^i  Prime ) 
C_  ( 2 ... A ) )  -> 
( ( 2 ... A )  i^i  Prime )  e.  Fin )
41, 2, 3sylancl 662 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 2 ... A )  i^i  Prime )  e.  Fin )
5 zre 10655 . . . . . . 7  |-  ( A  e.  ZZ  ->  A  e.  RR )
65adantr 465 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  e.  RR )
76ltp1d 10268 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  <  ( A  +  1 ) )
8 peano2z 10691 . . . . . . . 8  |-  ( A  e.  ZZ  ->  ( A  +  1 )  e.  ZZ )
98adantr 465 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  ZZ )
109zred 10752 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  RR )
116, 10ltnled 9526 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  <  ( A  +  1 )  <->  -.  ( A  +  1 )  <_  A )
)
127, 11mpbid 210 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  -.  ( A  + 
1 )  <_  A
)
132sseli 3357 . . . . 5  |-  ( ( A  +  1 )  e.  ( ( 2 ... A )  i^i 
Prime )  ->  ( A  +  1 )  e.  ( 2 ... A
) )
14 elfzle2 11460 . . . . 5  |-  ( ( A  +  1 )  e.  ( 2 ... A )  ->  ( A  +  1 )  <_  A )
1513, 14syl 16 . . . 4  |-  ( ( A  +  1 )  e.  ( ( 2 ... A )  i^i 
Prime )  ->  ( A  +  1 )  <_  A )
1612, 15nsyl 121 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  -.  ( A  + 
1 )  e.  ( ( 2 ... A
)  i^i  Prime ) )
17 ovex 6121 . . . 4  |-  ( A  +  1 )  e. 
_V
18 hashunsng 12159 . . . 4  |-  ( ( A  +  1 )  e.  _V  ->  (
( ( ( 2 ... A )  i^i 
Prime )  e.  Fin  /\ 
-.  ( A  + 
1 )  e.  ( ( 2 ... A
)  i^i  Prime ) )  ->  ( # `  (
( ( 2 ... A )  i^i  Prime )  u.  { ( A  +  1 ) } ) )  =  ( ( # `  (
( 2 ... A
)  i^i  Prime ) )  +  1 ) ) )
1917, 18ax-mp 5 . . 3  |-  ( ( ( ( 2 ... A )  i^i  Prime )  e.  Fin  /\  -.  ( A  +  1
)  e.  ( ( 2 ... A )  i^i  Prime ) )  -> 
( # `  ( ( ( 2 ... A
)  i^i  Prime )  u. 
{ ( A  + 
1 ) } ) )  =  ( (
# `  ( (
2 ... A )  i^i 
Prime ) )  +  1 ) )
204, 16, 19syl2anc 661 . 2  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( # `  (
( ( 2 ... A )  i^i  Prime )  u.  { ( A  +  1 ) } ) )  =  ( ( # `  (
( 2 ... A
)  i^i  Prime ) )  +  1 ) )
21 ppival2 22471 . . . 4  |-  ( ( A  +  1 )  e.  ZZ  ->  (π `  ( A  +  1 ) )  =  (
# `  ( (
2 ... ( A  + 
1 ) )  i^i 
Prime ) ) )
229, 21syl 16 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  (π `  ( A  + 
1 ) )  =  ( # `  (
( 2 ... ( A  +  1 ) )  i^i  Prime )
) )
23 2z 10683 . . . . . . . 8  |-  2  e.  ZZ
24 zcn 10656 . . . . . . . . . . . 12  |-  ( A  e.  ZZ  ->  A  e.  CC )
2524adantr 465 . . . . . . . . . . 11  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  e.  CC )
26 ax-1cn 9345 . . . . . . . . . . 11  |-  1  e.  CC
27 pncan 9621 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  + 
1 )  -  1 )  =  A )
2825, 26, 27sylancl 662 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( A  + 
1 )  -  1 )  =  A )
29 prmuz2 13786 . . . . . . . . . . . 12  |-  ( ( A  +  1 )  e.  Prime  ->  ( A  +  1 )  e.  ( ZZ>= `  2 )
)
3029adantl 466 . . . . . . . . . . 11  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  ( ZZ>= ` 
2 ) )
31 uz2m1nn 10934 . . . . . . . . . . 11  |-  ( ( A  +  1 )  e.  ( ZZ>= `  2
)  ->  ( ( A  +  1 )  -  1 )  e.  NN )
3230, 31syl 16 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( A  + 
1 )  -  1 )  e.  NN )
3328, 32eqeltrrd 2518 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  e.  NN )
34 nnuz 10901 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
35 2m1e1 10441 . . . . . . . . . . 11  |-  ( 2  -  1 )  =  1
3635fveq2i 5699 . . . . . . . . . 10  |-  ( ZZ>= `  ( 2  -  1 ) )  =  (
ZZ>= `  1 )
3734, 36eqtr4i 2466 . . . . . . . . 9  |-  NN  =  ( ZZ>= `  ( 2  -  1 ) )
3833, 37syl6eleq 2533 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  e.  ( ZZ>= `  ( 2  -  1 ) ) )
39 fzsuc2 11519 . . . . . . . 8  |-  ( ( 2  e.  ZZ  /\  A  e.  ( ZZ>= `  ( 2  -  1 ) ) )  -> 
( 2 ... ( A  +  1 ) )  =  ( ( 2 ... A )  u.  { ( A  +  1 ) } ) )
4023, 38, 39sylancr 663 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( 2 ... ( A  +  1 ) )  =  ( ( 2 ... A )  u.  { ( A  +  1 ) } ) )
4140ineq1d 3556 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  =  ( ( ( 2 ... A )  u.  { ( A  +  1 ) } )  i^i  Prime )
)
42 indir 3603 . . . . . 6  |-  ( ( ( 2 ... A
)  u.  { ( A  +  1 ) } )  i^i  Prime )  =  ( ( ( 2 ... A )  i^i  Prime )  u.  ( { ( A  + 
1 ) }  i^i  Prime
) )
4341, 42syl6eq 2491 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  =  ( ( ( 2 ... A )  i^i  Prime )  u.  ( { ( A  + 
1 ) }  i^i  Prime
) ) )
44 simpr 461 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  Prime )
4544snssd 4023 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  { ( A  + 
1 ) }  C_  Prime )
46 df-ss 3347 . . . . . . 7  |-  ( { ( A  +  1 ) }  C_  Prime  <->  ( { ( A  + 
1 ) }  i^i  Prime
)  =  { ( A  +  1 ) } )
4745, 46sylib 196 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( { ( A  +  1 ) }  i^i  Prime )  =  {
( A  +  1 ) } )
4847uneq2d 3515 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( ( 2 ... A )  i^i 
Prime )  u.  ( { ( A  + 
1 ) }  i^i  Prime
) )  =  ( ( ( 2 ... A )  i^i  Prime )  u.  { ( A  +  1 ) } ) )
4943, 48eqtrd 2475 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  =  ( ( ( 2 ... A )  i^i  Prime )  u.  {
( A  +  1 ) } ) )
5049fveq2d 5700 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( # `  (
( 2 ... ( A  +  1 ) )  i^i  Prime )
)  =  ( # `  ( ( ( 2 ... A )  i^i 
Prime )  u.  { ( A  +  1 ) } ) ) )
5122, 50eqtrd 2475 . 2  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  (π `  ( A  + 
1 ) )  =  ( # `  (
( ( 2 ... A )  i^i  Prime )  u.  { ( A  +  1 ) } ) ) )
52 ppival2 22471 . . . 4  |-  ( A  e.  ZZ  ->  (π `  A )  =  (
# `  ( (
2 ... A )  i^i 
Prime ) ) )
5352adantr 465 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  (π `  A )  =  ( # `  (
( 2 ... A
)  i^i  Prime ) ) )
5453oveq1d 6111 . 2  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( (π `  A )  +  1 )  =  ( ( # `  (
( 2 ... A
)  i^i  Prime ) )  +  1 ) )
5520, 51, 543eqtr4d 2485 1  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  (π `  ( A  + 
1 ) )  =  ( (π `  A )  +  1 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2977    u. cun 3331    i^i cin 3332    C_ wss 3333   {csn 3882   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   Fincfn 7315   CCcc 9285   RRcr 9286   1c1 9288    + caddc 9290    < clt 9423    <_ cle 9424    - cmin 9600   NNcn 10327   2c2 10376   ZZcz 10651   ZZ>=cuz 10866   ...cfz 11442   #chash 12108   Primecprime 13768  πcppi 22436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-n0 10585  df-z 10652  df-uz 10867  df-icc 11312  df-fz 11443  df-fl 11647  df-hash 12109  df-dvds 13541  df-prm 13769  df-ppi 22442
This theorem is referenced by:  ppip1le  22504  ppi1i  22511  bposlem5  22632
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