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Theorem ppiprm 20887
Description: The prime pi function 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 11267 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( 2 ... A
)  e.  Fin )
2 inss1 3521 . . . 4  |-  ( ( 2 ... A )  i^i  Prime )  C_  (
2 ... A )
3 ssfi 7288 . . . 4  |-  ( ( ( 2 ... A
)  e.  Fin  /\  ( ( 2 ... A )  i^i  Prime ) 
C_  ( 2 ... A ) )  -> 
( ( 2 ... A )  i^i  Prime )  e.  Fin )
41, 2, 3sylancl 644 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 2 ... A )  i^i  Prime )  e.  Fin )
5 zre 10242 . . . . . . 7  |-  ( A  e.  ZZ  ->  A  e.  RR )
65adantr 452 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  e.  RR )
76ltp1d 9897 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  <  ( A  +  1 ) )
8 peano2z 10274 . . . . . . . 8  |-  ( A  e.  ZZ  ->  ( A  +  1 )  e.  ZZ )
98adantr 452 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  ZZ )
109zred 10331 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  RR )
116, 10ltnled 9176 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  <  ( A  +  1 )  <->  -.  ( A  +  1 )  <_  A )
)
127, 11mpbid 202 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  -.  ( A  + 
1 )  <_  A
)
132sseli 3304 . . . . 5  |-  ( ( A  +  1 )  e.  ( ( 2 ... A )  i^i 
Prime )  ->  ( A  +  1 )  e.  ( 2 ... A
) )
14 elfzle2 11017 . . . . 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 115 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  -.  ( A  + 
1 )  e.  ( ( 2 ... A
)  i^i  Prime ) )
17 ovex 6065 . . . 4  |-  ( A  +  1 )  e. 
_V
18 hashunsng 11620 . . . 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 8 . . 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 643 . 2  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( # `  (
( ( 2 ... A )  i^i  Prime )  u.  { ( A  +  1 ) } ) )  =  ( ( # `  (
( 2 ... A
)  i^i  Prime ) )  +  1 ) )
21 ppival2 20864 . . . 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 10268 . . . . . . . 8  |-  2  e.  ZZ
24 zcn 10243 . . . . . . . . . . . 12  |-  ( A  e.  ZZ  ->  A  e.  CC )
2524adantr 452 . . . . . . . . . . 11  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  e.  CC )
26 ax-1cn 9004 . . . . . . . . . . 11  |-  1  e.  CC
27 pncan 9267 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  + 
1 )  -  1 )  =  A )
2825, 26, 27sylancl 644 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( A  + 
1 )  -  1 )  =  A )
29 prmuz2 13052 . . . . . . . . . . . 12  |-  ( ( A  +  1 )  e.  Prime  ->  ( A  +  1 )  e.  ( ZZ>= `  2 )
)
3029adantl 453 . . . . . . . . . . 11  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  ( ZZ>= ` 
2 ) )
31 uz2m1nn 10506 . . . . . . . . . . 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 2479 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  e.  NN )
34 nnuz 10477 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
35 2m1e1 10051 . . . . . . . . . . 11  |-  ( 2  -  1 )  =  1
3635fveq2i 5690 . . . . . . . . . 10  |-  ( ZZ>= `  ( 2  -  1 ) )  =  (
ZZ>= `  1 )
3734, 36eqtr4i 2427 . . . . . . . . 9  |-  NN  =  ( ZZ>= `  ( 2  -  1 ) )
3833, 37syl6eleq 2494 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  e.  ( ZZ>= `  ( 2  -  1 ) ) )
39 fzsuc2 11060 . . . . . . . 8  |-  ( ( 2  e.  ZZ  /\  A  e.  ( ZZ>= `  ( 2  -  1 ) ) )  -> 
( 2 ... ( A  +  1 ) )  =  ( ( 2 ... A )  u.  { ( A  +  1 ) } ) )
4023, 38, 39sylancr 645 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( 2 ... ( A  +  1 ) )  =  ( ( 2 ... A )  u.  { ( A  +  1 ) } ) )
4140ineq1d 3501 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  =  ( ( ( 2 ... A )  u.  { ( A  +  1 ) } )  i^i  Prime )
)
42 indir 3549 . . . . . 6  |-  ( ( ( 2 ... A
)  u.  { ( A  +  1 ) } )  i^i  Prime )  =  ( ( ( 2 ... A )  i^i  Prime )  u.  ( { ( A  + 
1 ) }  i^i  Prime
) )
4341, 42syl6eq 2452 . . . . 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 448 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  Prime )
4544snssd 3903 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  { ( A  + 
1 ) }  C_  Prime )
46 df-ss 3294 . . . . . . 7  |-  ( { ( A  +  1 ) }  C_  Prime  <->  ( { ( A  + 
1 ) }  i^i  Prime
)  =  { ( A  +  1 ) } )
4745, 46sylib 189 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( { ( A  +  1 ) }  i^i  Prime )  =  {
( A  +  1 ) } )
4847uneq2d 3461 . . . . 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 2436 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  =  ( ( ( 2 ... A )  i^i  Prime )  u.  {
( A  +  1 ) } ) )
5049fveq2d 5691 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( # `  (
( 2 ... ( A  +  1 ) )  i^i  Prime )
)  =  ( # `  ( ( ( 2 ... A )  i^i 
Prime )  u.  { ( A  +  1 ) } ) ) )
5122, 50eqtrd 2436 . 2  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  (π `  ( A  + 
1 ) )  =  ( # `  (
( ( 2 ... A )  i^i  Prime )  u.  { ( A  +  1 ) } ) ) )
52 ppival2 20864 . . . 4  |-  ( A  e.  ZZ  ->  (π `  A )  =  (
# `  ( (
2 ... A )  i^i 
Prime ) ) )
5352adantr 452 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  (π `  A )  =  ( # `  (
( 2 ... A
)  i^i  Prime ) ) )
5453oveq1d 6055 . 2  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( (π `  A )  +  1 )  =  ( ( # `  (
( 2 ... A
)  i^i  Prime ) )  +  1 ) )
5520, 51, 543eqtr4d 2446 1  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  (π `  ( A  + 
1 ) )  =  ( (π `  A )  +  1 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916    u. cun 3278    i^i cin 3279    C_ wss 3280   {csn 3774   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Fincfn 7068   CCcc 8944   RRcr 8945   1c1 8947    + caddc 8949    < clt 9076    <_ cle 9077    - cmin 9247   NNcn 9956   2c2 10005   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999   #chash 11573   Primecprime 13034  πcppi 20829
This theorem is referenced by:  ppip1le  20897  ppi1i  20904  bposlem5  21025
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-icc 10879  df-fz 11000  df-fl 11157  df-hash 11574  df-dvds 12808  df-prm 13035  df-ppi 20835
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