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Theorem ppiprm 23551
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 12086 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( 2 ... A
)  e.  Fin )
2 inss1 3714 . . . 4  |-  ( ( 2 ... A )  i^i  Prime )  C_  (
2 ... A )
3 ssfi 7759 . . . 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 10889 . . . . . . 7  |-  ( A  e.  ZZ  ->  A  e.  RR )
65adantr 465 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  e.  RR )
76ltp1d 10496 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  <  ( A  +  1 ) )
8 peano2z 10926 . . . . . . . 8  |-  ( A  e.  ZZ  ->  ( A  +  1 )  e.  ZZ )
98adantr 465 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  ZZ )
109zred 10990 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  RR )
116, 10ltnled 9749 . . . . 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 3495 . . . . 5  |-  ( ( A  +  1 )  e.  ( ( 2 ... A )  i^i 
Prime )  ->  ( A  +  1 )  e.  ( 2 ... A
) )
14 elfzle2 11715 . . . . 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 6324 . . . 4  |-  ( A  +  1 )  e. 
_V
18 hashunsng 12463 . . . 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 23528 . . . 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 10917 . . . . . . . 8  |-  2  e.  ZZ
24 zcn 10890 . . . . . . . . . . . 12  |-  ( A  e.  ZZ  ->  A  e.  CC )
2524adantr 465 . . . . . . . . . . 11  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  e.  CC )
26 ax-1cn 9567 . . . . . . . . . . 11  |-  1  e.  CC
27 pncan 9845 . . . . . . . . . . 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 14247 . . . . . . . . . . . 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 11181 . . . . . . . . . . 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 2546 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  e.  NN )
34 nnuz 11141 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
35 2m1e1 10671 . . . . . . . . . . 11  |-  ( 2  -  1 )  =  1
3635fveq2i 5875 . . . . . . . . . 10  |-  ( ZZ>= `  ( 2  -  1 ) )  =  (
ZZ>= `  1 )
3734, 36eqtr4i 2489 . . . . . . . . 9  |-  NN  =  ( ZZ>= `  ( 2  -  1 ) )
3833, 37syl6eleq 2555 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  e.  ( ZZ>= `  ( 2  -  1 ) ) )
39 fzsuc2 11763 . . . . . . . 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 3695 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  =  ( ( ( 2 ... A )  u.  { ( A  +  1 ) } )  i^i  Prime )
)
42 indir 3753 . . . . . 6  |-  ( ( ( 2 ... A
)  u.  { ( A  +  1 ) } )  i^i  Prime )  =  ( ( ( 2 ... A )  i^i  Prime )  u.  ( { ( A  + 
1 ) }  i^i  Prime
) )
4341, 42syl6eq 2514 . . . . 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 4177 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  { ( A  + 
1 ) }  C_  Prime )
46 df-ss 3485 . . . . . . 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 3654 . . . . 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 2498 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  =  ( ( ( 2 ... A )  i^i  Prime )  u.  {
( A  +  1 ) } ) )
5049fveq2d 5876 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( # `  (
( 2 ... ( A  +  1 ) )  i^i  Prime )
)  =  ( # `  ( ( ( 2 ... A )  i^i 
Prime )  u.  { ( A  +  1 ) } ) ) )
5122, 50eqtrd 2498 . 2  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  (π `  ( A  + 
1 ) )  =  ( # `  (
( ( 2 ... A )  i^i  Prime )  u.  { ( A  +  1 ) } ) ) )
52 ppival2 23528 . . . 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 6311 . 2  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( (π `  A )  +  1 )  =  ( ( # `  (
( 2 ... A
)  i^i  Prime ) )  +  1 ) )
5520, 51, 543eqtr4d 2508 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 1395    e. wcel 1819   _Vcvv 3109    u. cun 3469    i^i cin 3470    C_ wss 3471   {csn 4032   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   Fincfn 7535   CCcc 9507   RRcr 9508   1c1 9510    + caddc 9512    < clt 9645    <_ cle 9646    - cmin 9824   NNcn 10556   2c2 10606   ZZcz 10885   ZZ>=cuz 11106   ...cfz 11697   #chash 12408   Primecprime 14229  πcppi 23493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-icc 11561  df-fz 11698  df-fl 11932  df-hash 12409  df-dvds 13999  df-prm 14230  df-ppi 23499
This theorem is referenced by:  ppip1le  23561  ppi1i  23568  bposlem5  23689
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