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Theorem ppiprm 20495
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 11124 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( 2 ... A
)  e.  Fin )
2 inss1 3465 . . . 4  |-  ( ( 2 ... A )  i^i  Prime )  C_  (
2 ... A )
3 ssfi 7168 . . . 4  |-  ( ( ( 2 ... A
)  e.  Fin  /\  ( ( 2 ... A )  i^i  Prime ) 
C_  ( 2 ... A ) )  -> 
( ( 2 ... A )  i^i  Prime )  e.  Fin )
41, 2, 3sylancl 643 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 2 ... A )  i^i  Prime )  e.  Fin )
5 zre 10117 . . . . . . 7  |-  ( A  e.  ZZ  ->  A  e.  RR )
65adantr 451 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  e.  RR )
76ltp1d 9774 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  <  ( A  +  1 ) )
8 peano2z 10149 . . . . . . . 8  |-  ( A  e.  ZZ  ->  ( A  +  1 )  e.  ZZ )
98adantr 451 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  ZZ )
109zred 10206 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  RR )
116, 10ltnled 9053 . . . . 5  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  <  ( A  +  1 )  <->  -.  ( A  +  1 )  <_  A )
)
127, 11mpbid 201 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  -.  ( A  + 
1 )  <_  A
)
132sseli 3252 . . . . 5  |-  ( ( A  +  1 )  e.  ( ( 2 ... A )  i^i 
Prime )  ->  ( A  +  1 )  e.  ( 2 ... A
) )
14 elfzle2 10889 . . . . 5  |-  ( ( A  +  1 )  e.  ( 2 ... A )  ->  ( A  +  1 )  <_  A )
1513, 14syl 15 . . . 4  |-  ( ( A  +  1 )  e.  ( ( 2 ... A )  i^i 
Prime )  ->  ( A  +  1 )  <_  A )
1612, 15nsyl 113 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  -.  ( A  + 
1 )  e.  ( ( 2 ... A
)  i^i  Prime ) )
17 ovex 5967 . . . 4  |-  ( A  +  1 )  e. 
_V
18 hashunsng 11457 . . . 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 642 . 2  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( # `  (
( ( 2 ... A )  i^i  Prime )  u.  { ( A  +  1 ) } ) )  =  ( ( # `  (
( 2 ... A
)  i^i  Prime ) )  +  1 ) )
21 ppival2 20472 . . . 4  |-  ( ( A  +  1 )  e.  ZZ  ->  (π `  ( A  +  1 ) )  =  (
# `  ( (
2 ... ( A  + 
1 ) )  i^i 
Prime ) ) )
229, 21syl 15 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  (π `  ( A  + 
1 ) )  =  ( # `  (
( 2 ... ( A  +  1 ) )  i^i  Prime )
) )
23 2z 10143 . . . . . . . 8  |-  2  e.  ZZ
24 zcn 10118 . . . . . . . . . . . 12  |-  ( A  e.  ZZ  ->  A  e.  CC )
2524adantr 451 . . . . . . . . . . 11  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  e.  CC )
26 ax-1cn 8882 . . . . . . . . . . 11  |-  1  e.  CC
27 pncan 9144 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  + 
1 )  -  1 )  =  A )
2825, 26, 27sylancl 643 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( A  + 
1 )  -  1 )  =  A )
29 prmuz2 12867 . . . . . . . . . . . 12  |-  ( ( A  +  1 )  e.  Prime  ->  ( A  +  1 )  e.  ( ZZ>= `  2 )
)
3029adantl 452 . . . . . . . . . . 11  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  ( ZZ>= ` 
2 ) )
31 uz2m1nn 10381 . . . . . . . . . . 11  |-  ( ( A  +  1 )  e.  ( ZZ>= `  2
)  ->  ( ( A  +  1 )  -  1 )  e.  NN )
3230, 31syl 15 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( A  + 
1 )  -  1 )  e.  NN )
3328, 32eqeltrrd 2433 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  e.  NN )
34 nnuz 10352 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
35 2cn 9903 . . . . . . . . . . . 12  |-  2  e.  CC
36 1p1e2 9927 . . . . . . . . . . . 12  |-  ( 1  +  1 )  =  2
3735, 26, 26, 36subaddrii 9222 . . . . . . . . . . 11  |-  ( 2  -  1 )  =  1
3837fveq2i 5608 . . . . . . . . . 10  |-  ( ZZ>= `  ( 2  -  1 ) )  =  (
ZZ>= `  1 )
3934, 38eqtr4i 2381 . . . . . . . . 9  |-  NN  =  ( ZZ>= `  ( 2  -  1 ) )
4033, 39syl6eleq 2448 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  A  e.  ( ZZ>= `  ( 2  -  1 ) ) )
41 fzsuc2 10931 . . . . . . . 8  |-  ( ( 2  e.  ZZ  /\  A  e.  ( ZZ>= `  ( 2  -  1 ) ) )  -> 
( 2 ... ( A  +  1 ) )  =  ( ( 2 ... A )  u.  { ( A  +  1 ) } ) )
4223, 40, 41sylancr 644 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( 2 ... ( A  +  1 ) )  =  ( ( 2 ... A )  u.  { ( A  +  1 ) } ) )
4342ineq1d 3445 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  =  ( ( ( 2 ... A )  u.  { ( A  +  1 ) } )  i^i  Prime )
)
44 indir 3493 . . . . . 6  |-  ( ( ( 2 ... A
)  u.  { ( A  +  1 ) } )  i^i  Prime )  =  ( ( ( 2 ... A )  i^i  Prime )  u.  ( { ( A  + 
1 ) }  i^i  Prime
) )
4543, 44syl6eq 2406 . . . . 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
) ) )
46 simpr 447 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( A  +  1 )  e.  Prime )
4746snssd 3839 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  { ( A  + 
1 ) }  C_  Prime )
48 df-ss 3242 . . . . . . 7  |-  ( { ( A  +  1 ) }  C_  Prime  <->  ( { ( A  + 
1 ) }  i^i  Prime
)  =  { ( A  +  1 ) } )
4947, 48sylib 188 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( { ( A  +  1 ) }  i^i  Prime )  =  {
( A  +  1 ) } )
5049uneq2d 3405 . . . . 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 ) } ) )
5145, 50eqtrd 2390 . . . 4  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  =  ( ( ( 2 ... A )  i^i  Prime )  u.  {
( A  +  1 ) } ) )
5251fveq2d 5609 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( # `  (
( 2 ... ( A  +  1 ) )  i^i  Prime )
)  =  ( # `  ( ( ( 2 ... A )  i^i 
Prime )  u.  { ( A  +  1 ) } ) ) )
5322, 52eqtrd 2390 . 2  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  (π `  ( A  + 
1 ) )  =  ( # `  (
( ( 2 ... A )  i^i  Prime )  u.  { ( A  +  1 ) } ) ) )
54 ppival2 20472 . . . 4  |-  ( A  e.  ZZ  ->  (π `  A )  =  (
# `  ( (
2 ... A )  i^i 
Prime ) ) )
5554adantr 451 . . 3  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  (π `  A )  =  ( # `  (
( 2 ... A
)  i^i  Prime ) ) )
5655oveq1d 5957 . 2  |-  ( ( A  e.  ZZ  /\  ( A  +  1
)  e.  Prime )  ->  ( (π `  A )  +  1 )  =  ( ( # `  (
( 2 ... A
)  i^i  Prime ) )  +  1 ) )
5720, 53, 563eqtr4d 2400 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 358    = wceq 1642    e. wcel 1710   _Vcvv 2864    u. cun 3226    i^i cin 3227    C_ wss 3228   {csn 3716   class class class wbr 4102   ` cfv 5334  (class class class)co 5942   Fincfn 6948   CCcc 8822   RRcr 8823   1c1 8825    + caddc 8827    < clt 8954    <_ cle 8955    - cmin 9124   NNcn 9833   2c2 9882   ZZcz 10113   ZZ>=cuz 10319   ...cfz 10871   #chash 11427   Primecprime 12849  πcppi 20437
This theorem is referenced by:  ppip1le  20505  ppi1i  20512  bposlem5  20633
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-2o 6564  df-oadd 6567  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-sup 7281  df-card 7659  df-cda 7881  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-nn 9834  df-2 9891  df-n0 10055  df-z 10114  df-uz 10320  df-icc 10752  df-fz 10872  df-fl 11014  df-hash 11428  df-dvds 12623  df-prm 12850  df-ppi 20443
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