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Theorem ppinprm 20390
Description: The prime pi function at a non-prime. (Contributed by Mario Carneiro, 19-Sep-2014.)
Assertion
Ref Expression
ppinprm  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  (π `  ( A  + 
1 ) )  =  (π `  A ) )

Proof of Theorem ppinprm
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 inss2 3390 . . . . . . . . . . 11  |-  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  C_  Prime
2 simprr 733 . . . . . . . . . . 11  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime ) )
31, 2sseldi 3178 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  x  e.  Prime )
4 simprl 732 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  -.  ( A  +  1
)  e.  Prime )
5 nelne2 2536 . . . . . . . . . 10  |-  ( ( x  e.  Prime  /\  -.  ( A  +  1
)  e.  Prime )  ->  x  =/=  ( A  +  1 ) )
63, 4, 5syl2anc 642 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  x  =/=  ( A  +  1 ) )
7 elsn 3655 . . . . . . . . . 10  |-  ( x  e.  { ( A  +  1 ) }  <-> 
x  =  ( A  +  1 ) )
87necon3bbii 2477 . . . . . . . . 9  |-  ( -.  x  e.  { ( A  +  1 ) }  <->  x  =/=  ( A  +  1 ) )
96, 8sylibr 203 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  -.  x  e.  { ( A  +  1 ) } )
10 inss1 3389 . . . . . . . . . . . 12  |-  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  C_  (
2 ... ( A  + 
1 ) )
1110, 2sseldi 3178 . . . . . . . . . . 11  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  x  e.  ( 2 ... ( A  +  1 ) ) )
12 2z 10054 . . . . . . . . . . . 12  |-  2  e.  ZZ
13 zcn 10029 . . . . . . . . . . . . . . . 16  |-  ( A  e.  ZZ  ->  A  e.  CC )
1413adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  A  e.  CC )
15 ax-1cn 8795 . . . . . . . . . . . . . . 15  |-  1  e.  CC
16 pncan 9057 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  + 
1 )  -  1 )  =  A )
1714, 15, 16sylancl 643 . . . . . . . . . . . . . 14  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  (
( A  +  1 )  -  1 )  =  A )
18 elfzuz2 10801 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( 2 ... ( A  +  1 ) )  ->  ( A  +  1 )  e.  ( ZZ>= `  2
) )
19 uz2m1nn 10292 . . . . . . . . . . . . . . 15  |-  ( ( A  +  1 )  e.  ( ZZ>= `  2
)  ->  ( ( A  +  1 )  -  1 )  e.  NN )
2011, 18, 193syl 18 . . . . . . . . . . . . . 14  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  (
( A  +  1 )  -  1 )  e.  NN )
2117, 20eqeltrrd 2358 . . . . . . . . . . . . 13  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  A  e.  NN )
22 nnuz 10263 . . . . . . . . . . . . . 14  |-  NN  =  ( ZZ>= `  1 )
23 2cn 9816 . . . . . . . . . . . . . . . 16  |-  2  e.  CC
24 1p1e2 9840 . . . . . . . . . . . . . . . 16  |-  ( 1  +  1 )  =  2
2523, 15, 15, 24subaddrii 9135 . . . . . . . . . . . . . . 15  |-  ( 2  -  1 )  =  1
2625fveq2i 5528 . . . . . . . . . . . . . 14  |-  ( ZZ>= `  ( 2  -  1 ) )  =  (
ZZ>= `  1 )
2722, 26eqtr4i 2306 . . . . . . . . . . . . 13  |-  NN  =  ( ZZ>= `  ( 2  -  1 ) )
2821, 27syl6eleq 2373 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  A  e.  ( ZZ>= `  ( 2  -  1 ) ) )
29 fzsuc2 10842 . . . . . . . . . . . 12  |-  ( ( 2  e.  ZZ  /\  A  e.  ( ZZ>= `  ( 2  -  1 ) ) )  -> 
( 2 ... ( A  +  1 ) )  =  ( ( 2 ... A )  u.  { ( A  +  1 ) } ) )
3012, 28, 29sylancr 644 . . . . . . . . . . 11  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  (
2 ... ( A  + 
1 ) )  =  ( ( 2 ... A )  u.  {
( A  +  1 ) } ) )
3111, 30eleqtrd 2359 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  x  e.  ( ( 2 ... A )  u.  {
( A  +  1 ) } ) )
32 elun 3316 . . . . . . . . . 10  |-  ( x  e.  ( ( 2 ... A )  u. 
{ ( A  + 
1 ) } )  <-> 
( x  e.  ( 2 ... A )  \/  x  e.  {
( A  +  1 ) } ) )
3331, 32sylib 188 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  (
x  e.  ( 2 ... A )  \/  x  e.  { ( A  +  1 ) } ) )
3433ord 366 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  ( -.  x  e.  (
2 ... A )  ->  x  e.  { ( A  +  1 ) } ) )
359, 34mt3d 117 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  x  e.  ( 2 ... A
) )
36 elin 3358 . . . . . . 7  |-  ( x  e.  ( ( 2 ... A )  i^i 
Prime )  <->  ( x  e.  ( 2 ... A
)  /\  x  e.  Prime ) )
3735, 3, 36sylanbrc 645 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  x  e.  ( ( 2 ... A )  i^i  Prime ) )
3837expr 598 . . . . 5  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  ->  x  e.  ( ( 2 ... A )  i^i  Prime ) ) )
3938ssrdv 3185 . . . 4  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( ( 2 ... ( A  +  1 ) )  i^i  Prime ) 
C_  ( ( 2 ... A )  i^i 
Prime ) )
40 uzid 10242 . . . . . . 7  |-  ( A  e.  ZZ  ->  A  e.  ( ZZ>= `  A )
)
4140adantr 451 . . . . . 6  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  A  e.  ( ZZ>= `  A ) )
42 peano2uz 10272 . . . . . 6  |-  ( A  e.  ( ZZ>= `  A
)  ->  ( A  +  1 )  e.  ( ZZ>= `  A )
)
4341, 42syl 15 . . . . 5  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( A  +  1 )  e.  ( ZZ>= `  A ) )
44 fzss2 10831 . . . . 5  |-  ( ( A  +  1 )  e.  ( ZZ>= `  A
)  ->  ( 2 ... A )  C_  ( 2 ... ( A  +  1 ) ) )
45 ssrin 3394 . . . . 5  |-  ( ( 2 ... A ) 
C_  ( 2 ... ( A  +  1 ) )  ->  (
( 2 ... A
)  i^i  Prime )  C_  ( ( 2 ... ( A  +  1 ) )  i^i  Prime ) )
4643, 44, 453syl 18 . . . 4  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( ( 2 ... A )  i^i  Prime ) 
C_  ( ( 2 ... ( A  + 
1 ) )  i^i 
Prime ) )
4739, 46eqssd 3196 . . 3  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  =  ( ( 2 ... A )  i^i 
Prime ) )
4847fveq2d 5529 . 2  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( # `  (
( 2 ... ( A  +  1 ) )  i^i  Prime )
)  =  ( # `  ( ( 2 ... A )  i^i  Prime ) ) )
49 peano2z 10060 . . . 4  |-  ( A  e.  ZZ  ->  ( A  +  1 )  e.  ZZ )
5049adantr 451 . . 3  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( A  +  1 )  e.  ZZ )
51 ppival2 20366 . . 3  |-  ( ( A  +  1 )  e.  ZZ  ->  (π `  ( A  +  1 ) )  =  (
# `  ( (
2 ... ( A  + 
1 ) )  i^i 
Prime ) ) )
5250, 51syl 15 . 2  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  (π `  ( A  + 
1 ) )  =  ( # `  (
( 2 ... ( A  +  1 ) )  i^i  Prime )
) )
53 ppival2 20366 . . 3  |-  ( A  e.  ZZ  ->  (π `  A )  =  (
# `  ( (
2 ... A )  i^i 
Prime ) ) )
5453adantr 451 . 2  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  (π `  A )  =  ( # `  (
( 2 ... A
)  i^i  Prime ) ) )
5548, 52, 543eqtr4d 2325 1  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  (π `  ( A  + 
1 ) )  =  (π `  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    u. cun 3150    i^i cin 3151    C_ wss 3152   {csn 3640   ` cfv 5255  (class class class)co 5858   CCcc 8735   1c1 8738    + caddc 8740    - cmin 9037   NNcn 9746   2c2 9795   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782   #chash 11337   Primecprime 12758  πcppi 20331
This theorem is referenced by:  ppip1le  20399  ppi2i  20407  bposlem5  20527
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-icc 10663  df-fz 10783  df-fl 10925  df-dvds 12532  df-prm 12759  df-ppi 20337
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