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Theorem ppinprm 20406
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 3403 . . . . . . . . . . 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 3191 . . . . . . . . . 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 2549 . . . . . . . . . 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 3668 . . . . . . . . . 10  |-  ( x  e.  { ( A  +  1 ) }  <-> 
x  =  ( A  +  1 ) )
87necon3bbii 2490 . . . . . . . . 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 3402 . . . . . . . . . . . 12  |-  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  C_  (
2 ... ( A  + 
1 ) )
1110, 2sseldi 3191 . . . . . . . . . . 11  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  x  e.  ( 2 ... ( A  +  1 ) ) )
12 2z 10070 . . . . . . . . . . . 12  |-  2  e.  ZZ
13 zcn 10045 . . . . . . . . . . . . . . . 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 8811 . . . . . . . . . . . . . . 15  |-  1  e.  CC
16 pncan 9073 . . . . . . . . . . . . . . 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 10817 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( 2 ... ( A  +  1 ) )  ->  ( A  +  1 )  e.  ( ZZ>= `  2
) )
19 uz2m1nn 10308 . . . . . . . . . . . . . . 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 2371 . . . . . . . . . . . . 13  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  A  e.  NN )
22 nnuz 10279 . . . . . . . . . . . . . 14  |-  NN  =  ( ZZ>= `  1 )
23 2cn 9832 . . . . . . . . . . . . . . . 16  |-  2  e.  CC
24 1p1e2 9856 . . . . . . . . . . . . . . . 16  |-  ( 1  +  1 )  =  2
2523, 15, 15, 24subaddrii 9151 . . . . . . . . . . . . . . 15  |-  ( 2  -  1 )  =  1
2625fveq2i 5544 . . . . . . . . . . . . . 14  |-  ( ZZ>= `  ( 2  -  1 ) )  =  (
ZZ>= `  1 )
2722, 26eqtr4i 2319 . . . . . . . . . . . . 13  |-  NN  =  ( ZZ>= `  ( 2  -  1 ) )
2821, 27syl6eleq 2386 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  A  e.  ( ZZ>= `  ( 2  -  1 ) ) )
29 fzsuc2 10858 . . . . . . . . . . . 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 2372 . . . . . . . . . 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 3329 . . . . . . . . . 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 3371 . . . . . . 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 3198 . . . 4  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( ( 2 ... ( A  +  1 ) )  i^i  Prime ) 
C_  ( ( 2 ... A )  i^i 
Prime ) )
40 uzid 10258 . . . . . . 7  |-  ( A  e.  ZZ  ->  A  e.  ( ZZ>= `  A )
)
4140adantr 451 . . . . . 6  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  A  e.  ( ZZ>= `  A ) )
42 peano2uz 10288 . . . . . 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 10847 . . . . 5  |-  ( ( A  +  1 )  e.  ( ZZ>= `  A
)  ->  ( 2 ... A )  C_  ( 2 ... ( A  +  1 ) ) )
45 ssrin 3407 . . . . 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 3209 . . 3  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  =  ( ( 2 ... A )  i^i 
Prime ) )
4847fveq2d 5545 . 2  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( # `  (
( 2 ... ( A  +  1 ) )  i^i  Prime )
)  =  ( # `  ( ( 2 ... A )  i^i  Prime ) ) )
49 peano2z 10076 . . . 4  |-  ( A  e.  ZZ  ->  ( A  +  1 )  e.  ZZ )
5049adantr 451 . . 3  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( A  +  1 )  e.  ZZ )
51 ppival2 20382 . . 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 20382 . . 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 2338 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 1632    e. wcel 1696    =/= wne 2459    u. cun 3163    i^i cin 3164    C_ wss 3165   {csn 3653   ` cfv 5271  (class class class)co 5874   CCcc 8751   1c1 8754    + caddc 8756    - cmin 9053   NNcn 9762   2c2 9811   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798   #chash 11353   Primecprime 12774  πcppi 20347
This theorem is referenced by:  ppip1le  20415  ppi2i  20423  bposlem5  20543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-icc 10679  df-fz 10799  df-fl 10941  df-dvds 12548  df-prm 12775  df-ppi 20353
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