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Theorem ppinprm 22493
Description: The prime-counting function π 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 3574 . . . . . . . . . . 11  |-  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  C_  Prime
2 simprr 756 . . . . . . . . . . 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 3357 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  x  e.  Prime )
4 simprl 755 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  -.  ( A  +  1
)  e.  Prime )
5 nelne2 2705 . . . . . . . . . 10  |-  ( ( x  e.  Prime  /\  -.  ( A  +  1
)  e.  Prime )  ->  x  =/=  ( A  +  1 ) )
63, 4, 5syl2anc 661 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  x  =/=  ( A  +  1 ) )
7 elsn 3894 . . . . . . . . . 10  |-  ( x  e.  { ( A  +  1 ) }  <-> 
x  =  ( A  +  1 ) )
87necon3bbii 2642 . . . . . . . . 9  |-  ( -.  x  e.  { ( A  +  1 ) }  <->  x  =/=  ( A  +  1 ) )
96, 8sylibr 212 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  -.  x  e.  { ( A  +  1 ) } )
10 inss1 3573 . . . . . . . . . . . 12  |-  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  C_  (
2 ... ( A  + 
1 ) )
1110, 2sseldi 3357 . . . . . . . . . . 11  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  x  e.  ( 2 ... ( A  +  1 ) ) )
12 2z 10681 . . . . . . . . . . . 12  |-  2  e.  ZZ
13 zcn 10654 . . . . . . . . . . . . . . . 16  |-  ( A  e.  ZZ  ->  A  e.  CC )
1413adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  A  e.  CC )
15 ax-1cn 9343 . . . . . . . . . . . . . . 15  |-  1  e.  CC
16 pncan 9619 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  + 
1 )  -  1 )  =  A )
1714, 15, 16sylancl 662 . . . . . . . . . . . . . 14  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  (
( A  +  1 )  -  1 )  =  A )
18 elfzuz2 11459 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( 2 ... ( A  +  1 ) )  ->  ( A  +  1 )  e.  ( ZZ>= `  2
) )
19 uz2m1nn 10932 . . . . . . . . . . . . . . 15  |-  ( ( A  +  1 )  e.  ( ZZ>= `  2
)  ->  ( ( A  +  1 )  -  1 )  e.  NN )
2011, 18, 193syl 20 . . . . . . . . . . . . . 14  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  (
( A  +  1 )  -  1 )  e.  NN )
2117, 20eqeltrrd 2518 . . . . . . . . . . . . 13  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  A  e.  NN )
22 nnuz 10899 . . . . . . . . . . . . . 14  |-  NN  =  ( ZZ>= `  1 )
23 2m1e1 10439 . . . . . . . . . . . . . . 15  |-  ( 2  -  1 )  =  1
2423fveq2i 5697 . . . . . . . . . . . . . 14  |-  ( ZZ>= `  ( 2  -  1 ) )  =  (
ZZ>= `  1 )
2522, 24eqtr4i 2466 . . . . . . . . . . . . 13  |-  NN  =  ( ZZ>= `  ( 2  -  1 ) )
2621, 25syl6eleq 2533 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  A  e.  ( ZZ>= `  ( 2  -  1 ) ) )
27 fzsuc2 11517 . . . . . . . . . . . 12  |-  ( ( 2  e.  ZZ  /\  A  e.  ( ZZ>= `  ( 2  -  1 ) ) )  -> 
( 2 ... ( A  +  1 ) )  =  ( ( 2 ... A )  u.  { ( A  +  1 ) } ) )
2812, 26, 27sylancr 663 . . . . . . . . . . 11  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  (
2 ... ( A  + 
1 ) )  =  ( ( 2 ... A )  u.  {
( A  +  1 ) } ) )
2911, 28eleqtrd 2519 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  x  e.  ( ( 2 ... A )  u.  {
( A  +  1 ) } ) )
30 elun 3500 . . . . . . . . . 10  |-  ( x  e.  ( ( 2 ... A )  u. 
{ ( A  + 
1 ) } )  <-> 
( x  e.  ( 2 ... A )  \/  x  e.  {
( A  +  1 ) } ) )
3129, 30sylib 196 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  (
x  e.  ( 2 ... A )  \/  x  e.  { ( A  +  1 ) } ) )
3231ord 377 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  ( -.  x  e.  (
2 ... A )  ->  x  e.  { ( A  +  1 ) } ) )
339, 32mt3d 125 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  x  e.  ( 2 ... A
) )
3433, 3elind 3543 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  x  e.  ( ( 2 ... A )  i^i  Prime ) )
3534expr 615 . . . . 5  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  ->  x  e.  ( ( 2 ... A )  i^i  Prime ) ) )
3635ssrdv 3365 . . . 4  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( ( 2 ... ( A  +  1 ) )  i^i  Prime ) 
C_  ( ( 2 ... A )  i^i 
Prime ) )
37 uzid 10878 . . . . . 6  |-  ( A  e.  ZZ  ->  A  e.  ( ZZ>= `  A )
)
3837adantr 465 . . . . 5  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  A  e.  ( ZZ>= `  A ) )
39 peano2uz 10911 . . . . 5  |-  ( A  e.  ( ZZ>= `  A
)  ->  ( A  +  1 )  e.  ( ZZ>= `  A )
)
40 fzss2 11501 . . . . 5  |-  ( ( A  +  1 )  e.  ( ZZ>= `  A
)  ->  ( 2 ... A )  C_  ( 2 ... ( A  +  1 ) ) )
41 ssrin 3578 . . . . 5  |-  ( ( 2 ... A ) 
C_  ( 2 ... ( A  +  1 ) )  ->  (
( 2 ... A
)  i^i  Prime )  C_  ( ( 2 ... ( A  +  1 ) )  i^i  Prime ) )
4238, 39, 40, 414syl 21 . . . 4  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( ( 2 ... A )  i^i  Prime ) 
C_  ( ( 2 ... ( A  + 
1 ) )  i^i 
Prime ) )
4336, 42eqssd 3376 . . 3  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  =  ( ( 2 ... A )  i^i 
Prime ) )
4443fveq2d 5698 . 2  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( # `  (
( 2 ... ( A  +  1 ) )  i^i  Prime )
)  =  ( # `  ( ( 2 ... A )  i^i  Prime ) ) )
45 peano2z 10689 . . . 4  |-  ( A  e.  ZZ  ->  ( A  +  1 )  e.  ZZ )
4645adantr 465 . . 3  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( A  +  1 )  e.  ZZ )
47 ppival2 22469 . . 3  |-  ( ( A  +  1 )  e.  ZZ  ->  (π `  ( A  +  1 ) )  =  (
# `  ( (
2 ... ( A  + 
1 ) )  i^i 
Prime ) ) )
4846, 47syl 16 . 2  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  (π `  ( A  + 
1 ) )  =  ( # `  (
( 2 ... ( A  +  1 ) )  i^i  Prime )
) )
49 ppival2 22469 . . 3  |-  ( A  e.  ZZ  ->  (π `  A )  =  (
# `  ( (
2 ... A )  i^i 
Prime ) ) )
5049adantr 465 . 2  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  (π `  A )  =  ( # `  (
( 2 ... A
)  i^i  Prime ) ) )
5144, 48, 503eqtr4d 2485 1  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  (π `  ( A  + 
1 ) )  =  (π `  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2609    u. cun 3329    i^i cin 3330    C_ wss 3331   {csn 3880   ` cfv 5421  (class class class)co 6094   CCcc 9283   1c1 9286    + caddc 9288    - cmin 9598   NNcn 10325   2c2 10374   ZZcz 10649   ZZ>=cuz 10864   ...cfz 11440   #chash 12106   Primecprime 13766  πcppi 22434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362  ax-pre-sup 9363
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-om 6480  df-1st 6580  df-2nd 6581  df-recs 6835  df-rdg 6869  df-1o 6923  df-2o 6924  df-oadd 6927  df-er 7104  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-sup 7694  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-nn 10326  df-2 10383  df-n0 10583  df-z 10650  df-uz 10865  df-icc 11310  df-fz 11441  df-fl 11645  df-dvds 13539  df-prm 13767  df-ppi 22440
This theorem is referenced by:  ppip1le  22502  ppi2i  22510  bposlem5  22630
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