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Theorem ppinprm 23251
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 3719 . . . . . . . . . . 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 3502 . . . . . . . . . 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 2797 . . . . . . . . . 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 4041 . . . . . . . . . 10  |-  ( x  e.  { ( A  +  1 ) }  <-> 
x  =  ( A  +  1 ) )
87necon3bbii 2728 . . . . . . . . 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 3718 . . . . . . . . . . . 12  |-  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  C_  (
2 ... ( A  + 
1 ) )
1110, 2sseldi 3502 . . . . . . . . . . 11  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  x  e.  ( 2 ... ( A  +  1 ) ) )
12 2z 10897 . . . . . . . . . . . 12  |-  2  e.  ZZ
13 zcn 10870 . . . . . . . . . . . . . . . 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 9551 . . . . . . . . . . . . . . 15  |-  1  e.  CC
16 pncan 9827 . . . . . . . . . . . . . . 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 11692 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( 2 ... ( A  +  1 ) )  ->  ( A  +  1 )  e.  ( ZZ>= `  2
) )
19 uz2m1nn 11157 . . . . . . . . . . . . . . 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 2556 . . . . . . . . . . . . 13  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  A  e.  NN )
22 nnuz 11118 . . . . . . . . . . . . . 14  |-  NN  =  ( ZZ>= `  1 )
23 2m1e1 10651 . . . . . . . . . . . . . . 15  |-  ( 2  -  1 )  =  1
2423fveq2i 5869 . . . . . . . . . . . . . 14  |-  ( ZZ>= `  ( 2  -  1 ) )  =  (
ZZ>= `  1 )
2522, 24eqtr4i 2499 . . . . . . . . . . . . 13  |-  NN  =  ( ZZ>= `  ( 2  -  1 ) )
2621, 25syl6eleq 2565 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  A  e.  ( ZZ>= `  ( 2  -  1 ) ) )
27 fzsuc2 11738 . . . . . . . . . . . 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 2557 . . . . . . . . . 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 3645 . . . . . . . . . 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 3688 . . . . . 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 3510 . . . 4  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( ( 2 ... ( A  +  1 ) )  i^i  Prime ) 
C_  ( ( 2 ... A )  i^i 
Prime ) )
37 uzid 11097 . . . . . 6  |-  ( A  e.  ZZ  ->  A  e.  ( ZZ>= `  A )
)
3837adantr 465 . . . . 5  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  A  e.  ( ZZ>= `  A ) )
39 peano2uz 11135 . . . . 5  |-  ( A  e.  ( ZZ>= `  A
)  ->  ( A  +  1 )  e.  ( ZZ>= `  A )
)
40 fzss2 11724 . . . . 5  |-  ( ( A  +  1 )  e.  ( ZZ>= `  A
)  ->  ( 2 ... A )  C_  ( 2 ... ( A  +  1 ) ) )
41 ssrin 3723 . . . . 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 3521 . . 3  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  =  ( ( 2 ... A )  i^i 
Prime ) )
4443fveq2d 5870 . 2  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( # `  (
( 2 ... ( A  +  1 ) )  i^i  Prime )
)  =  ( # `  ( ( 2 ... A )  i^i  Prime ) ) )
45 peano2z 10905 . . . 4  |-  ( A  e.  ZZ  ->  ( A  +  1 )  e.  ZZ )
4645adantr 465 . . 3  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( A  +  1 )  e.  ZZ )
47 ppival2 23227 . . 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 23227 . . 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 2518 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 1379    e. wcel 1767    =/= wne 2662    u. cun 3474    i^i cin 3475    C_ wss 3476   {csn 4027   ` cfv 5588  (class class class)co 6285   CCcc 9491   1c1 9494    + caddc 9496    - cmin 9806   NNcn 10537   2c2 10586   ZZcz 10865   ZZ>=cuz 11083   ...cfz 11673   #chash 12374   Primecprime 14079  πcppi 23192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-2o 7132  df-oadd 7135  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-sup 7902  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-n0 10797  df-z 10866  df-uz 11084  df-icc 11537  df-fz 11674  df-fl 11898  df-dvds 13851  df-prm 14080  df-ppi 23198
This theorem is referenced by:  ppip1le  23260  ppi2i  23268  bposlem5  23388
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