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Theorem ppival 20365
Description: Value of the prime pi function. (Contributed by Mario Carneiro, 15-Sep-2014.)
Assertion
Ref Expression
ppival  |-  ( A  e.  RR  ->  (π `  A )  =  (
# `  ( (
0 [,] A )  i^i  Prime ) ) )

Proof of Theorem ppival
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 oveq2 5866 . . . 4  |-  ( x  =  A  ->  (
0 [,] x )  =  ( 0 [,] A ) )
21ineq1d 3369 . . 3  |-  ( x  =  A  ->  (
( 0 [,] x
)  i^i  Prime )  =  ( ( 0 [,] A )  i^i  Prime ) )
32fveq2d 5529 . 2  |-  ( x  =  A  ->  ( # `
 ( ( 0 [,] x )  i^i 
Prime ) )  =  (
# `  ( (
0 [,] A )  i^i  Prime ) ) )
4 df-ppi 20337 . 2  |- π  =  ( x  e.  RR  |->  (
# `  ( (
0 [,] x )  i^i  Prime ) ) )
5 fvex 5539 . 2  |-  ( # `  ( ( 0 [,] A )  i^i  Prime ) )  e.  _V
63, 4, 5fvmpt 5602 1  |-  ( A  e.  RR  ->  (π `  A )  =  (
# `  ( (
0 [,] A )  i^i  Prime ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    i^i cin 3151   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737   [,]cicc 10659   #chash 11337   Primecprime 12758  πcppi 20331
This theorem is referenced by:  ppival2  20366  ppival2g  20367  ppifl  20398  ppiwordi  20400  chtleppi  20449
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-ppi 20337
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