MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ppival Unicode version

Theorem ppival 20777
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 6028 . . . 4  |-  ( x  =  A  ->  (
0 [,] x )  =  ( 0 [,] A ) )
21ineq1d 3484 . . 3  |-  ( x  =  A  ->  (
( 0 [,] x
)  i^i  Prime )  =  ( ( 0 [,] A )  i^i  Prime ) )
32fveq2d 5672 . 2  |-  ( x  =  A  ->  ( # `
 ( ( 0 [,] x )  i^i 
Prime ) )  =  (
# `  ( (
0 [,] A )  i^i  Prime ) ) )
4 df-ppi 20749 . 2  |- π  =  ( x  e.  RR  |->  (
# `  ( (
0 [,] x )  i^i  Prime ) ) )
5 fvex 5682 . 2  |-  ( # `  ( ( 0 [,] A )  i^i  Prime ) )  e.  _V
63, 4, 5fvmpt 5745 1  |-  ( A  e.  RR  ->  (π `  A )  =  (
# `  ( (
0 [,] A )  i^i  Prime ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717    i^i cin 3262   ` cfv 5394  (class class class)co 6020   RRcr 8922   0cc0 8923   [,]cicc 10851   #chash 11545   Primecprime 13006  πcppi 20743
This theorem is referenced by:  ppival2  20778  ppival2g  20779  ppifl  20810  ppiwordi  20812  chtleppi  20861
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-ov 6023  df-ppi 20749
  Copyright terms: Public domain W3C validator