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Theorem ppival 23267
Description: Value of the prime-counting function pi. (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 6303 . . . 4  |-  ( x  =  A  ->  (
0 [,] x )  =  ( 0 [,] A ) )
21ineq1d 3704 . . 3  |-  ( x  =  A  ->  (
( 0 [,] x
)  i^i  Prime )  =  ( ( 0 [,] A )  i^i  Prime ) )
32fveq2d 5876 . 2  |-  ( x  =  A  ->  ( # `
 ( ( 0 [,] x )  i^i 
Prime ) )  =  (
# `  ( (
0 [,] A )  i^i  Prime ) ) )
4 df-ppi 23239 . 2  |- π  =  ( x  e.  RR  |->  (
# `  ( (
0 [,] x )  i^i  Prime ) ) )
5 fvex 5882 . 2  |-  ( # `  ( ( 0 [,] A )  i^i  Prime ) )  e.  _V
63, 4, 5fvmpt 5957 1  |-  ( A  e.  RR  ->  (π `  A )  =  (
# `  ( (
0 [,] A )  i^i  Prime ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767    i^i cin 3480   ` cfv 5594  (class class class)co 6295   RRcr 9503   0cc0 9504   [,]cicc 11544   #chash 12385   Primecprime 14093  πcppi 23233
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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-ppi 23239
This theorem is referenced by:  ppival2  23268  ppival2g  23269  ppifl  23300  ppiwordi  23302  chtleppi  23351
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