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Theorem ppival 23917
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 6313 . . . 4  |-  ( x  =  A  ->  (
0 [,] x )  =  ( 0 [,] A ) )
21ineq1d 3669 . . 3  |-  ( x  =  A  ->  (
( 0 [,] x
)  i^i  Prime )  =  ( ( 0 [,] A )  i^i  Prime ) )
32fveq2d 5885 . 2  |-  ( x  =  A  ->  ( # `
 ( ( 0 [,] x )  i^i 
Prime ) )  =  (
# `  ( (
0 [,] A )  i^i  Prime ) ) )
4 df-ppi 23889 . 2  |- π  =  ( x  e.  RR  |->  (
# `  ( (
0 [,] x )  i^i  Prime ) ) )
5 fvex 5891 . 2  |-  ( # `  ( ( 0 [,] A )  i^i  Prime ) )  e.  _V
63, 4, 5fvmpt 5964 1  |-  ( A  e.  RR  ->  (π `  A )  =  (
# `  ( (
0 [,] A )  i^i  Prime ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1870    i^i cin 3441   ` cfv 5601  (class class class)co 6305   RRcr 9537   0cc0 9538   [,]cicc 11638   #chash 12512   Primecprime 14593  πcppi 23883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-ppi 23889
This theorem is referenced by:  ppival2  23918  ppival2g  23919  ppifl  23950  ppiwordi  23952  chtleppi  24001
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