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Theorem ppival 20361
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 5828 . . . 4  |-  ( x  =  A  ->  (
0 [,] x )  =  ( 0 [,] A ) )
21ineq1d 3370 . . 3  |-  ( x  =  A  ->  (
( 0 [,] x
)  i^i  Prime )  =  ( ( 0 [,] A )  i^i  Prime ) )
32fveq2d 5490 . 2  |-  ( x  =  A  ->  ( # `
 ( ( 0 [,] x )  i^i 
Prime ) )  =  (
# `  ( (
0 [,] A )  i^i  Prime ) ) )
4 df-ppi 20333 . 2  |- π  =  ( x  e.  RR  |->  (
# `  ( (
0 [,] x )  i^i  Prime ) ) )
5 fvex 5500 . 2  |-  ( # `  ( ( 0 [,] A )  i^i  Prime ) )  e.  _V
63, 4, 5fvmpt 5564 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 1685    i^i cin 3152   ` cfv 5221  (class class class)co 5820   RRcr 8732   0cc0 8733   [,]cicc 10655   #chash 11333   Primecprime 12754  πcppi 20327
This theorem is referenced by:  ppival2  20362  ppival2g  20363  ppifl  20394  ppiwordi  20396  chtleppi  20445
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fv 5229  df-ov 5823  df-ppi 20333
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