Theorem chpval 24848

Description: Value of the second Chebyshev function, or summary von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016.)

Assertion

Ref	Expression
chpval	|- ( A e. RR -> (ψ ` A ) = sum_ n e. ( 1 ... ( |_ ` A ) ) (Λ ` n ) )

Distinct variable group:   A, n

Proof of Theorem chpval

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . 4 |- ( x = A -> ( |_ ` x ) = ( |_ ` A ) )
2	1	oveq2d 6666	. . 3 |- ( x = A -> ( 1 ... ( |_ ` x ) ) = ( 1 ... ( |_ ` A ) ) )
3	2	sumeq1d 14431	. 2 |- ( x = A -> sum_ n e. ( 1 ... ( |_ ` x ) ) (Λ ` n ) = sum_ n e. ( 1 ... ( |_ ` A ) ) (Λ ` n ) )
4		df-chp 24825	. 2 |- ψ = ( x e. RR |-> sum_ n e. ( 1 ... ( |_ ` x ) ) (Λ ` n ) )
5		sumex 14418	. 2 |- sum_ n e. ( 1 ... ( |_ ` A ) ) (Λ ` n ) e. _V
6	3, 4, 5	fvmpt 6282	1 |- ( A e. RR -> (ψ ` A ) = sum_ n e. ( 1 ... ( |_ ` A ) ) (Λ ` n ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   RRcr 9935   1c1 9937   ...cfz 12326   |_cfl 12591   sum_csu 14416  Λcvma 24818  ψcchp 24819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-iota 5851  df-fun 5890  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seq 12802  df-sum 14417  df-chp 24825

This theorem is referenced by:  efchpcl  24851  chpfl  24876  chpp1  24881  chpwordi  24883  chp1  24893  chtlepsi  24931  chpval2  24943  vmadivsum  25171  selberg  25237  selberg3lem1  25246  selberg4  25250  pntsval2  25265  chpvalz  30706