Theorem prmoval 15737

Description: Value of the primorial function for nonnegative integers. (Contributed by AV, 28-Aug-2020.)

Assertion

Ref	Expression
prmoval	|- ( N e. NN0 -> (#p ` N ) = prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) )

Distinct variable group:   k, N

Proof of Theorem prmoval

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-prmo 15736	. . 3 |- #p = ( n e. NN0 |-> prod_ k e. ( 1 ... n ) if ( k e. Prime , k , 1 ) )
2	1	a1i 11	. 2 |- ( N e. NN0 -> #p = ( n e. NN0 |-> prod_ k e. ( 1 ... n ) if ( k e. Prime , k , 1 ) ) )
3		oveq2 6658	. . . 4 |- ( n = N -> ( 1 ... n ) = ( 1 ... N ) )
4	3	prodeq1d 14651	. . 3 |- ( n = N -> prod_ k e. ( 1 ... n ) if ( k e. Prime , k , 1 ) = prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) )
5	4	adantl 482	. 2 |- ( ( N e. NN0 /\ n = N ) -> prod_ k e. ( 1 ... n ) if ( k e. Prime , k , 1 ) = prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) )
6		id 22	. 2 |- ( N e. NN0 -> N e. NN0 )
7		prodex 14637	. . 3 |- prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) e. _V
8	7	a1i 11	. 2 |- ( N e. NN0 -> prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) e. _V )
9	2, 5, 6, 8	fvmptd 6288	1 |- ( N e. NN0 -> (#p ` N ) = prod_ k e. ( 1 ... N ) if ( k e. Prime , k , 1 ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   ifcif 4086   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   1c1 9937   NN0cn0 11292   ...cfz 12326   prod_cprod 14635   Primecprime 15385  #_pcprmo 15735

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-csb 3534  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-prod 14636  df-prmo 15736

This theorem is referenced by:  prmocl  15738  prmo0  15740  prmo1  15741  prmop1  15742  prmdvdsprmo  15746  prmolefac  15750  prmodvdslcmf  15751  prmgapprmo  15766