Theorem muval2 24860

Description: The value of the Möbius function at a squarefree number. (Contributed by Mario Carneiro, 3-Oct-2014.)

Assertion

Ref	Expression
muval2	|- ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) -> ( mmu ` A ) = ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) )

Distinct variable group:   A, p

Proof of Theorem muval2

Step	Hyp	Ref	Expression
1		df-ne 2795	. . 3 |- ( ( mmu ` A ) =/= 0 <-> -. ( mmu ` A ) = 0 )
2		ifeqor 4132	. . . . 5 |- ( if ( E. p e. Prime ( p ^ 2 ) || A , 0 , ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) ) = 0 \/ if ( E. p e. Prime ( p ^ 2 ) || A , 0 , ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) ) = ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) )
3		muval 24858	. . . . . . 7 |- ( A e. NN -> ( mmu ` A ) = if ( E. p e. Prime ( p ^ 2 ) || A , 0 , ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) ) )
4	3	eqeq1d 2624	. . . . . 6 |- ( A e. NN -> ( ( mmu ` A ) = 0 <-> if ( E. p e. Prime ( p ^ 2 ) || A , 0 , ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) ) = 0 ) )
5	3	eqeq1d 2624	. . . . . 6 |- ( A e. NN -> ( ( mmu ` A ) = ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) <-> if ( E. p e. Prime ( p ^ 2 ) || A , 0 , ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) ) = ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) ) )
6	4, 5	orbi12d 746	. . . . 5 |- ( A e. NN -> ( ( ( mmu ` A ) = 0 \/ ( mmu ` A ) = ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) ) <-> ( if ( E. p e. Prime ( p ^ 2 ) || A , 0 , ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) ) = 0 \/ if ( E. p e. Prime ( p ^ 2 ) || A , 0 , ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) ) = ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) ) ) )
7	2, 6	mpbiri 248	. . . 4 |- ( A e. NN -> ( ( mmu ` A ) = 0 \/ ( mmu ` A ) = ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) ) )
8	7	ord 392	. . 3 |- ( A e. NN -> ( -. ( mmu ` A ) = 0 -> ( mmu ` A ) = ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) ) )
9	1, 8	syl5bi 232	. 2 |- ( A e. NN -> ( ( mmu ` A ) =/= 0 -> ( mmu ` A ) = ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) ) )
10	9	imp 445	1 |- ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) -> ( mmu ` A ) = ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   ifcif 4086   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   -ucneg 10267   NNcn 11020   2c2 11070   ^cexp 12860   #chash 13117   || cdvds 14983   Primecprime 15385   mmucmu 24821

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  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-mulcl 9998  ax-i2m1 10004

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-ne 2795  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-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-mu 24827

This theorem is referenced by:  mumul  24907  musum  24917