Theorem mumul 24907

Description: The Möbius function is a multiplicative function. This is one of the primary interests of the Möbius function as an arithmetic function. (Contributed by Mario Carneiro, 3-Oct-2014.)

Assertion

Ref	Expression
mumul	|- ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) -> ( mmu ` ( A x. B ) ) = ( ( mmu ` A ) x. ( mmu ` B ) ) )

Proof of Theorem mumul

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl2 1065	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( mmu ` A ) = 0 ) -> B e. NN )
2		mucl 24867	. . . . . 6 |- ( B e. NN -> ( mmu ` B ) e. ZZ )
3	1, 2	syl 17	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( mmu ` A ) = 0 ) -> ( mmu ` B ) e. ZZ )
4	3	zcnd 11483	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( mmu ` A ) = 0 ) -> ( mmu ` B ) e. CC )
5	4	mul02d 10234	. . 3 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( mmu ` A ) = 0 ) -> ( 0 x. ( mmu ` B ) ) = 0 )
6		simpr 477	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( mmu ` A ) = 0 ) -> ( mmu ` A ) = 0 )
7	6	oveq1d 6665	. . 3 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( mmu ` A ) = 0 ) -> ( ( mmu ` A ) x. ( mmu ` B ) ) = ( 0 x. ( mmu ` B ) ) )
8		mumullem1 24905	. . . 4 |- ( ( ( A e. NN /\ B e. NN ) /\ ( mmu ` A ) = 0 ) -> ( mmu ` ( A x. B ) ) = 0 )
9	8	3adantl3 1219	. . 3 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( mmu ` A ) = 0 ) -> ( mmu ` ( A x. B ) ) = 0 )
10	5, 7, 9	3eqtr4rd 2667	. 2 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( mmu ` A ) = 0 ) -> ( mmu ` ( A x. B ) ) = ( ( mmu ` A ) x. ( mmu ` B ) ) )
11		simpl1 1064	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( mmu ` B ) = 0 ) -> A e. NN )
12		mucl 24867	. . . . . 6 |- ( A e. NN -> ( mmu ` A ) e. ZZ )
13	11, 12	syl 17	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( mmu ` B ) = 0 ) -> ( mmu ` A ) e. ZZ )
14	13	zcnd 11483	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( mmu ` B ) = 0 ) -> ( mmu ` A ) e. CC )
15	14	mul01d 10235	. . 3 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( mmu ` B ) = 0 ) -> ( ( mmu ` A ) x. 0 ) = 0 )
16		simpr 477	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( mmu ` B ) = 0 ) -> ( mmu ` B ) = 0 )
17	16	oveq2d 6666	. . 3 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( mmu ` B ) = 0 ) -> ( ( mmu ` A ) x. ( mmu ` B ) ) = ( ( mmu ` A ) x. 0 ) )
18		nncn 11028	. . . . . . . 8 |- ( A e. NN -> A e. CC )
19		nncn 11028	. . . . . . . 8 |- ( B e. NN -> B e. CC )
20		mulcom 10022	. . . . . . . 8 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) = ( B x. A ) )
21	18, 19, 20	syl2an 494	. . . . . . 7 |- ( ( A e. NN /\ B e. NN ) -> ( A x. B ) = ( B x. A ) )
22	21	fveq2d 6195	. . . . . 6 |- ( ( A e. NN /\ B e. NN ) -> ( mmu ` ( A x. B ) ) = ( mmu ` ( B x. A ) ) )
23	22	adantr 481	. . . . 5 |- ( ( ( A e. NN /\ B e. NN ) /\ ( mmu ` B ) = 0 ) -> ( mmu ` ( A x. B ) ) = ( mmu ` ( B x. A ) ) )
24		mumullem1 24905	. . . . . 6 |- ( ( ( B e. NN /\ A e. NN ) /\ ( mmu ` B ) = 0 ) -> ( mmu ` ( B x. A ) ) = 0 )
25	24	ancom1s 847	. . . . 5 |- ( ( ( A e. NN /\ B e. NN ) /\ ( mmu ` B ) = 0 ) -> ( mmu ` ( B x. A ) ) = 0 )
26	23, 25	eqtrd 2656	. . . 4 |- ( ( ( A e. NN /\ B e. NN ) /\ ( mmu ` B ) = 0 ) -> ( mmu ` ( A x. B ) ) = 0 )
27	26	3adantl3 1219	. . 3 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( mmu ` B ) = 0 ) -> ( mmu ` ( A x. B ) ) = 0 )
28	15, 17, 27	3eqtr4rd 2667	. 2 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( mmu ` B ) = 0 ) -> ( mmu ` ( A x. B ) ) = ( ( mmu ` A ) x. ( mmu ` B ) ) )
29		simpl1 1064	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) -> A e. NN )
30		simpl2 1065	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) -> B e. NN )
31	29, 30	nnmulcld 11068	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) -> ( A x. B ) e. NN )
32		mumullem2 24906	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) -> ( mmu ` ( A x. B ) ) =/= 0 )
33		muval2 24860	. . . 4 |- ( ( ( A x. B ) e. NN /\ ( mmu ` ( A x. B ) ) =/= 0 ) -> ( mmu ` ( A x. B ) ) = ( -u 1 ^ ( # ` { p e. Prime | p || ( A x. B ) } ) ) )
34	31, 32, 33	syl2anc 693	. . 3 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) -> ( mmu ` ( A x. B ) ) = ( -u 1 ^ ( # ` { p e. Prime | p || ( A x. B ) } ) ) )
35		neg1cn 11124	. . . . . 6 |- -u 1 e. CC
36	35	a1i 11	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) -> -u 1 e. CC )
37		fzfi 12771	. . . . . . 7 |- ( 1 ... B ) e. Fin
38		prmnn 15388	. . . . . . . . . 10 |- ( p e. Prime -> p e. NN )
39	38	ssriv 3607	. . . . . . . . 9 |- Prime C_ NN
40		rabss2 3685	. . . . . . . . 9 |- ( Prime C_ NN -> { p e. Prime | p || B } C_ { p e. NN | p || B } )
41	39, 40	ax-mp 5	. . . . . . . 8 |- { p e. Prime | p || B } C_ { p e. NN | p || B }
42		dvdsssfz1 15040	. . . . . . . . 9 |- ( B e. NN -> { p e. NN | p || B } C_ ( 1 ... B ) )
43	30, 42	syl 17	. . . . . . . 8 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) -> { p e. NN | p || B } C_ ( 1 ... B ) )
44	41, 43	syl5ss 3614	. . . . . . 7 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) -> { p e. Prime | p || B } C_ ( 1 ... B ) )
45		ssfi 8180	. . . . . . 7 |- ( ( ( 1 ... B ) e. Fin /\ { p e. Prime | p || B } C_ ( 1 ... B ) ) -> { p e. Prime | p || B } e. Fin )
46	37, 44, 45	sylancr 695	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) -> { p e. Prime | p || B } e. Fin )
47		hashcl 13147	. . . . . 6 |- ( { p e. Prime | p || B } e. Fin -> ( # ` { p e. Prime | p || B } ) e. NN0 )
48	46, 47	syl 17	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) -> ( # ` { p e. Prime | p || B } ) e. NN0 )
49		fzfi 12771	. . . . . . 7 |- ( 1 ... A ) e. Fin
50		rabss2 3685	. . . . . . . . 9 |- ( Prime C_ NN -> { p e. Prime | p || A } C_ { p e. NN | p || A } )
51	39, 50	ax-mp 5	. . . . . . . 8 |- { p e. Prime | p || A } C_ { p e. NN | p || A }
52		dvdsssfz1 15040	. . . . . . . . 9 |- ( A e. NN -> { p e. NN | p || A } C_ ( 1 ... A ) )
53	29, 52	syl 17	. . . . . . . 8 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) -> { p e. NN | p || A } C_ ( 1 ... A ) )
54	51, 53	syl5ss 3614	. . . . . . 7 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) -> { p e. Prime | p || A } C_ ( 1 ... A ) )
55		ssfi 8180	. . . . . . 7 |- ( ( ( 1 ... A ) e. Fin /\ { p e. Prime | p || A } C_ ( 1 ... A ) ) -> { p e. Prime | p || A } e. Fin )
56	49, 54, 55	sylancr 695	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) -> { p e. Prime | p || A } e. Fin )
57		hashcl 13147	. . . . . 6 |- ( { p e. Prime | p || A } e. Fin -> ( # ` { p e. Prime | p || A } ) e. NN0 )
58	56, 57	syl 17	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) -> ( # ` { p e. Prime | p || A } ) e. NN0 )
59	36, 48, 58	expaddd 13010	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) -> ( -u 1 ^ ( ( # ` { p e. Prime | p || A } ) + ( # ` { p e. Prime | p || B } ) ) ) = ( ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) x. ( -u 1 ^ ( # ` { p e. Prime | p || B } ) ) ) )
60		simpr 477	. . . . . . . . . 10 |- ( ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> p e. Prime )
61		simpl1 1064	. . . . . . . . . . . 12 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> A e. NN )
62	61	nnzd 11481	. . . . . . . . . . 11 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> A e. ZZ )
63	62	adantlr 751	. . . . . . . . . 10 |- ( ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> A e. ZZ )
64		simpl2 1065	. . . . . . . . . . . 12 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> B e. NN )
65	64	nnzd 11481	. . . . . . . . . . 11 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> B e. ZZ )
66	65	adantlr 751	. . . . . . . . . 10 |- ( ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> B e. ZZ )
67		euclemma 15425	. . . . . . . . . 10 |- ( ( p e. Prime /\ A e. ZZ /\ B e. ZZ ) -> ( p || ( A x. B ) <-> ( p || A \/ p || B ) ) )
68	60, 63, 66, 67	syl3anc 1326	. . . . . . . . 9 |- ( ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> ( p || ( A x. B ) <-> ( p || A \/ p || B ) ) )
69	68	rabbidva 3188	. . . . . . . 8 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) -> { p e. Prime | p || ( A x. B ) } = { p e. Prime | ( p || A \/ p || B ) } )
70		unrab 3898	. . . . . . . 8 |- ( { p e. Prime | p || A } u. { p e. Prime | p || B } ) = { p e. Prime | ( p || A \/ p || B ) }
71	69, 70	syl6eqr 2674	. . . . . . 7 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) -> { p e. Prime | p || ( A x. B ) } = ( { p e. Prime | p || A } u. { p e. Prime | p || B } ) )
72	71	fveq2d 6195	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) -> ( # ` { p e. Prime | p || ( A x. B ) } ) = ( # ` ( { p e. Prime | p || A } u. { p e. Prime | p || B } ) ) )
73		inrab 3899	. . . . . . . 8 |- ( { p e. Prime | p || A } i^i { p e. Prime | p || B } ) = { p e. Prime | ( p || A /\ p || B ) }
74		nprmdvds1 15418	. . . . . . . . . . . 12 |- ( p e. Prime -> -. p || 1 )
75	74	adantl 482	. . . . . . . . . . 11 |- ( ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> -. p || 1 )
76		prmz 15389	. . . . . . . . . . . . . 14 |- ( p e. Prime -> p e. ZZ )
77	76	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> p e. ZZ )
78		dvdsgcd 15261	. . . . . . . . . . . . 13 |- ( ( p e. ZZ /\ A e. ZZ /\ B e. ZZ ) -> ( ( p || A /\ p || B ) -> p || ( A gcd B ) ) )
79	77, 63, 66, 78	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> ( ( p || A /\ p || B ) -> p || ( A gcd B ) ) )
80		simpll3 1102	. . . . . . . . . . . . 13 |- ( ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> ( A gcd B ) = 1 )
81	80	breq2d 4665	. . . . . . . . . . . 12 |- ( ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> ( p || ( A gcd B ) <-> p || 1 ) )
82	79, 81	sylibd 229	. . . . . . . . . . 11 |- ( ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> ( ( p || A /\ p || B ) -> p || 1 ) )
83	75, 82	mtod 189	. . . . . . . . . 10 |- ( ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> -. ( p || A /\ p || B ) )
84	83	ralrimiva 2966	. . . . . . . . 9 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) -> A. p e. Prime -. ( p || A /\ p || B ) )
85		rabeq0 3957	. . . . . . . . 9 |- ( { p e. Prime | ( p || A /\ p || B ) } = (/) <-> A. p e. Prime -. ( p || A /\ p || B ) )
86	84, 85	sylibr 224	. . . . . . . 8 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) -> { p e. Prime | ( p || A /\ p || B ) } = (/) )
87	73, 86	syl5eq 2668	. . . . . . 7 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) -> ( { p e. Prime | p || A } i^i { p e. Prime | p || B } ) = (/) )
88		hashun 13171	. . . . . . 7 |- ( ( { p e. Prime | p || A } e. Fin /\ { p e. Prime | p || B } e. Fin /\ ( { p e. Prime | p || A } i^i { p e. Prime | p || B } ) = (/) ) -> ( # ` ( { p e. Prime | p || A } u. { p e. Prime | p || B } ) ) = ( ( # ` { p e. Prime | p || A } ) + ( # ` { p e. Prime | p || B } ) ) )
89	56, 46, 87, 88	syl3anc 1326	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) -> ( # ` ( { p e. Prime | p || A } u. { p e. Prime | p || B } ) ) = ( ( # ` { p e. Prime | p || A } ) + ( # ` { p e. Prime | p || B } ) ) )
90	72, 89	eqtrd 2656	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) -> ( # ` { p e. Prime | p || ( A x. B ) } ) = ( ( # ` { p e. Prime | p || A } ) + ( # ` { p e. Prime | p || B } ) ) )
91	90	oveq2d 6666	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) -> ( -u 1 ^ ( # ` { p e. Prime | p || ( A x. B ) } ) ) = ( -u 1 ^ ( ( # ` { p e. Prime | p || A } ) + ( # ` { p e. Prime | p || B } ) ) ) )
92		simprl 794	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) -> ( mmu ` A ) =/= 0 )
93		muval2 24860	. . . . . 6 |- ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) -> ( mmu ` A ) = ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) )
94	29, 92, 93	syl2anc 693	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) -> ( mmu ` A ) = ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) )
95		simprr 796	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) -> ( mmu ` B ) =/= 0 )
96		muval2 24860	. . . . . 6 |- ( ( B e. NN /\ ( mmu ` B ) =/= 0 ) -> ( mmu ` B ) = ( -u 1 ^ ( # ` { p e. Prime | p || B } ) ) )
97	30, 95, 96	syl2anc 693	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) -> ( mmu ` B ) = ( -u 1 ^ ( # ` { p e. Prime | p || B } ) ) )
98	94, 97	oveq12d 6668	. . . 4 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) -> ( ( mmu ` A ) x. ( mmu ` B ) ) = ( ( -u 1 ^ ( # ` { p e. Prime | p || A } ) ) x. ( -u 1 ^ ( # ` { p e. Prime | p || B } ) ) ) )
99	59, 91, 98	3eqtr4rd 2667	. . 3 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) -> ( ( mmu ` A ) x. ( mmu ` B ) ) = ( -u 1 ^ ( # ` { p e. Prime | p || ( A x. B ) } ) ) )
100	34, 99	eqtr4d 2659	. 2 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) -> ( mmu ` ( A x. B ) ) = ( ( mmu ` A ) x. ( mmu ` B ) ) )
101	10, 28, 100	pm2.61da2ne 2882	1 |- ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) -> ( mmu ` ( A x. B ) ) = ( ( mmu ` A ) x. ( mmu ` B ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   -ucneg 10267   NNcn 11020   NN0cn0 11292   ZZcz 11377   ...cfz 12326   ^cexp 12860   #chash 13117   || cdvds 14983   gcd cgcd 15216   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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fz 12327  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-prm 15386  df-pc 15542  df-mu 24827

This theorem is referenced by: (None)