Theorem lgsdir 14103

Description: The Legendre symbol is completely multiplicative in its left argument. Generalization of theorem 9.9(a) in [ApostolNT] p. 188 (which assumes that A and B are odd positive integers). (Contributed by Mario Carneiro, 4-Feb-2015.)

Assertion

Ref	Expression
lgsdir	|- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> ( ( A x. B ) /L N ) = ( ( A /L N ) x. ( B /L N ) ) )

Proof of Theorem lgsdir

Dummy variables k n v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1cnd 7964	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) -> 1 e. CC )
2		0cnd 7941	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) -> 0 e. CC )
3		zsqcl 10576	. . . . . . . . . 10 |- ( B e. ZZ -> ( B ^ 2 ) e. ZZ )
4	3	3ad2ant2 1019	. . . . . . . . 9 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) -> ( B ^ 2 ) e. ZZ )
5		1z 9268	. . . . . . . . 9 |- 1 e. ZZ
6		zdceq 9317	. . . . . . . . 9 |- ( ( ( B ^ 2 ) e. ZZ /\ 1 e. ZZ ) -> DECID ( B ^ 2 ) = 1 )
7	4, 5, 6	sylancl 413	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) -> DECID ( B ^ 2 ) = 1 )
8	1, 2, 7	ifcldcd 3569	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) -> if ( ( B ^ 2 ) = 1 , 1 , 0 ) e. CC )
9	8	mulid2d 7966	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) -> ( 1 x. if ( ( B ^ 2 ) = 1 , 1 , 0 ) ) = if ( ( B ^ 2 ) = 1 , 1 , 0 ) )
10	9	ad3antrrr 492	. . . . 5 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) /\ ( A ^ 2 ) = 1 ) -> ( 1 x. if ( ( B ^ 2 ) = 1 , 1 , 0 ) ) = if ( ( B ^ 2 ) = 1 , 1 , 0 ) )
11		iftrue 3539	. . . . . . 7 |- ( ( A ^ 2 ) = 1 -> if ( ( A ^ 2 ) = 1 , 1 , 0 ) = 1 )
12	11	adantl 277	. . . . . 6 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) /\ ( A ^ 2 ) = 1 ) -> if ( ( A ^ 2 ) = 1 , 1 , 0 ) = 1 )
13	12	oveq1d 5884	. . . . 5 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) /\ ( A ^ 2 ) = 1 ) -> ( if ( ( A ^ 2 ) = 1 , 1 , 0 ) x. if ( ( B ^ 2 ) = 1 , 1 , 0 ) ) = ( 1 x. if ( ( B ^ 2 ) = 1 , 1 , 0 ) ) )
14		simpl1 1000	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> A e. ZZ )
15	14	zcnd 9365	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> A e. CC )
16	15	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) /\ ( A ^ 2 ) = 1 ) -> A e. CC )
17		simpl2 1001	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> B e. ZZ )
18	17	zcnd 9365	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> B e. CC )
19	18	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) /\ ( A ^ 2 ) = 1 ) -> B e. CC )
20	16, 19	sqmuld 10651	. . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) /\ ( A ^ 2 ) = 1 ) -> ( ( A x. B ) ^ 2 ) = ( ( A ^ 2 ) x. ( B ^ 2 ) ) )
21		simpr 110	. . . . . . . . 9 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) /\ ( A ^ 2 ) = 1 ) -> ( A ^ 2 ) = 1 )
22	21	oveq1d 5884	. . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) /\ ( A ^ 2 ) = 1 ) -> ( ( A ^ 2 ) x. ( B ^ 2 ) ) = ( 1 x. ( B ^ 2 ) ) )
23	18	sqcld 10637	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> ( B ^ 2 ) e. CC )
24	23	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) /\ ( A ^ 2 ) = 1 ) -> ( B ^ 2 ) e. CC )
25	24	mulid2d 7966	. . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) /\ ( A ^ 2 ) = 1 ) -> ( 1 x. ( B ^ 2 ) ) = ( B ^ 2 ) )
26	20, 22, 25	3eqtrd 2214	. . . . . . 7 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) /\ ( A ^ 2 ) = 1 ) -> ( ( A x. B ) ^ 2 ) = ( B ^ 2 ) )
27	26	eqeq1d 2186	. . . . . 6 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) /\ ( A ^ 2 ) = 1 ) -> ( ( ( A x. B ) ^ 2 ) = 1 <-> ( B ^ 2 ) = 1 ) )
28	27	ifbid 3555	. . . . 5 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) /\ ( A ^ 2 ) = 1 ) -> if ( ( ( A x. B ) ^ 2 ) = 1 , 1 , 0 ) = if ( ( B ^ 2 ) = 1 , 1 , 0 ) )
29	10, 13, 28	3eqtr4d 2220	. . . 4 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) /\ ( A ^ 2 ) = 1 ) -> ( if ( ( A ^ 2 ) = 1 , 1 , 0 ) x. if ( ( B ^ 2 ) = 1 , 1 , 0 ) ) = if ( ( ( A x. B ) ^ 2 ) = 1 , 1 , 0 ) )
30	8	mul02d 8339	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) -> ( 0 x. if ( ( B ^ 2 ) = 1 , 1 , 0 ) ) = 0 )
31	30	ad3antrrr 492	. . . . 5 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) /\ -. ( A ^ 2 ) = 1 ) -> ( 0 x. if ( ( B ^ 2 ) = 1 , 1 , 0 ) ) = 0 )
32		iffalse 3542	. . . . . . 7 |- ( -. ( A ^ 2 ) = 1 -> if ( ( A ^ 2 ) = 1 , 1 , 0 ) = 0 )
33	32	adantl 277	. . . . . 6 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) /\ -. ( A ^ 2 ) = 1 ) -> if ( ( A ^ 2 ) = 1 , 1 , 0 ) = 0 )
34	33	oveq1d 5884	. . . . 5 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) /\ -. ( A ^ 2 ) = 1 ) -> ( if ( ( A ^ 2 ) = 1 , 1 , 0 ) x. if ( ( B ^ 2 ) = 1 , 1 , 0 ) ) = ( 0 x. if ( ( B ^ 2 ) = 1 , 1 , 0 ) ) )
35		dvdsmul1 11804	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ B e. ZZ ) -> A || ( A x. B ) )
36	14, 17, 35	syl2anc 411	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> A || ( A x. B ) )
37	14, 17	zmulcld 9370	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> ( A x. B ) e. ZZ )
38		dvdssq 12015	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ ( A x. B ) e. ZZ ) -> ( A || ( A x. B ) <-> ( A ^ 2 ) || ( ( A x. B ) ^ 2 ) ) )
39	14, 37, 38	syl2anc 411	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> ( A || ( A x. B ) <-> ( A ^ 2 ) || ( ( A x. B ) ^ 2 ) ) )
40	36, 39	mpbid 147	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> ( A ^ 2 ) || ( ( A x. B ) ^ 2 ) )
41	40	adantr 276	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) -> ( A ^ 2 ) || ( ( A x. B ) ^ 2 ) )
42		breq2 4004	. . . . . . . . 9 |- ( ( ( A x. B ) ^ 2 ) = 1 -> ( ( A ^ 2 ) || ( ( A x. B ) ^ 2 ) <-> ( A ^ 2 ) || 1 ) )
43	41, 42	syl5ibcom 155	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) -> ( ( ( A x. B ) ^ 2 ) = 1 -> ( A ^ 2 ) || 1 ) )
44		simprl 529	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> A =/= 0 )
45	44	neneqd 2368	. . . . . . . . . . . . . . 15 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> -. A = 0 )
46		sqeq0 10569	. . . . . . . . . . . . . . . 16 |- ( A e. CC -> ( ( A ^ 2 ) = 0 <-> A = 0 ) )
47	15, 46	syl 14	. . . . . . . . . . . . . . 15 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> ( ( A ^ 2 ) = 0 <-> A = 0 ) )
48	45, 47	mtbird 673	. . . . . . . . . . . . . 14 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> -. ( A ^ 2 ) = 0 )
49		zsqcl2 10583	. . . . . . . . . . . . . . . 16 |- ( A e. ZZ -> ( A ^ 2 ) e. NN0 )
50	14, 49	syl 14	. . . . . . . . . . . . . . 15 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> ( A ^ 2 ) e. NN0 )
51		elnn0 9167	. . . . . . . . . . . . . . 15 |- ( ( A ^ 2 ) e. NN0 <-> ( ( A ^ 2 ) e. NN \/ ( A ^ 2 ) = 0 ) )
52	50, 51	sylib 122	. . . . . . . . . . . . . 14 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> ( ( A ^ 2 ) e. NN \/ ( A ^ 2 ) = 0 ) )
53	48, 52	ecased 1349	. . . . . . . . . . . . 13 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> ( A ^ 2 ) e. NN )
54	53	adantr 276	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) -> ( A ^ 2 ) e. NN )
55	54	nnzd 9363	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) -> ( A ^ 2 ) e. ZZ )
56		1nn 8919	. . . . . . . . . . 11 |- 1 e. NN
57		dvdsle 11833	. . . . . . . . . . 11 |- ( ( ( A ^ 2 ) e. ZZ /\ 1 e. NN ) -> ( ( A ^ 2 ) || 1 -> ( A ^ 2 ) <_ 1 ) )
58	55, 56, 57	sylancl 413	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) -> ( ( A ^ 2 ) || 1 -> ( A ^ 2 ) <_ 1 ) )
59	54	nnge1d 8951	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) -> 1 <_ ( A ^ 2 ) )
60	58, 59	jctird 317	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) -> ( ( A ^ 2 ) || 1 -> ( ( A ^ 2 ) <_ 1 /\ 1 <_ ( A ^ 2 ) ) ) )
61	54	nnred 8921	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) -> ( A ^ 2 ) e. RR )
62		1re 7947	. . . . . . . . . 10 |- 1 e. RR
63		letri3 8028	. . . . . . . . . 10 |- ( ( ( A ^ 2 ) e. RR /\ 1 e. RR ) -> ( ( A ^ 2 ) = 1 <-> ( ( A ^ 2 ) <_ 1 /\ 1 <_ ( A ^ 2 ) ) ) )
64	61, 62, 63	sylancl 413	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) -> ( ( A ^ 2 ) = 1 <-> ( ( A ^ 2 ) <_ 1 /\ 1 <_ ( A ^ 2 ) ) ) )
65	60, 64	sylibrd 169	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) -> ( ( A ^ 2 ) || 1 -> ( A ^ 2 ) = 1 ) )
66	43, 65	syld 45	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) -> ( ( ( A x. B ) ^ 2 ) = 1 -> ( A ^ 2 ) = 1 ) )
67	66	con3dimp 635	. . . . . 6 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) /\ -. ( A ^ 2 ) = 1 ) -> -. ( ( A x. B ) ^ 2 ) = 1 )
68	67	iffalsed 3544	. . . . 5 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) /\ -. ( A ^ 2 ) = 1 ) -> if ( ( ( A x. B ) ^ 2 ) = 1 , 1 , 0 ) = 0 )
69	31, 34, 68	3eqtr4d 2220	. . . 4 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) /\ -. ( A ^ 2 ) = 1 ) -> ( if ( ( A ^ 2 ) = 1 , 1 , 0 ) x. if ( ( B ^ 2 ) = 1 , 1 , 0 ) ) = if ( ( ( A x. B ) ^ 2 ) = 1 , 1 , 0 ) )
70		zdceq 9317	. . . . . 6 |- ( ( ( A ^ 2 ) e. ZZ /\ 1 e. ZZ ) -> DECID ( A ^ 2 ) = 1 )
71	55, 5, 70	sylancl 413	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) -> DECID ( A ^ 2 ) = 1 )
72		exmiddc 836	. . . . 5 |- (DECID ( A ^ 2 ) = 1 -> ( ( A ^ 2 ) = 1 \/ -. ( A ^ 2 ) = 1 ) )
73	71, 72	syl 14	. . . 4 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) -> ( ( A ^ 2 ) = 1 \/ -. ( A ^ 2 ) = 1 ) )
74	29, 69, 73	mpjaodan 798	. . 3 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) -> ( if ( ( A ^ 2 ) = 1 , 1 , 0 ) x. if ( ( B ^ 2 ) = 1 , 1 , 0 ) ) = if ( ( ( A x. B ) ^ 2 ) = 1 , 1 , 0 ) )
75		oveq2 5877	. . . . 5 |- ( N = 0 -> ( A /L N ) = ( A /L 0 ) )
76		lgs0 14081	. . . . . 6 |- ( A e. ZZ -> ( A /L 0 ) = if ( ( A ^ 2 ) = 1 , 1 , 0 ) )
77	14, 76	syl 14	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> ( A /L 0 ) = if ( ( A ^ 2 ) = 1 , 1 , 0 ) )
78	75, 77	sylan9eqr 2232	. . . 4 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) -> ( A /L N ) = if ( ( A ^ 2 ) = 1 , 1 , 0 ) )
79		oveq2 5877	. . . . 5 |- ( N = 0 -> ( B /L N ) = ( B /L 0 ) )
80		lgs0 14081	. . . . . 6 |- ( B e. ZZ -> ( B /L 0 ) = if ( ( B ^ 2 ) = 1 , 1 , 0 ) )
81	17, 80	syl 14	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> ( B /L 0 ) = if ( ( B ^ 2 ) = 1 , 1 , 0 ) )
82	79, 81	sylan9eqr 2232	. . . 4 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) -> ( B /L N ) = if ( ( B ^ 2 ) = 1 , 1 , 0 ) )
83	78, 82	oveq12d 5887	. . 3 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) -> ( ( A /L N ) x. ( B /L N ) ) = ( if ( ( A ^ 2 ) = 1 , 1 , 0 ) x. if ( ( B ^ 2 ) = 1 , 1 , 0 ) ) )
84		oveq2 5877	. . . 4 |- ( N = 0 -> ( ( A x. B ) /L N ) = ( ( A x. B ) /L 0 ) )
85		lgs0 14081	. . . . 5 |- ( ( A x. B ) e. ZZ -> ( ( A x. B ) /L 0 ) = if ( ( ( A x. B ) ^ 2 ) = 1 , 1 , 0 ) )
86	37, 85	syl 14	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> ( ( A x. B ) /L 0 ) = if ( ( ( A x. B ) ^ 2 ) = 1 , 1 , 0 ) )
87	84, 86	sylan9eqr 2232	. . 3 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) -> ( ( A x. B ) /L N ) = if ( ( ( A x. B ) ^ 2 ) = 1 , 1 , 0 ) )
88	74, 83, 87	3eqtr4rd 2221	. 2 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) -> ( ( A x. B ) /L N ) = ( ( A /L N ) x. ( B /L N ) ) )
89		lgsdilem 14095	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> if ( ( N < 0 /\ ( A x. B ) < 0 ) , -u 1 , 1 ) = ( if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) x. if ( ( N < 0 /\ B < 0 ) , -u 1 , 1 ) ) )
90	89	adantr 276	. . . 4 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> if ( ( N < 0 /\ ( A x. B ) < 0 ) , -u 1 , 1 ) = ( if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) x. if ( ( N < 0 /\ B < 0 ) , -u 1 , 1 ) ) )
91		simpl3 1002	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> N e. ZZ )
92		nnabscl 11093	. . . . . . 7 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. NN )
93	91, 92	sylan 283	. . . . . 6 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> ( abs ` N ) e. NN )
94		nnuz 9552	. . . . . 6 |- NN = ( ZZ>= ` 1 )
95	93, 94	eleqtrdi 2270	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> ( abs ` N ) e. ( ZZ>= ` 1 ) )
96		simpll1 1036	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> A e. ZZ )
97		simpll3 1038	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> N e. ZZ )
98		simpr 110	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> N =/= 0 )
99		eqid 2177	. . . . . . . . 9 |- ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) = ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) )
100	99	lgsfcl3 14089	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) : NN --> ZZ )
101	96, 97, 98, 100	syl3anc 1238	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) : NN --> ZZ )
102		elnnuz 9553	. . . . . . . 8 |- ( k e. NN <-> k e. ( ZZ>= ` 1 ) )
103	102	biimpri 133	. . . . . . 7 |- ( k e. ( ZZ>= ` 1 ) -> k e. NN )
104		ffvelcdm 5645	. . . . . . 7 |- ( ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) : NN --> ZZ /\ k e. NN ) -> ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ` k ) e. ZZ )
105	101, 103, 104	syl2an 289	. . . . . 6 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. ( ZZ>= ` 1 ) ) -> ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ` k ) e. ZZ )
106	105	zcnd 9365	. . . . 5 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. ( ZZ>= ` 1 ) ) -> ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ` k ) e. CC )
107		simpll2 1037	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> B e. ZZ )
108		eqid 2177	. . . . . . . . 9 |- ( n e. NN |-> if ( n e. Prime , ( ( B /L n ) ^ ( n pCnt N ) ) , 1 ) ) = ( n e. NN |-> if ( n e. Prime , ( ( B /L n ) ^ ( n pCnt N ) ) , 1 ) )
109	108	lgsfcl3 14089	. . . . . . . 8 |- ( ( B e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( n e. NN |-> if ( n e. Prime , ( ( B /L n ) ^ ( n pCnt N ) ) , 1 ) ) : NN --> ZZ )
110	107, 97, 98, 109	syl3anc 1238	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> ( n e. NN |-> if ( n e. Prime , ( ( B /L n ) ^ ( n pCnt N ) ) , 1 ) ) : NN --> ZZ )
111		ffvelcdm 5645	. . . . . . 7 |- ( ( ( n e. NN |-> if ( n e. Prime , ( ( B /L n ) ^ ( n pCnt N ) ) , 1 ) ) : NN --> ZZ /\ k e. NN ) -> ( ( n e. NN |-> if ( n e. Prime , ( ( B /L n ) ^ ( n pCnt N ) ) , 1 ) ) ` k ) e. ZZ )
112	110, 103, 111	syl2an 289	. . . . . 6 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. ( ZZ>= ` 1 ) ) -> ( ( n e. NN |-> if ( n e. Prime , ( ( B /L n ) ^ ( n pCnt N ) ) , 1 ) ) ` k ) e. ZZ )
113	112	zcnd 9365	. . . . 5 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. ( ZZ>= ` 1 ) ) -> ( ( n e. NN |-> if ( n e. Prime , ( ( B /L n ) ^ ( n pCnt N ) ) , 1 ) ) ` k ) e. CC )
114	96	adantr 276	. . . . . . . . . . . 12 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. Prime ) -> A e. ZZ )
115	107	adantr 276	. . . . . . . . . . . 12 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. Prime ) -> B e. ZZ )
116		simpr 110	. . . . . . . . . . . 12 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. Prime ) -> k e. Prime )
117		lgsdirprm 14102	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ B e. ZZ /\ k e. Prime ) -> ( ( A x. B ) /L k ) = ( ( A /L k ) x. ( B /L k ) ) )
118	114, 115, 116, 117	syl3anc 1238	. . . . . . . . . . 11 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. Prime ) -> ( ( A x. B ) /L k ) = ( ( A /L k ) x. ( B /L k ) ) )
119	118	oveq1d 5884	. . . . . . . . . 10 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. Prime ) -> ( ( ( A x. B ) /L k ) ^ ( k pCnt N ) ) = ( ( ( A /L k ) x. ( B /L k ) ) ^ ( k pCnt N ) ) )
120		prmz 12094	. . . . . . . . . . . . 13 |- ( k e. Prime -> k e. ZZ )
121		lgscl 14082	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ k e. ZZ ) -> ( A /L k ) e. ZZ )
122	96, 120, 121	syl2an 289	. . . . . . . . . . . 12 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. Prime ) -> ( A /L k ) e. ZZ )
123	122	zcnd 9365	. . . . . . . . . . 11 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. Prime ) -> ( A /L k ) e. CC )
124		lgscl 14082	. . . . . . . . . . . . 13 |- ( ( B e. ZZ /\ k e. ZZ ) -> ( B /L k ) e. ZZ )
125	107, 120, 124	syl2an 289	. . . . . . . . . . . 12 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. Prime ) -> ( B /L k ) e. ZZ )
126	125	zcnd 9365	. . . . . . . . . . 11 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. Prime ) -> ( B /L k ) e. CC )
127	97	adantr 276	. . . . . . . . . . . 12 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. Prime ) -> N e. ZZ )
128	98	adantr 276	. . . . . . . . . . . 12 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. Prime ) -> N =/= 0 )
129		pczcl 12281	. . . . . . . . . . . 12 |- ( ( k e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( k pCnt N ) e. NN0 )
130	116, 127, 128, 129	syl12anc 1236	. . . . . . . . . . 11 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. Prime ) -> ( k pCnt N ) e. NN0 )
131	123, 126, 130	mulexpd 10654	. . . . . . . . . 10 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. Prime ) -> ( ( ( A /L k ) x. ( B /L k ) ) ^ ( k pCnt N ) ) = ( ( ( A /L k ) ^ ( k pCnt N ) ) x. ( ( B /L k ) ^ ( k pCnt N ) ) ) )
132	119, 131	eqtrd 2210	. . . . . . . . 9 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. Prime ) -> ( ( ( A x. B ) /L k ) ^ ( k pCnt N ) ) = ( ( ( A /L k ) ^ ( k pCnt N ) ) x. ( ( B /L k ) ^ ( k pCnt N ) ) ) )
133		iftrue 3539	. . . . . . . . . 10 |- ( k e. Prime -> if ( k e. Prime , ( ( ( A x. B ) /L k ) ^ ( k pCnt N ) ) , 1 ) = ( ( ( A x. B ) /L k ) ^ ( k pCnt N ) ) )
134	133	adantl 277	. . . . . . . . 9 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. Prime ) -> if ( k e. Prime , ( ( ( A x. B ) /L k ) ^ ( k pCnt N ) ) , 1 ) = ( ( ( A x. B ) /L k ) ^ ( k pCnt N ) ) )
135		iftrue 3539	. . . . . . . . . . 11 |- ( k e. Prime -> if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt N ) ) , 1 ) = ( ( A /L k ) ^ ( k pCnt N ) ) )
136		iftrue 3539	. . . . . . . . . . 11 |- ( k e. Prime -> if ( k e. Prime , ( ( B /L k ) ^ ( k pCnt N ) ) , 1 ) = ( ( B /L k ) ^ ( k pCnt N ) ) )
137	135, 136	oveq12d 5887	. . . . . . . . . 10 |- ( k e. Prime -> ( if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt N ) ) , 1 ) x. if ( k e. Prime , ( ( B /L k ) ^ ( k pCnt N ) ) , 1 ) ) = ( ( ( A /L k ) ^ ( k pCnt N ) ) x. ( ( B /L k ) ^ ( k pCnt N ) ) ) )
138	137	adantl 277	. . . . . . . . 9 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. Prime ) -> ( if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt N ) ) , 1 ) x. if ( k e. Prime , ( ( B /L k ) ^ ( k pCnt N ) ) , 1 ) ) = ( ( ( A /L k ) ^ ( k pCnt N ) ) x. ( ( B /L k ) ^ ( k pCnt N ) ) ) )
139	132, 134, 138	3eqtr4d 2220	. . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. Prime ) -> if ( k e. Prime , ( ( ( A x. B ) /L k ) ^ ( k pCnt N ) ) , 1 ) = ( if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt N ) ) , 1 ) x. if ( k e. Prime , ( ( B /L k ) ^ ( k pCnt N ) ) , 1 ) ) )
140	139	adantlr 477	. . . . . . 7 |- ( ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> if ( k e. Prime , ( ( ( A x. B ) /L k ) ^ ( k pCnt N ) ) , 1 ) = ( if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt N ) ) , 1 ) x. if ( k e. Prime , ( ( B /L k ) ^ ( k pCnt N ) ) , 1 ) ) )
141		1t1e1 9060	. . . . . . . . . 10 |- ( 1 x. 1 ) = 1
142	141	eqcomi 2181	. . . . . . . . 9 |- 1 = ( 1 x. 1 )
143		iffalse 3542	. . . . . . . . 9 |- ( -. k e. Prime -> if ( k e. Prime , ( ( ( A x. B ) /L k ) ^ ( k pCnt N ) ) , 1 ) = 1 )
144		iffalse 3542	. . . . . . . . . 10 |- ( -. k e. Prime -> if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt N ) ) , 1 ) = 1 )
145		iffalse 3542	. . . . . . . . . 10 |- ( -. k e. Prime -> if ( k e. Prime , ( ( B /L k ) ^ ( k pCnt N ) ) , 1 ) = 1 )
146	144, 145	oveq12d 5887	. . . . . . . . 9 |- ( -. k e. Prime -> ( if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt N ) ) , 1 ) x. if ( k e. Prime , ( ( B /L k ) ^ ( k pCnt N ) ) , 1 ) ) = ( 1 x. 1 ) )
147	142, 143, 146	3eqtr4a 2236	. . . . . . . 8 |- ( -. k e. Prime -> if ( k e. Prime , ( ( ( A x. B ) /L k ) ^ ( k pCnt N ) ) , 1 ) = ( if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt N ) ) , 1 ) x. if ( k e. Prime , ( ( B /L k ) ^ ( k pCnt N ) ) , 1 ) ) )
148	147	adantl 277	. . . . . . 7 |- ( ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. ( ZZ>= ` 1 ) ) /\ -. k e. Prime ) -> if ( k e. Prime , ( ( ( A x. B ) /L k ) ^ ( k pCnt N ) ) , 1 ) = ( if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt N ) ) , 1 ) x. if ( k e. Prime , ( ( B /L k ) ^ ( k pCnt N ) ) , 1 ) ) )
149	103	adantl 277	. . . . . . . . 9 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. ( ZZ>= ` 1 ) ) -> k e. NN )
150		prmdc 12113	. . . . . . . . 9 |- ( k e. NN -> DECID k e. Prime )
151	149, 150	syl 14	. . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. ( ZZ>= ` 1 ) ) -> DECID k e. Prime )
152		exmiddc 836	. . . . . . . 8 |- (DECID k e. Prime -> ( k e. Prime \/ -. k e. Prime ) )
153	151, 152	syl 14	. . . . . . 7 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. ( ZZ>= ` 1 ) ) -> ( k e. Prime \/ -. k e. Prime ) )
154	140, 148, 153	mpjaodan 798	. . . . . 6 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. ( ZZ>= ` 1 ) ) -> if ( k e. Prime , ( ( ( A x. B ) /L k ) ^ ( k pCnt N ) ) , 1 ) = ( if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt N ) ) , 1 ) x. if ( k e. Prime , ( ( B /L k ) ^ ( k pCnt N ) ) , 1 ) ) )
155		eqid 2177	. . . . . . 7 |- ( n e. NN |-> if ( n e. Prime , ( ( ( A x. B ) /L n ) ^ ( n pCnt N ) ) , 1 ) ) = ( n e. NN |-> if ( n e. Prime , ( ( ( A x. B ) /L n ) ^ ( n pCnt N ) ) , 1 ) )
156		eleq1w 2238	. . . . . . . 8 |- ( n = k -> ( n e. Prime <-> k e. Prime ) )
157		oveq2 5877	. . . . . . . . 9 |- ( n = k -> ( ( A x. B ) /L n ) = ( ( A x. B ) /L k ) )
158		oveq1 5876	. . . . . . . . 9 |- ( n = k -> ( n pCnt N ) = ( k pCnt N ) )
159	157, 158	oveq12d 5887	. . . . . . . 8 |- ( n = k -> ( ( ( A x. B ) /L n ) ^ ( n pCnt N ) ) = ( ( ( A x. B ) /L k ) ^ ( k pCnt N ) ) )
160	156, 159	ifbieq1d 3556	. . . . . . 7 |- ( n = k -> if ( n e. Prime , ( ( ( A x. B ) /L n ) ^ ( n pCnt N ) ) , 1 ) = if ( k e. Prime , ( ( ( A x. B ) /L k ) ^ ( k pCnt N ) ) , 1 ) )
161	37	ad3antrrr 492	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> ( A x. B ) e. ZZ )
162	120	adantl 277	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> k e. ZZ )
163		lgscl 14082	. . . . . . . . . 10 |- ( ( ( A x. B ) e. ZZ /\ k e. ZZ ) -> ( ( A x. B ) /L k ) e. ZZ )
164	161, 162, 163	syl2anc 411	. . . . . . . . 9 |- ( ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> ( ( A x. B ) /L k ) e. ZZ )
165	130	adantlr 477	. . . . . . . . 9 |- ( ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> ( k pCnt N ) e. NN0 )
166		zexpcl 10521	. . . . . . . . 9 |- ( ( ( ( A x. B ) /L k ) e. ZZ /\ ( k pCnt N ) e. NN0 ) -> ( ( ( A x. B ) /L k ) ^ ( k pCnt N ) ) e. ZZ )
167	164, 165, 166	syl2anc 411	. . . . . . . 8 |- ( ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> ( ( ( A x. B ) /L k ) ^ ( k pCnt N ) ) e. ZZ )
168		1zzd 9269	. . . . . . . 8 |- ( ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. ( ZZ>= ` 1 ) ) /\ -. k e. Prime ) -> 1 e. ZZ )
169	167, 168, 151	ifcldadc 3563	. . . . . . 7 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. ( ZZ>= ` 1 ) ) -> if ( k e. Prime , ( ( ( A x. B ) /L k ) ^ ( k pCnt N ) ) , 1 ) e. ZZ )
170	155, 160, 149, 169	fvmptd3 5605	. . . . . 6 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. ( ZZ>= ` 1 ) ) -> ( ( n e. NN |-> if ( n e. Prime , ( ( ( A x. B ) /L n ) ^ ( n pCnt N ) ) , 1 ) ) ` k ) = if ( k e. Prime , ( ( ( A x. B ) /L k ) ^ ( k pCnt N ) ) , 1 ) )
171		oveq2 5877	. . . . . . . . . 10 |- ( n = k -> ( A /L n ) = ( A /L k ) )
172	171, 158	oveq12d 5887	. . . . . . . . 9 |- ( n = k -> ( ( A /L n ) ^ ( n pCnt N ) ) = ( ( A /L k ) ^ ( k pCnt N ) ) )
173	156, 172	ifbieq1d 3556	. . . . . . . 8 |- ( n = k -> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) = if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt N ) ) , 1 ) )
174	122	adantlr 477	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> ( A /L k ) e. ZZ )
175		zexpcl 10521	. . . . . . . . . 10 |- ( ( ( A /L k ) e. ZZ /\ ( k pCnt N ) e. NN0 ) -> ( ( A /L k ) ^ ( k pCnt N ) ) e. ZZ )
176	174, 165, 175	syl2anc 411	. . . . . . . . 9 |- ( ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> ( ( A /L k ) ^ ( k pCnt N ) ) e. ZZ )
177	176, 168, 151	ifcldadc 3563	. . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. ( ZZ>= ` 1 ) ) -> if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt N ) ) , 1 ) e. ZZ )
178	99, 173, 149, 177	fvmptd3 5605	. . . . . . 7 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. ( ZZ>= ` 1 ) ) -> ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ` k ) = if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt N ) ) , 1 ) )
179		oveq2 5877	. . . . . . . . . 10 |- ( n = k -> ( B /L n ) = ( B /L k ) )
180	179, 158	oveq12d 5887	. . . . . . . . 9 |- ( n = k -> ( ( B /L n ) ^ ( n pCnt N ) ) = ( ( B /L k ) ^ ( k pCnt N ) ) )
181	156, 180	ifbieq1d 3556	. . . . . . . 8 |- ( n = k -> if ( n e. Prime , ( ( B /L n ) ^ ( n pCnt N ) ) , 1 ) = if ( k e. Prime , ( ( B /L k ) ^ ( k pCnt N ) ) , 1 ) )
182	125	adantlr 477	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> ( B /L k ) e. ZZ )
183		zexpcl 10521	. . . . . . . . . 10 |- ( ( ( B /L k ) e. ZZ /\ ( k pCnt N ) e. NN0 ) -> ( ( B /L k ) ^ ( k pCnt N ) ) e. ZZ )
184	182, 165, 183	syl2anc 411	. . . . . . . . 9 |- ( ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> ( ( B /L k ) ^ ( k pCnt N ) ) e. ZZ )
185	184, 168, 151	ifcldadc 3563	. . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. ( ZZ>= ` 1 ) ) -> if ( k e. Prime , ( ( B /L k ) ^ ( k pCnt N ) ) , 1 ) e. ZZ )
186	108, 181, 149, 185	fvmptd3 5605	. . . . . . 7 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. ( ZZ>= ` 1 ) ) -> ( ( n e. NN |-> if ( n e. Prime , ( ( B /L n ) ^ ( n pCnt N ) ) , 1 ) ) ` k ) = if ( k e. Prime , ( ( B /L k ) ^ ( k pCnt N ) ) , 1 ) )
187	178, 186	oveq12d 5887	. . . . . 6 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. ( ZZ>= ` 1 ) ) -> ( ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ` k ) x. ( ( n e. NN |-> if ( n e. Prime , ( ( B /L n ) ^ ( n pCnt N ) ) , 1 ) ) ` k ) ) = ( if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt N ) ) , 1 ) x. if ( k e. Prime , ( ( B /L k ) ^ ( k pCnt N ) ) , 1 ) ) )
188	154, 170, 187	3eqtr4d 2220	. . . . 5 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. ( ZZ>= ` 1 ) ) -> ( ( n e. NN |-> if ( n e. Prime , ( ( ( A x. B ) /L n ) ^ ( n pCnt N ) ) , 1 ) ) ` k ) = ( ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ` k ) x. ( ( n e. NN |-> if ( n e. Prime , ( ( B /L n ) ^ ( n pCnt N ) ) , 1 ) ) ` k ) ) )
189	95, 106, 113, 188	prod3fmul 11533	. . . 4 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( ( A x. B ) /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) = ( ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) x. ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( B /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) ) )
190	90, 189	oveq12d 5887	. . 3 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> ( if ( ( N < 0 /\ ( A x. B ) < 0 ) , -u 1 , 1 ) x. ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( ( A x. B ) /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) ) = ( ( if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) x. if ( ( N < 0 /\ B < 0 ) , -u 1 , 1 ) ) x. ( ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) x. ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( B /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) ) ) )
191	37	adantr 276	. . . 4 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> ( A x. B ) e. ZZ )
192	155	lgsval4 14088	. . . 4 |- ( ( ( A x. B ) e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( ( A x. B ) /L N ) = ( if ( ( N < 0 /\ ( A x. B ) < 0 ) , -u 1 , 1 ) x. ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( ( A x. B ) /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) ) )
193	191, 97, 98, 192	syl3anc 1238	. . 3 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> ( ( A x. B ) /L N ) = ( if ( ( N < 0 /\ ( A x. B ) < 0 ) , -u 1 , 1 ) x. ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( ( A x. B ) /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) ) )
194	99	lgsval4 14088	. . . . . 6 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( A /L N ) = ( if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) x. ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) ) )
195	96, 97, 98, 194	syl3anc 1238	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> ( A /L N ) = ( if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) x. ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) ) )
196	108	lgsval4 14088	. . . . . 6 |- ( ( B e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( B /L N ) = ( if ( ( N < 0 /\ B < 0 ) , -u 1 , 1 ) x. ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( B /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) ) )
197	107, 97, 98, 196	syl3anc 1238	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> ( B /L N ) = ( if ( ( N < 0 /\ B < 0 ) , -u 1 , 1 ) x. ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( B /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) ) )
198	195, 197	oveq12d 5887	. . . 4 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> ( ( A /L N ) x. ( B /L N ) ) = ( ( if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) x. ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) ) x. ( if ( ( N < 0 /\ B < 0 ) , -u 1 , 1 ) x. ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( B /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) ) ) )
199		neg1cn 9013	. . . . . . 7 |- -u 1 e. CC
200	199	a1i 9	. . . . . 6 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> -u 1 e. CC )
201		1cnd 7964	. . . . . 6 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> 1 e. CC )
202		0z 9253	. . . . . . . 8 |- 0 e. ZZ
203		zdclt 9319	. . . . . . . 8 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N < 0 )
204	97, 202, 203	sylancl 413	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> DECID N < 0 )
205		zdclt 9319	. . . . . . . 8 |- ( ( A e. ZZ /\ 0 e. ZZ ) -> DECID A < 0 )
206	96, 202, 205	sylancl 413	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> DECID A < 0 )
207		dcan2 934	. . . . . . 7 |- (DECID N < 0 -> (DECID A < 0 -> DECID ( N < 0 /\ A < 0 ) ) )
208	204, 206, 207	sylc 62	. . . . . 6 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> DECID ( N < 0 /\ A < 0 ) )
209	200, 201, 208	ifcldcd 3569	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) e. CC )
210		1zzd 9269	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> 1 e. ZZ )
211	101	ffvelcdmda 5647	. . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. NN ) -> ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ` k ) e. ZZ )
212		zmulcl 9295	. . . . . . . . 9 |- ( ( k e. ZZ /\ v e. ZZ ) -> ( k x. v ) e. ZZ )
213	212	adantl 277	. . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ ( k e. ZZ /\ v e. ZZ ) ) -> ( k x. v ) e. ZZ )
214	94, 210, 211, 213	seqf 10447	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) : NN --> ZZ )
215	214, 93	ffvelcdmd 5648	. . . . . 6 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) e. ZZ )
216	215	zcnd 9365	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) e. CC )
217		neg1z 9274	. . . . . . . 8 |- -u 1 e. ZZ
218	217	a1i 9	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> -u 1 e. ZZ )
219		zdclt 9319	. . . . . . . . 9 |- ( ( B e. ZZ /\ 0 e. ZZ ) -> DECID B < 0 )
220	107, 202, 219	sylancl 413	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> DECID B < 0 )
221		dcan2 934	. . . . . . . 8 |- (DECID N < 0 -> (DECID B < 0 -> DECID ( N < 0 /\ B < 0 ) ) )
222	204, 220, 221	sylc 62	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> DECID ( N < 0 /\ B < 0 ) )
223	218, 210, 222	ifcldcd 3569	. . . . . 6 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> if ( ( N < 0 /\ B < 0 ) , -u 1 , 1 ) e. ZZ )
224	223	zcnd 9365	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> if ( ( N < 0 /\ B < 0 ) , -u 1 , 1 ) e. CC )
225	110	ffvelcdmda 5647	. . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. NN ) -> ( ( n e. NN |-> if ( n e. Prime , ( ( B /L n ) ^ ( n pCnt N ) ) , 1 ) ) ` k ) e. ZZ )
226	94, 210, 225, 213	seqf 10447	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( B /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) : NN --> ZZ )
227	226, 93	ffvelcdmd 5648	. . . . . 6 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( B /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) e. ZZ )
228	227	zcnd 9365	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( B /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) e. CC )
229	209, 216, 224, 228	mul4d 8102	. . . 4 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> ( ( if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) x. ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) ) x. ( if ( ( N < 0 /\ B < 0 ) , -u 1 , 1 ) x. ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( B /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) ) ) = ( ( if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) x. if ( ( N < 0 /\ B < 0 ) , -u 1 , 1 ) ) x. ( ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) x. ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( B /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) ) ) )
230	198, 229	eqtrd 2210	. . 3 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> ( ( A /L N ) x. ( B /L N ) ) = ( ( if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) x. if ( ( N < 0 /\ B < 0 ) , -u 1 , 1 ) ) x. ( ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) x. ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( B /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) ) ) )
231	190, 193, 230	3eqtr4d 2220	. 2 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> ( ( A x. B ) /L N ) = ( ( A /L N ) x. ( B /L N ) ) )
232		zdceq 9317	. . . 4 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
233	91, 202, 232	sylancl 413	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> DECID N = 0 )
234		dcne 2358	. . 3 |- (DECID N = 0 <-> ( N = 0 \/ N =/= 0 ) )
235	233, 234	sylib 122	. 2 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> ( N = 0 \/ N =/= 0 ) )
236	88, 231, 235	mpjaodan 798	1 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> ( ( A x. B ) /L N ) = ( ( A /L N ) x. ( B /L N ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708  DECID wdc 834   /\ w3a 978   = wceq 1353   e. wcel 2148   =/= wne 2347   ifcif 3534   class class class wbr 4000   |-> cmpt 4061   -->wf 5208   ` cfv 5212  (class class class)co 5869   CCcc 7800   RRcr 7801   0cc0 7802   1c1 7803   x. cmul 7807   < clt 7982   <_ cle 7983   -ucneg 8119   NNcn 8908   2c2 8959   NN0cn0 9165   ZZcz 9242   ZZ>=cuz 9517   seqcseq 10431   ^cexp 10505   abscabs 10990   || cdvds 11778   Primecprime 12090   pCnt cpc 12267   /Lclgs 14065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-xor 1376  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-frec 6386  df-1o 6411  df-2o 6412  df-oadd 6415  df-er 6529  df-en 6735  df-dom 6736  df-fin 6737  df-sup 6977  df-inf 6978  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-5 8970  df-6 8971  df-7 8972  df-8 8973  df-9 8974  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-fl 10256  df-mod 10309  df-seqfrec 10432  df-exp 10506  df-ihash 10740  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-clim 11271  df-proddc 11543  df-dvds 11779  df-gcd 11927  df-prm 12091  df-phi 12194  df-pc 12268  df-lgs 14066

This theorem is referenced by:  lgssq  14108  lgsmulsqcoprm  14114  lgsdirnn0  14115