Theorem lgsdir 13536

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 7911	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) -> 1 e. CC )
2		0cnd 7888	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) -> 0 e. CC )
3		zsqcl 10521	. . . . . . . . . 10 |- ( B e. ZZ -> ( B ^ 2 ) e. ZZ )
4	3	3ad2ant2 1009	. . . . . . . . 9 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) -> ( B ^ 2 ) e. ZZ )
5		1z 9213	. . . . . . . . 9 |- 1 e. ZZ
6		zdceq 9262	. . . . . . . . 9 |- ( ( ( B ^ 2 ) e. ZZ /\ 1 e. ZZ ) -> DECID ( B ^ 2 ) = 1 )
7	4, 5, 6	sylancl 410	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) -> DECID ( B ^ 2 ) = 1 )
8	1, 2, 7	ifcldcd 3554	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) -> if ( ( B ^ 2 ) = 1 , 1 , 0 ) e. CC )
9	8	mulid2d 7913	. . . . . 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 484	. . . . 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 3524	. . . . . . 7 |- ( ( A ^ 2 ) = 1 -> if ( ( A ^ 2 ) = 1 , 1 , 0 ) = 1 )
12	11	adantl 275	. . . . . 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 5856	. . . . 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 990	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> A e. ZZ )
15	14	zcnd 9310	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> A e. CC )
16	15	ad2antrr 480	. . . . . . . . 9 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) /\ ( A ^ 2 ) = 1 ) -> A e. CC )
17		simpl2 991	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> B e. ZZ )
18	17	zcnd 9310	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> B e. CC )
19	18	ad2antrr 480	. . . . . . . . 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 10596	. . . . . . . 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 109	. . . . . . . . 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 5856	. . . . . . . 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 10582	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> ( B ^ 2 ) e. CC )
24	23	ad2antrr 480	. . . . . . . . 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 7913	. . . . . . . 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 2202	. . . . . . 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 2174	. . . . . 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 3540	. . . . 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 2208	. . . 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 8286	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) -> ( 0 x. if ( ( B ^ 2 ) = 1 , 1 , 0 ) ) = 0 )
31	30	ad3antrrr 484	. . . . 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 3527	. . . . . . 7 |- ( -. ( A ^ 2 ) = 1 -> if ( ( A ^ 2 ) = 1 , 1 , 0 ) = 0 )
33	32	adantl 275	. . . . . 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 5856	. . . . 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 11749	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ B e. ZZ ) -> A || ( A x. B ) )
36	14, 17, 35	syl2anc 409	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> A || ( A x. B ) )
37	14, 17	zmulcld 9315	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> ( A x. B ) e. ZZ )
38		dvdssq 11960	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ ( A x. B ) e. ZZ ) -> ( A || ( A x. B ) <-> ( A ^ 2 ) || ( ( A x. B ) ^ 2 ) ) )
39	14, 37, 38	syl2anc 409	. . . . . . . . . . 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 146	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> ( A ^ 2 ) || ( ( A x. B ) ^ 2 ) )
41	40	adantr 274	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) -> ( A ^ 2 ) || ( ( A x. B ) ^ 2 ) )
42		breq2 3985	. . . . . . . . 9 |- ( ( ( A x. B ) ^ 2 ) = 1 -> ( ( A ^ 2 ) || ( ( A x. B ) ^ 2 ) <-> ( A ^ 2 ) || 1 ) )
43	41, 42	syl5ibcom 154	. . . . . . . 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 521	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> A =/= 0 )
45	44	neneqd 2356	. . . . . . . . . . . . . . 15 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> -. A = 0 )
46		sqeq0 10514	. . . . . . . . . . . . . . . 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 663	. . . . . . . . . . . . . 14 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> -. ( A ^ 2 ) = 0 )
49		zsqcl2 10528	. . . . . . . . . . . . . . . 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 9112	. . . . . . . . . . . . . . 15 |- ( ( A ^ 2 ) e. NN0 <-> ( ( A ^ 2 ) e. NN \/ ( A ^ 2 ) = 0 ) )
52	50, 51	sylib 121	. . . . . . . . . . . . . 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 1339	. . . . . . . . . . . . 13 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> ( A ^ 2 ) e. NN )
54	53	adantr 274	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) -> ( A ^ 2 ) e. NN )
55	54	nnzd 9308	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) -> ( A ^ 2 ) e. ZZ )
56		1nn 8864	. . . . . . . . . . 11 |- 1 e. NN
57		dvdsle 11778	. . . . . . . . . . 11 |- ( ( ( A ^ 2 ) e. ZZ /\ 1 e. NN ) -> ( ( A ^ 2 ) || 1 -> ( A ^ 2 ) <_ 1 ) )
58	55, 56, 57	sylancl 410	. . . . . . . . . 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 8896	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) -> 1 <_ ( A ^ 2 ) )
60	58, 59	jctird 315	. . . . . . . . 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 8866	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) -> ( A ^ 2 ) e. RR )
62		1re 7894	. . . . . . . . . 10 |- 1 e. RR
63		letri3 7975	. . . . . . . . . 10 |- ( ( ( A ^ 2 ) e. RR /\ 1 e. RR ) -> ( ( A ^ 2 ) = 1 <-> ( ( A ^ 2 ) <_ 1 /\ 1 <_ ( A ^ 2 ) ) ) )
64	61, 62, 63	sylancl 410	. . . . . . . . 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 168	. . . . . . . 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 625	. . . . . 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 3529	. . . . 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 2208	. . . 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 9262	. . . . . 6 |- ( ( ( A ^ 2 ) e. ZZ /\ 1 e. ZZ ) -> DECID ( A ^ 2 ) = 1 )
71	55, 5, 70	sylancl 410	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N = 0 ) -> DECID ( A ^ 2 ) = 1 )
72		exmiddc 826	. . . . 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 788	. . 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 5849	. . . . 5 |- ( N = 0 -> ( A /L N ) = ( A /L 0 ) )
76		lgs0 13514	. . . . . 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 2220	. . . 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 5849	. . . . 5 |- ( N = 0 -> ( B /L N ) = ( B /L 0 ) )
80		lgs0 13514	. . . . . 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 2220	. . . 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 5859	. . 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 5849	. . . 4 |- ( N = 0 -> ( ( A x. B ) /L N ) = ( ( A x. B ) /L 0 ) )
85		lgs0 13514	. . . . 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 2220	. . 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 2209	. 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 13528	. . . . 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 274	. . . 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 992	. . . . . . 7 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> N e. ZZ )
92		nnabscl 11038	. . . . . . 7 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. NN )
93	91, 92	sylan 281	. . . . . 6 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> ( abs ` N ) e. NN )
94		nnuz 9497	. . . . . 6 |- NN = ( ZZ>= ` 1 )
95	93, 94	eleqtrdi 2258	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> ( abs ` N ) e. ( ZZ>= ` 1 ) )
96		simpll1 1026	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> A e. ZZ )
97		simpll3 1028	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> N e. ZZ )
98		simpr 109	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> N =/= 0 )
99		eqid 2165	. . . . . . . . 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 13522	. . . . . . . 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 1228	. . . . . . 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 9498	. . . . . . . 8 |- ( k e. NN <-> k e. ( ZZ>= ` 1 ) )
103	102	biimpri 132	. . . . . . 7 |- ( k e. ( ZZ>= ` 1 ) -> k e. NN )
104		ffvelrn 5617	. . . . . . 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 287	. . . . . 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 9310	. . . . 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 1027	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> B e. ZZ )
108		eqid 2165	. . . . . . . . 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 13522	. . . . . . . 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 1228	. . . . . . 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		ffvelrn 5617	. . . . . . 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 287	. . . . . 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 9310	. . . . 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 274	. . . . . . . . . . . 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 274	. . . . . . . . . . . 12 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. Prime ) -> B e. ZZ )
116		simpr 109	. . . . . . . . . . . 12 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. Prime ) -> k e. Prime )
117		lgsdirprm 13535	. . . . . . . . . . . 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 1228	. . . . . . . . . . 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 5856	. . . . . . . . . 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 12039	. . . . . . . . . . . . 13 |- ( k e. Prime -> k e. ZZ )
121		lgscl 13515	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ k e. ZZ ) -> ( A /L k ) e. ZZ )
122	96, 120, 121	syl2an 287	. . . . . . . . . . . 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 9310	. . . . . . . . . . 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 13515	. . . . . . . . . . . . 13 |- ( ( B e. ZZ /\ k e. ZZ ) -> ( B /L k ) e. ZZ )
125	107, 120, 124	syl2an 287	. . . . . . . . . . . 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 9310	. . . . . . . . . . 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 274	. . . . . . . . . . . 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 274	. . . . . . . . . . . 12 |- ( ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) /\ k e. Prime ) -> N =/= 0 )
129		pczcl 12226	. . . . . . . . . . . 12 |- ( ( k e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( k pCnt N ) e. NN0 )
130	116, 127, 128, 129	syl12anc 1226	. . . . . . . . . . 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 10599	. . . . . . . . . 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 2198	. . . . . . . . 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 3524	. . . . . . . . . 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 275	. . . . . . . . 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 3524	. . . . . . . . . . 11 |- ( k e. Prime -> if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt N ) ) , 1 ) = ( ( A /L k ) ^ ( k pCnt N ) ) )
136		iftrue 3524	. . . . . . . . . . 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 5859	. . . . . . . . . 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 275	. . . . . . . . 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 2208	. . . . . . . 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 469	. . . . . . 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 9005	. . . . . . . . . 10 |- ( 1 x. 1 ) = 1
142	141	eqcomi 2169	. . . . . . . . 9 |- 1 = ( 1 x. 1 )
143		iffalse 3527	. . . . . . . . 9 |- ( -. k e. Prime -> if ( k e. Prime , ( ( ( A x. B ) /L k ) ^ ( k pCnt N ) ) , 1 ) = 1 )
144		iffalse 3527	. . . . . . . . . 10 |- ( -. k e. Prime -> if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt N ) ) , 1 ) = 1 )
145		iffalse 3527	. . . . . . . . . 10 |- ( -. k e. Prime -> if ( k e. Prime , ( ( B /L k ) ^ ( k pCnt N ) ) , 1 ) = 1 )
146	144, 145	oveq12d 5859	. . . . . . . . 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 2224	. . . . . . . 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 275	. . . . . . 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 275	. . . . . . . . 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 12058	. . . . . . . . 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 826	. . . . . . . 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 788	. . . . . 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 2165	. . . . . . 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 2226	. . . . . . . 8 |- ( n = k -> ( n e. Prime <-> k e. Prime ) )
157		oveq2 5849	. . . . . . . . 9 |- ( n = k -> ( ( A x. B ) /L n ) = ( ( A x. B ) /L k ) )
158		oveq1 5848	. . . . . . . . 9 |- ( n = k -> ( n pCnt N ) = ( k pCnt N ) )
159	157, 158	oveq12d 5859	. . . . . . . 8 |- ( n = k -> ( ( ( A x. B ) /L n ) ^ ( n pCnt N ) ) = ( ( ( A x. B ) /L k ) ^ ( k pCnt N ) ) )
160	156, 159	ifbieq1d 3541	. . . . . . 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 484	. . . . . . . . . 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 275	. . . . . . . . . 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 13515	. . . . . . . . . 10 |- ( ( ( A x. B ) e. ZZ /\ k e. ZZ ) -> ( ( A x. B ) /L k ) e. ZZ )
164	161, 162, 163	syl2anc 409	. . . . . . . . 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 469	. . . . . . . . 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 10466	. . . . . . . . 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 409	. . . . . . . 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 9214	. . . . . . . 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 3548	. . . . . . 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 5578	. . . . . 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 5849	. . . . . . . . . 10 |- ( n = k -> ( A /L n ) = ( A /L k ) )
172	171, 158	oveq12d 5859	. . . . . . . . 9 |- ( n = k -> ( ( A /L n ) ^ ( n pCnt N ) ) = ( ( A /L k ) ^ ( k pCnt N ) ) )
173	156, 172	ifbieq1d 3541	. . . . . . . 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 469	. . . . . . . . . 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 10466	. . . . . . . . . 10 |- ( ( ( A /L k ) e. ZZ /\ ( k pCnt N ) e. NN0 ) -> ( ( A /L k ) ^ ( k pCnt N ) ) e. ZZ )
176	174, 165, 175	syl2anc 409	. . . . . . . . 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 3548	. . . . . . . 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 5578	. . . . . . 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 5849	. . . . . . . . . 10 |- ( n = k -> ( B /L n ) = ( B /L k ) )
180	179, 158	oveq12d 5859	. . . . . . . . 9 |- ( n = k -> ( ( B /L n ) ^ ( n pCnt N ) ) = ( ( B /L k ) ^ ( k pCnt N ) ) )
181	156, 180	ifbieq1d 3541	. . . . . . . 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 469	. . . . . . . . . 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 10466	. . . . . . . . . 10 |- ( ( ( B /L k ) e. ZZ /\ ( k pCnt N ) e. NN0 ) -> ( ( B /L k ) ^ ( k pCnt N ) ) e. ZZ )
184	182, 165, 183	syl2anc 409	. . . . . . . . 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 3548	. . . . . . . 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 5578	. . . . . . 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 5859	. . . . . 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 2208	. . . . 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 11478	. . . 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 5859	. . 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 274	. . . 4 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> ( A x. B ) e. ZZ )
192	155	lgsval4 13521	. . . 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 1228	. . 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 13521	. . . . . 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 1228	. . . . 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 13521	. . . . . 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 1228	. . . . 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 5859	. . . 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 8958	. . . . . . 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 7911	. . . . . 6 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> 1 e. CC )
202		0z 9198	. . . . . . . 8 |- 0 e. ZZ
203		zdclt 9264	. . . . . . . 8 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N < 0 )
204	97, 202, 203	sylancl 410	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> DECID N < 0 )
205		zdclt 9264	. . . . . . . 8 |- ( ( A e. ZZ /\ 0 e. ZZ ) -> DECID A < 0 )
206	96, 202, 205	sylancl 410	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> DECID A < 0 )
207		dcan2 924	. . . . . . 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 3554	. . . . 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 9214	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> 1 e. ZZ )
211	101	ffvelrnda 5619	. . . . . . . 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 9240	. . . . . . . . 9 |- ( ( k e. ZZ /\ v e. ZZ ) -> ( k x. v ) e. ZZ )
213	212	adantl 275	. . . . . . . 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 10392	. . . . . . 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	ffvelrnd 5620	. . . . . 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 9310	. . . . 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 9219	. . . . . . . 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 9264	. . . . . . . . 9 |- ( ( B e. ZZ /\ 0 e. ZZ ) -> DECID B < 0 )
220	107, 202, 219	sylancl 410	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) /\ N =/= 0 ) -> DECID B < 0 )
221		dcan2 924	. . . . . . . 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 3554	. . . . . 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 9310	. . . . 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	ffvelrnda 5619	. . . . . . . 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 10392	. . . . . . 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	ffvelrnd 5620	. . . . . 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 9310	. . . . 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 8049	. . . 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 2198	. . 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 2208	. 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 9262	. . . 4 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
233	91, 202, 232	sylancl 410	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> DECID N = 0 )
234		dcne 2346	. . 3 |- (DECID N = 0 <-> ( N = 0 \/ N =/= 0 ) )
235	233, 234	sylib 121	. 2 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> ( N = 0 \/ N =/= 0 ) )
236	88, 231, 235	mpjaodan 788	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 103   <-> wb 104   \/ wo 698  DECID wdc 824   /\ w3a 968   = wceq 1343   e. wcel 2136   =/= wne 2335   ifcif 3519   class class class wbr 3981   |-> cmpt 4042   -->wf 5183   ` cfv 5187  (class class class)co 5841   CCcc 7747   RRcr 7748   0cc0 7749   1c1 7750   x. cmul 7754   < clt 7929   <_ cle 7930   -ucneg 8066   NNcn 8853   2c2 8904   NN0cn0 9110   ZZcz 9187   ZZ>=cuz 9462   seqcseq 10376   ^cexp 10450   abscabs 10935   || cdvds 11723   Primecprime 12035   pCnt cpc 12212   /Lclgs 13498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4096  ax-sep 4099  ax-nul 4107  ax-pow 4152  ax-pr 4186  ax-un 4410  ax-setind 4513  ax-iinf 4564  ax-cnex 7840  ax-resscn 7841  ax-1cn 7842  ax-1re 7843  ax-icn 7844  ax-addcl 7845  ax-addrcl 7846  ax-mulcl 7847  ax-mulrcl 7848  ax-addcom 7849  ax-mulcom 7850  ax-addass 7851  ax-mulass 7852  ax-distr 7853  ax-i2m1 7854  ax-0lt1 7855  ax-1rid 7856  ax-0id 7857  ax-rnegex 7858  ax-precex 7859  ax-cnre 7860  ax-pre-ltirr 7861  ax-pre-ltwlin 7862  ax-pre-lttrn 7863  ax-pre-apti 7864  ax-pre-ltadd 7865  ax-pre-mulgt0 7866  ax-pre-mulext 7867  ax-arch 7868  ax-caucvg 7869

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-xor 1366  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ne 2336  df-nel 2431  df-ral 2448  df-rex 2449  df-reu 2450  df-rmo 2451  df-rab 2452  df-v 2727  df-sbc 2951  df-csb 3045  df-dif 3117  df-un 3119  df-in 3121  df-ss 3128  df-nul 3409  df-if 3520  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-int 3824  df-iun 3867  df-br 3982  df-opab 4043  df-mpt 4044  df-tr 4080  df-id 4270  df-po 4273  df-iso 4274  df-iord 4343  df-on 4345  df-ilim 4346  df-suc 4348  df-iom 4567  df-xp 4609  df-rel 4610  df-cnv 4611  df-co 4612  df-dm 4613  df-rn 4614  df-res 4615  df-ima 4616  df-iota 5152  df-fun 5189  df-fn 5190  df-f 5191  df-f1 5192  df-fo 5193  df-f1o 5194  df-fv 5195  df-isom 5196  df-riota 5797  df-ov 5844  df-oprab 5845  df-mpo 5846  df-1st 6105  df-2nd 6106  df-recs 6269  df-irdg 6334  df-frec 6355  df-1o 6380  df-2o 6381  df-oadd 6384  df-er 6497  df-en 6703  df-dom 6704  df-fin 6705  df-sup 6945  df-inf 6946  df-pnf 7931  df-mnf 7932  df-xr 7933  df-ltxr 7934  df-le 7935  df-sub 8067  df-neg 8068  df-reap 8469  df-ap 8476  df-div 8565  df-inn 8854  df-2 8912  df-3 8913  df-4 8914  df-5 8915  df-6 8916  df-7 8917  df-8 8918  df-9 8919  df-n0 9111  df-z 9188  df-uz 9463  df-q 9554  df-rp 9586  df-fz 9941  df-fzo 10074  df-fl 10201  df-mod 10254  df-seqfrec 10377  df-exp 10451  df-ihash 10685  df-cj 10780  df-re 10781  df-im 10782  df-rsqrt 10936  df-abs 10937  df-clim 11216  df-proddc 11488  df-dvds 11724  df-gcd 11872  df-prm 12036  df-phi 12139  df-pc 12213  df-lgs 13499

This theorem is referenced by:  lgssq  13541  lgsmulsqcoprm  13547  lgsdirnn0  13548