Theorem lgsneg 13640

Description: The Legendre symbol is either even or odd under negation with respect to the second parameter according to the sign of the first. (Contributed by Mario Carneiro, 4-Feb-2015.)

Assertion

Ref	Expression
lgsneg	|- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( A /L -u N ) = ( if ( A < 0 , -u 1 , 1 ) x. ( A /L N ) ) )

Proof of Theorem lgsneg

Dummy variables n x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iftrue 3530	. . . . . . . . 9 |- ( A < 0 -> if ( A < 0 , -u 1 , 1 ) = -u 1 )
2	1	adantl 275	. . . . . . . 8 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> if ( A < 0 , -u 1 , 1 ) = -u 1 )
3	2	oveq1d 5865	. . . . . . 7 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> ( if ( A < 0 , -u 1 , 1 ) x. if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) ) = ( -u 1 x. if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) ) )
4		simpl2 996	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> N e. ZZ )
5		0z 9210	. . . . . . . . . . 11 |- 0 e. ZZ
6		zdclt 9276	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N < 0 )
7	5, 6	mpan2 423	. . . . . . . . . 10 |- ( N e. ZZ -> DECID N < 0 )
8		oveq2 5858	. . . . . . . . . . . 12 |- ( if ( N < 0 , -u 1 , 1 ) = -u 1 -> ( -u 1 x. if ( N < 0 , -u 1 , 1 ) ) = ( -u 1 x. -u 1 ) )
9		neg1mulneg1e1 9077	. . . . . . . . . . . 12 |- ( -u 1 x. -u 1 ) = 1
10	8, 9	eqtrdi 2219	. . . . . . . . . . 11 |- ( if ( N < 0 , -u 1 , 1 ) = -u 1 -> ( -u 1 x. if ( N < 0 , -u 1 , 1 ) ) = 1 )
11		oveq2 5858	. . . . . . . . . . . 12 |- ( if ( N < 0 , -u 1 , 1 ) = 1 -> ( -u 1 x. if ( N < 0 , -u 1 , 1 ) ) = ( -u 1 x. 1 ) )
12		ax-1cn 7854	. . . . . . . . . . . . 13 |- 1 e. CC
13	12	mulm1i 8309	. . . . . . . . . . . 12 |- ( -u 1 x. 1 ) = -u 1
14	11, 13	eqtrdi 2219	. . . . . . . . . . 11 |- ( if ( N < 0 , -u 1 , 1 ) = 1 -> ( -u 1 x. if ( N < 0 , -u 1 , 1 ) ) = -u 1 )
15	10, 14	ifsbdc 3537	. . . . . . . . . 10 |- (DECID N < 0 -> ( -u 1 x. if ( N < 0 , -u 1 , 1 ) ) = if ( N < 0 , 1 , -u 1 ) )
16	7, 15	syl 14	. . . . . . . . 9 |- ( N e. ZZ -> ( -u 1 x. if ( N < 0 , -u 1 , 1 ) ) = if ( N < 0 , 1 , -u 1 ) )
17	4, 16	syl 14	. . . . . . . 8 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> ( -u 1 x. if ( N < 0 , -u 1 , 1 ) ) = if ( N < 0 , 1 , -u 1 ) )
18		simpr 109	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> A < 0 )
19	18	biantrud 302	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> ( N < 0 <-> ( N < 0 /\ A < 0 ) ) )
20	19	ifbid 3546	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> if ( N < 0 , -u 1 , 1 ) = if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) )
21	20	oveq2d 5866	. . . . . . . 8 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> ( -u 1 x. if ( N < 0 , -u 1 , 1 ) ) = ( -u 1 x. if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) ) )
22		simpl3 997	. . . . . . . . . . . . 13 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> N =/= 0 )
23	22	necomd 2426	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> 0 =/= N )
24		zltlen 9277	. . . . . . . . . . . . 13 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> ( N < 0 <-> ( N <_ 0 /\ 0 =/= N ) ) )
25	4, 5, 24	sylancl 411	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> ( N < 0 <-> ( N <_ 0 /\ 0 =/= N ) ) )
26	23, 25	mpbiran2d 440	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> ( N < 0 <-> N <_ 0 ) )
27	4	zred 9321	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> N e. RR )
28	27	le0neg1d 8423	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> ( N <_ 0 <-> 0 <_ -u N ) )
29		0re 7907	. . . . . . . . . . . 12 |- 0 e. RR
30	27	renegcld 8286	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> -u N e. RR )
31		lenlt 7982	. . . . . . . . . . . 12 |- ( ( 0 e. RR /\ -u N e. RR ) -> ( 0 <_ -u N <-> -. -u N < 0 ) )
32	29, 30, 31	sylancr 412	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> ( 0 <_ -u N <-> -. -u N < 0 ) )
33	26, 28, 32	3bitrd 213	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> ( N < 0 <-> -. -u N < 0 ) )
34	33	ifbid 3546	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> if ( N < 0 , 1 , -u 1 ) = if ( -. -u N < 0 , 1 , -u 1 ) )
35		znegcl 9230	. . . . . . . . . . . 12 |- ( N e. ZZ -> -u N e. ZZ )
36		zdclt 9276	. . . . . . . . . . . 12 |- ( ( -u N e. ZZ /\ 0 e. ZZ ) -> DECID -u N < 0 )
37	35, 5, 36	sylancl 411	. . . . . . . . . . 11 |- ( N e. ZZ -> DECID -u N < 0 )
38		ifnotdc 3561	. . . . . . . . . . 11 |- (DECID -u N < 0 -> if ( -. -u N < 0 , 1 , -u 1 ) = if ( -u N < 0 , -u 1 , 1 ) )
39	37, 38	syl 14	. . . . . . . . . 10 |- ( N e. ZZ -> if ( -. -u N < 0 , 1 , -u 1 ) = if ( -u N < 0 , -u 1 , 1 ) )
40	4, 39	syl 14	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> if ( -. -u N < 0 , 1 , -u 1 ) = if ( -u N < 0 , -u 1 , 1 ) )
41	34, 40	eqtrd 2203	. . . . . . . 8 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> if ( N < 0 , 1 , -u 1 ) = if ( -u N < 0 , -u 1 , 1 ) )
42	17, 21, 41	3eqtr3d 2211	. . . . . . 7 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> ( -u 1 x. if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) ) = if ( -u N < 0 , -u 1 , 1 ) )
43	18	biantrud 302	. . . . . . . 8 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> ( -u N < 0 <-> ( -u N < 0 /\ A < 0 ) ) )
44	43	ifbid 3546	. . . . . . 7 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> if ( -u N < 0 , -u 1 , 1 ) = if ( ( -u N < 0 /\ A < 0 ) , -u 1 , 1 ) )
45	3, 42, 44	3eqtrd 2207	. . . . . 6 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> ( if ( A < 0 , -u 1 , 1 ) x. if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) ) = if ( ( -u N < 0 /\ A < 0 ) , -u 1 , 1 ) )
46		1t1e1 9017	. . . . . . 7 |- ( 1 x. 1 ) = 1
47		iffalse 3533	. . . . . . . . 9 |- ( -. A < 0 -> if ( A < 0 , -u 1 , 1 ) = 1 )
48	47	adantl 275	. . . . . . . 8 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ -. A < 0 ) -> if ( A < 0 , -u 1 , 1 ) = 1 )
49		simpr 109	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ -. A < 0 ) -> -. A < 0 )
50	49	intnand 926	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ -. A < 0 ) -> -. ( N < 0 /\ A < 0 ) )
51	50	iffalsed 3535	. . . . . . . 8 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ -. A < 0 ) -> if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) = 1 )
52	48, 51	oveq12d 5868	. . . . . . 7 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ -. A < 0 ) -> ( if ( A < 0 , -u 1 , 1 ) x. if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) ) = ( 1 x. 1 ) )
53	49	intnand 926	. . . . . . . 8 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ -. A < 0 ) -> -. ( -u N < 0 /\ A < 0 ) )
54	53	iffalsed 3535	. . . . . . 7 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ -. A < 0 ) -> if ( ( -u N < 0 /\ A < 0 ) , -u 1 , 1 ) = 1 )
55	46, 52, 54	3eqtr4a 2229	. . . . . 6 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ -. A < 0 ) -> ( if ( A < 0 , -u 1 , 1 ) x. if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) ) = if ( ( -u N < 0 /\ A < 0 ) , -u 1 , 1 ) )
56		simp1 992	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> A e. ZZ )
57		zdclt 9276	. . . . . . . 8 |- ( ( A e. ZZ /\ 0 e. ZZ ) -> DECID A < 0 )
58	56, 5, 57	sylancl 411	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> DECID A < 0 )
59		exmiddc 831	. . . . . . 7 |- (DECID A < 0 -> ( A < 0 \/ -. A < 0 ) )
60	58, 59	syl 14	. . . . . 6 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( A < 0 \/ -. A < 0 ) )
61	45, 55, 60	mpjaodan 793	. . . . 5 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( if ( A < 0 , -u 1 , 1 ) x. if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) ) = if ( ( -u N < 0 /\ A < 0 ) , -u 1 , 1 ) )
62	61	eqcomd 2176	. . . 4 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> if ( ( -u N < 0 /\ A < 0 ) , -u 1 , 1 ) = ( if ( A < 0 , -u 1 , 1 ) x. if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) ) )
63		simpr 109	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. Prime ) -> n e. Prime )
64		simpl2 996	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. Prime ) -> N e. ZZ )
65		zq 9572	. . . . . . . . . . . 12 |- ( N e. ZZ -> N e. QQ )
66	64, 65	syl 14	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. Prime ) -> N e. QQ )
67		pcneg 12265	. . . . . . . . . . 11 |- ( ( n e. Prime /\ N e. QQ ) -> ( n pCnt -u N ) = ( n pCnt N ) )
68	63, 66, 67	syl2anc 409	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. Prime ) -> ( n pCnt -u N ) = ( n pCnt N ) )
69	68	oveq2d 5866	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. Prime ) -> ( ( A /L n ) ^ ( n pCnt -u N ) ) = ( ( A /L n ) ^ ( n pCnt N ) ) )
70	69	adantlr 474	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) /\ n e. Prime ) -> ( ( A /L n ) ^ ( n pCnt -u N ) ) = ( ( A /L n ) ^ ( n pCnt N ) ) )
71		prmdc 12071	. . . . . . . . 9 |- ( n e. NN -> DECID n e. Prime )
72	71	adantl 275	. . . . . . . 8 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) -> DECID n e. Prime )
73	70, 72	ifeq1dadc 3555	. . . . . . 7 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) -> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt -u N ) ) , 1 ) = if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) )
74	73	mpteq2dva 4077	. . . . . 6 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt -u N ) ) , 1 ) ) = ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) )
75	74	seqeq3d 10396	. . . . 5 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt -u N ) ) , 1 ) ) ) = seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) )
76		zcn 9204	. . . . . . 7 |- ( N e. ZZ -> N e. CC )
77	76	3ad2ant2 1014	. . . . . 6 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> N e. CC )
78	77	absnegd 11140	. . . . 5 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( abs ` -u N ) = ( abs ` N ) )
79	75, 78	fveq12d 5501	. . . 4 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt -u N ) ) , 1 ) ) ) ` ( abs ` -u N ) ) = ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) )
80	62, 79	oveq12d 5868	. . 3 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( if ( ( -u N < 0 /\ A < 0 ) , -u 1 , 1 ) x. ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt -u N ) ) , 1 ) ) ) ` ( abs ` -u N ) ) ) = ( ( if ( A < 0 , -u 1 , 1 ) x. 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 ) ) ) )
81		neg1cn 8970	. . . . . 6 |- -u 1 e. CC
82	81	a1i 9	. . . . 5 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> -u 1 e. CC )
83	12	a1i 9	. . . . 5 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> 1 e. CC )
84	82, 83, 58	ifcldcd 3560	. . . 4 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> if ( A < 0 , -u 1 , 1 ) e. CC )
85	7	3ad2ant2 1014	. . . . . 6 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> DECID N < 0 )
86		dcan2 929	. . . . . 6 |- (DECID N < 0 -> (DECID A < 0 -> DECID ( N < 0 /\ A < 0 ) ) )
87	85, 58, 86	sylc 62	. . . . 5 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> DECID ( N < 0 /\ A < 0 ) )
88	82, 83, 87	ifcldcd 3560	. . . 4 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) e. CC )
89		nnuz 9509	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
90		1zzd 9226	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> 1 e. ZZ )
91		eqid 2170	. . . . . . . . 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 ) )
92	91	lgsfcl3 13637	. . . . . . . 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 )
93	92	ffvelrnda 5628	. . . . . . 7 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ x e. NN ) -> ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ` x ) e. ZZ )
94		zmulcl 9252	. . . . . . . 8 |- ( ( x e. ZZ /\ y e. ZZ ) -> ( x x. y ) e. ZZ )
95	94	adantl 275	. . . . . . 7 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( x x. y ) e. ZZ )
96	89, 90, 93, 95	seqf 10404	. . . . . 6 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) : NN --> ZZ )
97		nnabscl 11051	. . . . . . 7 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. NN )
98	97	3adant1 1010	. . . . . 6 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. NN )
99	96, 98	ffvelrnd 5629	. . . . 5 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) e. ZZ )
100	99	zcnd 9322	. . . 4 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) e. CC )
101	84, 88, 100	mulassd 7930	. . 3 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( ( if ( A < 0 , -u 1 , 1 ) x. 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 ) ) ) = ( if ( A < 0 , -u 1 , 1 ) x. ( 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 ) ) ) ) )
102	80, 101	eqtrd 2203	. 2 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( if ( ( -u N < 0 /\ A < 0 ) , -u 1 , 1 ) x. ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt -u N ) ) , 1 ) ) ) ` ( abs ` -u N ) ) ) = ( if ( A < 0 , -u 1 , 1 ) x. ( 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 ) ) ) ) )
103	35	3ad2ant2 1014	. . 3 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> -u N e. ZZ )
104		simp3 994	. . . 4 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> N =/= 0 )
105	77, 104	negne0d 8215	. . 3 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> -u N =/= 0 )
106		eqid 2170	. . . 4 |- ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt -u N ) ) , 1 ) ) = ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt -u N ) ) , 1 ) )
107	106	lgsval4 13636	. . 3 |- ( ( A e. ZZ /\ -u N e. ZZ /\ -u N =/= 0 ) -> ( A /L -u N ) = ( if ( ( -u N < 0 /\ A < 0 ) , -u 1 , 1 ) x. ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt -u N ) ) , 1 ) ) ) ` ( abs ` -u N ) ) ) )
108	56, 103, 105, 107	syl3anc 1233	. 2 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( A /L -u N ) = ( if ( ( -u N < 0 /\ A < 0 ) , -u 1 , 1 ) x. ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt -u N ) ) , 1 ) ) ) ` ( abs ` -u N ) ) ) )
109	91	lgsval4 13636	. . 3 |- ( ( 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 ) ) ) )
110	109	oveq2d 5866	. 2 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( if ( A < 0 , -u 1 , 1 ) x. ( A /L N ) ) = ( if ( A < 0 , -u 1 , 1 ) x. ( 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 ) ) ) ) )
111	102, 108, 110	3eqtr4d 2213	1 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( A /L -u N ) = ( if ( A < 0 , -u 1 , 1 ) x. ( A /L N ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 703  DECID wdc 829   /\ w3a 973   = wceq 1348   e. wcel 2141   =/= wne 2340   ifcif 3525   class class class wbr 3987   |-> cmpt 4048   ` cfv 5196  (class class class)co 5850   CCcc 7759   RRcr 7760   0cc0 7761   1c1 7762   x. cmul 7766   < clt 7941   <_ cle 7942   -ucneg 8078   NNcn 8865   ZZcz 9199   QQcq 9565   seqcseq 10388   ^cexp 10462   abscabs 10948   Primecprime 12048   pCnt cpc 12225   /Lclgs 13613

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-mulrcl 7860  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-mulass 7864  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-1rid 7868  ax-0id 7869  ax-rnegex 7870  ax-precex 7871  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-apti 7876  ax-pre-ltadd 7877  ax-pre-mulgt0 7878  ax-pre-mulext 7879  ax-arch 7880  ax-caucvg 7881

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-xor 1371  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-isom 5205  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-irdg 6346  df-frec 6367  df-1o 6392  df-2o 6393  df-oadd 6396  df-er 6509  df-en 6715  df-dom 6716  df-fin 6717  df-sup 6957  df-inf 6958  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-reap 8481  df-ap 8488  df-div 8577  df-inn 8866  df-2 8924  df-3 8925  df-4 8926  df-5 8927  df-6 8928  df-7 8929  df-8 8930  df-n0 9123  df-z 9200  df-uz 9475  df-q 9566  df-rp 9598  df-fz 9953  df-fzo 10086  df-fl 10213  df-mod 10266  df-seqfrec 10389  df-exp 10463  df-ihash 10697  df-cj 10793  df-re 10794  df-im 10795  df-rsqrt 10949  df-abs 10950  df-clim 11229  df-proddc 11501  df-dvds 11737  df-gcd 11885  df-prm 12049  df-phi 12152  df-pc 12226  df-lgs 13614

This theorem is referenced by:  lgsneg1  13641