Theorem lgsneg 16023

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 3631	. . . . . . . . 9 |- ( A < 0 -> if ( A < 0 , -u 1 , 1 ) = -u 1 )
2	1	adantl 277	. . . . . . . 8 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> if ( A < 0 , -u 1 , 1 ) = -u 1 )
3	2	oveq1d 6073	. . . . . . 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 1028	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> N e. ZZ )
5		0z 9605	. . . . . . . . . . 11 |- 0 e. ZZ
6		zdclt 9672	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N < 0 )
7	5, 6	mpan2 425	. . . . . . . . . 10 |- ( N e. ZZ -> DECID N < 0 )
8		oveq2 6066	. . . . . . . . . . . 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 9467	. . . . . . . . . . . 12 |- ( -u 1 x. -u 1 ) = 1
10	8, 9	eqtrdi 2283	. . . . . . . . . . 11 |- ( if ( N < 0 , -u 1 , 1 ) = -u 1 -> ( -u 1 x. if ( N < 0 , -u 1 , 1 ) ) = 1 )
11		oveq2 6066	. . . . . . . . . . . 12 |- ( if ( N < 0 , -u 1 , 1 ) = 1 -> ( -u 1 x. if ( N < 0 , -u 1 , 1 ) ) = ( -u 1 x. 1 ) )
12		ax-1cn 8236	. . . . . . . . . . . . 13 |- 1 e. CC
13	12	mulm1i 8693	. . . . . . . . . . . 12 |- ( -u 1 x. 1 ) = -u 1
14	11, 13	eqtrdi 2283	. . . . . . . . . . 11 |- ( if ( N < 0 , -u 1 , 1 ) = 1 -> ( -u 1 x. if ( N < 0 , -u 1 , 1 ) ) = -u 1 )
15	10, 14	ifsbdc 3639	. . . . . . . . . 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 110	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> A < 0 )
19	18	biantrud 304	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> ( N < 0 <-> ( N < 0 /\ A < 0 ) ) )
20	19	ifbid 3648	. . . . . . . . 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 6074	. . . . . . . 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 1029	. . . . . . . . . . . . 13 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> N =/= 0 )
23	22	necomd 2500	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> 0 =/= N )
24		zltlen 9674	. . . . . . . . . . . . 13 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> ( N < 0 <-> ( N <_ 0 /\ 0 =/= N ) ) )
25	4, 5, 24	sylancl 413	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> ( N < 0 <-> ( N <_ 0 /\ 0 =/= N ) ) )
26	23, 25	mpbiran2d 442	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> ( N < 0 <-> N <_ 0 ) )
27	4	zred 9718	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> N e. RR )
28	27	le0neg1d 8808	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> ( N <_ 0 <-> 0 <_ -u N ) )
29		0re 8290	. . . . . . . . . . . 12 |- 0 e. RR
30	27	renegcld 8670	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> -u N e. RR )
31		lenlt 8365	. . . . . . . . . . . 12 |- ( ( 0 e. RR /\ -u N e. RR ) -> ( 0 <_ -u N <-> -. -u N < 0 ) )
32	29, 30, 31	sylancr 414	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> ( 0 <_ -u N <-> -. -u N < 0 ) )
33	26, 28, 32	3bitrd 214	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> ( N < 0 <-> -. -u N < 0 ) )
34	33	ifbid 3648	. . . . . . . . 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 9625	. . . . . . . . . . . 12 |- ( N e. ZZ -> -u N e. ZZ )
36		zdclt 9672	. . . . . . . . . . . 12 |- ( ( -u N e. ZZ /\ 0 e. ZZ ) -> DECID -u N < 0 )
37	35, 5, 36	sylancl 413	. . . . . . . . . . 11 |- ( N e. ZZ -> DECID -u N < 0 )
38		ifnotdc 3665	. . . . . . . . . . 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 2267	. . . . . . . 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 2275	. . . . . . 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 304	. . . . . . . 8 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> ( -u N < 0 <-> ( -u N < 0 /\ A < 0 ) ) )
44	43	ifbid 3648	. . . . . . 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 2271	. . . . . 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 9407	. . . . . . 7 |- ( 1 x. 1 ) = 1
47		iffalse 3634	. . . . . . . . 9 |- ( -. A < 0 -> if ( A < 0 , -u 1 , 1 ) = 1 )
48	47	adantl 277	. . . . . . . 8 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ -. A < 0 ) -> if ( A < 0 , -u 1 , 1 ) = 1 )
49		simpr 110	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ -. A < 0 ) -> -. A < 0 )
50	49	intnand 939	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ -. A < 0 ) -> -. ( N < 0 /\ A < 0 ) )
51	50	iffalsed 3636	. . . . . . . 8 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ -. A < 0 ) -> if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) = 1 )
52	48, 51	oveq12d 6076	. . . . . . 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 939	. . . . . . . 8 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ -. A < 0 ) -> -. ( -u N < 0 /\ A < 0 ) )
54	53	iffalsed 3636	. . . . . . 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 2293	. . . . . 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 1024	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> A e. ZZ )
57		zdclt 9672	. . . . . . . 8 |- ( ( A e. ZZ /\ 0 e. ZZ ) -> DECID A < 0 )
58	56, 5, 57	sylancl 413	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> DECID A < 0 )
59		exmiddc 844	. . . . . . 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 806	. . . . 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 2240	. . . 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 110	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. Prime ) -> n e. Prime )
64		simpl2 1028	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. Prime ) -> N e. ZZ )
65		zq 9976	. . . . . . . . . . . 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 13048	. . . . . . . . . . 11 |- ( ( n e. Prime /\ N e. QQ ) -> ( n pCnt -u N ) = ( n pCnt N ) )
68	63, 66, 67	syl2anc 411	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. Prime ) -> ( n pCnt -u N ) = ( n pCnt N ) )
69	68	oveq2d 6074	. . . . . . . . 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 477	. . . . . . . 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 12852	. . . . . . . . 9 |- ( n e. NN -> DECID n e. Prime )
72	71	adantl 277	. . . . . . . 8 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. NN ) -> DECID n e. Prime )
73	70, 72	ifeq1dadc 3657	. . . . . . 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 4205	. . . . . 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 10841	. . . . 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 9599	. . . . . . 7 |- ( N e. ZZ -> N e. CC )
77	76	3ad2ant2 1046	. . . . . 6 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> N e. CC )
78	77	absnegd 11899	. . . . 5 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( abs ` -u N ) = ( abs ` N ) )
79	75, 78	fveq12d 5682	. . . 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 6076	. . 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 9359	. . . . . 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 3664	. . . 4 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> if ( A < 0 , -u 1 , 1 ) e. CC )
85	7	3ad2ant2 1046	. . . . . 6 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> DECID N < 0 )
86		dcan2 943	. . . . . 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 3664	. . . 4 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) e. CC )
89		nnuz 9908	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
90		1zzd 9621	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> 1 e. ZZ )
91		eqid 2234	. . . . . . . . 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 16020	. . . . . . . 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	ffvelcdmda 5817	. . . . . . 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 9648	. . . . . . . 8 |- ( ( x e. ZZ /\ y e. ZZ ) -> ( x x. y ) e. ZZ )
95	94	adantl 277	. . . . . . 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 10850	. . . . . 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 11810	. . . . . . 7 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. NN )
98	97	3adant1 1042	. . . . . 6 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. NN )
99	96, 98	ffvelcdmd 5818	. . . . 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 9719	. . . 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 8313	. . 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 2267	. 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 1046	. . 3 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> -u N e. ZZ )
104		simp3 1026	. . . 4 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> N =/= 0 )
105	77, 104	negne0d 8598	. . 3 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> -u N =/= 0 )
106		eqid 2234	. . . 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 16019	. . 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 1274	. 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 16019	. . 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 6074	. 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 2277	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 104   <-> wb 105   \/ wo 716  DECID wdc 842   /\ w3a 1005   = wceq 1398   e. wcel 2205   =/= wne 2414   ifcif 3624   class class class wbr 4114   |-> cmpt 4176   ` cfv 5357  (class class class)co 6058   CCcc 8141   RRcr 8142   0cc0 8143   1c1 8144   x. cmul 8148   < clt 8324   <_ cle 8325   -ucneg 8461   NNcn 9254   ZZcz 9594   QQcq 9969   seqcseq 10833   ^cexp 10924   abscabs 11707   Primecprime 12829   pCnt cpc 13007   /Lclgs 15996

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-xor 1421  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-fl 10654  df-mod 10709  df-seqfrec 10834  df-exp 10925  df-ihash 11164  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-clim 11989  df-proddc 12262  df-dvds 12499  df-gcd 12675  df-prm 12830  df-phi 12933  df-pc 13008  df-lgs 15997

This theorem is referenced by:  lgsneg1  16024