Theorem lgsneg 16057

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 3642	. . . . . . . . 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 6090	. . . . . . 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 1032	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> N e. ZZ )
5		0z 9634	. . . . . . . . . . 11 |- 0 e. ZZ
6		zdclt 9701	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N < 0 )
7	5, 6	mpan2 429	. . . . . . . . . 10 |- ( N e. ZZ -> DECID N < 0 )
8		oveq2 6083	. . . . . . . . . . . 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 9496	. . . . . . . . . . . 12 |- ( -u 1 x. -u 1 ) = 1
10	8, 9	eqtrdi 2287	. . . . . . . . . . 11 |- ( if ( N < 0 , -u 1 , 1 ) = -u 1 -> ( -u 1 x. if ( N < 0 , -u 1 , 1 ) ) = 1 )
11		oveq2 6083	. . . . . . . . . . . 12 |- ( if ( N < 0 , -u 1 , 1 ) = 1 -> ( -u 1 x. if ( N < 0 , -u 1 , 1 ) ) = ( -u 1 x. 1 ) )
12		ax-1cn 8262	. . . . . . . . . . . . 13 |- 1 e. CC
13	12	mulm1i 8720	. . . . . . . . . . . 12 |- ( -u 1 x. 1 ) = -u 1
14	11, 13	eqtrdi 2287	. . . . . . . . . . 11 |- ( if ( N < 0 , -u 1 , 1 ) = 1 -> ( -u 1 x. if ( N < 0 , -u 1 , 1 ) ) = -u 1 )
15	10, 14	ifsbdc 3650	. . . . . . . . . 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 3659	. . . . . . . . 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 6091	. . . . . . . 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 1033	. . . . . . . . . . . . 13 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> N =/= 0 )
23	22	necomd 2506	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> 0 =/= N )
24		zltlen 9703	. . . . . . . . . . . . 13 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> ( N < 0 <-> ( N <_ 0 /\ 0 =/= N ) ) )
25	4, 5, 24	sylancl 417	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> ( N < 0 <-> ( N <_ 0 /\ 0 =/= N ) ) )
26	23, 25	mpbiran2d 446	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> ( N < 0 <-> N <_ 0 ) )
27	4	zred 9747	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> N e. RR )
28	27	le0neg1d 8835	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> ( N <_ 0 <-> 0 <_ -u N ) )
29		0re 8316	. . . . . . . . . . . 12 |- 0 e. RR
30	27	renegcld 8697	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> -u N e. RR )
31		lenlt 8391	. . . . . . . . . . . 12 |- ( ( 0 e. RR /\ -u N e. RR ) -> ( 0 <_ -u N <-> -. -u N < 0 ) )
32	29, 30, 31	sylancr 418	. . . . . . . . . . 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 3659	. . . . . . . . 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 9654	. . . . . . . . . . . 12 |- ( N e. ZZ -> -u N e. ZZ )
36		zdclt 9701	. . . . . . . . . . . 12 |- ( ( -u N e. ZZ /\ 0 e. ZZ ) -> DECID -u N < 0 )
37	35, 5, 36	sylancl 417	. . . . . . . . . . 11 |- ( N e. ZZ -> DECID -u N < 0 )
38		ifnotdc 3676	. . . . . . . . . . 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 2271	. . . . . . . 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 2279	. . . . . . 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 3659	. . . . . . 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 2275	. . . . . 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 9436	. . . . . . 7 |- ( 1 x. 1 ) = 1
47		iffalse 3645	. . . . . . . . 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 943	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ -. A < 0 ) -> -. ( N < 0 /\ A < 0 ) )
51	50	iffalsed 3647	. . . . . . . 8 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ -. A < 0 ) -> if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) = 1 )
52	48, 51	oveq12d 6093	. . . . . . 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 943	. . . . . . . 8 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ -. A < 0 ) -> -. ( -u N < 0 /\ A < 0 ) )
54	53	iffalsed 3647	. . . . . . 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 2297	. . . . . 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 1028	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> A e. ZZ )
57		zdclt 9701	. . . . . . . 8 |- ( ( A e. ZZ /\ 0 e. ZZ ) -> DECID A < 0 )
58	56, 5, 57	sylancl 417	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> DECID A < 0 )
59		exmiddc 848	. . . . . . 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 810	. . . . 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 2244	. . . 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 1032	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. Prime ) -> N e. ZZ )
65		zq 10005	. . . . . . . . . . . 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 13082	. . . . . . . . . . 11 |- ( ( n e. Prime /\ N e. QQ ) -> ( n pCnt -u N ) = ( n pCnt N ) )
68	63, 66, 67	syl2anc 415	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. Prime ) -> ( n pCnt -u N ) = ( n pCnt N ) )
69	68	oveq2d 6091	. . . . . . . . 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 481	. . . . . . . 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 12886	. . . . . . . . 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 3668	. . . . . . 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 4216	. . . . . 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 10870	. . . . 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 9628	. . . . . . 7 |- ( N e. ZZ -> N e. CC )
77	76	3ad2ant2 1050	. . . . . 6 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> N e. CC )
78	77	absnegd 11933	. . . . 5 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( abs ` -u N ) = ( abs ` N ) )
79	75, 78	fveq12d 5697	. . . 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 6093	. . 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 9388	. . . . . 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 3675	. . . 4 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> if ( A < 0 , -u 1 , 1 ) e. CC )
85	7	3ad2ant2 1050	. . . . . 6 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> DECID N < 0 )
86		dcan2 947	. . . . . 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 3675	. . . 4 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) e. CC )
89		nnuz 9937	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
90		1zzd 9650	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> 1 e. ZZ )
91		eqid 2238	. . . . . . . . 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 16054	. . . . . . . 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 5834	. . . . . . 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 9677	. . . . . . . 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 10879	. . . . . 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 11844	. . . . . . 7 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. NN )
98	97	3adant1 1046	. . . . . 6 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. NN )
99	96, 98	ffvelcdmd 5835	. . . . 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 9748	. . . 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 8339	. . 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 2271	. 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 1050	. . 3 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> -u N e. ZZ )
104		simp3 1030	. . . 4 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> N =/= 0 )
105	77, 104	negne0d 8625	. . 3 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> -u N =/= 0 )
106		eqid 2238	. . . 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 16053	. . 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 1278	. 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 16053	. . 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 6091	. 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 2281	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 720  DECID wdc 846   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   ifcif 3635   class class class wbr 4125   |-> cmpt 4187   ` cfv 5372  (class class class)co 6075   CCcc 8167   RRcr 8168   0cc0 8169   1c1 8170   x. cmul 8174   < clt 8350   <_ cle 8351   -ucneg 8488   NNcn 9283   ZZcz 9623   QQcq 9998   seqcseq 10862   ^cexp 10953   abscabs 11741   Primecprime 12863   pCnt cpc 13041   /Lclgs 16030

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-xor 1425  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-isom 5381  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-frec 6652  df-1o 6677  df-2o 6678  df-oadd 6681  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-sup 7314  df-inf 7315  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-fz 10391  df-fzo 10528  df-fl 10683  df-mod 10738  df-seqfrec 10863  df-exp 10954  df-ihash 11193  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-clim 12023  df-proddc 12296  df-dvds 12533  df-gcd 12709  df-prm 12864  df-phi 12967  df-pc 13042  df-lgs 16031

This theorem is referenced by:  lgsneg1  16058