Theorem lgsneg 25046

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 4092	. . . . . . . . 9 |- ( A < 0 -> if ( A < 0 , -u 1 , 1 ) = -u 1 )
2	1	adantl 482	. . . . . . . 8 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> if ( A < 0 , -u 1 , 1 ) = -u 1 )
3	2	oveq1d 6665	. . . . . . 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		oveq2 6658	. . . . . . . . . 10 |- ( if ( N < 0 , -u 1 , 1 ) = -u 1 -> ( -u 1 x. if ( N < 0 , -u 1 , 1 ) ) = ( -u 1 x. -u 1 ) )
5		neg1mulneg1e1 11245	. . . . . . . . . 10 |- ( -u 1 x. -u 1 ) = 1
6	4, 5	syl6eq 2672	. . . . . . . . 9 |- ( if ( N < 0 , -u 1 , 1 ) = -u 1 -> ( -u 1 x. if ( N < 0 , -u 1 , 1 ) ) = 1 )
7		oveq2 6658	. . . . . . . . . 10 |- ( if ( N < 0 , -u 1 , 1 ) = 1 -> ( -u 1 x. if ( N < 0 , -u 1 , 1 ) ) = ( -u 1 x. 1 ) )
8		ax-1cn 9994	. . . . . . . . . . 11 |- 1 e. CC
9	8	mulm1i 10475	. . . . . . . . . 10 |- ( -u 1 x. 1 ) = -u 1
10	7, 9	syl6eq 2672	. . . . . . . . 9 |- ( if ( N < 0 , -u 1 , 1 ) = 1 -> ( -u 1 x. if ( N < 0 , -u 1 , 1 ) ) = -u 1 )
11	6, 10	ifsb 4099	. . . . . . . 8 |- ( -u 1 x. if ( N < 0 , -u 1 , 1 ) ) = if ( N < 0 , 1 , -u 1 )
12		simpr 477	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> A < 0 )
13	12	biantrud 528	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> ( N < 0 <-> ( N < 0 /\ A < 0 ) ) )
14	13	ifbid 4108	. . . . . . . . 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 ) )
15	14	oveq2d 6666	. . . . . . . 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 ) ) )
16		simpl2 1065	. . . . . . . . . . . . . 14 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> N e. ZZ )
17	16	zred 11482	. . . . . . . . . . . . 13 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> N e. RR )
18		0re 10040	. . . . . . . . . . . . 13 |- 0 e. RR
19		ltlen 10138	. . . . . . . . . . . . 13 |- ( ( N e. RR /\ 0 e. RR ) -> ( N < 0 <-> ( N <_ 0 /\ 0 =/= N ) ) )
20	17, 18, 19	sylancl 694	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> ( N < 0 <-> ( N <_ 0 /\ 0 =/= N ) ) )
21		simpl3 1066	. . . . . . . . . . . . . 14 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> N =/= 0 )
22	21	necomd 2849	. . . . . . . . . . . . 13 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> 0 =/= N )
23	22	biantrud 528	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> ( N <_ 0 <-> ( N <_ 0 /\ 0 =/= N ) ) )
24	20, 23	bitr4d 271	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> ( N < 0 <-> N <_ 0 ) )
25	17	le0neg1d 10599	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> ( N <_ 0 <-> 0 <_ -u N ) )
26	17	renegcld 10457	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> -u N e. RR )
27		lenlt 10116	. . . . . . . . . . . 12 |- ( ( 0 e. RR /\ -u N e. RR ) -> ( 0 <_ -u N <-> -. -u N < 0 ) )
28	18, 26, 27	sylancr 695	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> ( 0 <_ -u N <-> -. -u N < 0 ) )
29	24, 25, 28	3bitrd 294	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> ( N < 0 <-> -. -u N < 0 ) )
30	29	ifbid 4108	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> if ( N < 0 , 1 , -u 1 ) = if ( -. -u N < 0 , 1 , -u 1 ) )
31		ifnot 4133	. . . . . . . . 9 |- if ( -. -u N < 0 , 1 , -u 1 ) = if ( -u N < 0 , -u 1 , 1 )
32	30, 31	syl6eq 2672	. . . . . . . 8 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> if ( N < 0 , 1 , -u 1 ) = if ( -u N < 0 , -u 1 , 1 ) )
33	11, 15, 32	3eqtr3a 2680	. . . . . . 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 ) )
34	12	biantrud 528	. . . . . . . 8 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ A < 0 ) -> ( -u N < 0 <-> ( -u N < 0 /\ A < 0 ) ) )
35	34	ifbid 4108	. . . . . . 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 ) )
36	3, 33, 35	3eqtrd 2660	. . . . . 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 ) )
37		1t1e1 11175	. . . . . . 7 |- ( 1 x. 1 ) = 1
38		iffalse 4095	. . . . . . . . 9 |- ( -. A < 0 -> if ( A < 0 , -u 1 , 1 ) = 1 )
39	38	adantl 482	. . . . . . . 8 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ -. A < 0 ) -> if ( A < 0 , -u 1 , 1 ) = 1 )
40		simpr 477	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ -. A < 0 ) -> -. A < 0 )
41	40	intnand 962	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ -. A < 0 ) -> -. ( N < 0 /\ A < 0 ) )
42	41	iffalsed 4097	. . . . . . . 8 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ -. A < 0 ) -> if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) = 1 )
43	39, 42	oveq12d 6668	. . . . . . 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 ) )
44	40	intnand 962	. . . . . . . 8 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ -. A < 0 ) -> -. ( -u N < 0 /\ A < 0 ) )
45	44	iffalsed 4097	. . . . . . 7 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ -. A < 0 ) -> if ( ( -u N < 0 /\ A < 0 ) , -u 1 , 1 ) = 1 )
46	37, 43, 45	3eqtr4a 2682	. . . . . 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 ) )
47	36, 46	pm2.61dan 832	. . . . 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 ) )
48	47	eqcomd 2628	. . . 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 ) ) )
49		simpr 477	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. Prime ) -> n e. Prime )
50		simpl2 1065	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. Prime ) -> N e. ZZ )
51		zq 11794	. . . . . . . . . . 11 |- ( N e. ZZ -> N e. QQ )
52	50, 51	syl 17	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. Prime ) -> N e. QQ )
53		pcneg 15578	. . . . . . . . . 10 |- ( ( n e. Prime /\ N e. QQ ) -> ( n pCnt -u N ) = ( n pCnt N ) )
54	49, 52, 53	syl2anc 693	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. Prime ) -> ( n pCnt -u N ) = ( n pCnt N ) )
55	54	oveq2d 6666	. . . . . . . 8 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ n e. Prime ) -> ( ( A /L n ) ^ ( n pCnt -u N ) ) = ( ( A /L n ) ^ ( n pCnt N ) ) )
56	55	ifeq1da 4116	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt -u N ) ) , 1 ) = if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) )
57	56	mpteq2dv 4745	. . . . . 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 ) ) )
58	57	seqeq3d 12809	. . . . 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 ) ) ) )
59		zcn 11382	. . . . . . 7 |- ( N e. ZZ -> N e. CC )
60	59	3ad2ant2 1083	. . . . . 6 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> N e. CC )
61	60	absnegd 14188	. . . . 5 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( abs ` -u N ) = ( abs ` N ) )
62	58, 61	fveq12d 6197	. . . 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 ) ) )
63	48, 62	oveq12d 6668	. . 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 ) ) ) )
64		neg1cn 11124	. . . . . 6 |- -u 1 e. CC
65	64, 8	keepel 4155	. . . . 5 |- if ( A < 0 , -u 1 , 1 ) e. CC
66	65	a1i 11	. . . 4 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> if ( A < 0 , -u 1 , 1 ) e. CC )
67	64, 8	keepel 4155	. . . . 5 |- if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) e. CC
68	67	a1i 11	. . . 4 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) e. CC )
69		nnabscl 14065	. . . . . . . 8 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. NN )
70	69	3adant1 1079	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. NN )
71		nnuz 11723	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
72	70, 71	syl6eleq 2711	. . . . . 6 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. ( ZZ>= ` 1 ) )
73		eqid 2622	. . . . . . . 8 |- ( 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 ) )
74	73	lgsfcl3 25043	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) : NN --> ZZ )
75		elfznn 12370	. . . . . . 7 |- ( x e. ( 1 ... ( abs ` N ) ) -> x e. NN )
76		ffvelrn 6357	. . . . . . 7 |- ( ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) : NN --> ZZ /\ x e. NN ) -> ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ` x ) e. ZZ )
77	74, 75, 76	syl2an 494	. . . . . 6 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ x e. ( 1 ... ( abs ` N ) ) ) -> ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ` x ) e. ZZ )
78		zmulcl 11426	. . . . . . 7 |- ( ( x e. ZZ /\ y e. ZZ ) -> ( x x. y ) e. ZZ )
79	78	adantl 482	. . . . . 6 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( x x. y ) e. ZZ )
80	72, 77, 79	seqcl 12821	. . . . 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 )
81	80	zcnd 11483	. . . 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 )
82	66, 68, 81	mulassd 10063	. . 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 ) ) ) ) )
83	63, 82	eqtrd 2656	. 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 ) ) ) ) )
84		simp1 1061	. . 3 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> A e. ZZ )
85		znegcl 11412	. . . 4 |- ( N e. ZZ -> -u N e. ZZ )
86	85	3ad2ant2 1083	. . 3 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> -u N e. ZZ )
87		simp3 1063	. . . 4 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> N =/= 0 )
88	60, 87	negne0d 10390	. . 3 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> -u N =/= 0 )
89		eqid 2622	. . . 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 ) )
90	89	lgsval4 25042	. . 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 ) ) ) )
91	84, 86, 88, 90	syl3anc 1326	. 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 ) ) ) )
92	73	lgsval4 25042	. . 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 ) ) ) )
93	92	oveq2d 6666	. 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 ) ) ) ) )
94	83, 91, 93	3eqtr4d 2666	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   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   < clt 10074   <_ cle 10075   -ucneg 10267   NNcn 11020   ZZcz 11377   ZZ>=cuz 11687   QQcq 11788   ...cfz 12326   seqcseq 12801   ^cexp 12860   abscabs 13974   Primecprime 15385   pCnt cpc 15541   /Lclgs 25019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-prm 15386  df-phi 15471  df-pc 15542  df-lgs 25020

This theorem is referenced by:  lgsneg1  25047