Theorem lgsne0 25060

Description: The Legendre symbol is nonzero (and hence equal to 1 or -u 1) precisely when the arguments are coprime. (Contributed by Mario Carneiro, 5-Feb-2015.)

Assertion

Ref	Expression
lgsne0	|- ( ( A e. ZZ /\ N e. ZZ ) -> ( ( A /L N ) =/= 0 <-> ( A gcd N ) = 1 ) )

Proof of Theorem lgsne0

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

Step	Hyp	Ref	Expression
1		iffalse 4095	. . . . . 6 |- ( -. ( A ^ 2 ) = 1 -> if ( ( A ^ 2 ) = 1 , 1 , 0 ) = 0 )
2	1	necon1ai 2821	. . . . 5 |- ( if ( ( A ^ 2 ) = 1 , 1 , 0 ) =/= 0 -> ( A ^ 2 ) = 1 )
3		iftrue 4092	. . . . . 6 |- ( ( A ^ 2 ) = 1 -> if ( ( A ^ 2 ) = 1 , 1 , 0 ) = 1 )
4		ax-1ne0 10005	. . . . . . 7 |- 1 =/= 0
5	4	a1i 11	. . . . . 6 |- ( ( A ^ 2 ) = 1 -> 1 =/= 0 )
6	3, 5	eqnetrd 2861	. . . . 5 |- ( ( A ^ 2 ) = 1 -> if ( ( A ^ 2 ) = 1 , 1 , 0 ) =/= 0 )
7	2, 6	impbii 199	. . . 4 |- ( if ( ( A ^ 2 ) = 1 , 1 , 0 ) =/= 0 <-> ( A ^ 2 ) = 1 )
8		zre 11381	. . . . . . . 8 |- ( A e. ZZ -> A e. RR )
9	8	ad2antrr 762	. . . . . . 7 |- ( ( ( A e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> A e. RR )
10		absresq 14042	. . . . . . 7 |- ( A e. RR -> ( ( abs ` A ) ^ 2 ) = ( A ^ 2 ) )
11	9, 10	syl 17	. . . . . 6 |- ( ( ( A e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( ( abs ` A ) ^ 2 ) = ( A ^ 2 ) )
12		sq1 12958	. . . . . . 7 |- ( 1 ^ 2 ) = 1
13	12	a1i 11	. . . . . 6 |- ( ( ( A e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( 1 ^ 2 ) = 1 )
14	11, 13	eqeq12d 2637	. . . . 5 |- ( ( ( A e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( ( ( abs ` A ) ^ 2 ) = ( 1 ^ 2 ) <-> ( A ^ 2 ) = 1 ) )
15	9	recnd 10068	. . . . . . 7 |- ( ( ( A e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> A e. CC )
16	15	abscld 14175	. . . . . 6 |- ( ( ( A e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( abs ` A ) e. RR )
17	15	absge0d 14183	. . . . . 6 |- ( ( ( A e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> 0 <_ ( abs ` A ) )
18		1re 10039	. . . . . . 7 |- 1 e. RR
19		0le1 10551	. . . . . . 7 |- 0 <_ 1
20		sq11 12936	. . . . . . 7 |- ( ( ( ( abs ` A ) e. RR /\ 0 <_ ( abs ` A ) ) /\ ( 1 e. RR /\ 0 <_ 1 ) ) -> ( ( ( abs ` A ) ^ 2 ) = ( 1 ^ 2 ) <-> ( abs ` A ) = 1 ) )
21	18, 19, 20	mpanr12 721	. . . . . 6 |- ( ( ( abs ` A ) e. RR /\ 0 <_ ( abs ` A ) ) -> ( ( ( abs ` A ) ^ 2 ) = ( 1 ^ 2 ) <-> ( abs ` A ) = 1 ) )
22	16, 17, 21	syl2anc 693	. . . . 5 |- ( ( ( A e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( ( ( abs ` A ) ^ 2 ) = ( 1 ^ 2 ) <-> ( abs ` A ) = 1 ) )
23	14, 22	bitr3d 270	. . . 4 |- ( ( ( A e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( ( A ^ 2 ) = 1 <-> ( abs ` A ) = 1 ) )
24	7, 23	syl5bb 272	. . 3 |- ( ( ( A e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( if ( ( A ^ 2 ) = 1 , 1 , 0 ) =/= 0 <-> ( abs ` A ) = 1 ) )
25		oveq2 6658	. . . . 5 |- ( N = 0 -> ( A /L N ) = ( A /L 0 ) )
26		lgs0 25035	. . . . . 6 |- ( A e. ZZ -> ( A /L 0 ) = if ( ( A ^ 2 ) = 1 , 1 , 0 ) )
27	26	adantr 481	. . . . 5 |- ( ( A e. ZZ /\ N e. ZZ ) -> ( A /L 0 ) = if ( ( A ^ 2 ) = 1 , 1 , 0 ) )
28	25, 27	sylan9eqr 2678	. . . 4 |- ( ( ( A e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( A /L N ) = if ( ( A ^ 2 ) = 1 , 1 , 0 ) )
29	28	neeq1d 2853	. . 3 |- ( ( ( A e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( ( A /L N ) =/= 0 <-> if ( ( A ^ 2 ) = 1 , 1 , 0 ) =/= 0 ) )
30		oveq2 6658	. . . . 5 |- ( N = 0 -> ( A gcd N ) = ( A gcd 0 ) )
31		gcdid0 15241	. . . . . 6 |- ( A e. ZZ -> ( A gcd 0 ) = ( abs ` A ) )
32	31	adantr 481	. . . . 5 |- ( ( A e. ZZ /\ N e. ZZ ) -> ( A gcd 0 ) = ( abs ` A ) )
33	30, 32	sylan9eqr 2678	. . . 4 |- ( ( ( A e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( A gcd N ) = ( abs ` A ) )
34	33	eqeq1d 2624	. . 3 |- ( ( ( A e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( ( A gcd N ) = 1 <-> ( abs ` A ) = 1 ) )
35	24, 29, 34	3bitr4d 300	. 2 |- ( ( ( A e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( ( A /L N ) =/= 0 <-> ( A gcd N ) = 1 ) )
36		eqid 2622	. . . . . 6 |- ( 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 ) )
37	36	lgsval4 25042	. . . . 5 |- ( ( 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 ) ) ) )
38	37	neeq1d 2853	. . . 4 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( ( A /L 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 ) ) ) =/= 0 ) )
39		neeq1 2856	. . . . . . 7 |- ( -u 1 = if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) -> ( -u 1 =/= 0 <-> if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) =/= 0 ) )
40		neeq1 2856	. . . . . . 7 |- ( 1 = if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) -> ( 1 =/= 0 <-> if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) =/= 0 ) )
41		neg1ne0 11126	. . . . . . 7 |- -u 1 =/= 0
42	39, 40, 41, 4	keephyp 4152	. . . . . 6 |- if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) =/= 0
43	42	biantrur 527	. . . . 5 |- ( ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) =/= 0 <-> ( if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) =/= 0 /\ ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) =/= 0 ) )
44		neg1cn 11124	. . . . . . . 8 |- -u 1 e. CC
45		ax-1cn 9994	. . . . . . . 8 |- 1 e. CC
46	44, 45	keepel 4155	. . . . . . 7 |- if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) e. CC
47	46	a1i 11	. . . . . 6 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) e. CC )
48		nnabscl 14065	. . . . . . . . 9 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. NN )
49	48	3adant1 1079	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. NN )
50		nnuz 11723	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
51	49, 50	syl6eleq 2711	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. ( ZZ>= ` 1 ) )
52	36	lgsfcl3 25043	. . . . . . . . 9 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) : NN --> ZZ )
53		elfznn 12370	. . . . . . . . 9 |- ( k e. ( 1 ... ( abs ` N ) ) -> k e. NN )
54		ffvelrn 6357	. . . . . . . . 9 |- ( ( ( 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 )
55	52, 53, 54	syl2an 494	. . . . . . . 8 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ k e. ( 1 ... ( abs ` N ) ) ) -> ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ` k ) e. ZZ )
56	55	zcnd 11483	. . . . . . 7 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ k e. ( 1 ... ( abs ` N ) ) ) -> ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ` k ) e. CC )
57		mulcl 10020	. . . . . . . 8 |- ( ( k e. CC /\ x e. CC ) -> ( k x. x ) e. CC )
58	57	adantl 482	. . . . . . 7 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( k e. CC /\ x e. CC ) ) -> ( k x. x ) e. CC )
59	51, 56, 58	seqcl 12821	. . . . . 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 ) ) ) ` ( abs ` N ) ) e. CC )
60	47, 59	mulne0bd 10678	. . . . 5 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( ( if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) =/= 0 /\ ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` 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 ) ) ) =/= 0 ) )
61	43, 60	syl5rbb 273	. . . 4 |- ( ( A e. ZZ /\ N e. ZZ /\ 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 ) ) ) =/= 0 <-> ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) =/= 0 ) )
62		simpr 477	. . . . . . . . . 10 |- ( ( A = 0 /\ N = 0 ) -> N = 0 )
63	62	necon3ai 2819	. . . . . . . . 9 |- ( N =/= 0 -> -. ( A = 0 /\ N = 0 ) )
64		gcdn0cl 15224	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ N e. ZZ ) /\ -. ( A = 0 /\ N = 0 ) ) -> ( A gcd N ) e. NN )
65	63, 64	sylan2 491	. . . . . . . 8 |- ( ( ( A e. ZZ /\ N e. ZZ ) /\ N =/= 0 ) -> ( A gcd N ) e. NN )
66	65	3impa 1259	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( A gcd N ) e. NN )
67		eluz2b3 11762	. . . . . . . . 9 |- ( ( A gcd N ) e. ( ZZ>= ` 2 ) <-> ( ( A gcd N ) e. NN /\ ( A gcd N ) =/= 1 ) )
68		exprmfct 15416	. . . . . . . . 9 |- ( ( A gcd N ) e. ( ZZ>= ` 2 ) -> E. p e. Prime p || ( A gcd N ) )
69	67, 68	sylbir 225	. . . . . . . 8 |- ( ( ( A gcd N ) e. NN /\ ( A gcd N ) =/= 1 ) -> E. p e. Prime p || ( A gcd N ) )
70	57	adantl 482	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ ( k e. CC /\ x e. CC ) ) -> ( k x. x ) e. CC )
71	56	adantlr 751	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ k e. ( 1 ... ( abs ` N ) ) ) -> ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ` k ) e. CC )
72		mul02 10214	. . . . . . . . . . 11 |- ( k e. CC -> ( 0 x. k ) = 0 )
73	72	adantl 482	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ k e. CC ) -> ( 0 x. k ) = 0 )
74		mul01 10215	. . . . . . . . . . 11 |- ( k e. CC -> ( k x. 0 ) = 0 )
75	74	adantl 482	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ k e. CC ) -> ( k x. 0 ) = 0 )
76		simprr 796	. . . . . . . . . . . . . . 15 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) -> p || ( A gcd N ) )
77		prmz 15389	. . . . . . . . . . . . . . . . 17 |- ( p e. Prime -> p e. ZZ )
78	77	ad2antrl 764	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) -> p e. ZZ )
79		simpl1 1064	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) -> A e. ZZ )
80		simpl2 1065	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) -> N e. ZZ )
81		dvdsgcdb 15262	. . . . . . . . . . . . . . . 16 |- ( ( p e. ZZ /\ A e. ZZ /\ N e. ZZ ) -> ( ( p || A /\ p || N ) <-> p || ( A gcd N ) ) )
82	78, 79, 80, 81	syl3anc 1326	. . . . . . . . . . . . . . 15 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) -> ( ( p || A /\ p || N ) <-> p || ( A gcd N ) ) )
83	76, 82	mpbird 247	. . . . . . . . . . . . . 14 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) -> ( p || A /\ p || N ) )
84	83	simprd 479	. . . . . . . . . . . . 13 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) -> p || N )
85		dvdsabsb 15001	. . . . . . . . . . . . . 14 |- ( ( p e. ZZ /\ N e. ZZ ) -> ( p || N <-> p || ( abs ` N ) ) )
86	78, 80, 85	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) -> ( p || N <-> p || ( abs ` N ) ) )
87	84, 86	mpbid 222	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) -> p || ( abs ` N ) )
88	49	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) -> ( abs ` N ) e. NN )
89		dvdsle 15032	. . . . . . . . . . . . 13 |- ( ( p e. ZZ /\ ( abs ` N ) e. NN ) -> ( p || ( abs ` N ) -> p <_ ( abs ` N ) ) )
90	78, 88, 89	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) -> ( p || ( abs ` N ) -> p <_ ( abs ` N ) ) )
91	87, 90	mpd 15	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) -> p <_ ( abs ` N ) )
92		prmnn 15388	. . . . . . . . . . . . . 14 |- ( p e. Prime -> p e. NN )
93	92	ad2antrl 764	. . . . . . . . . . . . 13 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) -> p e. NN )
94	93, 50	syl6eleq 2711	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) -> p e. ( ZZ>= ` 1 ) )
95	88	nnzd 11481	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) -> ( abs ` N ) e. ZZ )
96		elfz5 12334	. . . . . . . . . . . 12 |- ( ( p e. ( ZZ>= ` 1 ) /\ ( abs ` N ) e. ZZ ) -> ( p e. ( 1 ... ( abs ` N ) ) <-> p <_ ( abs ` N ) ) )
97	94, 95, 96	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) -> ( p e. ( 1 ... ( abs ` N ) ) <-> p <_ ( abs ` N ) ) )
98	91, 97	mpbird 247	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) -> p e. ( 1 ... ( abs ` N ) ) )
99		eleq1 2689	. . . . . . . . . . . . . 14 |- ( n = p -> ( n e. Prime <-> p e. Prime ) )
100		oveq2 6658	. . . . . . . . . . . . . . 15 |- ( n = p -> ( A /L n ) = ( A /L p ) )
101		oveq1 6657	. . . . . . . . . . . . . . 15 |- ( n = p -> ( n pCnt N ) = ( p pCnt N ) )
102	100, 101	oveq12d 6668	. . . . . . . . . . . . . 14 |- ( n = p -> ( ( A /L n ) ^ ( n pCnt N ) ) = ( ( A /L p ) ^ ( p pCnt N ) ) )
103	99, 102	ifbieq1d 4109	. . . . . . . . . . . . 13 |- ( n = p -> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) = if ( p e. Prime , ( ( A /L p ) ^ ( p pCnt N ) ) , 1 ) )
104		ovex 6678	. . . . . . . . . . . . . 14 |- ( ( A /L p ) ^ ( p pCnt N ) ) e. _V
105		1ex 10035	. . . . . . . . . . . . . 14 |- 1 e. _V
106	104, 105	ifex 4156	. . . . . . . . . . . . 13 |- if ( p e. Prime , ( ( A /L p ) ^ ( p pCnt N ) ) , 1 ) e. _V
107	103, 36, 106	fvmpt 6282	. . . . . . . . . . . 12 |- ( p e. NN -> ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ` p ) = if ( p e. Prime , ( ( A /L p ) ^ ( p pCnt N ) ) , 1 ) )
108	93, 107	syl 17	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) -> ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ` p ) = if ( p e. Prime , ( ( A /L p ) ^ ( p pCnt N ) ) , 1 ) )
109		iftrue 4092	. . . . . . . . . . . 12 |- ( p e. Prime -> if ( p e. Prime , ( ( A /L p ) ^ ( p pCnt N ) ) , 1 ) = ( ( A /L p ) ^ ( p pCnt N ) ) )
110	109	ad2antrl 764	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) -> if ( p e. Prime , ( ( A /L p ) ^ ( p pCnt N ) ) , 1 ) = ( ( A /L p ) ^ ( p pCnt N ) ) )
111		oveq2 6658	. . . . . . . . . . . . . . . 16 |- ( p = 2 -> ( A /L p ) = ( A /L 2 ) )
112		lgs2 25039	. . . . . . . . . . . . . . . . 17 |- ( A e. ZZ -> ( A /L 2 ) = if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) )
113	79, 112	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) -> ( A /L 2 ) = if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) )
114	111, 113	sylan9eqr 2678	. . . . . . . . . . . . . . 15 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p = 2 ) -> ( A /L p ) = if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) )
115		simpr 477	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p = 2 ) -> p = 2 )
116	83	simpld 475	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) -> p || A )
117	116	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p = 2 ) -> p || A )
118	115, 117	eqbrtrrd 4677	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p = 2 ) -> 2 || A )
119	118	iftrued 4094	. . . . . . . . . . . . . . 15 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p = 2 ) -> if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) = 0 )
120	114, 119	eqtrd 2656	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p = 2 ) -> ( A /L p ) = 0 )
121		simpll1 1100	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> A e. ZZ )
122		simprl 794	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) -> p e. Prime )
123	122	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> p e. Prime )
124		simpr 477	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> p =/= 2 )
125		eldifsn 4317	. . . . . . . . . . . . . . . . 17 |- ( p e. ( Prime \ { 2 } ) <-> ( p e. Prime /\ p =/= 2 ) )
126	123, 124, 125	sylanbrc 698	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> p e. ( Prime \ { 2 } ) )
127		lgsval3 25040	. . . . . . . . . . . . . . . 16 |- ( ( A e. ZZ /\ p e. ( Prime \ { 2 } ) ) -> ( A /L p ) = ( ( ( ( A ^ ( ( p - 1 ) / 2 ) ) + 1 ) mod p ) - 1 ) )
128	121, 126, 127	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> ( A /L p ) = ( ( ( ( A ^ ( ( p - 1 ) / 2 ) ) + 1 ) mod p ) - 1 ) )
129		oddprm 15515	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( p e. ( Prime \ { 2 } ) -> ( ( p - 1 ) / 2 ) e. NN )
130	126, 129	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> ( ( p - 1 ) / 2 ) e. NN )
131	130	nnnn0d 11351	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> ( ( p - 1 ) / 2 ) e. NN0 )
132		zexpcl 12875	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( A e. ZZ /\ ( ( p - 1 ) / 2 ) e. NN0 ) -> ( A ^ ( ( p - 1 ) / 2 ) ) e. ZZ )
133	121, 131, 132	syl2anc 693	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> ( A ^ ( ( p - 1 ) / 2 ) ) e. ZZ )
134	133	zred 11482	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> ( A ^ ( ( p - 1 ) / 2 ) ) e. RR )
135		0red 10041	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> 0 e. RR )
136	18	a1i 11	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> 1 e. RR )
137	123, 92	syl 17	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> p e. NN )
138	137	nnrpd 11870	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> p e. RR+ )
139		0zd 11389	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> 0 e. ZZ )
140	116	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> p || A )
141		dvdsval3 14987	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( p e. NN /\ A e. ZZ ) -> ( p || A <-> ( A mod p ) = 0 ) )
142	137, 121, 141	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> ( p || A <-> ( A mod p ) = 0 ) )
143	140, 142	mpbid 222	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> ( A mod p ) = 0 )
144		0mod 12701	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( p e. RR+ -> ( 0 mod p ) = 0 )
145	138, 144	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> ( 0 mod p ) = 0 )
146	143, 145	eqtr4d 2659	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> ( A mod p ) = ( 0 mod p ) )
147		modexp 12999	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( A e. ZZ /\ 0 e. ZZ ) /\ ( ( ( p - 1 ) / 2 ) e. NN0 /\ p e. RR+ ) /\ ( A mod p ) = ( 0 mod p ) ) -> ( ( A ^ ( ( p - 1 ) / 2 ) ) mod p ) = ( ( 0 ^ ( ( p - 1 ) / 2 ) ) mod p ) )
148	121, 139, 131, 138, 146, 147	syl221anc 1337	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> ( ( A ^ ( ( p - 1 ) / 2 ) ) mod p ) = ( ( 0 ^ ( ( p - 1 ) / 2 ) ) mod p ) )
149	130	0expd 13024	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> ( 0 ^ ( ( p - 1 ) / 2 ) ) = 0 )
150	149	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> ( ( 0 ^ ( ( p - 1 ) / 2 ) ) mod p ) = ( 0 mod p ) )
151	148, 150	eqtrd 2656	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> ( ( A ^ ( ( p - 1 ) / 2 ) ) mod p ) = ( 0 mod p ) )
152		modadd1 12707	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( A ^ ( ( p - 1 ) / 2 ) ) e. RR /\ 0 e. RR ) /\ ( 1 e. RR /\ p e. RR+ ) /\ ( ( A ^ ( ( p - 1 ) / 2 ) ) mod p ) = ( 0 mod p ) ) -> ( ( ( A ^ ( ( p - 1 ) / 2 ) ) + 1 ) mod p ) = ( ( 0 + 1 ) mod p ) )
153	134, 135, 136, 138, 151, 152	syl221anc 1337	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> ( ( ( A ^ ( ( p - 1 ) / 2 ) ) + 1 ) mod p ) = ( ( 0 + 1 ) mod p ) )
154		0p1e1 11132	. . . . . . . . . . . . . . . . . . . 20 |- ( 0 + 1 ) = 1
155	154	oveq1i 6660	. . . . . . . . . . . . . . . . . . 19 |- ( ( 0 + 1 ) mod p ) = ( 1 mod p )
156	153, 155	syl6eq 2672	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> ( ( ( A ^ ( ( p - 1 ) / 2 ) ) + 1 ) mod p ) = ( 1 mod p ) )
157	137	nnred 11035	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> p e. RR )
158		prmuz2 15408	. . . . . . . . . . . . . . . . . . . . . 22 |- ( p e. Prime -> p e. ( ZZ>= ` 2 ) )
159	123, 158	syl 17	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> p e. ( ZZ>= ` 2 ) )
160		eluz2b2 11761	. . . . . . . . . . . . . . . . . . . . 21 |- ( p e. ( ZZ>= ` 2 ) <-> ( p e. NN /\ 1 < p ) )
161	159, 160	sylib 208	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> ( p e. NN /\ 1 < p ) )
162	161	simprd 479	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> 1 < p )
163		1mod 12702	. . . . . . . . . . . . . . . . . . 19 |- ( ( p e. RR /\ 1 < p ) -> ( 1 mod p ) = 1 )
164	157, 162, 163	syl2anc 693	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> ( 1 mod p ) = 1 )
165	156, 164	eqtrd 2656	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> ( ( ( A ^ ( ( p - 1 ) / 2 ) ) + 1 ) mod p ) = 1 )
166	165	oveq1d 6665	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> ( ( ( ( A ^ ( ( p - 1 ) / 2 ) ) + 1 ) mod p ) - 1 ) = ( 1 - 1 ) )
167		1m1e0 11089	. . . . . . . . . . . . . . . 16 |- ( 1 - 1 ) = 0
168	166, 167	syl6eq 2672	. . . . . . . . . . . . . . 15 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> ( ( ( ( A ^ ( ( p - 1 ) / 2 ) ) + 1 ) mod p ) - 1 ) = 0 )
169	128, 168	eqtrd 2656	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) /\ p =/= 2 ) -> ( A /L p ) = 0 )
170	120, 169	pm2.61dane 2881	. . . . . . . . . . . . 13 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) -> ( A /L p ) = 0 )
171	170	oveq1d 6665	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) -> ( ( A /L p ) ^ ( p pCnt N ) ) = ( 0 ^ ( p pCnt N ) ) )
172		zq 11794	. . . . . . . . . . . . . . . 16 |- ( N e. ZZ -> N e. QQ )
173	80, 172	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) -> N e. QQ )
174		pcabs 15579	. . . . . . . . . . . . . . 15 |- ( ( p e. Prime /\ N e. QQ ) -> ( p pCnt ( abs ` N ) ) = ( p pCnt N ) )
175	122, 173, 174	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) -> ( p pCnt ( abs ` N ) ) = ( p pCnt N ) )
176		pcelnn 15574	. . . . . . . . . . . . . . . 16 |- ( ( p e. Prime /\ ( abs ` N ) e. NN ) -> ( ( p pCnt ( abs ` N ) ) e. NN <-> p || ( abs ` N ) ) )
177	122, 88, 176	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) -> ( ( p pCnt ( abs ` N ) ) e. NN <-> p || ( abs ` N ) ) )
178	87, 177	mpbird 247	. . . . . . . . . . . . . 14 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) -> ( p pCnt ( abs ` N ) ) e. NN )
179	175, 178	eqeltrrd 2702	. . . . . . . . . . . . 13 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) -> ( p pCnt N ) e. NN )
180	179	0expd 13024	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) -> ( 0 ^ ( p pCnt N ) ) = 0 )
181	171, 180	eqtrd 2656	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) -> ( ( A /L p ) ^ ( p pCnt N ) ) = 0 )
182	108, 110, 181	3eqtrd 2660	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) -> ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ` p ) = 0 )
183	70, 71, 73, 75, 98, 88, 182	seqz 12849	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( p e. Prime /\ p || ( A gcd N ) ) ) -> ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) = 0 )
184	183	rexlimdvaa 3032	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( E. p e. Prime p || ( A gcd N ) -> ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) = 0 ) )
185	69, 184	syl5 34	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( ( ( A gcd N ) e. NN /\ ( A gcd N ) =/= 1 ) -> ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) = 0 ) )
186	66, 185	mpand 711	. . . . . 6 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( ( A gcd N ) =/= 1 -> ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) = 0 ) )
187	186	necon1d 2816	. . . . 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 ) ) =/= 0 -> ( A gcd N ) = 1 ) )
188	51	adantr 481	. . . . . . . 8 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) -> ( abs ` N ) e. ( ZZ>= ` 1 ) )
189	53	adantl 482	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. ( 1 ... ( abs ` N ) ) ) -> k e. NN )
190		eleq1 2689	. . . . . . . . . . . 12 |- ( n = k -> ( n e. Prime <-> k e. Prime ) )
191		oveq2 6658	. . . . . . . . . . . . 13 |- ( n = k -> ( A /L n ) = ( A /L k ) )
192		oveq1 6657	. . . . . . . . . . . . 13 |- ( n = k -> ( n pCnt N ) = ( k pCnt N ) )
193	191, 192	oveq12d 6668	. . . . . . . . . . . 12 |- ( n = k -> ( ( A /L n ) ^ ( n pCnt N ) ) = ( ( A /L k ) ^ ( k pCnt N ) ) )
194	190, 193	ifbieq1d 4109	. . . . . . . . . . 11 |- ( n = k -> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) = if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt N ) ) , 1 ) )
195		ovex 6678	. . . . . . . . . . . 12 |- ( ( A /L k ) ^ ( k pCnt N ) ) e. _V
196	195, 105	ifex 4156	. . . . . . . . . . 11 |- if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt N ) ) , 1 ) e. _V
197	194, 36, 196	fvmpt 6282	. . . . . . . . . 10 |- ( k e. NN -> ( ( 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 ) )
198	189, 197	syl 17	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. ( 1 ... ( abs ` N ) ) ) -> ( ( 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 ) )
199		simpll1 1100	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) -> A e. ZZ )
200		prmz 15389	. . . . . . . . . . . . . . . . 17 |- ( k e. Prime -> k e. ZZ )
201	200	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) -> k e. ZZ )
202		lgscl 25036	. . . . . . . . . . . . . . . 16 |- ( ( A e. ZZ /\ k e. ZZ ) -> ( A /L k ) e. ZZ )
203	199, 201, 202	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) -> ( A /L k ) e. ZZ )
204	203	zcnd 11483	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) -> ( A /L k ) e. CC )
205	204	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) -> ( A /L k ) e. CC )
206		oveq2 6658	. . . . . . . . . . . . . . . . 17 |- ( k = 2 -> ( A /L k ) = ( A /L 2 ) )
207	199	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) -> A e. ZZ )
208	207, 112	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) -> ( A /L 2 ) = if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) )
209	206, 208	sylan9eqr 2678	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k = 2 ) -> ( A /L k ) = if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) )
210		nprmdvds1 15418	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( k e. Prime -> -. k || 1 )
211	210	adantl 482	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) -> -. k || 1 )
212		simpll2 1101	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) -> N e. ZZ )
213		dvdsgcdb 15262	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( k e. ZZ /\ A e. ZZ /\ N e. ZZ ) -> ( ( k || A /\ k || N ) <-> k || ( A gcd N ) ) )
214	201, 199, 212, 213	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) -> ( ( k || A /\ k || N ) <-> k || ( A gcd N ) ) )
215		simplr 792	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) -> ( A gcd N ) = 1 )
216	215	breq2d 4665	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) -> ( k || ( A gcd N ) <-> k || 1 ) )
217	214, 216	bitrd 268	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) -> ( ( k || A /\ k || N ) <-> k || 1 ) )
218	211, 217	mtbird 315	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) -> -. ( k || A /\ k || N ) )
219		imnan 438	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( k || A -> -. k || N ) <-> -. ( k || A /\ k || N ) )
220	218, 219	sylibr 224	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) -> ( k || A -> -. k || N ) )
221	220	con2d 129	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) -> ( k || N -> -. k || A ) )
222	221	imp 445	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) -> -. k || A )
223		breq1 4656	. . . . . . . . . . . . . . . . . . . 20 |- ( k = 2 -> ( k || A <-> 2 || A ) )
224	223	notbid 308	. . . . . . . . . . . . . . . . . . 19 |- ( k = 2 -> ( -. k || A <-> -. 2 || A ) )
225	222, 224	syl5ibcom 235	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) -> ( k = 2 -> -. 2 || A ) )
226	225	imp 445	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k = 2 ) -> -. 2 || A )
227	226	iffalsed 4097	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k = 2 ) -> if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) = if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) )
228	209, 227	eqtrd 2656	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k = 2 ) -> ( A /L k ) = if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) )
229		neeq1 2856	. . . . . . . . . . . . . . . . 17 |- ( 1 = if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) -> ( 1 =/= 0 <-> if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) =/= 0 ) )
230		neeq1 2856	. . . . . . . . . . . . . . . . 17 |- ( -u 1 = if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) -> ( -u 1 =/= 0 <-> if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) =/= 0 ) )
231	229, 230, 4, 41	keephyp 4152	. . . . . . . . . . . . . . . 16 |- if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) =/= 0
232	231	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k = 2 ) -> if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) =/= 0 )
233	228, 232	eqnetrd 2861	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k = 2 ) -> ( A /L k ) =/= 0 )
234		simpr 477	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) -> k e. Prime )
235	234	ad2antrr 762	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> k e. Prime )
236	235, 210	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> -. k || 1 )
237		simplr 792	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> k || N )
238	235, 200	syl 17	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> k e. ZZ )
239	207	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> A e. ZZ )
240		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> k =/= 2 )
241		eldifsn 4317	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( k e. ( Prime \ { 2 } ) <-> ( k e. Prime /\ k =/= 2 ) )
242	235, 240, 241	sylanbrc 698	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> k e. ( Prime \ { 2 } ) )
243		oddprm 15515	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( k e. ( Prime \ { 2 } ) -> ( ( k - 1 ) / 2 ) e. NN )
244	242, 243	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> ( ( k - 1 ) / 2 ) e. NN )
245	244	nnnn0d 11351	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> ( ( k - 1 ) / 2 ) e. NN0 )
246		zexpcl 12875	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( A e. ZZ /\ ( ( k - 1 ) / 2 ) e. NN0 ) -> ( A ^ ( ( k - 1 ) / 2 ) ) e. ZZ )
247	239, 245, 246	syl2anc 693	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> ( A ^ ( ( k - 1 ) / 2 ) ) e. ZZ )
248	212	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> N e. ZZ )
249		dvdsgcd 15261	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( k e. ZZ /\ ( A ^ ( ( k - 1 ) / 2 ) ) e. ZZ /\ N e. ZZ ) -> ( ( k || ( A ^ ( ( k - 1 ) / 2 ) ) /\ k || N ) -> k || ( ( A ^ ( ( k - 1 ) / 2 ) ) gcd N ) ) )
250	238, 247, 248, 249	syl3anc 1326	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> ( ( k || ( A ^ ( ( k - 1 ) / 2 ) ) /\ k || N ) -> k || ( ( A ^ ( ( k - 1 ) / 2 ) ) gcd N ) ) )
251	237, 250	mpan2d 710	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> ( k || ( A ^ ( ( k - 1 ) / 2 ) ) -> k || ( ( A ^ ( ( k - 1 ) / 2 ) ) gcd N ) ) )
252	239	zcnd 11483	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> A e. CC )
253	252, 245	absexpd 14191	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> ( abs ` ( A ^ ( ( k - 1 ) / 2 ) ) ) = ( ( abs ` A ) ^ ( ( k - 1 ) / 2 ) ) )
254	253	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> ( ( abs ` ( A ^ ( ( k - 1 ) / 2 ) ) ) gcd ( abs ` N ) ) = ( ( ( abs ` A ) ^ ( ( k - 1 ) / 2 ) ) gcd ( abs ` N ) ) )
255		gcdabs 15250	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( A ^ ( ( k - 1 ) / 2 ) ) e. ZZ /\ N e. ZZ ) -> ( ( abs ` ( A ^ ( ( k - 1 ) / 2 ) ) ) gcd ( abs ` N ) ) = ( ( A ^ ( ( k - 1 ) / 2 ) ) gcd N ) )
256	247, 248, 255	syl2anc 693	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> ( ( abs ` ( A ^ ( ( k - 1 ) / 2 ) ) ) gcd ( abs ` N ) ) = ( ( A ^ ( ( k - 1 ) / 2 ) ) gcd N ) )
257		gcdabs 15250	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( A e. ZZ /\ N e. ZZ ) -> ( ( abs ` A ) gcd ( abs ` N ) ) = ( A gcd N ) )
258	239, 248, 257	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> ( ( abs ` A ) gcd ( abs ` N ) ) = ( A gcd N ) )
259	215	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> ( A gcd N ) = 1 )
260	258, 259	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> ( ( abs ` A ) gcd ( abs ` N ) ) = 1 )
261	222	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> -. k || A )
262		dvds0 14997	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( k e. ZZ -> k || 0 )
263	238, 262	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> k || 0 )
264		breq2 4657	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( A = 0 -> ( k || A <-> k || 0 ) )
265	263, 264	syl5ibrcom 237	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> ( A = 0 -> k || A ) )
266	265	necon3bd 2808	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> ( -. k || A -> A =/= 0 ) )
267	261, 266	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> A =/= 0 )
268		nnabscl 14065	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( A e. ZZ /\ A =/= 0 ) -> ( abs ` A ) e. NN )
269	239, 267, 268	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> ( abs ` A ) e. NN )
270		simpll3 1102	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) -> N =/= 0 )
271	212, 270, 48	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) -> ( abs ` N ) e. NN )
272	271	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> ( abs ` N ) e. NN )
273		rplpwr 15276	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( abs ` A ) e. NN /\ ( abs ` N ) e. NN /\ ( ( k - 1 ) / 2 ) e. NN ) -> ( ( ( abs ` A ) gcd ( abs ` N ) ) = 1 -> ( ( ( abs ` A ) ^ ( ( k - 1 ) / 2 ) ) gcd ( abs ` N ) ) = 1 ) )
274	269, 272, 244, 273	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> ( ( ( abs ` A ) gcd ( abs ` N ) ) = 1 -> ( ( ( abs ` A ) ^ ( ( k - 1 ) / 2 ) ) gcd ( abs ` N ) ) = 1 ) )
275	260, 274	mpd 15	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> ( ( ( abs ` A ) ^ ( ( k - 1 ) / 2 ) ) gcd ( abs ` N ) ) = 1 )
276	254, 256, 275	3eqtr3d 2664	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> ( ( A ^ ( ( k - 1 ) / 2 ) ) gcd N ) = 1 )
277	276	breq2d 4665	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> ( k || ( ( A ^ ( ( k - 1 ) / 2 ) ) gcd N ) <-> k || 1 ) )
278	251, 277	sylibd 229	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> ( k || ( A ^ ( ( k - 1 ) / 2 ) ) -> k || 1 ) )
279	236, 278	mtod 189	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> -. k || ( A ^ ( ( k - 1 ) / 2 ) ) )
280		prmnn 15388	. . . . . . . . . . . . . . . . . . . . 21 |- ( k e. Prime -> k e. NN )
281	280	adantl 482	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) -> k e. NN )
282	281	ad2antrr 762	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> k e. NN )
283		dvdsval3 14987	. . . . . . . . . . . . . . . . . . 19 |- ( ( k e. NN /\ ( A ^ ( ( k - 1 ) / 2 ) ) e. ZZ ) -> ( k || ( A ^ ( ( k - 1 ) / 2 ) ) <-> ( ( A ^ ( ( k - 1 ) / 2 ) ) mod k ) = 0 ) )
284	282, 247, 283	syl2anc 693	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> ( k || ( A ^ ( ( k - 1 ) / 2 ) ) <-> ( ( A ^ ( ( k - 1 ) / 2 ) ) mod k ) = 0 ) )
285	284	necon3bbid 2831	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> ( -. k || ( A ^ ( ( k - 1 ) / 2 ) ) <-> ( ( A ^ ( ( k - 1 ) / 2 ) ) mod k ) =/= 0 ) )
286	279, 285	mpbid 222	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> ( ( A ^ ( ( k - 1 ) / 2 ) ) mod k ) =/= 0 )
287		lgsvalmod 25041	. . . . . . . . . . . . . . . . 17 |- ( ( A e. ZZ /\ k e. ( Prime \ { 2 } ) ) -> ( ( A /L k ) mod k ) = ( ( A ^ ( ( k - 1 ) / 2 ) ) mod k ) )
288	239, 242, 287	syl2anc 693	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> ( ( A /L k ) mod k ) = ( ( A ^ ( ( k - 1 ) / 2 ) ) mod k ) )
289	282	nnrpd 11870	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> k e. RR+ )
290		0mod 12701	. . . . . . . . . . . . . . . . 17 |- ( k e. RR+ -> ( 0 mod k ) = 0 )
291	289, 290	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> ( 0 mod k ) = 0 )
292	286, 288, 291	3netr4d 2871	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> ( ( A /L k ) mod k ) =/= ( 0 mod k ) )
293		oveq1 6657	. . . . . . . . . . . . . . . 16 |- ( ( A /L k ) = 0 -> ( ( A /L k ) mod k ) = ( 0 mod k ) )
294	293	necon3i 2826	. . . . . . . . . . . . . . 15 |- ( ( ( A /L k ) mod k ) =/= ( 0 mod k ) -> ( A /L k ) =/= 0 )
295	292, 294	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) /\ k =/= 2 ) -> ( A /L k ) =/= 0 )
296	233, 295	pm2.61dane 2881	. . . . . . . . . . . . 13 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) -> ( A /L k ) =/= 0 )
297		pczcl 15553	. . . . . . . . . . . . . . . 16 |- ( ( k e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( k pCnt N ) e. NN0 )
298	234, 212, 270, 297	syl12anc 1324	. . . . . . . . . . . . . . 15 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) -> ( k pCnt N ) e. NN0 )
299	298	nn0zd 11480	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) -> ( k pCnt N ) e. ZZ )
300	299	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) -> ( k pCnt N ) e. ZZ )
301		expclz 12885	. . . . . . . . . . . . . 14 |- ( ( ( A /L k ) e. CC /\ ( A /L k ) =/= 0 /\ ( k pCnt N ) e. ZZ ) -> ( ( A /L k ) ^ ( k pCnt N ) ) e. CC )
302		expne0i 12892	. . . . . . . . . . . . . 14 |- ( ( ( A /L k ) e. CC /\ ( A /L k ) =/= 0 /\ ( k pCnt N ) e. ZZ ) -> ( ( A /L k ) ^ ( k pCnt N ) ) =/= 0 )
303		neeq1 2856	. . . . . . . . . . . . . . 15 |- ( x = ( ( A /L k ) ^ ( k pCnt N ) ) -> ( x =/= 0 <-> ( ( A /L k ) ^ ( k pCnt N ) ) =/= 0 ) )
304	303	elrab 3363	. . . . . . . . . . . . . 14 |- ( ( ( A /L k ) ^ ( k pCnt N ) ) e. { x e. CC | x =/= 0 } <-> ( ( ( A /L k ) ^ ( k pCnt N ) ) e. CC /\ ( ( A /L k ) ^ ( k pCnt N ) ) =/= 0 ) )
305	301, 302, 304	sylanbrc 698	. . . . . . . . . . . . 13 |- ( ( ( A /L k ) e. CC /\ ( A /L k ) =/= 0 /\ ( k pCnt N ) e. ZZ ) -> ( ( A /L k ) ^ ( k pCnt N ) ) e. { x e. CC | x =/= 0 } )
306	205, 296, 300, 305	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ k || N ) -> ( ( A /L k ) ^ ( k pCnt N ) ) e. { x e. CC | x =/= 0 } )
307		dvdsabsb 15001	. . . . . . . . . . . . . . . . . . 19 |- ( ( k e. ZZ /\ N e. ZZ ) -> ( k || N <-> k || ( abs ` N ) ) )
308	201, 212, 307	syl2anc 693	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) -> ( k || N <-> k || ( abs ` N ) ) )
309	308	notbid 308	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) -> ( -. k || N <-> -. k || ( abs ` N ) ) )
310		pceq0 15575	. . . . . . . . . . . . . . . . . 18 |- ( ( k e. Prime /\ ( abs ` N ) e. NN ) -> ( ( k pCnt ( abs ` N ) ) = 0 <-> -. k || ( abs ` N ) ) )
311	234, 271, 310	syl2anc 693	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) -> ( ( k pCnt ( abs ` N ) ) = 0 <-> -. k || ( abs ` N ) ) )
312	212, 172	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) -> N e. QQ )
313		pcabs 15579	. . . . . . . . . . . . . . . . . . 19 |- ( ( k e. Prime /\ N e. QQ ) -> ( k pCnt ( abs ` N ) ) = ( k pCnt N ) )
314	234, 312, 313	syl2anc 693	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) -> ( k pCnt ( abs ` N ) ) = ( k pCnt N ) )
315	314	eqeq1d 2624	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) -> ( ( k pCnt ( abs ` N ) ) = 0 <-> ( k pCnt N ) = 0 ) )
316	309, 311, 315	3bitr2rd 297	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) -> ( ( k pCnt N ) = 0 <-> -. k || N ) )
317	316	biimpar 502	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ -. k || N ) -> ( k pCnt N ) = 0 )
318	317	oveq2d 6666	. . . . . . . . . . . . . 14 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ -. k || N ) -> ( ( A /L k ) ^ ( k pCnt N ) ) = ( ( A /L k ) ^ 0 ) )
319	204	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ -. k || N ) -> ( A /L k ) e. CC )
320	319	exp0d 13002	. . . . . . . . . . . . . 14 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ -. k || N ) -> ( ( A /L k ) ^ 0 ) = 1 )
321	318, 320	eqtrd 2656	. . . . . . . . . . . . 13 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ -. k || N ) -> ( ( A /L k ) ^ ( k pCnt N ) ) = 1 )
322		neeq1 2856	. . . . . . . . . . . . . . 15 |- ( x = 1 -> ( x =/= 0 <-> 1 =/= 0 ) )
323	322	elrab 3363	. . . . . . . . . . . . . 14 |- ( 1 e. { x e. CC | x =/= 0 } <-> ( 1 e. CC /\ 1 =/= 0 ) )
324	45, 4, 323	mpbir2an 955	. . . . . . . . . . . . 13 |- 1 e. { x e. CC | x =/= 0 }
325	321, 324	syl6eqel 2709	. . . . . . . . . . . 12 |- ( ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) /\ -. k || N ) -> ( ( A /L k ) ^ ( k pCnt N ) ) e. { x e. CC | x =/= 0 } )
326	306, 325	pm2.61dan 832	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. Prime ) -> ( ( A /L k ) ^ ( k pCnt N ) ) e. { x e. CC | x =/= 0 } )
327	324	a1i 11	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ -. k e. Prime ) -> 1 e. { x e. CC | x =/= 0 } )
328	326, 327	ifclda 4120	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) -> if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt N ) ) , 1 ) e. { x e. CC | x =/= 0 } )
329	328	adantr 481	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. ( 1 ... ( abs ` N ) ) ) -> if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt N ) ) , 1 ) e. { x e. CC | x =/= 0 } )
330	198, 329	eqeltrd 2701	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ k e. ( 1 ... ( abs ` N ) ) ) -> ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ` k ) e. { x e. CC | x =/= 0 } )
331		neeq1 2856	. . . . . . . . . . . 12 |- ( x = k -> ( x =/= 0 <-> k =/= 0 ) )
332	331	elrab 3363	. . . . . . . . . . 11 |- ( k e. { x e. CC | x =/= 0 } <-> ( k e. CC /\ k =/= 0 ) )
333		neeq1 2856	. . . . . . . . . . . 12 |- ( x = y -> ( x =/= 0 <-> y =/= 0 ) )
334	333	elrab 3363	. . . . . . . . . . 11 |- ( y e. { x e. CC | x =/= 0 } <-> ( y e. CC /\ y =/= 0 ) )
335		mulcl 10020	. . . . . . . . . . . . 13 |- ( ( k e. CC /\ y e. CC ) -> ( k x. y ) e. CC )
336	335	ad2ant2r 783	. . . . . . . . . . . 12 |- ( ( ( k e. CC /\ k =/= 0 ) /\ ( y e. CC /\ y =/= 0 ) ) -> ( k x. y ) e. CC )
337		mulne0 10669	. . . . . . . . . . . 12 |- ( ( ( k e. CC /\ k =/= 0 ) /\ ( y e. CC /\ y =/= 0 ) ) -> ( k x. y ) =/= 0 )
338	336, 337	jca 554	. . . . . . . . . . 11 |- ( ( ( k e. CC /\ k =/= 0 ) /\ ( y e. CC /\ y =/= 0 ) ) -> ( ( k x. y ) e. CC /\ ( k x. y ) =/= 0 ) )
339	332, 334, 338	syl2anb 496	. . . . . . . . . 10 |- ( ( k e. { x e. CC | x =/= 0 } /\ y e. { x e. CC | x =/= 0 } ) -> ( ( k x. y ) e. CC /\ ( k x. y ) =/= 0 ) )
340		neeq1 2856	. . . . . . . . . . 11 |- ( x = ( k x. y ) -> ( x =/= 0 <-> ( k x. y ) =/= 0 ) )
341	340	elrab 3363	. . . . . . . . . 10 |- ( ( k x. y ) e. { x e. CC | x =/= 0 } <-> ( ( k x. y ) e. CC /\ ( k x. y ) =/= 0 ) )
342	339, 341	sylibr 224	. . . . . . . . 9 |- ( ( k e. { x e. CC | x =/= 0 } /\ y e. { x e. CC | x =/= 0 } ) -> ( k x. y ) e. { x e. CC | x =/= 0 } )
343	342	adantl 482	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) /\ ( k e. { x e. CC | x =/= 0 } /\ y e. { x e. CC | x =/= 0 } ) ) -> ( k x. y ) e. { x e. CC | x =/= 0 } )
344	188, 330, 343	seqcl 12821	. . . . . . 7 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) -> ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) e. { x e. CC | x =/= 0 } )
345		neeq1 2856	. . . . . . . . 9 |- ( x = ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) -> ( x =/= 0 <-> ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) =/= 0 ) )
346	345	elrab 3363	. . . . . . . 8 |- ( ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) e. { x e. CC | x =/= 0 } <-> ( ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) e. CC /\ ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) =/= 0 ) )
347	346	simprbi 480	. . . . . . 7 |- ( ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) e. { x e. CC | x =/= 0 } -> ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) =/= 0 )
348	344, 347	syl 17	. . . . . 6 |- ( ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) /\ ( A gcd N ) = 1 ) -> ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) =/= 0 )
349	348	ex 450	. . . . 5 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( ( A gcd N ) = 1 -> ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) =/= 0 ) )
350	187, 349	impbid 202	. . . 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 ) ) =/= 0 <-> ( A gcd N ) = 1 ) )
351	38, 61, 350	3bitrd 294	. . 3 |- ( ( A e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( ( A /L N ) =/= 0 <-> ( A gcd N ) = 1 ) )
352	351	3expa 1265	. 2 |- ( ( ( A e. ZZ /\ N e. ZZ ) /\ N =/= 0 ) -> ( ( A /L N ) =/= 0 <-> ( A gcd N ) = 1 ) )
353	35, 352	pm2.61dane 2881	1 |- ( ( A e. ZZ /\ N e. ZZ ) -> ( ( A /L N ) =/= 0 <-> ( A gcd N ) = 1 ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   \ cdif 3571   ifcif 4086   {csn 4177   {cpr 4179   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   2c2 11070   7c7 11075   8c8 11076   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   QQcq 11788   RR+crp 11832   ...cfz 12326   mod cmo 12668   seqcseq 12801   ^cexp 12860   abscabs 13974   || cdvds 14983   gcd cgcd 15216   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:  lgsabs1  25061  lgsprme0  25064  lgsdirnn0  25069  lgsqr  25076  lgsdchr  25080  lgsquad3  25112  2lgsoddprm  25141