Theorem lgsdi 13693

Description: The Legendre symbol is completely multiplicative in its right argument. Generalization of theorem 9.9(b) in [ApostolNT] p. 188 (which assumes that M and N are odd positive integers). (Contributed by Mario Carneiro, 5-Feb-2015.)

Assertion

Ref	Expression
lgsdi	|- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )

Proof of Theorem lgsdi

Dummy variables k n v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		3anrot 978	. . . . 5 |- ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) <-> ( M e. ZZ /\ N e. ZZ /\ A e. ZZ ) )
2		lgsdilem 13683	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ /\ A e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> if ( ( A < 0 /\ ( M x. N ) < 0 ) , -u 1 , 1 ) = ( if ( ( A < 0 /\ M < 0 ) , -u 1 , 1 ) x. if ( ( A < 0 /\ N < 0 ) , -u 1 , 1 ) ) )
3	1, 2	sylanb 282	. . . 4 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> if ( ( A < 0 /\ ( M x. N ) < 0 ) , -u 1 , 1 ) = ( if ( ( A < 0 /\ M < 0 ) , -u 1 , 1 ) x. if ( ( A < 0 /\ N < 0 ) , -u 1 , 1 ) ) )
4		ancom 264	. . . . 5 |- ( ( ( M x. N ) < 0 /\ A < 0 ) <-> ( A < 0 /\ ( M x. N ) < 0 ) )
5		ifbi 3545	. . . . 5 |- ( ( ( ( M x. N ) < 0 /\ A < 0 ) <-> ( A < 0 /\ ( M x. N ) < 0 ) ) -> if ( ( ( M x. N ) < 0 /\ A < 0 ) , -u 1 , 1 ) = if ( ( A < 0 /\ ( M x. N ) < 0 ) , -u 1 , 1 ) )
6	4, 5	ax-mp 5	. . . 4 |- if ( ( ( M x. N ) < 0 /\ A < 0 ) , -u 1 , 1 ) = if ( ( A < 0 /\ ( M x. N ) < 0 ) , -u 1 , 1 )
7		ancom 264	. . . . . 6 |- ( ( M < 0 /\ A < 0 ) <-> ( A < 0 /\ M < 0 ) )
8		ifbi 3545	. . . . . 6 |- ( ( ( M < 0 /\ A < 0 ) <-> ( A < 0 /\ M < 0 ) ) -> if ( ( M < 0 /\ A < 0 ) , -u 1 , 1 ) = if ( ( A < 0 /\ M < 0 ) , -u 1 , 1 ) )
9	7, 8	ax-mp 5	. . . . 5 |- if ( ( M < 0 /\ A < 0 ) , -u 1 , 1 ) = if ( ( A < 0 /\ M < 0 ) , -u 1 , 1 )
10		ancom 264	. . . . . 6 |- ( ( N < 0 /\ A < 0 ) <-> ( A < 0 /\ N < 0 ) )
11		ifbi 3545	. . . . . 6 |- ( ( ( N < 0 /\ A < 0 ) <-> ( A < 0 /\ N < 0 ) ) -> if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) = if ( ( A < 0 /\ N < 0 ) , -u 1 , 1 ) )
12	10, 11	ax-mp 5	. . . . 5 |- if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) = if ( ( A < 0 /\ N < 0 ) , -u 1 , 1 )
13	9, 12	oveq12i 5863	. . . 4 |- ( if ( ( M < 0 /\ A < 0 ) , -u 1 , 1 ) x. if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) ) = ( if ( ( A < 0 /\ M < 0 ) , -u 1 , 1 ) x. if ( ( A < 0 /\ N < 0 ) , -u 1 , 1 ) )
14	3, 6, 13	3eqtr4g 2228	. . 3 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> if ( ( ( M x. N ) < 0 /\ A < 0 ) , -u 1 , 1 ) = ( if ( ( M < 0 /\ A < 0 ) , -u 1 , 1 ) x. if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) ) )
15		simpl2 996	. . . . . . . 8 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> M e. ZZ )
16		simpl3 997	. . . . . . . 8 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> N e. ZZ )
17	15, 16	zmulcld 9329	. . . . . . 7 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( M x. N ) e. ZZ )
18	15	zcnd 9324	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> M e. CC )
19	16	zcnd 9324	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> N e. CC )
20		simprl 526	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> M =/= 0 )
21		0z 9212	. . . . . . . . . . 11 |- 0 e. ZZ
22		zapne 9275	. . . . . . . . . . 11 |- ( ( M e. ZZ /\ 0 e. ZZ ) -> ( M # 0 <-> M =/= 0 ) )
23	15, 21, 22	sylancl 411	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( M # 0 <-> M =/= 0 ) )
24	20, 23	mpbird 166	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> M # 0 )
25		simprr 527	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> N =/= 0 )
26		zapne 9275	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> ( N # 0 <-> N =/= 0 ) )
27	16, 21, 26	sylancl 411	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( N # 0 <-> N =/= 0 ) )
28	25, 27	mpbird 166	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> N # 0 )
29	18, 19, 24, 28	mulap0d 8565	. . . . . . . 8 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( M x. N ) # 0 )
30		zapne 9275	. . . . . . . . 9 |- ( ( ( M x. N ) e. ZZ /\ 0 e. ZZ ) -> ( ( M x. N ) # 0 <-> ( M x. N ) =/= 0 ) )
31	17, 21, 30	sylancl 411	. . . . . . . 8 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( ( M x. N ) # 0 <-> ( M x. N ) =/= 0 ) )
32	29, 31	mpbid 146	. . . . . . 7 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( M x. N ) =/= 0 )
33		nnabscl 11053	. . . . . . 7 |- ( ( ( M x. N ) e. ZZ /\ ( M x. N ) =/= 0 ) -> ( abs ` ( M x. N ) ) e. NN )
34	17, 32, 33	syl2anc 409	. . . . . 6 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( abs ` ( M x. N ) ) e. NN )
35		nnuz 9511	. . . . . 6 |- NN = ( ZZ>= ` 1 )
36	34, 35	eleqtrdi 2263	. . . . 5 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( abs ` ( M x. N ) ) e. ( ZZ>= ` 1 ) )
37		simpl1 995	. . . . . . . 8 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> A e. ZZ )
38		eqid 2170	. . . . . . . . 9 |- ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt M ) ) , 1 ) ) = ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt M ) ) , 1 ) )
39	38	lgsfcl3 13677	. . . . . . . 8 |- ( ( A e. ZZ /\ M e. ZZ /\ M =/= 0 ) -> ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt M ) ) , 1 ) ) : NN --> ZZ )
40	37, 15, 20, 39	syl3anc 1233	. . . . . . 7 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt M ) ) , 1 ) ) : NN --> ZZ )
41		elnnuz 9512	. . . . . . . 8 |- ( k e. NN <-> k e. ( ZZ>= ` 1 ) )
42	41	biimpri 132	. . . . . . 7 |- ( k e. ( ZZ>= ` 1 ) -> k e. NN )
43		ffvelrn 5627	. . . . . . 7 |- ( ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt M ) ) , 1 ) ) : NN --> ZZ /\ k e. NN ) -> ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt M ) ) , 1 ) ) ` k ) e. ZZ )
44	40, 42, 43	syl2an 287	. . . . . 6 |- ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) -> ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt M ) ) , 1 ) ) ` k ) e. ZZ )
45	44	zcnd 9324	. . . . 5 |- ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) -> ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt M ) ) , 1 ) ) ` k ) e. CC )
46		eqid 2170	. . . . . . . . 9 |- ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) = ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) )
47	46	lgsfcl3 13677	. . . . . . . 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 )
48	37, 16, 25, 47	syl3anc 1233	. . . . . . 7 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) : NN --> ZZ )
49		ffvelrn 5627	. . . . . . 7 |- ( ( ( 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 )
50	48, 42, 49	syl2an 287	. . . . . 6 |- ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) -> ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ` k ) e. ZZ )
51	50	zcnd 9324	. . . . 5 |- ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) -> ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ` k ) e. CC )
52		simpr 109	. . . . . . . . . . 11 |- ( ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> k e. Prime )
53	15	ad2antrr 485	. . . . . . . . . . 11 |- ( ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> M e. ZZ )
54	20	ad2antrr 485	. . . . . . . . . . 11 |- ( ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> M =/= 0 )
55	16	ad2antrr 485	. . . . . . . . . . 11 |- ( ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> N e. ZZ )
56	25	ad2antrr 485	. . . . . . . . . . 11 |- ( ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> N =/= 0 )
57		pcmul 12244	. . . . . . . . . . 11 |- ( ( k e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( k pCnt ( M x. N ) ) = ( ( k pCnt M ) + ( k pCnt N ) ) )
58	52, 53, 54, 55, 56, 57	syl122anc 1242	. . . . . . . . . 10 |- ( ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> ( k pCnt ( M x. N ) ) = ( ( k pCnt M ) + ( k pCnt N ) ) )
59	58	oveq2d 5867	. . . . . . . . 9 |- ( ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> ( ( A /L k ) ^ ( k pCnt ( M x. N ) ) ) = ( ( A /L k ) ^ ( ( k pCnt M ) + ( k pCnt N ) ) ) )
60	37	ad2antrr 485	. . . . . . . . . . . 12 |- ( ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> A e. ZZ )
61		prmz 12054	. . . . . . . . . . . . 13 |- ( k e. Prime -> k e. ZZ )
62	61	adantl 275	. . . . . . . . . . . 12 |- ( ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> k e. ZZ )
63		lgscl 13670	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ k e. ZZ ) -> ( A /L k ) e. ZZ )
64	60, 62, 63	syl2anc 409	. . . . . . . . . . 11 |- ( ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> ( A /L k ) e. ZZ )
65	64	zcnd 9324	. . . . . . . . . 10 |- ( ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> ( A /L k ) e. CC )
66		pczcl 12241	. . . . . . . . . . 11 |- ( ( k e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( k pCnt N ) e. NN0 )
67	52, 55, 56, 66	syl12anc 1231	. . . . . . . . . 10 |- ( ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> ( k pCnt N ) e. NN0 )
68		pczcl 12241	. . . . . . . . . . 11 |- ( ( k e. Prime /\ ( M e. ZZ /\ M =/= 0 ) ) -> ( k pCnt M ) e. NN0 )
69	52, 53, 54, 68	syl12anc 1231	. . . . . . . . . 10 |- ( ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> ( k pCnt M ) e. NN0 )
70	65, 67, 69	expaddd 10600	. . . . . . . . 9 |- ( ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> ( ( A /L k ) ^ ( ( k pCnt M ) + ( k pCnt N ) ) ) = ( ( ( A /L k ) ^ ( k pCnt M ) ) x. ( ( A /L k ) ^ ( k pCnt N ) ) ) )
71	59, 70	eqtrd 2203	. . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> ( ( A /L k ) ^ ( k pCnt ( M x. N ) ) ) = ( ( ( A /L k ) ^ ( k pCnt M ) ) x. ( ( A /L k ) ^ ( k pCnt N ) ) ) )
72		iftrue 3530	. . . . . . . . 9 |- ( k e. Prime -> if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt ( M x. N ) ) ) , 1 ) = ( ( A /L k ) ^ ( k pCnt ( M x. N ) ) ) )
73	72	adantl 275	. . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt ( M x. N ) ) ) , 1 ) = ( ( A /L k ) ^ ( k pCnt ( M x. N ) ) ) )
74		iftrue 3530	. . . . . . . . . 10 |- ( k e. Prime -> if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt M ) ) , 1 ) = ( ( A /L k ) ^ ( k pCnt M ) ) )
75		iftrue 3530	. . . . . . . . . 10 |- ( k e. Prime -> if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt N ) ) , 1 ) = ( ( A /L k ) ^ ( k pCnt N ) ) )
76	74, 75	oveq12d 5869	. . . . . . . . 9 |- ( k e. Prime -> ( if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt M ) ) , 1 ) x. if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt N ) ) , 1 ) ) = ( ( ( A /L k ) ^ ( k pCnt M ) ) x. ( ( A /L k ) ^ ( k pCnt N ) ) ) )
77	76	adantl 275	. . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> ( if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt M ) ) , 1 ) x. if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt N ) ) , 1 ) ) = ( ( ( A /L k ) ^ ( k pCnt M ) ) x. ( ( A /L k ) ^ ( k pCnt N ) ) ) )
78	71, 73, 77	3eqtr4rd 2214	. . . . . . 7 |- ( ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> ( if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt M ) ) , 1 ) x. if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt N ) ) , 1 ) ) = if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt ( M x. N ) ) ) , 1 ) )
79		1t1e1 9019	. . . . . . . . 9 |- ( 1 x. 1 ) = 1
80		iffalse 3533	. . . . . . . . . 10 |- ( -. k e. Prime -> if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt M ) ) , 1 ) = 1 )
81		iffalse 3533	. . . . . . . . . 10 |- ( -. k e. Prime -> if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt N ) ) , 1 ) = 1 )
82	80, 81	oveq12d 5869	. . . . . . . . 9 |- ( -. k e. Prime -> ( if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt M ) ) , 1 ) x. if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt N ) ) , 1 ) ) = ( 1 x. 1 ) )
83		iffalse 3533	. . . . . . . . 9 |- ( -. k e. Prime -> if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt ( M x. N ) ) ) , 1 ) = 1 )
84	79, 82, 83	3eqtr4a 2229	. . . . . . . 8 |- ( -. k e. Prime -> ( if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt M ) ) , 1 ) x. if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt N ) ) , 1 ) ) = if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt ( M x. N ) ) ) , 1 ) )
85	84	adantl 275	. . . . . . 7 |- ( ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) /\ -. k e. Prime ) -> ( if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt M ) ) , 1 ) x. if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt N ) ) , 1 ) ) = if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt ( M x. N ) ) ) , 1 ) )
86		prmdc 12073	. . . . . . . . . 10 |- ( k e. NN -> DECID k e. Prime )
87		exmiddc 831	. . . . . . . . . 10 |- (DECID k e. Prime -> ( k e. Prime \/ -. k e. Prime ) )
88	86, 87	syl 14	. . . . . . . . 9 |- ( k e. NN -> ( k e. Prime \/ -. k e. Prime ) )
89	42, 88	syl 14	. . . . . . . 8 |- ( k e. ( ZZ>= ` 1 ) -> ( k e. Prime \/ -. k e. Prime ) )
90	89	adantl 275	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) -> ( k e. Prime \/ -. k e. Prime ) )
91	78, 85, 90	mpjaodan 793	. . . . . 6 |- ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) -> ( if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt M ) ) , 1 ) x. if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt N ) ) , 1 ) ) = if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt ( M x. N ) ) ) , 1 ) )
92		eleq1w 2231	. . . . . . . . 9 |- ( n = k -> ( n e. Prime <-> k e. Prime ) )
93		oveq2 5859	. . . . . . . . . 10 |- ( n = k -> ( A /L n ) = ( A /L k ) )
94		oveq1 5858	. . . . . . . . . 10 |- ( n = k -> ( n pCnt M ) = ( k pCnt M ) )
95	93, 94	oveq12d 5869	. . . . . . . . 9 |- ( n = k -> ( ( A /L n ) ^ ( n pCnt M ) ) = ( ( A /L k ) ^ ( k pCnt M ) ) )
96	92, 95	ifbieq1d 3547	. . . . . . . 8 |- ( n = k -> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt M ) ) , 1 ) = if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt M ) ) , 1 ) )
97	42	adantl 275	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) -> k e. NN )
98		zexpcl 10480	. . . . . . . . . 10 |- ( ( ( A /L k ) e. ZZ /\ ( k pCnt M ) e. NN0 ) -> ( ( A /L k ) ^ ( k pCnt M ) ) e. ZZ )
99	64, 69, 98	syl2anc 409	. . . . . . . . 9 |- ( ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> ( ( A /L k ) ^ ( k pCnt M ) ) e. ZZ )
100		1zzd 9228	. . . . . . . . 9 |- ( ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) /\ -. k e. Prime ) -> 1 e. ZZ )
101	97, 86	syl 14	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) -> DECID k e. Prime )
102	99, 100, 101	ifcldadc 3554	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) -> if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt M ) ) , 1 ) e. ZZ )
103	38, 96, 97, 102	fvmptd3 5587	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) -> ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt M ) ) , 1 ) ) ` k ) = if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt M ) ) , 1 ) )
104		oveq1 5858	. . . . . . . . . 10 |- ( n = k -> ( n pCnt N ) = ( k pCnt N ) )
105	93, 104	oveq12d 5869	. . . . . . . . 9 |- ( n = k -> ( ( A /L n ) ^ ( n pCnt N ) ) = ( ( A /L k ) ^ ( k pCnt N ) ) )
106	92, 105	ifbieq1d 3547	. . . . . . . 8 |- ( n = k -> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) = if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt N ) ) , 1 ) )
107		zexpcl 10480	. . . . . . . . . 10 |- ( ( ( A /L k ) e. ZZ /\ ( k pCnt N ) e. NN0 ) -> ( ( A /L k ) ^ ( k pCnt N ) ) e. ZZ )
108	64, 67, 107	syl2anc 409	. . . . . . . . 9 |- ( ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> ( ( A /L k ) ^ ( k pCnt N ) ) e. ZZ )
109	108, 100, 101	ifcldadc 3554	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) -> if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt N ) ) , 1 ) e. ZZ )
110	46, 106, 97, 109	fvmptd3 5587	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) -> ( ( 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 ) )
111	103, 110	oveq12d 5869	. . . . . 6 |- ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) -> ( ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt M ) ) , 1 ) ) ` k ) x. ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ` k ) ) = ( if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt M ) ) , 1 ) x. if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt N ) ) , 1 ) ) )
112		eqid 2170	. . . . . . 7 |- ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt ( M x. N ) ) ) , 1 ) ) = ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt ( M x. N ) ) ) , 1 ) )
113		oveq1 5858	. . . . . . . . 9 |- ( n = k -> ( n pCnt ( M x. N ) ) = ( k pCnt ( M x. N ) ) )
114	93, 113	oveq12d 5869	. . . . . . . 8 |- ( n = k -> ( ( A /L n ) ^ ( n pCnt ( M x. N ) ) ) = ( ( A /L k ) ^ ( k pCnt ( M x. N ) ) ) )
115	92, 114	ifbieq1d 3547	. . . . . . 7 |- ( n = k -> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt ( M x. N ) ) ) , 1 ) = if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt ( M x. N ) ) ) , 1 ) )
116	17	ad2antrr 485	. . . . . . . . . 10 |- ( ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> ( M x. N ) e. ZZ )
117	32	ad2antrr 485	. . . . . . . . . 10 |- ( ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> ( M x. N ) =/= 0 )
118		pczcl 12241	. . . . . . . . . 10 |- ( ( k e. Prime /\ ( ( M x. N ) e. ZZ /\ ( M x. N ) =/= 0 ) ) -> ( k pCnt ( M x. N ) ) e. NN0 )
119	52, 116, 117, 118	syl12anc 1231	. . . . . . . . 9 |- ( ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> ( k pCnt ( M x. N ) ) e. NN0 )
120		zexpcl 10480	. . . . . . . . 9 |- ( ( ( A /L k ) e. ZZ /\ ( k pCnt ( M x. N ) ) e. NN0 ) -> ( ( A /L k ) ^ ( k pCnt ( M x. N ) ) ) e. ZZ )
121	64, 119, 120	syl2anc 409	. . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) /\ k e. Prime ) -> ( ( A /L k ) ^ ( k pCnt ( M x. N ) ) ) e. ZZ )
122	121, 100, 101	ifcldadc 3554	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) -> if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt ( M x. N ) ) ) , 1 ) e. ZZ )
123	112, 115, 97, 122	fvmptd3 5587	. . . . . 6 |- ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) -> ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt ( M x. N ) ) ) , 1 ) ) ` k ) = if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt ( M x. N ) ) ) , 1 ) )
124	91, 111, 123	3eqtr4rd 2214	. . . . 5 |- ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) -> ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt ( M x. N ) ) ) , 1 ) ) ` k ) = ( ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt M ) ) , 1 ) ) ` k ) x. ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ` k ) ) )
125	36, 45, 51, 124	prod3fmul 11493	. . . 4 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt ( M x. N ) ) ) , 1 ) ) ) ` ( abs ` ( M x. N ) ) ) = ( ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt M ) ) , 1 ) ) ) ` ( abs ` ( M x. N ) ) ) x. ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` ( M x. N ) ) ) ) )
126	37, 15, 16, 20, 25, 38	lgsdilem2 13692	. . . . 5 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt M ) ) , 1 ) ) ) ` ( abs ` M ) ) = ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt M ) ) , 1 ) ) ) ` ( abs ` ( M x. N ) ) ) )
127	37, 16, 15, 25, 20, 46	lgsdilem2 13692	. . . . . 6 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) = ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` ( N x. M ) ) ) )
128	18, 19	mulcomd 7930	. . . . . . . 8 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( M x. N ) = ( N x. M ) )
129	128	fveq2d 5498	. . . . . . 7 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( abs ` ( M x. N ) ) = ( abs ` ( N x. M ) ) )
130	129	fveq2d 5498	. . . . . 6 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` ( M x. N ) ) ) = ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` ( N x. M ) ) ) )
131	127, 130	eqtr4d 2206	. . . . 5 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) = ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` ( M x. N ) ) ) )
132	126, 131	oveq12d 5869	. . . 4 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt M ) ) , 1 ) ) ) ` ( abs ` M ) ) x. ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) ) = ( ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt M ) ) , 1 ) ) ) ` ( abs ` ( M x. N ) ) ) x. ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` ( M x. N ) ) ) ) )
133	125, 132	eqtr4d 2206	. . 3 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt ( M x. N ) ) ) , 1 ) ) ) ` ( abs ` ( M x. N ) ) ) = ( ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt M ) ) , 1 ) ) ) ` ( abs ` M ) ) x. ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) ) )
134	14, 133	oveq12d 5869	. 2 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( if ( ( ( M x. N ) < 0 /\ A < 0 ) , -u 1 , 1 ) x. ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt ( M x. N ) ) ) , 1 ) ) ) ` ( abs ` ( M x. N ) ) ) ) = ( ( if ( ( M < 0 /\ 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 M ) ) , 1 ) ) ) ` ( abs ` M ) ) x. ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) ) ) )
135	112	lgsval4 13676	. . 3 |- ( ( A e. ZZ /\ ( M x. N ) e. ZZ /\ ( M x. N ) =/= 0 ) -> ( A /L ( M x. N ) ) = ( if ( ( ( M x. N ) < 0 /\ A < 0 ) , -u 1 , 1 ) x. ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt ( M x. N ) ) ) , 1 ) ) ) ` ( abs ` ( M x. N ) ) ) ) )
136	37, 17, 32, 135	syl3anc 1233	. 2 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( A /L ( M x. N ) ) = ( if ( ( ( M x. N ) < 0 /\ A < 0 ) , -u 1 , 1 ) x. ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt ( M x. N ) ) ) , 1 ) ) ) ` ( abs ` ( M x. N ) ) ) ) )
137	38	lgsval4 13676	. . . . 5 |- ( ( A e. ZZ /\ M e. ZZ /\ M =/= 0 ) -> ( A /L M ) = ( if ( ( M < 0 /\ A < 0 ) , -u 1 , 1 ) x. ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt M ) ) , 1 ) ) ) ` ( abs ` M ) ) ) )
138	37, 15, 20, 137	syl3anc 1233	. . . 4 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( A /L M ) = ( if ( ( M < 0 /\ A < 0 ) , -u 1 , 1 ) x. ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt M ) ) , 1 ) ) ) ` ( abs ` M ) ) ) )
139	46	lgsval4 13676	. . . . 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 ) ) ) )
140	37, 16, 25, 139	syl3anc 1233	. . . 4 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ 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 ) ) ) )
141	138, 140	oveq12d 5869	. . 3 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( ( A /L M ) x. ( A /L N ) ) = ( ( if ( ( M < 0 /\ A < 0 ) , -u 1 , 1 ) x. ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt M ) ) , 1 ) ) ) ` ( abs ` M ) ) ) 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 ) ) ) ) )
142		neg1z 9233	. . . . . . 7 |- -u 1 e. ZZ
143	142	a1i 9	. . . . . 6 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> -u 1 e. ZZ )
144		1zzd 9228	. . . . . 6 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> 1 e. ZZ )
145		zdclt 9278	. . . . . . . 8 |- ( ( M e. ZZ /\ 0 e. ZZ ) -> DECID M < 0 )
146	15, 21, 145	sylancl 411	. . . . . . 7 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> DECID M < 0 )
147		zdclt 9278	. . . . . . . 8 |- ( ( A e. ZZ /\ 0 e. ZZ ) -> DECID A < 0 )
148	37, 21, 147	sylancl 411	. . . . . . 7 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> DECID A < 0 )
149		dcan2 929	. . . . . . 7 |- (DECID M < 0 -> (DECID A < 0 -> DECID ( M < 0 /\ A < 0 ) ) )
150	146, 148, 149	sylc 62	. . . . . 6 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> DECID ( M < 0 /\ A < 0 ) )
151	143, 144, 150	ifcldcd 3560	. . . . 5 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> if ( ( M < 0 /\ A < 0 ) , -u 1 , 1 ) e. ZZ )
152	151	zcnd 9324	. . . 4 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> if ( ( M < 0 /\ A < 0 ) , -u 1 , 1 ) e. CC )
153	40	ffvelrnda 5629	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. NN ) -> ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt M ) ) , 1 ) ) ` k ) e. ZZ )
154		zmulcl 9254	. . . . . . . 8 |- ( ( k e. ZZ /\ v e. ZZ ) -> ( k x. v ) e. ZZ )
155	154	adantl 275	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ ( k e. ZZ /\ v e. ZZ ) ) -> ( k x. v ) e. ZZ )
156	35, 144, 153, 155	seqf 10406	. . . . . 6 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt M ) ) , 1 ) ) ) : NN --> ZZ )
157		nnabscl 11053	. . . . . . 7 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` M ) e. NN )
158	15, 20, 157	syl2anc 409	. . . . . 6 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( abs ` M ) e. NN )
159	156, 158	ffvelrnd 5630	. . . . 5 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt M ) ) , 1 ) ) ) ` ( abs ` M ) ) e. ZZ )
160	159	zcnd 9324	. . . 4 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt M ) ) , 1 ) ) ) ` ( abs ` M ) ) e. CC )
161		zdclt 9278	. . . . . . . 8 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N < 0 )
162	16, 21, 161	sylancl 411	. . . . . . 7 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> DECID N < 0 )
163		dcan2 929	. . . . . . 7 |- (DECID N < 0 -> (DECID A < 0 -> DECID ( N < 0 /\ A < 0 ) ) )
164	162, 148, 163	sylc 62	. . . . . 6 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> DECID ( N < 0 /\ A < 0 ) )
165	143, 144, 164	ifcldcd 3560	. . . . 5 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) e. ZZ )
166	165	zcnd 9324	. . . 4 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> if ( ( N < 0 /\ A < 0 ) , -u 1 , 1 ) e. CC )
167	48	ffvelrnda 5629	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. NN ) -> ( ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ` k ) e. ZZ )
168	35, 144, 167, 155	seqf 10406	. . . . . 6 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) : NN --> ZZ )
169		nnabscl 11053	. . . . . . 7 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. NN )
170	16, 25, 169	syl2anc 409	. . . . . 6 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( abs ` N ) e. NN )
171	168, 170	ffvelrnd 5630	. . . . 5 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) e. ZZ )
172	171	zcnd 9324	. . . 4 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) e. CC )
173	152, 160, 166, 172	mul4d 8063	. . 3 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( ( if ( ( M < 0 /\ A < 0 ) , -u 1 , 1 ) x. ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt M ) ) , 1 ) ) ) ` ( abs ` M ) ) ) 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 ( ( M < 0 /\ 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 M ) ) , 1 ) ) ) ` ( abs ` M ) ) x. ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) ) ) )
174	141, 173	eqtrd 2203	. 2 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( ( A /L M ) x. ( A /L N ) ) = ( ( if ( ( M < 0 /\ 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 M ) ) , 1 ) ) ) ` ( abs ` M ) ) x. ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( ( A /L n ) ^ ( n pCnt N ) ) , 1 ) ) ) ` ( abs ` N ) ) ) ) )
175	134, 136, 174	3eqtr4d 2213	1 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 703  DECID wdc 829   /\ w3a 973   = wceq 1348   e. wcel 2141   =/= wne 2340   ifcif 3525   class class class wbr 3987   |-> cmpt 4048   -->wf 5192   ` cfv 5196  (class class class)co 5851   0cc0 7763   1c1 7764   + caddc 7766   x. cmul 7768   < clt 7943   -ucneg 8080   # cap 8489   NNcn 8867   NN0cn0 9124   ZZcz 9201   ZZ>=cuz 9476   seqcseq 10390   ^cexp 10464   abscabs 10950   Primecprime 12050   pCnt cpc 12227   /Lclgs 13653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7854  ax-resscn 7855  ax-1cn 7856  ax-1re 7857  ax-icn 7858  ax-addcl 7859  ax-addrcl 7860  ax-mulcl 7861  ax-mulrcl 7862  ax-addcom 7863  ax-mulcom 7864  ax-addass 7865  ax-mulass 7866  ax-distr 7867  ax-i2m1 7868  ax-0lt1 7869  ax-1rid 7870  ax-0id 7871  ax-rnegex 7872  ax-precex 7873  ax-cnre 7874  ax-pre-ltirr 7875  ax-pre-ltwlin 7876  ax-pre-lttrn 7877  ax-pre-apti 7878  ax-pre-ltadd 7879  ax-pre-mulgt0 7880  ax-pre-mulext 7881  ax-arch 7882  ax-caucvg 7883

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-xor 1371  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-isom 5205  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-recs 6282  df-irdg 6347  df-frec 6368  df-1o 6393  df-2o 6394  df-oadd 6397  df-er 6510  df-en 6716  df-dom 6717  df-fin 6718  df-sup 6958  df-inf 6959  df-pnf 7945  df-mnf 7946  df-xr 7947  df-ltxr 7948  df-le 7949  df-sub 8081  df-neg 8082  df-reap 8483  df-ap 8490  df-div 8579  df-inn 8868  df-2 8926  df-3 8927  df-4 8928  df-5 8929  df-6 8930  df-7 8931  df-8 8932  df-n0 9125  df-z 9202  df-uz 9477  df-q 9568  df-rp 9600  df-fz 9955  df-fzo 10088  df-fl 10215  df-mod 10268  df-seqfrec 10391  df-exp 10465  df-ihash 10699  df-cj 10795  df-re 10796  df-im 10797  df-rsqrt 10951  df-abs 10952  df-clim 11231  df-proddc 11503  df-dvds 11739  df-gcd 11887  df-prm 12051  df-phi 12154  df-pc 12228  df-lgs 13654

This theorem is referenced by:  lgssq2  13697  lgsdinn0  13704