Theorem lgsdi 15278

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 985	. . . . 5 |- ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) <-> ( M e. ZZ /\ N e. ZZ /\ A e. ZZ ) )
2		lgsdilem 15268	. . . . 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 284	. . . 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 266	. . . . 5 |- ( ( ( M x. N ) < 0 /\ A < 0 ) <-> ( A < 0 /\ ( M x. N ) < 0 ) )
5		ifbi 3581	. . . . 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 266	. . . . . 6 |- ( ( M < 0 /\ A < 0 ) <-> ( A < 0 /\ M < 0 ) )
8		ifbi 3581	. . . . . 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 266	. . . . . 6 |- ( ( N < 0 /\ A < 0 ) <-> ( A < 0 /\ N < 0 ) )
11		ifbi 3581	. . . . . 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 5934	. . . 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 2254	. . 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 1003	. . . . . . . 8 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> M e. ZZ )
16		simpl3 1004	. . . . . . . 8 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> N e. ZZ )
17	15, 16	zmulcld 9454	. . . . . . 7 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( M x. N ) e. ZZ )
18	15	zcnd 9449	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> M e. CC )
19	16	zcnd 9449	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> N e. CC )
20		simprl 529	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> M =/= 0 )
21		0z 9337	. . . . . . . . . . 11 |- 0 e. ZZ
22		zapne 9400	. . . . . . . . . . 11 |- ( ( M e. ZZ /\ 0 e. ZZ ) -> ( M # 0 <-> M =/= 0 ) )
23	15, 21, 22	sylancl 413	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( M # 0 <-> M =/= 0 ) )
24	20, 23	mpbird 167	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> M # 0 )
25		simprr 531	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> N =/= 0 )
26		zapne 9400	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> ( N # 0 <-> N =/= 0 ) )
27	16, 21, 26	sylancl 413	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( N # 0 <-> N =/= 0 ) )
28	25, 27	mpbird 167	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> N # 0 )
29	18, 19, 24, 28	mulap0d 8685	. . . . . . . 8 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( M x. N ) # 0 )
30		zapne 9400	. . . . . . . . 9 |- ( ( ( M x. N ) e. ZZ /\ 0 e. ZZ ) -> ( ( M x. N ) # 0 <-> ( M x. N ) =/= 0 ) )
31	17, 21, 30	sylancl 413	. . . . . . . 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 147	. . . . . . 7 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( M x. N ) =/= 0 )
33		nnabscl 11265	. . . . . . 7 |- ( ( ( M x. N ) e. ZZ /\ ( M x. N ) =/= 0 ) -> ( abs ` ( M x. N ) ) e. NN )
34	17, 32, 33	syl2anc 411	. . . . . 6 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( abs ` ( M x. N ) ) e. NN )
35		nnuz 9637	. . . . . 6 |- NN = ( ZZ>= ` 1 )
36	34, 35	eleqtrdi 2289	. . . . 5 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( abs ` ( M x. N ) ) e. ( ZZ>= ` 1 ) )
37		simpl1 1002	. . . . . . . 8 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> A e. ZZ )
38		eqid 2196	. . . . . . . . 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 15262	. . . . . . . 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 1249	. . . . . . 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 9638	. . . . . . . 8 |- ( k e. NN <-> k e. ( ZZ>= ` 1 ) )
42	41	biimpri 133	. . . . . . 7 |- ( k e. ( ZZ>= ` 1 ) -> k e. NN )
43		ffvelcdm 5695	. . . . . . 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 289	. . . . . 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 9449	. . . . 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 2196	. . . . . . . . 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 15262	. . . . . . . 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 1249	. . . . . . 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		ffvelcdm 5695	. . . . . . 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 289	. . . . . 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 9449	. . . . 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 110	. . . . . . . . . . 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 488	. . . . . . . . . . 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 488	. . . . . . . . . . 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 488	. . . . . . . . . . 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 488	. . . . . . . . . . 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 12470	. . . . . . . . . . 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 1258	. . . . . . . . . 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 5938	. . . . . . . . 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 488	. . . . . . . . . . . 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 12279	. . . . . . . . . . . . 13 |- ( k e. Prime -> k e. ZZ )
62	61	adantl 277	. . . . . . . . . . . 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 15255	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ k e. ZZ ) -> ( A /L k ) e. ZZ )
64	60, 62, 63	syl2anc 411	. . . . . . . . . . 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 9449	. . . . . . . . . 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 12467	. . . . . . . . . . 11 |- ( ( k e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( k pCnt N ) e. NN0 )
67	52, 55, 56, 66	syl12anc 1247	. . . . . . . . . 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 12467	. . . . . . . . . . 11 |- ( ( k e. Prime /\ ( M e. ZZ /\ M =/= 0 ) ) -> ( k pCnt M ) e. NN0 )
69	52, 53, 54, 68	syl12anc 1247	. . . . . . . . . 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 10767	. . . . . . . . 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 2229	. . . . . . . 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 3566	. . . . . . . . 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 277	. . . . . . . 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 3566	. . . . . . . . . 10 |- ( k e. Prime -> if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt M ) ) , 1 ) = ( ( A /L k ) ^ ( k pCnt M ) ) )
75		iftrue 3566	. . . . . . . . . 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 5940	. . . . . . . . 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 277	. . . . . . . 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 2240	. . . . . . 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 9143	. . . . . . . . 9 |- ( 1 x. 1 ) = 1
80		iffalse 3569	. . . . . . . . . 10 |- ( -. k e. Prime -> if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt M ) ) , 1 ) = 1 )
81		iffalse 3569	. . . . . . . . . 10 |- ( -. k e. Prime -> if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt N ) ) , 1 ) = 1 )
82	80, 81	oveq12d 5940	. . . . . . . . 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 3569	. . . . . . . . 9 |- ( -. k e. Prime -> if ( k e. Prime , ( ( A /L k ) ^ ( k pCnt ( M x. N ) ) ) , 1 ) = 1 )
84	79, 82, 83	3eqtr4a 2255	. . . . . . . 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 277	. . . . . . 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 12298	. . . . . . . . . 10 |- ( k e. NN -> DECID k e. Prime )
87		exmiddc 837	. . . . . . . . . 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 277	. . . . . . 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 799	. . . . . 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 2257	. . . . . . . . 9 |- ( n = k -> ( n e. Prime <-> k e. Prime ) )
93		oveq2 5930	. . . . . . . . . 10 |- ( n = k -> ( A /L n ) = ( A /L k ) )
94		oveq1 5929	. . . . . . . . . 10 |- ( n = k -> ( n pCnt M ) = ( k pCnt M ) )
95	93, 94	oveq12d 5940	. . . . . . . . 9 |- ( n = k -> ( ( A /L n ) ^ ( n pCnt M ) ) = ( ( A /L k ) ^ ( k pCnt M ) ) )
96	92, 95	ifbieq1d 3583	. . . . . . . 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 277	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) /\ k e. ( ZZ>= ` 1 ) ) -> k e. NN )
98		zexpcl 10646	. . . . . . . . . 10 |- ( ( ( A /L k ) e. ZZ /\ ( k pCnt M ) e. NN0 ) -> ( ( A /L k ) ^ ( k pCnt M ) ) e. ZZ )
99	64, 69, 98	syl2anc 411	. . . . . . . . 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 9353	. . . . . . . . 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 3590	. . . . . . . 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 5655	. . . . . . 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 5929	. . . . . . . . . 10 |- ( n = k -> ( n pCnt N ) = ( k pCnt N ) )
105	93, 104	oveq12d 5940	. . . . . . . . 9 |- ( n = k -> ( ( A /L n ) ^ ( n pCnt N ) ) = ( ( A /L k ) ^ ( k pCnt N ) ) )
106	92, 105	ifbieq1d 3583	. . . . . . . 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 10646	. . . . . . . . . 10 |- ( ( ( A /L k ) e. ZZ /\ ( k pCnt N ) e. NN0 ) -> ( ( A /L k ) ^ ( k pCnt N ) ) e. ZZ )
108	64, 67, 107	syl2anc 411	. . . . . . . . 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 3590	. . . . . . . 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 5655	. . . . . . 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 5940	. . . . . 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 2196	. . . . . . 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 5929	. . . . . . . . 9 |- ( n = k -> ( n pCnt ( M x. N ) ) = ( k pCnt ( M x. N ) ) )
114	93, 113	oveq12d 5940	. . . . . . . 8 |- ( n = k -> ( ( A /L n ) ^ ( n pCnt ( M x. N ) ) ) = ( ( A /L k ) ^ ( k pCnt ( M x. N ) ) ) )
115	92, 114	ifbieq1d 3583	. . . . . . 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 488	. . . . . . . . . 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 488	. . . . . . . . . 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 12467	. . . . . . . . . 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 1247	. . . . . . . . 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 10646	. . . . . . . . 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 411	. . . . . . . 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 3590	. . . . . . 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 5655	. . . . . 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 2240	. . . . 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 11706	. . . 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 15277	. . . . 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 15277	. . . . . 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 8048	. . . . . . . 8 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( M x. N ) = ( N x. M ) )
129	128	fveq2d 5562	. . . . . . 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 5562	. . . . . 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 2232	. . . . 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 5940	. . . 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 2232	. . 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 5940	. 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 15261	. . 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 1249	. 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 15261	. . . . 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 1249	. . . 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 15261	. . . . 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 1249	. . . 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 5940	. . 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 9358	. . . . . . 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 9353	. . . . . 6 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> 1 e. ZZ )
145		zdclt 9403	. . . . . . . 8 |- ( ( M e. ZZ /\ 0 e. ZZ ) -> DECID M < 0 )
146	15, 21, 145	sylancl 413	. . . . . . 7 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> DECID M < 0 )
147		zdclt 9403	. . . . . . . 8 |- ( ( A e. ZZ /\ 0 e. ZZ ) -> DECID A < 0 )
148	37, 21, 147	sylancl 413	. . . . . . 7 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> DECID A < 0 )
149		dcan2 936	. . . . . . 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 3597	. . . . 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 9449	. . . 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	ffvelcdmda 5697	. . . . . . 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 9379	. . . . . . . 8 |- ( ( k e. ZZ /\ v e. ZZ ) -> ( k x. v ) e. ZZ )
155	154	adantl 277	. . . . . . 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 10556	. . . . . 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 11265	. . . . . . 7 |- ( ( M e. ZZ /\ M =/= 0 ) -> ( abs ` M ) e. NN )
158	15, 20, 157	syl2anc 411	. . . . . 6 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( abs ` M ) e. NN )
159	156, 158	ffvelcdmd 5698	. . . . 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 9449	. . . 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 9403	. . . . . . . 8 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N < 0 )
162	16, 21, 161	sylancl 413	. . . . . . 7 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> DECID N < 0 )
163		dcan2 936	. . . . . . 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 3597	. . . . 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 9449	. . . 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	ffvelcdmda 5697	. . . . . . 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 10556	. . . . . 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 11265	. . . . . . 7 |- ( ( N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. NN )
170	16, 25, 169	syl2anc 411	. . . . . 6 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( abs ` N ) e. NN )
171	168, 170	ffvelcdmd 5698	. . . . 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 9449	. . . 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 8181	. . 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 2229	. 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 2239	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 104   <-> wb 105   \/ wo 709  DECID wdc 835   /\ w3a 980   = wceq 1364   e. wcel 2167   =/= wne 2367   ifcif 3561   class class class wbr 4033   |-> cmpt 4094   -->wf 5254   ` cfv 5258  (class class class)co 5922   0cc0 7879   1c1 7880   + caddc 7882   x. cmul 7884   < clt 8061   -ucneg 8198   # cap 8608   NNcn 8990   NN0cn0 9249   ZZcz 9326   ZZ>=cuz 9601   seqcseq 10539   ^cexp 10630   abscabs 11162   Primecprime 12275   pCnt cpc 12453   /Lclgs 15238

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-xor 1387  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-frec 6449  df-1o 6474  df-2o 6475  df-oadd 6478  df-er 6592  df-en 6800  df-dom 6801  df-fin 6802  df-sup 7050  df-inf 7051  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-5 9052  df-6 9053  df-7 9054  df-8 9055  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-fz 10084  df-fzo 10218  df-fl 10360  df-mod 10415  df-seqfrec 10540  df-exp 10631  df-ihash 10868  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-clim 11444  df-proddc 11716  df-dvds 11953  df-gcd 12121  df-prm 12276  df-phi 12379  df-pc 12454  df-lgs 15239

This theorem is referenced by:  lgssq2  15282  lgsdinn0  15289  lgsquad2lem1  15322