Step |
Hyp |
Ref |
Expression |
1 |
|
oveq2 |
|- ( x = N -> ( A /L x ) = ( A /L N ) ) |
2 |
1
|
oveq1d |
|- ( x = N -> ( ( A /L x ) x. ( A /L 0 ) ) = ( ( A /L N ) x. ( A /L 0 ) ) ) |
3 |
2
|
eqeq2d |
|- ( x = N -> ( ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) <-> ( A /L 0 ) = ( ( A /L N ) x. ( A /L 0 ) ) ) ) |
4 |
|
sq1 |
|- ( 1 ^ 2 ) = 1 |
5 |
4
|
eqeq2i |
|- ( ( A ^ 2 ) = ( 1 ^ 2 ) <-> ( A ^ 2 ) = 1 ) |
6 |
|
nn0re |
|- ( A e. NN0 -> A e. RR ) |
7 |
|
nn0ge0 |
|- ( A e. NN0 -> 0 <_ A ) |
8 |
|
1re |
|- 1 e. RR |
9 |
|
0le1 |
|- 0 <_ 1 |
10 |
|
sq11 |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( 1 e. RR /\ 0 <_ 1 ) ) -> ( ( A ^ 2 ) = ( 1 ^ 2 ) <-> A = 1 ) ) |
11 |
8 9 10
|
mpanr12 |
|- ( ( A e. RR /\ 0 <_ A ) -> ( ( A ^ 2 ) = ( 1 ^ 2 ) <-> A = 1 ) ) |
12 |
6 7 11
|
syl2anc |
|- ( A e. NN0 -> ( ( A ^ 2 ) = ( 1 ^ 2 ) <-> A = 1 ) ) |
13 |
12
|
adantr |
|- ( ( A e. NN0 /\ x e. ZZ ) -> ( ( A ^ 2 ) = ( 1 ^ 2 ) <-> A = 1 ) ) |
14 |
5 13
|
bitr3id |
|- ( ( A e. NN0 /\ x e. ZZ ) -> ( ( A ^ 2 ) = 1 <-> A = 1 ) ) |
15 |
14
|
biimpa |
|- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> A = 1 ) |
16 |
15
|
oveq1d |
|- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( A /L x ) = ( 1 /L x ) ) |
17 |
|
1lgs |
|- ( x e. ZZ -> ( 1 /L x ) = 1 ) |
18 |
17
|
ad2antlr |
|- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( 1 /L x ) = 1 ) |
19 |
16 18
|
eqtrd |
|- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( A /L x ) = 1 ) |
20 |
19
|
oveq1d |
|- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( ( A /L x ) x. ( A /L 0 ) ) = ( 1 x. ( A /L 0 ) ) ) |
21 |
|
nn0z |
|- ( A e. NN0 -> A e. ZZ ) |
22 |
21
|
ad2antrr |
|- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> A e. ZZ ) |
23 |
|
0z |
|- 0 e. ZZ |
24 |
|
lgscl |
|- ( ( A e. ZZ /\ 0 e. ZZ ) -> ( A /L 0 ) e. ZZ ) |
25 |
22 23 24
|
sylancl |
|- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( A /L 0 ) e. ZZ ) |
26 |
25
|
zcnd |
|- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( A /L 0 ) e. CC ) |
27 |
26
|
mulid2d |
|- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( 1 x. ( A /L 0 ) ) = ( A /L 0 ) ) |
28 |
20 27
|
eqtr2d |
|- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) ) |
29 |
|
lgscl |
|- ( ( A e. ZZ /\ x e. ZZ ) -> ( A /L x ) e. ZZ ) |
30 |
21 29
|
sylan |
|- ( ( A e. NN0 /\ x e. ZZ ) -> ( A /L x ) e. ZZ ) |
31 |
30
|
zcnd |
|- ( ( A e. NN0 /\ x e. ZZ ) -> ( A /L x ) e. CC ) |
32 |
31
|
adantr |
|- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) =/= 1 ) -> ( A /L x ) e. CC ) |
33 |
32
|
mul01d |
|- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) =/= 1 ) -> ( ( A /L x ) x. 0 ) = 0 ) |
34 |
21
|
adantr |
|- ( ( A e. NN0 /\ x e. ZZ ) -> A e. ZZ ) |
35 |
|
lgs0 |
|- ( A e. ZZ -> ( A /L 0 ) = if ( ( A ^ 2 ) = 1 , 1 , 0 ) ) |
36 |
34 35
|
syl |
|- ( ( A e. NN0 /\ x e. ZZ ) -> ( A /L 0 ) = if ( ( A ^ 2 ) = 1 , 1 , 0 ) ) |
37 |
|
ifnefalse |
|- ( ( A ^ 2 ) =/= 1 -> if ( ( A ^ 2 ) = 1 , 1 , 0 ) = 0 ) |
38 |
36 37
|
sylan9eq |
|- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) =/= 1 ) -> ( A /L 0 ) = 0 ) |
39 |
38
|
oveq2d |
|- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) =/= 1 ) -> ( ( A /L x ) x. ( A /L 0 ) ) = ( ( A /L x ) x. 0 ) ) |
40 |
33 39 38
|
3eqtr4rd |
|- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) =/= 1 ) -> ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) ) |
41 |
28 40
|
pm2.61dane |
|- ( ( A e. NN0 /\ x e. ZZ ) -> ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) ) |
42 |
41
|
ralrimiva |
|- ( A e. NN0 -> A. x e. ZZ ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) ) |
43 |
42
|
3ad2ant1 |
|- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> A. x e. ZZ ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) ) |
44 |
|
simp3 |
|- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> N e. ZZ ) |
45 |
3 43 44
|
rspcdva |
|- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L 0 ) = ( ( A /L N ) x. ( A /L 0 ) ) ) |
46 |
45
|
adantr |
|- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L 0 ) = ( ( A /L N ) x. ( A /L 0 ) ) ) |
47 |
21
|
3ad2ant1 |
|- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> A e. ZZ ) |
48 |
47 23 24
|
sylancl |
|- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L 0 ) e. ZZ ) |
49 |
48
|
zcnd |
|- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L 0 ) e. CC ) |
50 |
49
|
adantr |
|- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L 0 ) e. CC ) |
51 |
|
lgscl |
|- ( ( A e. ZZ /\ N e. ZZ ) -> ( A /L N ) e. ZZ ) |
52 |
47 44 51
|
syl2anc |
|- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L N ) e. ZZ ) |
53 |
52
|
zcnd |
|- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L N ) e. CC ) |
54 |
53
|
adantr |
|- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L N ) e. CC ) |
55 |
50 54
|
mulcomd |
|- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( ( A /L 0 ) x. ( A /L N ) ) = ( ( A /L N ) x. ( A /L 0 ) ) ) |
56 |
46 55
|
eqtr4d |
|- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L 0 ) = ( ( A /L 0 ) x. ( A /L N ) ) ) |
57 |
|
oveq1 |
|- ( M = 0 -> ( M x. N ) = ( 0 x. N ) ) |
58 |
44
|
zcnd |
|- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> N e. CC ) |
59 |
58
|
mul02d |
|- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( 0 x. N ) = 0 ) |
60 |
57 59
|
sylan9eqr |
|- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( M x. N ) = 0 ) |
61 |
60
|
oveq2d |
|- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L ( M x. N ) ) = ( A /L 0 ) ) |
62 |
|
simpr |
|- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> M = 0 ) |
63 |
62
|
oveq2d |
|- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L M ) = ( A /L 0 ) ) |
64 |
63
|
oveq1d |
|- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( ( A /L M ) x. ( A /L N ) ) = ( ( A /L 0 ) x. ( A /L N ) ) ) |
65 |
56 61 64
|
3eqtr4d |
|- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) ) |
66 |
|
oveq2 |
|- ( x = M -> ( A /L x ) = ( A /L M ) ) |
67 |
66
|
oveq1d |
|- ( x = M -> ( ( A /L x ) x. ( A /L 0 ) ) = ( ( A /L M ) x. ( A /L 0 ) ) ) |
68 |
67
|
eqeq2d |
|- ( x = M -> ( ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) <-> ( A /L 0 ) = ( ( A /L M ) x. ( A /L 0 ) ) ) ) |
69 |
|
simp2 |
|- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> M e. ZZ ) |
70 |
68 43 69
|
rspcdva |
|- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L 0 ) = ( ( A /L M ) x. ( A /L 0 ) ) ) |
71 |
70
|
adantr |
|- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( A /L 0 ) = ( ( A /L M ) x. ( A /L 0 ) ) ) |
72 |
|
oveq2 |
|- ( N = 0 -> ( M x. N ) = ( M x. 0 ) ) |
73 |
69
|
zcnd |
|- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> M e. CC ) |
74 |
73
|
mul01d |
|- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( M x. 0 ) = 0 ) |
75 |
72 74
|
sylan9eqr |
|- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( M x. N ) = 0 ) |
76 |
75
|
oveq2d |
|- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( A /L ( M x. N ) ) = ( A /L 0 ) ) |
77 |
|
simpr |
|- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> N = 0 ) |
78 |
77
|
oveq2d |
|- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( A /L N ) = ( A /L 0 ) ) |
79 |
78
|
oveq2d |
|- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( ( A /L M ) x. ( A /L N ) ) = ( ( A /L M ) x. ( A /L 0 ) ) ) |
80 |
71 76 79
|
3eqtr4d |
|- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) ) |
81 |
|
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 ) ) ) |
82 |
21 81
|
syl3anl1 |
|- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) ) |
83 |
65 80 82
|
pm2.61da2ne |
|- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) ) |