Step |
Hyp |
Ref |
Expression |
1 |
|
sqdeccom12.a |
|- A e. NN0 |
2 |
|
sqdeccom12.b |
|- B e. NN0 |
3 |
1 1
|
nn0mulcli |
|- ( A x. A ) e. NN0 |
4 |
|
0nn0 |
|- 0 e. NN0 |
5 |
3 4
|
deccl |
|- ; ( A x. A ) 0 e. NN0 |
6 |
5 4
|
deccl |
|- ; ; ( A x. A ) 0 0 e. NN0 |
7 |
6
|
nn0cni |
|- ; ; ( A x. A ) 0 0 e. CC |
8 |
2 2
|
nn0mulcli |
|- ( B x. B ) e. NN0 |
9 |
8 4
|
deccl |
|- ; ( B x. B ) 0 e. NN0 |
10 |
9 4
|
deccl |
|- ; ; ( B x. B ) 0 0 e. NN0 |
11 |
10
|
nn0cni |
|- ; ; ( B x. B ) 0 0 e. CC |
12 |
1
|
nn0cni |
|- A e. CC |
13 |
12 12
|
mulcli |
|- ( A x. A ) e. CC |
14 |
2
|
nn0cni |
|- B e. CC |
15 |
14 14
|
mulcli |
|- ( B x. B ) e. CC |
16 |
|
subadd4 |
|- ( ( ( ; ; ( A x. A ) 0 0 e. CC /\ ; ; ( B x. B ) 0 0 e. CC ) /\ ( ( A x. A ) e. CC /\ ( B x. B ) e. CC ) ) -> ( ( ; ; ( A x. A ) 0 0 - ; ; ( B x. B ) 0 0 ) - ( ( A x. A ) - ( B x. B ) ) ) = ( ( ; ; ( A x. A ) 0 0 + ( B x. B ) ) - ( ; ; ( B x. B ) 0 0 + ( A x. A ) ) ) ) |
17 |
7 11 13 15 16
|
mp4an |
|- ( ( ; ; ( A x. A ) 0 0 - ; ; ( B x. B ) 0 0 ) - ( ( A x. A ) - ( B x. B ) ) ) = ( ( ; ; ( A x. A ) 0 0 + ( B x. B ) ) - ( ; ; ( B x. B ) 0 0 + ( A x. A ) ) ) |
18 |
|
eqid |
|- ; ; ( A x. A ) 0 0 = ; ; ( A x. A ) 0 0 |
19 |
15
|
addid2i |
|- ( 0 + ( B x. B ) ) = ( B x. B ) |
20 |
5 4 8 18 19
|
decaddi |
|- ( ; ; ( A x. A ) 0 0 + ( B x. B ) ) = ; ; ( A x. A ) 0 ( B x. B ) |
21 |
|
eqid |
|- ; ; ( B x. B ) 0 0 = ; ; ( B x. B ) 0 0 |
22 |
13
|
addid2i |
|- ( 0 + ( A x. A ) ) = ( A x. A ) |
23 |
9 4 3 21 22
|
decaddi |
|- ( ; ; ( B x. B ) 0 0 + ( A x. A ) ) = ; ; ( B x. B ) 0 ( A x. A ) |
24 |
20 23
|
oveq12i |
|- ( ( ; ; ( A x. A ) 0 0 + ( B x. B ) ) - ( ; ; ( B x. B ) 0 0 + ( A x. A ) ) ) = ( ; ; ( A x. A ) 0 ( B x. B ) - ; ; ( B x. B ) 0 ( A x. A ) ) |
25 |
17 24
|
eqtr2i |
|- ( ; ; ( A x. A ) 0 ( B x. B ) - ; ; ( B x. B ) 0 ( A x. A ) ) = ( ( ; ; ( A x. A ) 0 0 - ; ; ( B x. B ) 0 0 ) - ( ( A x. A ) - ( B x. B ) ) ) |
26 |
|
eqid |
|- ; ; ( ( B x. B ) + ( A x. A ) ) ( ( ( A x. B ) + ( B x. A ) ) + 0 ) ( ( A x. A ) + ( B x. B ) ) = ; ; ( ( B x. B ) + ( A x. A ) ) ( ( ( A x. B ) + ( B x. A ) ) + 0 ) ( ( A x. A ) + ( B x. B ) ) |
27 |
2 1
|
nn0mulcli |
|- ( B x. A ) e. NN0 |
28 |
1 2 27
|
numcl |
|- ( ( A x. B ) + ( B x. A ) ) e. NN0 |
29 |
3 28
|
deccl |
|- ; ( A x. A ) ( ( A x. B ) + ( B x. A ) ) e. NN0 |
30 |
|
eqid |
|- ; ; ( A x. A ) ( ( A x. B ) + ( B x. A ) ) ( B x. B ) = ; ; ( A x. A ) ( ( A x. B ) + ( B x. A ) ) ( B x. B ) |
31 |
|
eqid |
|- ; ; ( B x. B ) 0 ( A x. A ) = ; ; ( B x. B ) 0 ( A x. A ) |
32 |
|
eqid |
|- ; ( A x. A ) ( ( A x. B ) + ( B x. A ) ) = ; ( A x. A ) ( ( A x. B ) + ( B x. A ) ) |
33 |
|
eqid |
|- ; ( B x. B ) 0 = ; ( B x. B ) 0 |
34 |
13 15
|
addcomi |
|- ( ( A x. A ) + ( B x. B ) ) = ( ( B x. B ) + ( A x. A ) ) |
35 |
|
eqid |
|- ( ( ( A x. B ) + ( B x. A ) ) + 0 ) = ( ( ( A x. B ) + ( B x. A ) ) + 0 ) |
36 |
3 28 8 4 32 33 34 35
|
decadd |
|- ( ; ( A x. A ) ( ( A x. B ) + ( B x. A ) ) + ; ( B x. B ) 0 ) = ; ( ( B x. B ) + ( A x. A ) ) ( ( ( A x. B ) + ( B x. A ) ) + 0 ) |
37 |
15 13
|
addcomi |
|- ( ( B x. B ) + ( A x. A ) ) = ( ( A x. A ) + ( B x. B ) ) |
38 |
29 8 9 3 30 31 36 37
|
decadd |
|- ( ; ; ( A x. A ) ( ( A x. B ) + ( B x. A ) ) ( B x. B ) + ; ; ( B x. B ) 0 ( A x. A ) ) = ; ; ( ( B x. B ) + ( A x. A ) ) ( ( ( A x. B ) + ( B x. A ) ) + 0 ) ( ( A x. A ) + ( B x. B ) ) |
39 |
8 28
|
deccl |
|- ; ( B x. B ) ( ( A x. B ) + ( B x. A ) ) e. NN0 |
40 |
|
eqid |
|- ; ; ( B x. B ) ( ( A x. B ) + ( B x. A ) ) ( A x. A ) = ; ; ( B x. B ) ( ( A x. B ) + ( B x. A ) ) ( A x. A ) |
41 |
|
eqid |
|- ; ; ( A x. A ) 0 ( B x. B ) = ; ; ( A x. A ) 0 ( B x. B ) |
42 |
|
eqid |
|- ; ( B x. B ) ( ( A x. B ) + ( B x. A ) ) = ; ( B x. B ) ( ( A x. B ) + ( B x. A ) ) |
43 |
|
eqid |
|- ; ( A x. A ) 0 = ; ( A x. A ) 0 |
44 |
|
eqid |
|- ( ( B x. B ) + ( A x. A ) ) = ( ( B x. B ) + ( A x. A ) ) |
45 |
8 28 3 4 42 43 44 35
|
decadd |
|- ( ; ( B x. B ) ( ( A x. B ) + ( B x. A ) ) + ; ( A x. A ) 0 ) = ; ( ( B x. B ) + ( A x. A ) ) ( ( ( A x. B ) + ( B x. A ) ) + 0 ) |
46 |
|
eqid |
|- ( ( A x. A ) + ( B x. B ) ) = ( ( A x. A ) + ( B x. B ) ) |
47 |
39 3 5 8 40 41 45 46
|
decadd |
|- ( ; ; ( B x. B ) ( ( A x. B ) + ( B x. A ) ) ( A x. A ) + ; ; ( A x. A ) 0 ( B x. B ) ) = ; ; ( ( B x. B ) + ( A x. A ) ) ( ( ( A x. B ) + ( B x. A ) ) + 0 ) ( ( A x. A ) + ( B x. B ) ) |
48 |
26 38 47
|
3eqtr4i |
|- ( ; ; ( A x. A ) ( ( A x. B ) + ( B x. A ) ) ( B x. B ) + ; ; ( B x. B ) 0 ( A x. A ) ) = ( ; ; ( B x. B ) ( ( A x. B ) + ( B x. A ) ) ( A x. A ) + ; ; ( A x. A ) 0 ( B x. B ) ) |
49 |
29 8
|
deccl |
|- ; ; ( A x. A ) ( ( A x. B ) + ( B x. A ) ) ( B x. B ) e. NN0 |
50 |
49
|
nn0cni |
|- ; ; ( A x. A ) ( ( A x. B ) + ( B x. A ) ) ( B x. B ) e. CC |
51 |
9 3
|
deccl |
|- ; ; ( B x. B ) 0 ( A x. A ) e. NN0 |
52 |
51
|
nn0cni |
|- ; ; ( B x. B ) 0 ( A x. A ) e. CC |
53 |
39 3
|
deccl |
|- ; ; ( B x. B ) ( ( A x. B ) + ( B x. A ) ) ( A x. A ) e. NN0 |
54 |
53
|
nn0cni |
|- ; ; ( B x. B ) ( ( A x. B ) + ( B x. A ) ) ( A x. A ) e. CC |
55 |
5 8
|
deccl |
|- ; ; ( A x. A ) 0 ( B x. B ) e. NN0 |
56 |
55
|
nn0cni |
|- ; ; ( A x. A ) 0 ( B x. B ) e. CC |
57 |
|
addsubeq4com |
|- ( ( ( ; ; ( A x. A ) ( ( A x. B ) + ( B x. A ) ) ( B x. B ) e. CC /\ ; ; ( B x. B ) 0 ( A x. A ) e. CC ) /\ ( ; ; ( B x. B ) ( ( A x. B ) + ( B x. A ) ) ( A x. A ) e. CC /\ ; ; ( A x. A ) 0 ( B x. B ) e. CC ) ) -> ( ( ; ; ( A x. A ) ( ( A x. B ) + ( B x. A ) ) ( B x. B ) + ; ; ( B x. B ) 0 ( A x. A ) ) = ( ; ; ( B x. B ) ( ( A x. B ) + ( B x. A ) ) ( A x. A ) + ; ; ( A x. A ) 0 ( B x. B ) ) <-> ( ; ; ( A x. A ) ( ( A x. B ) + ( B x. A ) ) ( B x. B ) - ; ; ( B x. B ) ( ( A x. B ) + ( B x. A ) ) ( A x. A ) ) = ( ; ; ( A x. A ) 0 ( B x. B ) - ; ; ( B x. B ) 0 ( A x. A ) ) ) ) |
58 |
50 52 54 56 57
|
mp4an |
|- ( ( ; ; ( A x. A ) ( ( A x. B ) + ( B x. A ) ) ( B x. B ) + ; ; ( B x. B ) 0 ( A x. A ) ) = ( ; ; ( B x. B ) ( ( A x. B ) + ( B x. A ) ) ( A x. A ) + ; ; ( A x. A ) 0 ( B x. B ) ) <-> ( ; ; ( A x. A ) ( ( A x. B ) + ( B x. A ) ) ( B x. B ) - ; ; ( B x. B ) ( ( A x. B ) + ( B x. A ) ) ( A x. A ) ) = ( ; ; ( A x. A ) 0 ( B x. B ) - ; ; ( B x. B ) 0 ( A x. A ) ) ) |
59 |
48 58
|
mpbi |
|- ( ; ; ( A x. A ) ( ( A x. B ) + ( B x. A ) ) ( B x. B ) - ; ; ( B x. B ) ( ( A x. B ) + ( B x. A ) ) ( A x. A ) ) = ( ; ; ( A x. A ) 0 ( B x. B ) - ; ; ( B x. B ) 0 ( A x. A ) ) |
60 |
|
10nn0 |
|- ; 1 0 e. NN0 |
61 |
60 4
|
deccl |
|- ; ; 1 0 0 e. NN0 |
62 |
61
|
nn0cni |
|- ; ; 1 0 0 e. CC |
63 |
|
ax-1cn |
|- 1 e. CC |
64 |
13 15
|
subcli |
|- ( ( A x. A ) - ( B x. B ) ) e. CC |
65 |
62 63 64
|
subdiri |
|- ( ( ; ; 1 0 0 - 1 ) x. ( ( A x. A ) - ( B x. B ) ) ) = ( ( ; ; 1 0 0 x. ( ( A x. A ) - ( B x. B ) ) ) - ( 1 x. ( ( A x. A ) - ( B x. B ) ) ) ) |
66 |
62 13 15
|
subdii |
|- ( ; ; 1 0 0 x. ( ( A x. A ) - ( B x. B ) ) ) = ( ( ; ; 1 0 0 x. ( A x. A ) ) - ( ; ; 1 0 0 x. ( B x. B ) ) ) |
67 |
|
eqid |
|- ; ; 1 0 0 = ; ; 1 0 0 |
68 |
3
|
dec0u |
|- ( ; 1 0 x. ( A x. A ) ) = ; ( A x. A ) 0 |
69 |
13
|
mul02i |
|- ( 0 x. ( A x. A ) ) = 0 |
70 |
3 60 4 67 68 69
|
decmul1 |
|- ( ; ; 1 0 0 x. ( A x. A ) ) = ; ; ( A x. A ) 0 0 |
71 |
8
|
dec0u |
|- ( ; 1 0 x. ( B x. B ) ) = ; ( B x. B ) 0 |
72 |
15
|
mul02i |
|- ( 0 x. ( B x. B ) ) = 0 |
73 |
8 60 4 67 71 72
|
decmul1 |
|- ( ; ; 1 0 0 x. ( B x. B ) ) = ; ; ( B x. B ) 0 0 |
74 |
70 73
|
oveq12i |
|- ( ( ; ; 1 0 0 x. ( A x. A ) ) - ( ; ; 1 0 0 x. ( B x. B ) ) ) = ( ; ; ( A x. A ) 0 0 - ; ; ( B x. B ) 0 0 ) |
75 |
66 74
|
eqtri |
|- ( ; ; 1 0 0 x. ( ( A x. A ) - ( B x. B ) ) ) = ( ; ; ( A x. A ) 0 0 - ; ; ( B x. B ) 0 0 ) |
76 |
64
|
mulid2i |
|- ( 1 x. ( ( A x. A ) - ( B x. B ) ) ) = ( ( A x. A ) - ( B x. B ) ) |
77 |
75 76
|
oveq12i |
|- ( ( ; ; 1 0 0 x. ( ( A x. A ) - ( B x. B ) ) ) - ( 1 x. ( ( A x. A ) - ( B x. B ) ) ) ) = ( ( ; ; ( A x. A ) 0 0 - ; ; ( B x. B ) 0 0 ) - ( ( A x. A ) - ( B x. B ) ) ) |
78 |
65 77
|
eqtri |
|- ( ( ; ; 1 0 0 - 1 ) x. ( ( A x. A ) - ( B x. B ) ) ) = ( ( ; ; ( A x. A ) 0 0 - ; ; ( B x. B ) 0 0 ) - ( ( A x. A ) - ( B x. B ) ) ) |
79 |
25 59 78
|
3eqtr4i |
|- ( ; ; ( A x. A ) ( ( A x. B ) + ( B x. A ) ) ( B x. B ) - ; ; ( B x. B ) ( ( A x. B ) + ( B x. A ) ) ( A x. A ) ) = ( ( ; ; 1 0 0 - 1 ) x. ( ( A x. A ) - ( B x. B ) ) ) |
80 |
|
eqid |
|- ( A x. A ) = ( A x. A ) |
81 |
|
eqid |
|- ( ( A x. B ) + ( B x. A ) ) = ( ( A x. B ) + ( B x. A ) ) |
82 |
|
eqid |
|- ( B x. B ) = ( B x. B ) |
83 |
1 2 1 2 80 81 82
|
decpmulnc |
|- ( ; A B x. ; A B ) = ; ; ( A x. A ) ( ( A x. B ) + ( B x. A ) ) ( B x. B ) |
84 |
14 12
|
mulcli |
|- ( B x. A ) e. CC |
85 |
12 14
|
mulcli |
|- ( A x. B ) e. CC |
86 |
84 85
|
addcomi |
|- ( ( B x. A ) + ( A x. B ) ) = ( ( A x. B ) + ( B x. A ) ) |
87 |
2 1 2 1 82 86 80
|
decpmulnc |
|- ( ; B A x. ; B A ) = ; ; ( B x. B ) ( ( A x. B ) + ( B x. A ) ) ( A x. A ) |
88 |
83 87
|
oveq12i |
|- ( ( ; A B x. ; A B ) - ( ; B A x. ; B A ) ) = ( ; ; ( A x. A ) ( ( A x. B ) + ( B x. A ) ) ( B x. B ) - ; ; ( B x. B ) ( ( A x. B ) + ( B x. A ) ) ( A x. A ) ) |
89 |
|
9nn0 |
|- 9 e. NN0 |
90 |
89 89
|
deccl |
|- ; 9 9 e. NN0 |
91 |
90
|
nn0cni |
|- ; 9 9 e. CC |
92 |
|
9p1e10 |
|- ( 9 + 1 ) = ; 1 0 |
93 |
|
eqid |
|- ; 9 9 = ; 9 9 |
94 |
89 92 93
|
decsucc |
|- ( ; 9 9 + 1 ) = ; ; 1 0 0 |
95 |
91 63 94
|
addcomli |
|- ( 1 + ; 9 9 ) = ; ; 1 0 0 |
96 |
63 91 95
|
mvlladdi |
|- ; 9 9 = ( ; ; 1 0 0 - 1 ) |
97 |
96
|
oveq1i |
|- ( ; 9 9 x. ( ( A x. A ) - ( B x. B ) ) ) = ( ( ; ; 1 0 0 - 1 ) x. ( ( A x. A ) - ( B x. B ) ) ) |
98 |
79 88 97
|
3eqtr4i |
|- ( ( ; A B x. ; A B ) - ( ; B A x. ; B A ) ) = ( ; 9 9 x. ( ( A x. A ) - ( B x. B ) ) ) |