Step |
Hyp |
Ref |
Expression |
1 |
|
ax-1cn |
|- 1 e. CC |
2 |
|
mulsub |
|- ( ( ( A e. CC /\ 1 e. CC ) /\ ( B e. CC /\ 1 e. CC ) ) -> ( ( A - 1 ) x. ( B - 1 ) ) = ( ( ( A x. B ) + ( 1 x. 1 ) ) - ( ( A x. 1 ) + ( B x. 1 ) ) ) ) |
3 |
1 2
|
mpanr2 |
|- ( ( ( A e. CC /\ 1 e. CC ) /\ B e. CC ) -> ( ( A - 1 ) x. ( B - 1 ) ) = ( ( ( A x. B ) + ( 1 x. 1 ) ) - ( ( A x. 1 ) + ( B x. 1 ) ) ) ) |
4 |
1 3
|
mpanl2 |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A - 1 ) x. ( B - 1 ) ) = ( ( ( A x. B ) + ( 1 x. 1 ) ) - ( ( A x. 1 ) + ( B x. 1 ) ) ) ) |
5 |
1
|
mulid1i |
|- ( 1 x. 1 ) = 1 |
6 |
5
|
oveq2i |
|- ( ( A x. B ) + ( 1 x. 1 ) ) = ( ( A x. B ) + 1 ) |
7 |
6
|
a1i |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) + ( 1 x. 1 ) ) = ( ( A x. B ) + 1 ) ) |
8 |
|
mulid1 |
|- ( A e. CC -> ( A x. 1 ) = A ) |
9 |
|
mulid1 |
|- ( B e. CC -> ( B x. 1 ) = B ) |
10 |
8 9
|
oveqan12d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A x. 1 ) + ( B x. 1 ) ) = ( A + B ) ) |
11 |
7 10
|
oveq12d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A x. B ) + ( 1 x. 1 ) ) - ( ( A x. 1 ) + ( B x. 1 ) ) ) = ( ( ( A x. B ) + 1 ) - ( A + B ) ) ) |
12 |
|
mulcl |
|- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC ) |
13 |
|
addcl |
|- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC ) |
14 |
|
addsub |
|- ( ( ( A x. B ) e. CC /\ 1 e. CC /\ ( A + B ) e. CC ) -> ( ( ( A x. B ) + 1 ) - ( A + B ) ) = ( ( ( A x. B ) - ( A + B ) ) + 1 ) ) |
15 |
1 14
|
mp3an2 |
|- ( ( ( A x. B ) e. CC /\ ( A + B ) e. CC ) -> ( ( ( A x. B ) + 1 ) - ( A + B ) ) = ( ( ( A x. B ) - ( A + B ) ) + 1 ) ) |
16 |
12 13 15
|
syl2anc |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A x. B ) + 1 ) - ( A + B ) ) = ( ( ( A x. B ) - ( A + B ) ) + 1 ) ) |
17 |
4 11 16
|
3eqtrd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A - 1 ) x. ( B - 1 ) ) = ( ( ( A x. B ) - ( A + B ) ) + 1 ) ) |
18 |
17
|
eqeq1d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A - 1 ) x. ( B - 1 ) ) = 1 <-> ( ( ( A x. B ) - ( A + B ) ) + 1 ) = 1 ) ) |
19 |
12 13
|
subcld |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) - ( A + B ) ) e. CC ) |
20 |
|
0cn |
|- 0 e. CC |
21 |
|
addcan2 |
|- ( ( ( ( A x. B ) - ( A + B ) ) e. CC /\ 0 e. CC /\ 1 e. CC ) -> ( ( ( ( A x. B ) - ( A + B ) ) + 1 ) = ( 0 + 1 ) <-> ( ( A x. B ) - ( A + B ) ) = 0 ) ) |
22 |
20 1 21
|
mp3an23 |
|- ( ( ( A x. B ) - ( A + B ) ) e. CC -> ( ( ( ( A x. B ) - ( A + B ) ) + 1 ) = ( 0 + 1 ) <-> ( ( A x. B ) - ( A + B ) ) = 0 ) ) |
23 |
19 22
|
syl |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( A x. B ) - ( A + B ) ) + 1 ) = ( 0 + 1 ) <-> ( ( A x. B ) - ( A + B ) ) = 0 ) ) |
24 |
1
|
addid2i |
|- ( 0 + 1 ) = 1 |
25 |
24
|
eqeq2i |
|- ( ( ( ( A x. B ) - ( A + B ) ) + 1 ) = ( 0 + 1 ) <-> ( ( ( A x. B ) - ( A + B ) ) + 1 ) = 1 ) |
26 |
23 25
|
bitr3di |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A x. B ) - ( A + B ) ) = 0 <-> ( ( ( A x. B ) - ( A + B ) ) + 1 ) = 1 ) ) |
27 |
12 13
|
subeq0ad |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A x. B ) - ( A + B ) ) = 0 <-> ( A x. B ) = ( A + B ) ) ) |
28 |
18 26 27
|
3bitr2rd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) = ( A + B ) <-> ( ( A - 1 ) x. ( B - 1 ) ) = 1 ) ) |