Step |
Hyp |
Ref |
Expression |
1 |
|
simplr |
|- ( ( ( A e. CC /\ B e. RR ) /\ ( abs ` A ) < B ) -> B e. RR ) |
2 |
1
|
recnd |
|- ( ( ( A e. CC /\ B e. RR ) /\ ( abs ` A ) < B ) -> B e. CC ) |
3 |
|
simpll |
|- ( ( ( A e. CC /\ B e. RR ) /\ ( abs ` A ) < B ) -> A e. CC ) |
4 |
|
abscl |
|- ( A e. CC -> ( abs ` A ) e. RR ) |
5 |
3 4
|
syl |
|- ( ( ( A e. CC /\ B e. RR ) /\ ( abs ` A ) < B ) -> ( abs ` A ) e. RR ) |
6 |
|
abscl |
|- ( B e. CC -> ( abs ` B ) e. RR ) |
7 |
2 6
|
syl |
|- ( ( ( A e. CC /\ B e. RR ) /\ ( abs ` A ) < B ) -> ( abs ` B ) e. RR ) |
8 |
|
simpr |
|- ( ( ( A e. CC /\ B e. RR ) /\ ( abs ` A ) < B ) -> ( abs ` A ) < B ) |
9 |
|
leabs |
|- ( B e. RR -> B <_ ( abs ` B ) ) |
10 |
1 9
|
syl |
|- ( ( ( A e. CC /\ B e. RR ) /\ ( abs ` A ) < B ) -> B <_ ( abs ` B ) ) |
11 |
5 1 7 8 10
|
ltletrd |
|- ( ( ( A e. CC /\ B e. RR ) /\ ( abs ` A ) < B ) -> ( abs ` A ) < ( abs ` B ) ) |
12 |
5 11
|
gtned |
|- ( ( ( A e. CC /\ B e. RR ) /\ ( abs ` A ) < B ) -> ( abs ` B ) =/= ( abs ` A ) ) |
13 |
|
fveq2 |
|- ( B = A -> ( abs ` B ) = ( abs ` A ) ) |
14 |
13
|
necon3i |
|- ( ( abs ` B ) =/= ( abs ` A ) -> B =/= A ) |
15 |
12 14
|
syl |
|- ( ( ( A e. CC /\ B e. RR ) /\ ( abs ` A ) < B ) -> B =/= A ) |
16 |
2 3 15
|
subne0d |
|- ( ( ( A e. CC /\ B e. RR ) /\ ( abs ` A ) < B ) -> ( B - A ) =/= 0 ) |
17 |
16
|
3impa |
|- ( ( A e. CC /\ B e. RR /\ ( abs ` A ) < B ) -> ( B - A ) =/= 0 ) |