Step |
Hyp |
Ref |
Expression |
1 |
|
oveq1 |
|- ( ( A + C ) = ( B + C ) -> ( ( A + C ) + ( 0 -R C ) ) = ( ( B + C ) + ( 0 -R C ) ) ) |
2 |
1
|
adantl |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A + C ) = ( B + C ) ) -> ( ( A + C ) + ( 0 -R C ) ) = ( ( B + C ) + ( 0 -R C ) ) ) |
3 |
|
simpl |
|- ( ( A e. RR /\ C e. RR ) -> A e. RR ) |
4 |
3
|
recnd |
|- ( ( A e. RR /\ C e. RR ) -> A e. CC ) |
5 |
|
simpr |
|- ( ( A e. RR /\ C e. RR ) -> C e. RR ) |
6 |
5
|
recnd |
|- ( ( A e. RR /\ C e. RR ) -> C e. CC ) |
7 |
|
rernegcl |
|- ( C e. RR -> ( 0 -R C ) e. RR ) |
8 |
7
|
adantl |
|- ( ( A e. RR /\ C e. RR ) -> ( 0 -R C ) e. RR ) |
9 |
8
|
recnd |
|- ( ( A e. RR /\ C e. RR ) -> ( 0 -R C ) e. CC ) |
10 |
4 6 9
|
addassd |
|- ( ( A e. RR /\ C e. RR ) -> ( ( A + C ) + ( 0 -R C ) ) = ( A + ( C + ( 0 -R C ) ) ) ) |
11 |
|
renegid |
|- ( C e. RR -> ( C + ( 0 -R C ) ) = 0 ) |
12 |
11
|
oveq2d |
|- ( C e. RR -> ( A + ( C + ( 0 -R C ) ) ) = ( A + 0 ) ) |
13 |
12
|
adantl |
|- ( ( A e. RR /\ C e. RR ) -> ( A + ( C + ( 0 -R C ) ) ) = ( A + 0 ) ) |
14 |
|
readdid1 |
|- ( A e. RR -> ( A + 0 ) = A ) |
15 |
14
|
adantr |
|- ( ( A e. RR /\ C e. RR ) -> ( A + 0 ) = A ) |
16 |
10 13 15
|
3eqtrd |
|- ( ( A e. RR /\ C e. RR ) -> ( ( A + C ) + ( 0 -R C ) ) = A ) |
17 |
16
|
3adant2 |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + C ) + ( 0 -R C ) ) = A ) |
18 |
17
|
adantr |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A + C ) = ( B + C ) ) -> ( ( A + C ) + ( 0 -R C ) ) = A ) |
19 |
|
simpl |
|- ( ( B e. RR /\ C e. RR ) -> B e. RR ) |
20 |
19
|
recnd |
|- ( ( B e. RR /\ C e. RR ) -> B e. CC ) |
21 |
|
simpr |
|- ( ( B e. RR /\ C e. RR ) -> C e. RR ) |
22 |
21
|
recnd |
|- ( ( B e. RR /\ C e. RR ) -> C e. CC ) |
23 |
7
|
adantl |
|- ( ( B e. RR /\ C e. RR ) -> ( 0 -R C ) e. RR ) |
24 |
23
|
recnd |
|- ( ( B e. RR /\ C e. RR ) -> ( 0 -R C ) e. CC ) |
25 |
20 22 24
|
addassd |
|- ( ( B e. RR /\ C e. RR ) -> ( ( B + C ) + ( 0 -R C ) ) = ( B + ( C + ( 0 -R C ) ) ) ) |
26 |
11
|
oveq2d |
|- ( C e. RR -> ( B + ( C + ( 0 -R C ) ) ) = ( B + 0 ) ) |
27 |
26
|
adantl |
|- ( ( B e. RR /\ C e. RR ) -> ( B + ( C + ( 0 -R C ) ) ) = ( B + 0 ) ) |
28 |
|
readdid1 |
|- ( B e. RR -> ( B + 0 ) = B ) |
29 |
28
|
adantr |
|- ( ( B e. RR /\ C e. RR ) -> ( B + 0 ) = B ) |
30 |
25 27 29
|
3eqtrd |
|- ( ( B e. RR /\ C e. RR ) -> ( ( B + C ) + ( 0 -R C ) ) = B ) |
31 |
30
|
3adant1 |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( B + C ) + ( 0 -R C ) ) = B ) |
32 |
31
|
adantr |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A + C ) = ( B + C ) ) -> ( ( B + C ) + ( 0 -R C ) ) = B ) |
33 |
2 18 32
|
3eqtr3d |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A + C ) = ( B + C ) ) -> A = B ) |
34 |
33
|
ex |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + C ) = ( B + C ) -> A = B ) ) |
35 |
|
oveq1 |
|- ( A = B -> ( A + C ) = ( B + C ) ) |
36 |
34 35
|
impbid1 |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + C ) = ( B + C ) <-> A = B ) ) |