Step |
Hyp |
Ref |
Expression |
1 |
|
omopth.1 |
|- A e. _om |
2 |
|
omopth.2 |
|- B e. _om |
3 |
|
omopth.3 |
|- C e. _om |
4 |
|
omopth.4 |
|- D e. _om |
5 |
1 2
|
nnacli |
|- ( A +o B ) e. _om |
6 |
5
|
nnoni |
|- ( A +o B ) e. On |
7 |
6
|
onordi |
|- Ord ( A +o B ) |
8 |
3 4
|
nnacli |
|- ( C +o D ) e. _om |
9 |
8
|
nnoni |
|- ( C +o D ) e. On |
10 |
9
|
onordi |
|- Ord ( C +o D ) |
11 |
|
ordtri3 |
|- ( ( Ord ( A +o B ) /\ Ord ( C +o D ) ) -> ( ( A +o B ) = ( C +o D ) <-> -. ( ( A +o B ) e. ( C +o D ) \/ ( C +o D ) e. ( A +o B ) ) ) ) |
12 |
7 10 11
|
mp2an |
|- ( ( A +o B ) = ( C +o D ) <-> -. ( ( A +o B ) e. ( C +o D ) \/ ( C +o D ) e. ( A +o B ) ) ) |
13 |
12
|
con2bii |
|- ( ( ( A +o B ) e. ( C +o D ) \/ ( C +o D ) e. ( A +o B ) ) <-> -. ( A +o B ) = ( C +o D ) ) |
14 |
1 2 8 4
|
omopthlem2 |
|- ( ( A +o B ) e. ( C +o D ) -> -. ( ( ( C +o D ) .o ( C +o D ) ) +o D ) = ( ( ( A +o B ) .o ( A +o B ) ) +o B ) ) |
15 |
|
eqcom |
|- ( ( ( ( C +o D ) .o ( C +o D ) ) +o D ) = ( ( ( A +o B ) .o ( A +o B ) ) +o B ) <-> ( ( ( A +o B ) .o ( A +o B ) ) +o B ) = ( ( ( C +o D ) .o ( C +o D ) ) +o D ) ) |
16 |
14 15
|
sylnib |
|- ( ( A +o B ) e. ( C +o D ) -> -. ( ( ( A +o B ) .o ( A +o B ) ) +o B ) = ( ( ( C +o D ) .o ( C +o D ) ) +o D ) ) |
17 |
3 4 5 2
|
omopthlem2 |
|- ( ( C +o D ) e. ( A +o B ) -> -. ( ( ( A +o B ) .o ( A +o B ) ) +o B ) = ( ( ( C +o D ) .o ( C +o D ) ) +o D ) ) |
18 |
16 17
|
jaoi |
|- ( ( ( A +o B ) e. ( C +o D ) \/ ( C +o D ) e. ( A +o B ) ) -> -. ( ( ( A +o B ) .o ( A +o B ) ) +o B ) = ( ( ( C +o D ) .o ( C +o D ) ) +o D ) ) |
19 |
13 18
|
sylbir |
|- ( -. ( A +o B ) = ( C +o D ) -> -. ( ( ( A +o B ) .o ( A +o B ) ) +o B ) = ( ( ( C +o D ) .o ( C +o D ) ) +o D ) ) |
20 |
19
|
con4i |
|- ( ( ( ( A +o B ) .o ( A +o B ) ) +o B ) = ( ( ( C +o D ) .o ( C +o D ) ) +o D ) -> ( A +o B ) = ( C +o D ) ) |
21 |
|
id |
|- ( ( ( ( A +o B ) .o ( A +o B ) ) +o B ) = ( ( ( C +o D ) .o ( C +o D ) ) +o D ) -> ( ( ( A +o B ) .o ( A +o B ) ) +o B ) = ( ( ( C +o D ) .o ( C +o D ) ) +o D ) ) |
22 |
20 20
|
oveq12d |
|- ( ( ( ( A +o B ) .o ( A +o B ) ) +o B ) = ( ( ( C +o D ) .o ( C +o D ) ) +o D ) -> ( ( A +o B ) .o ( A +o B ) ) = ( ( C +o D ) .o ( C +o D ) ) ) |
23 |
22
|
oveq1d |
|- ( ( ( ( A +o B ) .o ( A +o B ) ) +o B ) = ( ( ( C +o D ) .o ( C +o D ) ) +o D ) -> ( ( ( A +o B ) .o ( A +o B ) ) +o D ) = ( ( ( C +o D ) .o ( C +o D ) ) +o D ) ) |
24 |
21 23
|
eqtr4d |
|- ( ( ( ( A +o B ) .o ( A +o B ) ) +o B ) = ( ( ( C +o D ) .o ( C +o D ) ) +o D ) -> ( ( ( A +o B ) .o ( A +o B ) ) +o B ) = ( ( ( A +o B ) .o ( A +o B ) ) +o D ) ) |
25 |
5 5
|
nnmcli |
|- ( ( A +o B ) .o ( A +o B ) ) e. _om |
26 |
|
nnacan |
|- ( ( ( ( A +o B ) .o ( A +o B ) ) e. _om /\ B e. _om /\ D e. _om ) -> ( ( ( ( A +o B ) .o ( A +o B ) ) +o B ) = ( ( ( A +o B ) .o ( A +o B ) ) +o D ) <-> B = D ) ) |
27 |
25 2 4 26
|
mp3an |
|- ( ( ( ( A +o B ) .o ( A +o B ) ) +o B ) = ( ( ( A +o B ) .o ( A +o B ) ) +o D ) <-> B = D ) |
28 |
24 27
|
sylib |
|- ( ( ( ( A +o B ) .o ( A +o B ) ) +o B ) = ( ( ( C +o D ) .o ( C +o D ) ) +o D ) -> B = D ) |
29 |
28
|
oveq2d |
|- ( ( ( ( A +o B ) .o ( A +o B ) ) +o B ) = ( ( ( C +o D ) .o ( C +o D ) ) +o D ) -> ( C +o B ) = ( C +o D ) ) |
30 |
20 29
|
eqtr4d |
|- ( ( ( ( A +o B ) .o ( A +o B ) ) +o B ) = ( ( ( C +o D ) .o ( C +o D ) ) +o D ) -> ( A +o B ) = ( C +o B ) ) |
31 |
|
nnacom |
|- ( ( B e. _om /\ A e. _om ) -> ( B +o A ) = ( A +o B ) ) |
32 |
2 1 31
|
mp2an |
|- ( B +o A ) = ( A +o B ) |
33 |
|
nnacom |
|- ( ( B e. _om /\ C e. _om ) -> ( B +o C ) = ( C +o B ) ) |
34 |
2 3 33
|
mp2an |
|- ( B +o C ) = ( C +o B ) |
35 |
30 32 34
|
3eqtr4g |
|- ( ( ( ( A +o B ) .o ( A +o B ) ) +o B ) = ( ( ( C +o D ) .o ( C +o D ) ) +o D ) -> ( B +o A ) = ( B +o C ) ) |
36 |
|
nnacan |
|- ( ( B e. _om /\ A e. _om /\ C e. _om ) -> ( ( B +o A ) = ( B +o C ) <-> A = C ) ) |
37 |
2 1 3 36
|
mp3an |
|- ( ( B +o A ) = ( B +o C ) <-> A = C ) |
38 |
35 37
|
sylib |
|- ( ( ( ( A +o B ) .o ( A +o B ) ) +o B ) = ( ( ( C +o D ) .o ( C +o D ) ) +o D ) -> A = C ) |
39 |
38 28
|
jca |
|- ( ( ( ( A +o B ) .o ( A +o B ) ) +o B ) = ( ( ( C +o D ) .o ( C +o D ) ) +o D ) -> ( A = C /\ B = D ) ) |
40 |
|
oveq12 |
|- ( ( A = C /\ B = D ) -> ( A +o B ) = ( C +o D ) ) |
41 |
40 40
|
oveq12d |
|- ( ( A = C /\ B = D ) -> ( ( A +o B ) .o ( A +o B ) ) = ( ( C +o D ) .o ( C +o D ) ) ) |
42 |
|
simpr |
|- ( ( A = C /\ B = D ) -> B = D ) |
43 |
41 42
|
oveq12d |
|- ( ( A = C /\ B = D ) -> ( ( ( A +o B ) .o ( A +o B ) ) +o B ) = ( ( ( C +o D ) .o ( C +o D ) ) +o D ) ) |
44 |
39 43
|
impbii |
|- ( ( ( ( A +o B ) .o ( A +o B ) ) +o B ) = ( ( ( C +o D ) .o ( C +o D ) ) +o D ) <-> ( A = C /\ B = D ) ) |