Step |
Hyp |
Ref |
Expression |
1 |
|
oveq1 |
|- ( x = A -> ( x +o B ) = ( A +o B ) ) |
2 |
|
oveq2 |
|- ( x = A -> ( B +o x ) = ( B +o A ) ) |
3 |
1 2
|
eqeq12d |
|- ( x = A -> ( ( x +o B ) = ( B +o x ) <-> ( A +o B ) = ( B +o A ) ) ) |
4 |
3
|
imbi2d |
|- ( x = A -> ( ( B e. _om -> ( x +o B ) = ( B +o x ) ) <-> ( B e. _om -> ( A +o B ) = ( B +o A ) ) ) ) |
5 |
|
oveq1 |
|- ( x = (/) -> ( x +o B ) = ( (/) +o B ) ) |
6 |
|
oveq2 |
|- ( x = (/) -> ( B +o x ) = ( B +o (/) ) ) |
7 |
5 6
|
eqeq12d |
|- ( x = (/) -> ( ( x +o B ) = ( B +o x ) <-> ( (/) +o B ) = ( B +o (/) ) ) ) |
8 |
|
oveq1 |
|- ( x = y -> ( x +o B ) = ( y +o B ) ) |
9 |
|
oveq2 |
|- ( x = y -> ( B +o x ) = ( B +o y ) ) |
10 |
8 9
|
eqeq12d |
|- ( x = y -> ( ( x +o B ) = ( B +o x ) <-> ( y +o B ) = ( B +o y ) ) ) |
11 |
|
oveq1 |
|- ( x = suc y -> ( x +o B ) = ( suc y +o B ) ) |
12 |
|
oveq2 |
|- ( x = suc y -> ( B +o x ) = ( B +o suc y ) ) |
13 |
11 12
|
eqeq12d |
|- ( x = suc y -> ( ( x +o B ) = ( B +o x ) <-> ( suc y +o B ) = ( B +o suc y ) ) ) |
14 |
|
nna0r |
|- ( B e. _om -> ( (/) +o B ) = B ) |
15 |
|
nna0 |
|- ( B e. _om -> ( B +o (/) ) = B ) |
16 |
14 15
|
eqtr4d |
|- ( B e. _om -> ( (/) +o B ) = ( B +o (/) ) ) |
17 |
|
suceq |
|- ( ( y +o B ) = ( B +o y ) -> suc ( y +o B ) = suc ( B +o y ) ) |
18 |
|
oveq2 |
|- ( x = B -> ( suc y +o x ) = ( suc y +o B ) ) |
19 |
|
oveq2 |
|- ( x = B -> ( y +o x ) = ( y +o B ) ) |
20 |
|
suceq |
|- ( ( y +o x ) = ( y +o B ) -> suc ( y +o x ) = suc ( y +o B ) ) |
21 |
19 20
|
syl |
|- ( x = B -> suc ( y +o x ) = suc ( y +o B ) ) |
22 |
18 21
|
eqeq12d |
|- ( x = B -> ( ( suc y +o x ) = suc ( y +o x ) <-> ( suc y +o B ) = suc ( y +o B ) ) ) |
23 |
22
|
imbi2d |
|- ( x = B -> ( ( y e. _om -> ( suc y +o x ) = suc ( y +o x ) ) <-> ( y e. _om -> ( suc y +o B ) = suc ( y +o B ) ) ) ) |
24 |
|
oveq2 |
|- ( x = (/) -> ( suc y +o x ) = ( suc y +o (/) ) ) |
25 |
|
oveq2 |
|- ( x = (/) -> ( y +o x ) = ( y +o (/) ) ) |
26 |
|
suceq |
|- ( ( y +o x ) = ( y +o (/) ) -> suc ( y +o x ) = suc ( y +o (/) ) ) |
27 |
25 26
|
syl |
|- ( x = (/) -> suc ( y +o x ) = suc ( y +o (/) ) ) |
28 |
24 27
|
eqeq12d |
|- ( x = (/) -> ( ( suc y +o x ) = suc ( y +o x ) <-> ( suc y +o (/) ) = suc ( y +o (/) ) ) ) |
29 |
|
oveq2 |
|- ( x = z -> ( suc y +o x ) = ( suc y +o z ) ) |
30 |
|
oveq2 |
|- ( x = z -> ( y +o x ) = ( y +o z ) ) |
31 |
|
suceq |
|- ( ( y +o x ) = ( y +o z ) -> suc ( y +o x ) = suc ( y +o z ) ) |
32 |
30 31
|
syl |
|- ( x = z -> suc ( y +o x ) = suc ( y +o z ) ) |
33 |
29 32
|
eqeq12d |
|- ( x = z -> ( ( suc y +o x ) = suc ( y +o x ) <-> ( suc y +o z ) = suc ( y +o z ) ) ) |
34 |
|
oveq2 |
|- ( x = suc z -> ( suc y +o x ) = ( suc y +o suc z ) ) |
35 |
|
oveq2 |
|- ( x = suc z -> ( y +o x ) = ( y +o suc z ) ) |
36 |
|
suceq |
|- ( ( y +o x ) = ( y +o suc z ) -> suc ( y +o x ) = suc ( y +o suc z ) ) |
37 |
35 36
|
syl |
|- ( x = suc z -> suc ( y +o x ) = suc ( y +o suc z ) ) |
38 |
34 37
|
eqeq12d |
|- ( x = suc z -> ( ( suc y +o x ) = suc ( y +o x ) <-> ( suc y +o suc z ) = suc ( y +o suc z ) ) ) |
39 |
|
peano2 |
|- ( y e. _om -> suc y e. _om ) |
40 |
|
nna0 |
|- ( suc y e. _om -> ( suc y +o (/) ) = suc y ) |
41 |
39 40
|
syl |
|- ( y e. _om -> ( suc y +o (/) ) = suc y ) |
42 |
|
nna0 |
|- ( y e. _om -> ( y +o (/) ) = y ) |
43 |
|
suceq |
|- ( ( y +o (/) ) = y -> suc ( y +o (/) ) = suc y ) |
44 |
42 43
|
syl |
|- ( y e. _om -> suc ( y +o (/) ) = suc y ) |
45 |
41 44
|
eqtr4d |
|- ( y e. _om -> ( suc y +o (/) ) = suc ( y +o (/) ) ) |
46 |
|
suceq |
|- ( ( suc y +o z ) = suc ( y +o z ) -> suc ( suc y +o z ) = suc suc ( y +o z ) ) |
47 |
|
nnasuc |
|- ( ( suc y e. _om /\ z e. _om ) -> ( suc y +o suc z ) = suc ( suc y +o z ) ) |
48 |
39 47
|
sylan |
|- ( ( y e. _om /\ z e. _om ) -> ( suc y +o suc z ) = suc ( suc y +o z ) ) |
49 |
|
nnasuc |
|- ( ( y e. _om /\ z e. _om ) -> ( y +o suc z ) = suc ( y +o z ) ) |
50 |
|
suceq |
|- ( ( y +o suc z ) = suc ( y +o z ) -> suc ( y +o suc z ) = suc suc ( y +o z ) ) |
51 |
49 50
|
syl |
|- ( ( y e. _om /\ z e. _om ) -> suc ( y +o suc z ) = suc suc ( y +o z ) ) |
52 |
48 51
|
eqeq12d |
|- ( ( y e. _om /\ z e. _om ) -> ( ( suc y +o suc z ) = suc ( y +o suc z ) <-> suc ( suc y +o z ) = suc suc ( y +o z ) ) ) |
53 |
46 52
|
syl5ibr |
|- ( ( y e. _om /\ z e. _om ) -> ( ( suc y +o z ) = suc ( y +o z ) -> ( suc y +o suc z ) = suc ( y +o suc z ) ) ) |
54 |
53
|
expcom |
|- ( z e. _om -> ( y e. _om -> ( ( suc y +o z ) = suc ( y +o z ) -> ( suc y +o suc z ) = suc ( y +o suc z ) ) ) ) |
55 |
28 33 38 45 54
|
finds2 |
|- ( x e. _om -> ( y e. _om -> ( suc y +o x ) = suc ( y +o x ) ) ) |
56 |
23 55
|
vtoclga |
|- ( B e. _om -> ( y e. _om -> ( suc y +o B ) = suc ( y +o B ) ) ) |
57 |
56
|
imp |
|- ( ( B e. _om /\ y e. _om ) -> ( suc y +o B ) = suc ( y +o B ) ) |
58 |
|
nnasuc |
|- ( ( B e. _om /\ y e. _om ) -> ( B +o suc y ) = suc ( B +o y ) ) |
59 |
57 58
|
eqeq12d |
|- ( ( B e. _om /\ y e. _om ) -> ( ( suc y +o B ) = ( B +o suc y ) <-> suc ( y +o B ) = suc ( B +o y ) ) ) |
60 |
17 59
|
syl5ibr |
|- ( ( B e. _om /\ y e. _om ) -> ( ( y +o B ) = ( B +o y ) -> ( suc y +o B ) = ( B +o suc y ) ) ) |
61 |
60
|
expcom |
|- ( y e. _om -> ( B e. _om -> ( ( y +o B ) = ( B +o y ) -> ( suc y +o B ) = ( B +o suc y ) ) ) ) |
62 |
7 10 13 16 61
|
finds2 |
|- ( x e. _om -> ( B e. _om -> ( x +o B ) = ( B +o x ) ) ) |
63 |
4 62
|
vtoclga |
|- ( A e. _om -> ( B e. _om -> ( A +o B ) = ( B +o A ) ) ) |
64 |
63
|
imp |
|- ( ( A e. _om /\ B e. _om ) -> ( A +o B ) = ( B +o A ) ) |