Step |
Hyp |
Ref |
Expression |
1 |
|
nn0mnd.g |
|- M = { <. ( Base ` ndx ) , NN0 >. , <. ( +g ` ndx ) , + >. } |
2 |
|
nn0addcl |
|- ( ( x e. NN0 /\ y e. NN0 ) -> ( x + y ) e. NN0 ) |
3 |
|
nn0cn |
|- ( x e. NN0 -> x e. CC ) |
4 |
|
nn0cn |
|- ( y e. NN0 -> y e. CC ) |
5 |
|
nn0cn |
|- ( z e. NN0 -> z e. CC ) |
6 |
3 4 5
|
3anim123i |
|- ( ( x e. NN0 /\ y e. NN0 /\ z e. NN0 ) -> ( x e. CC /\ y e. CC /\ z e. CC ) ) |
7 |
6
|
3expa |
|- ( ( ( x e. NN0 /\ y e. NN0 ) /\ z e. NN0 ) -> ( x e. CC /\ y e. CC /\ z e. CC ) ) |
8 |
|
addass |
|- ( ( x e. CC /\ y e. CC /\ z e. CC ) -> ( ( x + y ) + z ) = ( x + ( y + z ) ) ) |
9 |
7 8
|
syl |
|- ( ( ( x e. NN0 /\ y e. NN0 ) /\ z e. NN0 ) -> ( ( x + y ) + z ) = ( x + ( y + z ) ) ) |
10 |
9
|
ralrimiva |
|- ( ( x e. NN0 /\ y e. NN0 ) -> A. z e. NN0 ( ( x + y ) + z ) = ( x + ( y + z ) ) ) |
11 |
2 10
|
jca |
|- ( ( x e. NN0 /\ y e. NN0 ) -> ( ( x + y ) e. NN0 /\ A. z e. NN0 ( ( x + y ) + z ) = ( x + ( y + z ) ) ) ) |
12 |
11
|
rgen2 |
|- A. x e. NN0 A. y e. NN0 ( ( x + y ) e. NN0 /\ A. z e. NN0 ( ( x + y ) + z ) = ( x + ( y + z ) ) ) |
13 |
|
c0ex |
|- 0 e. _V |
14 |
|
eleq1 |
|- ( e = 0 -> ( e e. NN0 <-> 0 e. NN0 ) ) |
15 |
|
oveq1 |
|- ( e = 0 -> ( e + x ) = ( 0 + x ) ) |
16 |
15
|
eqeq1d |
|- ( e = 0 -> ( ( e + x ) = x <-> ( 0 + x ) = x ) ) |
17 |
|
oveq2 |
|- ( e = 0 -> ( x + e ) = ( x + 0 ) ) |
18 |
17
|
eqeq1d |
|- ( e = 0 -> ( ( x + e ) = x <-> ( x + 0 ) = x ) ) |
19 |
16 18
|
anbi12d |
|- ( e = 0 -> ( ( ( e + x ) = x /\ ( x + e ) = x ) <-> ( ( 0 + x ) = x /\ ( x + 0 ) = x ) ) ) |
20 |
19
|
ralbidv |
|- ( e = 0 -> ( A. x e. NN0 ( ( e + x ) = x /\ ( x + e ) = x ) <-> A. x e. NN0 ( ( 0 + x ) = x /\ ( x + 0 ) = x ) ) ) |
21 |
14 20
|
anbi12d |
|- ( e = 0 -> ( ( e e. NN0 /\ A. x e. NN0 ( ( e + x ) = x /\ ( x + e ) = x ) ) <-> ( 0 e. NN0 /\ A. x e. NN0 ( ( 0 + x ) = x /\ ( x + 0 ) = x ) ) ) ) |
22 |
|
0nn0 |
|- 0 e. NN0 |
23 |
3
|
addid2d |
|- ( x e. NN0 -> ( 0 + x ) = x ) |
24 |
3
|
addid1d |
|- ( x e. NN0 -> ( x + 0 ) = x ) |
25 |
23 24
|
jca |
|- ( x e. NN0 -> ( ( 0 + x ) = x /\ ( x + 0 ) = x ) ) |
26 |
25
|
rgen |
|- A. x e. NN0 ( ( 0 + x ) = x /\ ( x + 0 ) = x ) |
27 |
22 26
|
pm3.2i |
|- ( 0 e. NN0 /\ A. x e. NN0 ( ( 0 + x ) = x /\ ( x + 0 ) = x ) ) |
28 |
13 21 27
|
ceqsexv2d |
|- E. e ( e e. NN0 /\ A. x e. NN0 ( ( e + x ) = x /\ ( x + e ) = x ) ) |
29 |
|
df-rex |
|- ( E. e e. NN0 A. x e. NN0 ( ( e + x ) = x /\ ( x + e ) = x ) <-> E. e ( e e. NN0 /\ A. x e. NN0 ( ( e + x ) = x /\ ( x + e ) = x ) ) ) |
30 |
28 29
|
mpbir |
|- E. e e. NN0 A. x e. NN0 ( ( e + x ) = x /\ ( x + e ) = x ) |
31 |
12 30
|
pm3.2i |
|- ( A. x e. NN0 A. y e. NN0 ( ( x + y ) e. NN0 /\ A. z e. NN0 ( ( x + y ) + z ) = ( x + ( y + z ) ) ) /\ E. e e. NN0 A. x e. NN0 ( ( e + x ) = x /\ ( x + e ) = x ) ) |
32 |
|
nn0ex |
|- NN0 e. _V |
33 |
1
|
grpbase |
|- ( NN0 e. _V -> NN0 = ( Base ` M ) ) |
34 |
32 33
|
ax-mp |
|- NN0 = ( Base ` M ) |
35 |
|
addex |
|- + e. _V |
36 |
1
|
grpplusg |
|- ( + e. _V -> + = ( +g ` M ) ) |
37 |
35 36
|
ax-mp |
|- + = ( +g ` M ) |
38 |
34 37
|
ismnd |
|- ( M e. Mnd <-> ( A. x e. NN0 A. y e. NN0 ( ( x + y ) e. NN0 /\ A. z e. NN0 ( ( x + y ) + z ) = ( x + ( y + z ) ) ) /\ E. e e. NN0 A. x e. NN0 ( ( e + x ) = x /\ ( x + e ) = x ) ) ) |
39 |
31 38
|
mpbir |
|- M e. Mnd |