| Step |
Hyp |
Ref |
Expression |
| 1 |
|
smndex1ibas.m |
|- M = ( EndoFMnd ` NN0 ) |
| 2 |
|
smndex1ibas.n |
|- N e. NN |
| 3 |
|
smndex1ibas.i |
|- I = ( x e. NN0 |-> ( x mod N ) ) |
| 4 |
|
smndex1ibas.g |
|- G = ( n e. ( 0 ..^ N ) |-> ( x e. NN0 |-> n ) ) |
| 5 |
|
elfzonn0 |
|- ( K e. ( 0 ..^ N ) -> K e. NN0 ) |
| 6 |
5
|
adantr |
|- ( ( K e. ( 0 ..^ N ) /\ x e. NN0 ) -> K e. NN0 ) |
| 7 |
6
|
ralrimiva |
|- ( K e. ( 0 ..^ N ) -> A. x e. NN0 K e. NN0 ) |
| 8 |
|
eqid |
|- ( x e. NN0 |-> K ) = ( x e. NN0 |-> K ) |
| 9 |
8
|
fmpt |
|- ( A. x e. NN0 K e. NN0 <-> ( x e. NN0 |-> K ) : NN0 --> NN0 ) |
| 10 |
7 9
|
sylib |
|- ( K e. ( 0 ..^ N ) -> ( x e. NN0 |-> K ) : NN0 --> NN0 ) |
| 11 |
|
nn0ex |
|- NN0 e. _V |
| 12 |
11 11
|
elmap |
|- ( ( x e. NN0 |-> K ) e. ( NN0 ^m NN0 ) <-> ( x e. NN0 |-> K ) : NN0 --> NN0 ) |
| 13 |
10 12
|
sylibr |
|- ( K e. ( 0 ..^ N ) -> ( x e. NN0 |-> K ) e. ( NN0 ^m NN0 ) ) |
| 14 |
4
|
a1i |
|- ( K e. ( 0 ..^ N ) -> G = ( n e. ( 0 ..^ N ) |-> ( x e. NN0 |-> n ) ) ) |
| 15 |
|
id |
|- ( n = K -> n = K ) |
| 16 |
15
|
mpteq2dv |
|- ( n = K -> ( x e. NN0 |-> n ) = ( x e. NN0 |-> K ) ) |
| 17 |
16
|
adantl |
|- ( ( K e. ( 0 ..^ N ) /\ n = K ) -> ( x e. NN0 |-> n ) = ( x e. NN0 |-> K ) ) |
| 18 |
|
id |
|- ( K e. ( 0 ..^ N ) -> K e. ( 0 ..^ N ) ) |
| 19 |
11
|
mptex |
|- ( x e. NN0 |-> K ) e. _V |
| 20 |
19
|
a1i |
|- ( K e. ( 0 ..^ N ) -> ( x e. NN0 |-> K ) e. _V ) |
| 21 |
14 17 18 20
|
fvmptd |
|- ( K e. ( 0 ..^ N ) -> ( G ` K ) = ( x e. NN0 |-> K ) ) |
| 22 |
|
eqid |
|- ( Base ` M ) = ( Base ` M ) |
| 23 |
1 22
|
efmndbas |
|- ( Base ` M ) = ( NN0 ^m NN0 ) |
| 24 |
23
|
a1i |
|- ( K e. ( 0 ..^ N ) -> ( Base ` M ) = ( NN0 ^m NN0 ) ) |
| 25 |
13 21 24
|
3eltr4d |
|- ( K e. ( 0 ..^ N ) -> ( G ` K ) e. ( Base ` M ) ) |