Step |
Hyp |
Ref |
Expression |
1 |
|
df-2 |
|- 2 = ( 1 + 1 ) |
2 |
1
|
oveq1i |
|- ( 2 .h A ) = ( ( 1 + 1 ) .h A ) |
3 |
|
ax-1cn |
|- 1 e. CC |
4 |
|
ax-hvdistr2 |
|- ( ( 1 e. CC /\ 1 e. CC /\ A e. ~H ) -> ( ( 1 + 1 ) .h A ) = ( ( 1 .h A ) +h ( 1 .h A ) ) ) |
5 |
3 3 4
|
mp3an12 |
|- ( A e. ~H -> ( ( 1 + 1 ) .h A ) = ( ( 1 .h A ) +h ( 1 .h A ) ) ) |
6 |
2 5
|
syl5eq |
|- ( A e. ~H -> ( 2 .h A ) = ( ( 1 .h A ) +h ( 1 .h A ) ) ) |
7 |
|
ax-hvdistr1 |
|- ( ( 1 e. CC /\ A e. ~H /\ A e. ~H ) -> ( 1 .h ( A +h A ) ) = ( ( 1 .h A ) +h ( 1 .h A ) ) ) |
8 |
3 7
|
mp3an1 |
|- ( ( A e. ~H /\ A e. ~H ) -> ( 1 .h ( A +h A ) ) = ( ( 1 .h A ) +h ( 1 .h A ) ) ) |
9 |
8
|
anidms |
|- ( A e. ~H -> ( 1 .h ( A +h A ) ) = ( ( 1 .h A ) +h ( 1 .h A ) ) ) |
10 |
|
hvaddcl |
|- ( ( A e. ~H /\ A e. ~H ) -> ( A +h A ) e. ~H ) |
11 |
10
|
anidms |
|- ( A e. ~H -> ( A +h A ) e. ~H ) |
12 |
|
ax-hvmulid |
|- ( ( A +h A ) e. ~H -> ( 1 .h ( A +h A ) ) = ( A +h A ) ) |
13 |
11 12
|
syl |
|- ( A e. ~H -> ( 1 .h ( A +h A ) ) = ( A +h A ) ) |
14 |
6 9 13
|
3eqtr2d |
|- ( A e. ~H -> ( 2 .h A ) = ( A +h A ) ) |