Step |
Hyp |
Ref |
Expression |
1 |
|
polid.1 |
|- A e. ~H |
2 |
|
polid.2 |
|- B e. ~H |
3 |
1 2 2 1
|
polid2i |
|- ( A .ih B ) = ( ( ( ( ( A +h B ) .ih ( A +h B ) ) - ( ( A -h B ) .ih ( A -h B ) ) ) + ( _i x. ( ( ( A +h ( _i .h B ) ) .ih ( A +h ( _i .h B ) ) ) - ( ( A -h ( _i .h B ) ) .ih ( A -h ( _i .h B ) ) ) ) ) ) / 4 ) |
4 |
1 2
|
hvaddcli |
|- ( A +h B ) e. ~H |
5 |
4
|
normsqi |
|- ( ( normh ` ( A +h B ) ) ^ 2 ) = ( ( A +h B ) .ih ( A +h B ) ) |
6 |
1 2
|
hvsubcli |
|- ( A -h B ) e. ~H |
7 |
6
|
normsqi |
|- ( ( normh ` ( A -h B ) ) ^ 2 ) = ( ( A -h B ) .ih ( A -h B ) ) |
8 |
5 7
|
oveq12i |
|- ( ( ( normh ` ( A +h B ) ) ^ 2 ) - ( ( normh ` ( A -h B ) ) ^ 2 ) ) = ( ( ( A +h B ) .ih ( A +h B ) ) - ( ( A -h B ) .ih ( A -h B ) ) ) |
9 |
|
ax-icn |
|- _i e. CC |
10 |
9 2
|
hvmulcli |
|- ( _i .h B ) e. ~H |
11 |
1 10
|
hvaddcli |
|- ( A +h ( _i .h B ) ) e. ~H |
12 |
11
|
normsqi |
|- ( ( normh ` ( A +h ( _i .h B ) ) ) ^ 2 ) = ( ( A +h ( _i .h B ) ) .ih ( A +h ( _i .h B ) ) ) |
13 |
1 10
|
hvsubcli |
|- ( A -h ( _i .h B ) ) e. ~H |
14 |
13
|
normsqi |
|- ( ( normh ` ( A -h ( _i .h B ) ) ) ^ 2 ) = ( ( A -h ( _i .h B ) ) .ih ( A -h ( _i .h B ) ) ) |
15 |
12 14
|
oveq12i |
|- ( ( ( normh ` ( A +h ( _i .h B ) ) ) ^ 2 ) - ( ( normh ` ( A -h ( _i .h B ) ) ) ^ 2 ) ) = ( ( ( A +h ( _i .h B ) ) .ih ( A +h ( _i .h B ) ) ) - ( ( A -h ( _i .h B ) ) .ih ( A -h ( _i .h B ) ) ) ) |
16 |
15
|
oveq2i |
|- ( _i x. ( ( ( normh ` ( A +h ( _i .h B ) ) ) ^ 2 ) - ( ( normh ` ( A -h ( _i .h B ) ) ) ^ 2 ) ) ) = ( _i x. ( ( ( A +h ( _i .h B ) ) .ih ( A +h ( _i .h B ) ) ) - ( ( A -h ( _i .h B ) ) .ih ( A -h ( _i .h B ) ) ) ) ) |
17 |
8 16
|
oveq12i |
|- ( ( ( ( normh ` ( A +h B ) ) ^ 2 ) - ( ( normh ` ( A -h B ) ) ^ 2 ) ) + ( _i x. ( ( ( normh ` ( A +h ( _i .h B ) ) ) ^ 2 ) - ( ( normh ` ( A -h ( _i .h B ) ) ) ^ 2 ) ) ) ) = ( ( ( ( A +h B ) .ih ( A +h B ) ) - ( ( A -h B ) .ih ( A -h B ) ) ) + ( _i x. ( ( ( A +h ( _i .h B ) ) .ih ( A +h ( _i .h B ) ) ) - ( ( A -h ( _i .h B ) ) .ih ( A -h ( _i .h B ) ) ) ) ) ) |
18 |
17
|
oveq1i |
|- ( ( ( ( ( normh ` ( A +h B ) ) ^ 2 ) - ( ( normh ` ( A -h B ) ) ^ 2 ) ) + ( _i x. ( ( ( normh ` ( A +h ( _i .h B ) ) ) ^ 2 ) - ( ( normh ` ( A -h ( _i .h B ) ) ) ^ 2 ) ) ) ) / 4 ) = ( ( ( ( ( A +h B ) .ih ( A +h B ) ) - ( ( A -h B ) .ih ( A -h B ) ) ) + ( _i x. ( ( ( A +h ( _i .h B ) ) .ih ( A +h ( _i .h B ) ) ) - ( ( A -h ( _i .h B ) ) .ih ( A -h ( _i .h B ) ) ) ) ) ) / 4 ) |
19 |
3 18
|
eqtr4i |
|- ( A .ih B ) = ( ( ( ( ( normh ` ( A +h B ) ) ^ 2 ) - ( ( normh ` ( A -h B ) ) ^ 2 ) ) + ( _i x. ( ( ( normh ` ( A +h ( _i .h B ) ) ) ^ 2 ) - ( ( normh ` ( A -h ( _i .h B ) ) ) ^ 2 ) ) ) ) / 4 ) |