# Metamath Proof Explorer

## Theorem normlem8

Description: Lemma used to derive properties of norm. (Contributed by NM, 30-Jun-2005) (New usage is discouraged.)

Ref Expression
Hypotheses normlem8.1
`|- A e. ~H`
normlem8.2
`|- B e. ~H`
normlem8.3
`|- C e. ~H`
normlem8.4
`|- D e. ~H`
Assertion normlem8
`|- ( ( A +h B ) .ih ( C +h D ) ) = ( ( ( A .ih C ) + ( B .ih D ) ) + ( ( A .ih D ) + ( B .ih C ) ) )`

### Proof

Step Hyp Ref Expression
1 normlem8.1
` |-  A e. ~H`
2 normlem8.2
` |-  B e. ~H`
3 normlem8.3
` |-  C e. ~H`
4 normlem8.4
` |-  D e. ~H`
5 his7
` |-  ( ( A e. ~H /\ C e. ~H /\ D e. ~H ) -> ( A .ih ( C +h D ) ) = ( ( A .ih C ) + ( A .ih D ) ) )`
6 1 3 4 5 mp3an
` |-  ( A .ih ( C +h D ) ) = ( ( A .ih C ) + ( A .ih D ) )`
7 his7
` |-  ( ( B e. ~H /\ C e. ~H /\ D e. ~H ) -> ( B .ih ( C +h D ) ) = ( ( B .ih C ) + ( B .ih D ) ) )`
8 2 3 4 7 mp3an
` |-  ( B .ih ( C +h D ) ) = ( ( B .ih C ) + ( B .ih D ) )`
9 6 8 oveq12i
` |-  ( ( A .ih ( C +h D ) ) + ( B .ih ( C +h D ) ) ) = ( ( ( A .ih C ) + ( A .ih D ) ) + ( ( B .ih C ) + ( B .ih D ) ) )`
` |-  ( C +h D ) e. ~H`
11 ax-his2
` |-  ( ( A e. ~H /\ B e. ~H /\ ( C +h D ) e. ~H ) -> ( ( A +h B ) .ih ( C +h D ) ) = ( ( A .ih ( C +h D ) ) + ( B .ih ( C +h D ) ) ) )`
12 1 2 10 11 mp3an
` |-  ( ( A +h B ) .ih ( C +h D ) ) = ( ( A .ih ( C +h D ) ) + ( B .ih ( C +h D ) ) )`
13 1 3 hicli
` |-  ( A .ih C ) e. CC`
14 2 4 hicli
` |-  ( B .ih D ) e. CC`
15 1 4 hicli
` |-  ( A .ih D ) e. CC`
16 2 3 hicli
` |-  ( B .ih C ) e. CC`
17 13 14 15 16 add42i
` |-  ( ( ( A .ih C ) + ( B .ih D ) ) + ( ( A .ih D ) + ( B .ih C ) ) ) = ( ( ( A .ih C ) + ( A .ih D ) ) + ( ( B .ih C ) + ( B .ih D ) ) )`
18 9 12 17 3eqtr4i
` |-  ( ( A +h B ) .ih ( C +h D ) ) = ( ( ( A .ih C ) + ( B .ih D ) ) + ( ( A .ih D ) + ( B .ih C ) ) )`