Step |
Hyp |
Ref |
Expression |
1 |
|
dih1rn.h |
|- H = ( LHyp ` K ) |
2 |
|
dih1rn.i |
|- I = ( ( DIsoH ` K ) ` W ) |
3 |
|
dih1rn.u |
|- U = ( ( DVecH ` K ) ` W ) |
4 |
|
dih1rn.v |
|- V = ( Base ` U ) |
5 |
|
eqid |
|- ( 1. ` K ) = ( 1. ` K ) |
6 |
5 1 2 3 4
|
dih1 |
|- ( ( K e. HL /\ W e. H ) -> ( I ` ( 1. ` K ) ) = V ) |
7 |
|
hlop |
|- ( K e. HL -> K e. OP ) |
8 |
7
|
adantr |
|- ( ( K e. HL /\ W e. H ) -> K e. OP ) |
9 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
10 |
9 5
|
op1cl |
|- ( K e. OP -> ( 1. ` K ) e. ( Base ` K ) ) |
11 |
8 10
|
syl |
|- ( ( K e. HL /\ W e. H ) -> ( 1. ` K ) e. ( Base ` K ) ) |
12 |
9 1 2
|
dihcl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( 1. ` K ) e. ( Base ` K ) ) -> ( I ` ( 1. ` K ) ) e. ran I ) |
13 |
11 12
|
mpdan |
|- ( ( K e. HL /\ W e. H ) -> ( I ` ( 1. ` K ) ) e. ran I ) |
14 |
6 13
|
eqeltrrd |
|- ( ( K e. HL /\ W e. H ) -> V e. ran I ) |