Step |
Hyp |
Ref |
Expression |
1 |
|
diam.m |
|- ./\ = ( meet ` K ) |
2 |
|
diam.h |
|- H = ( LHyp ` K ) |
3 |
|
diam.i |
|- I = ( ( DIsoA ` K ) ` W ) |
4 |
|
simpl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. ran I /\ Y e. ran I ) ) -> ( K e. HL /\ W e. H ) ) |
5 |
2 3
|
diacnvclN |
|- ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( `' I ` X ) e. dom I ) |
6 |
5
|
adantrr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. ran I /\ Y e. ran I ) ) -> ( `' I ` X ) e. dom I ) |
7 |
2 3
|
diacnvclN |
|- ( ( ( K e. HL /\ W e. H ) /\ Y e. ran I ) -> ( `' I ` Y ) e. dom I ) |
8 |
7
|
adantrl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. ran I /\ Y e. ran I ) ) -> ( `' I ` Y ) e. dom I ) |
9 |
1 2 3
|
diameetN |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( `' I ` X ) e. dom I /\ ( `' I ` Y ) e. dom I ) ) -> ( I ` ( ( `' I ` X ) ./\ ( `' I ` Y ) ) ) = ( ( I ` ( `' I ` X ) ) i^i ( I ` ( `' I ` Y ) ) ) ) |
10 |
4 6 8 9
|
syl12anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. ran I /\ Y e. ran I ) ) -> ( I ` ( ( `' I ` X ) ./\ ( `' I ` Y ) ) ) = ( ( I ` ( `' I ` X ) ) i^i ( I ` ( `' I ` Y ) ) ) ) |
11 |
2 3
|
diaf11N |
|- ( ( K e. HL /\ W e. H ) -> I : dom I -1-1-onto-> ran I ) |
12 |
11
|
adantr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. ran I /\ Y e. ran I ) ) -> I : dom I -1-1-onto-> ran I ) |
13 |
|
simprl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. ran I /\ Y e. ran I ) ) -> X e. ran I ) |
14 |
|
f1ocnvfv2 |
|- ( ( I : dom I -1-1-onto-> ran I /\ X e. ran I ) -> ( I ` ( `' I ` X ) ) = X ) |
15 |
12 13 14
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. ran I /\ Y e. ran I ) ) -> ( I ` ( `' I ` X ) ) = X ) |
16 |
|
simprr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. ran I /\ Y e. ran I ) ) -> Y e. ran I ) |
17 |
|
f1ocnvfv2 |
|- ( ( I : dom I -1-1-onto-> ran I /\ Y e. ran I ) -> ( I ` ( `' I ` Y ) ) = Y ) |
18 |
12 16 17
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. ran I /\ Y e. ran I ) ) -> ( I ` ( `' I ` Y ) ) = Y ) |
19 |
15 18
|
ineq12d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. ran I /\ Y e. ran I ) ) -> ( ( I ` ( `' I ` X ) ) i^i ( I ` ( `' I ` Y ) ) ) = ( X i^i Y ) ) |
20 |
10 19
|
eqtr2d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. ran I /\ Y e. ran I ) ) -> ( X i^i Y ) = ( I ` ( ( `' I ` X ) ./\ ( `' I ` Y ) ) ) ) |