Step |
Hyp |
Ref |
Expression |
1 |
|
hl0lt1.s |
|- .< = ( lt ` K ) |
2 |
|
hl0lt1.z |
|- .0. = ( 0. ` K ) |
3 |
|
hl0lt1.u |
|- .1. = ( 1. ` K ) |
4 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
5 |
4 1 2 3
|
hlhgt2 |
|- ( K e. HL -> E. x e. ( Base ` K ) ( .0. .< x /\ x .< .1. ) ) |
6 |
|
hlpos |
|- ( K e. HL -> K e. Poset ) |
7 |
6
|
adantr |
|- ( ( K e. HL /\ x e. ( Base ` K ) ) -> K e. Poset ) |
8 |
|
hlop |
|- ( K e. HL -> K e. OP ) |
9 |
8
|
adantr |
|- ( ( K e. HL /\ x e. ( Base ` K ) ) -> K e. OP ) |
10 |
4 2
|
op0cl |
|- ( K e. OP -> .0. e. ( Base ` K ) ) |
11 |
9 10
|
syl |
|- ( ( K e. HL /\ x e. ( Base ` K ) ) -> .0. e. ( Base ` K ) ) |
12 |
|
simpr |
|- ( ( K e. HL /\ x e. ( Base ` K ) ) -> x e. ( Base ` K ) ) |
13 |
4 3
|
op1cl |
|- ( K e. OP -> .1. e. ( Base ` K ) ) |
14 |
9 13
|
syl |
|- ( ( K e. HL /\ x e. ( Base ` K ) ) -> .1. e. ( Base ` K ) ) |
15 |
4 1
|
plttr |
|- ( ( K e. Poset /\ ( .0. e. ( Base ` K ) /\ x e. ( Base ` K ) /\ .1. e. ( Base ` K ) ) ) -> ( ( .0. .< x /\ x .< .1. ) -> .0. .< .1. ) ) |
16 |
7 11 12 14 15
|
syl13anc |
|- ( ( K e. HL /\ x e. ( Base ` K ) ) -> ( ( .0. .< x /\ x .< .1. ) -> .0. .< .1. ) ) |
17 |
16
|
rexlimdva |
|- ( K e. HL -> ( E. x e. ( Base ` K ) ( .0. .< x /\ x .< .1. ) -> .0. .< .1. ) ) |
18 |
5 17
|
mpd |
|- ( K e. HL -> .0. .< .1. ) |