| Step |
Hyp |
Ref |
Expression |
| 1 |
|
olm1.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
| 2 |
|
olm1.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
| 3 |
|
olm1.u |
⊢ 1 = ( 1. ‘ 𝐾 ) |
| 4 |
|
olop |
⊢ ( 𝐾 ∈ OL → 𝐾 ∈ OP ) |
| 5 |
4
|
adantr |
⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → 𝐾 ∈ OP ) |
| 6 |
|
eqid |
⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 ) |
| 7 |
|
eqid |
⊢ ( oc ‘ 𝐾 ) = ( oc ‘ 𝐾 ) |
| 8 |
6 3 7
|
opoc1 |
⊢ ( 𝐾 ∈ OP → ( ( oc ‘ 𝐾 ) ‘ 1 ) = ( 0. ‘ 𝐾 ) ) |
| 9 |
5 8
|
syl |
⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → ( ( oc ‘ 𝐾 ) ‘ 1 ) = ( 0. ‘ 𝐾 ) ) |
| 10 |
9
|
oveq2d |
⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 1 ) ) = ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( join ‘ 𝐾 ) ( 0. ‘ 𝐾 ) ) ) |
| 11 |
1 7
|
opoccl |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∈ 𝐵 ) |
| 12 |
4 11
|
sylan |
⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∈ 𝐵 ) |
| 13 |
|
eqid |
⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 ) |
| 14 |
1 13 6
|
olj01 |
⊢ ( ( 𝐾 ∈ OL ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∈ 𝐵 ) → ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( join ‘ 𝐾 ) ( 0. ‘ 𝐾 ) ) = ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ) |
| 15 |
12 14
|
syldan |
⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( join ‘ 𝐾 ) ( 0. ‘ 𝐾 ) ) = ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ) |
| 16 |
10 15
|
eqtrd |
⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 1 ) ) = ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ) |
| 17 |
16
|
fveq2d |
⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → ( ( oc ‘ 𝐾 ) ‘ ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 1 ) ) ) = ( ( oc ‘ 𝐾 ) ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ) ) |
| 18 |
1 3
|
op1cl |
⊢ ( 𝐾 ∈ OP → 1 ∈ 𝐵 ) |
| 19 |
5 18
|
syl |
⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → 1 ∈ 𝐵 ) |
| 20 |
1 13 2 7
|
oldmj4 |
⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 1 ∈ 𝐵 ) → ( ( oc ‘ 𝐾 ) ‘ ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 1 ) ) ) = ( 𝑋 ∧ 1 ) ) |
| 21 |
19 20
|
mpd3an3 |
⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → ( ( oc ‘ 𝐾 ) ‘ ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 1 ) ) ) = ( 𝑋 ∧ 1 ) ) |
| 22 |
1 7
|
opococ |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → ( ( oc ‘ 𝐾 ) ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ) = 𝑋 ) |
| 23 |
4 22
|
sylan |
⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → ( ( oc ‘ 𝐾 ) ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ) = 𝑋 ) |
| 24 |
17 21 23
|
3eqtr3d |
⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∧ 1 ) = 𝑋 ) |