Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme43.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
cdleme43.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
cdleme43.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
cdleme43.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
5 |
|
cdleme43.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
6 |
|
cdleme43.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
7 |
|
cdleme43.u |
⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) |
8 |
|
cdleme43.x |
⊢ 𝑋 = ( ( 𝑄 ∨ 𝑃 ) ∧ 𝑊 ) |
9 |
|
cdleme43.c |
⊢ 𝐶 = ( ( 𝑆 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ) |
10 |
|
cdleme43.f |
⊢ 𝑍 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐶 ∨ ( ( 𝑅 ∨ 𝑆 ) ∧ 𝑊 ) ) ) |
11 |
|
cdleme43.d |
⊢ 𝐷 = ( ( 𝑆 ∨ 𝑋 ) ∧ ( 𝑃 ∨ ( ( 𝑄 ∨ 𝑆 ) ∧ 𝑊 ) ) ) |
12 |
|
cdleme43.g |
⊢ 𝐺 = ( ( 𝑄 ∨ 𝑃 ) ∧ ( 𝐷 ∨ ( ( 𝑍 ∨ 𝑆 ) ∧ 𝑊 ) ) ) |
13 |
|
cdleme43.e |
⊢ 𝐸 = ( ( 𝐷 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝐷 ) ∧ 𝑊 ) ) ) |
14 |
|
cdleme43.v |
⊢ 𝑉 = ( ( 𝑍 ∨ 𝑆 ) ∧ 𝑊 ) |
15 |
|
cdleme43.y |
⊢ 𝑌 = ( ( 𝑅 ∨ 𝐷 ) ∧ 𝑊 ) |
16 |
3 5
|
hlatjcom |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑄 ) = ( 𝑄 ∨ 𝑃 ) ) |
17 |
14
|
oveq2i |
⊢ ( 𝐷 ∨ 𝑉 ) = ( 𝐷 ∨ ( ( 𝑍 ∨ 𝑆 ) ∧ 𝑊 ) ) |
18 |
17
|
a1i |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝐷 ∨ 𝑉 ) = ( 𝐷 ∨ ( ( 𝑍 ∨ 𝑆 ) ∧ 𝑊 ) ) ) |
19 |
16 18
|
oveq12d |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐷 ∨ 𝑉 ) ) = ( ( 𝑄 ∨ 𝑃 ) ∧ ( 𝐷 ∨ ( ( 𝑍 ∨ 𝑆 ) ∧ 𝑊 ) ) ) ) |
20 |
12 19
|
eqtr4id |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝐺 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐷 ∨ 𝑉 ) ) ) |