Step |
Hyp |
Ref |
Expression |
1 |
|
paddass.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
2 |
|
paddass.p |
⊢ + = ( +𝑃 ‘ 𝐾 ) |
3 |
|
hllat |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat ) |
4 |
3
|
adantr |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) → 𝐾 ∈ Lat ) |
5 |
|
simpr1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) → 𝑋 ⊆ 𝐴 ) |
6 |
|
simpr2 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) → 𝑌 ⊆ 𝐴 ) |
7 |
1 2
|
paddcom |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) → ( 𝑋 + 𝑌 ) = ( 𝑌 + 𝑋 ) ) |
8 |
4 5 6 7
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) → ( 𝑋 + 𝑌 ) = ( 𝑌 + 𝑋 ) ) |
9 |
8
|
oveq1d |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) → ( ( 𝑋 + 𝑌 ) + 𝑍 ) = ( ( 𝑌 + 𝑋 ) + 𝑍 ) ) |
10 |
1 2
|
paddass |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) → ( ( 𝑋 + 𝑌 ) + 𝑍 ) = ( 𝑋 + ( 𝑌 + 𝑍 ) ) ) |
11 |
|
simpl |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) → 𝐾 ∈ HL ) |
12 |
|
simpr3 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) → 𝑍 ⊆ 𝐴 ) |
13 |
1 2
|
paddass |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) → ( ( 𝑌 + 𝑋 ) + 𝑍 ) = ( 𝑌 + ( 𝑋 + 𝑍 ) ) ) |
14 |
11 6 5 12 13
|
syl13anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) → ( ( 𝑌 + 𝑋 ) + 𝑍 ) = ( 𝑌 + ( 𝑋 + 𝑍 ) ) ) |
15 |
9 10 14
|
3eqtr3d |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴 ) ) → ( 𝑋 + ( 𝑌 + 𝑍 ) ) = ( 𝑌 + ( 𝑋 + 𝑍 ) ) ) |