Step |
Hyp |
Ref |
Expression |
1 |
|
meetval2.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
meetval2.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
meetval2.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
4 |
|
meetval2.k |
⊢ ( 𝜑 → 𝐾 ∈ 𝑉 ) |
5 |
|
meetval2.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
6 |
|
meetval2.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
7 |
|
meetlem.e |
⊢ ( 𝜑 → 〈 𝑋 , 𝑌 〉 ∈ dom ∧ ) |
8 |
1 2 3 4 5 6 7
|
meeteu |
⊢ ( 𝜑 → ∃! 𝑥 ∈ 𝐵 ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ) ) |
9 |
|
riotasbc |
⊢ ( ∃! 𝑥 ∈ 𝐵 ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ) → [ ( ℩ 𝑥 ∈ 𝐵 ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ) ) / 𝑥 ] ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ) ) |
10 |
8 9
|
syl |
⊢ ( 𝜑 → [ ( ℩ 𝑥 ∈ 𝐵 ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ) ) / 𝑥 ] ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ) ) |
11 |
1 2 3 4 5 6
|
meetval2 |
⊢ ( 𝜑 → ( 𝑋 ∧ 𝑌 ) = ( ℩ 𝑥 ∈ 𝐵 ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ) ) ) |
12 |
11
|
sbceq1d |
⊢ ( 𝜑 → ( [ ( 𝑋 ∧ 𝑌 ) / 𝑥 ] ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ) ↔ [ ( ℩ 𝑥 ∈ 𝐵 ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ) ) / 𝑥 ] ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ) ) ) |
13 |
10 12
|
mpbird |
⊢ ( 𝜑 → [ ( 𝑋 ∧ 𝑌 ) / 𝑥 ] ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ) ) |
14 |
|
ovex |
⊢ ( 𝑋 ∧ 𝑌 ) ∈ V |
15 |
|
breq1 |
⊢ ( 𝑥 = ( 𝑋 ∧ 𝑌 ) → ( 𝑥 ≤ 𝑋 ↔ ( 𝑋 ∧ 𝑌 ) ≤ 𝑋 ) ) |
16 |
|
breq1 |
⊢ ( 𝑥 = ( 𝑋 ∧ 𝑌 ) → ( 𝑥 ≤ 𝑌 ↔ ( 𝑋 ∧ 𝑌 ) ≤ 𝑌 ) ) |
17 |
15 16
|
anbi12d |
⊢ ( 𝑥 = ( 𝑋 ∧ 𝑌 ) → ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ↔ ( ( 𝑋 ∧ 𝑌 ) ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑌 ) ) ) |
18 |
|
breq2 |
⊢ ( 𝑥 = ( 𝑋 ∧ 𝑌 ) → ( 𝑧 ≤ 𝑥 ↔ 𝑧 ≤ ( 𝑋 ∧ 𝑌 ) ) ) |
19 |
18
|
imbi2d |
⊢ ( 𝑥 = ( 𝑋 ∧ 𝑌 ) → ( ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ↔ ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ ( 𝑋 ∧ 𝑌 ) ) ) ) |
20 |
19
|
ralbidv |
⊢ ( 𝑥 = ( 𝑋 ∧ 𝑌 ) → ( ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ↔ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ ( 𝑋 ∧ 𝑌 ) ) ) ) |
21 |
17 20
|
anbi12d |
⊢ ( 𝑥 = ( 𝑋 ∧ 𝑌 ) → ( ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ) ↔ ( ( ( 𝑋 ∧ 𝑌 ) ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ ( 𝑋 ∧ 𝑌 ) ) ) ) ) |
22 |
14 21
|
sbcie |
⊢ ( [ ( 𝑋 ∧ 𝑌 ) / 𝑥 ] ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ) ↔ ( ( ( 𝑋 ∧ 𝑌 ) ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ ( 𝑋 ∧ 𝑌 ) ) ) ) |
23 |
13 22
|
sylib |
⊢ ( 𝜑 → ( ( ( 𝑋 ∧ 𝑌 ) ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ ( 𝑋 ∧ 𝑌 ) ) ) ) |