Metamath Proof Explorer
Description: Any element is less than or equal to a poset's upper bound (if defined).
(Contributed by NM, 22-Oct-2011) (Revised by NM, 13-Sep-2018)
|
|
Ref |
Expression |
|
Hypotheses |
p0le.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
|
|
p0le.g |
⊢ 𝐺 = ( glb ‘ 𝐾 ) |
|
|
p0le.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
|
|
p0le.0 |
⊢ 0 = ( 0. ‘ 𝐾 ) |
|
|
p0le.k |
⊢ ( 𝜑 → 𝐾 ∈ 𝑉 ) |
|
|
p0le.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
|
|
p0le.d |
⊢ ( 𝜑 → 𝐵 ∈ dom 𝐺 ) |
|
Assertion |
p0le |
⊢ ( 𝜑 → 0 ≤ 𝑋 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
p0le.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
p0le.g |
⊢ 𝐺 = ( glb ‘ 𝐾 ) |
3 |
|
p0le.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
4 |
|
p0le.0 |
⊢ 0 = ( 0. ‘ 𝐾 ) |
5 |
|
p0le.k |
⊢ ( 𝜑 → 𝐾 ∈ 𝑉 ) |
6 |
|
p0le.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
7 |
|
p0le.d |
⊢ ( 𝜑 → 𝐵 ∈ dom 𝐺 ) |
8 |
1 2 4
|
p0val |
⊢ ( 𝐾 ∈ 𝑉 → 0 = ( 𝐺 ‘ 𝐵 ) ) |
9 |
5 8
|
syl |
⊢ ( 𝜑 → 0 = ( 𝐺 ‘ 𝐵 ) ) |
10 |
1 3 2 5 7 6
|
glble |
⊢ ( 𝜑 → ( 𝐺 ‘ 𝐵 ) ≤ 𝑋 ) |
11 |
9 10
|
eqbrtrd |
⊢ ( 𝜑 → 0 ≤ 𝑋 ) |