Step |
Hyp |
Ref |
Expression |
1 |
|
p0val.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
p0val.g |
⊢ 𝐺 = ( glb ‘ 𝐾 ) |
3 |
|
p0val.z |
⊢ 0 = ( 0. ‘ 𝐾 ) |
4 |
|
elex |
⊢ ( 𝐾 ∈ 𝑉 → 𝐾 ∈ V ) |
5 |
|
fveq2 |
⊢ ( 𝑝 = 𝐾 → ( glb ‘ 𝑝 ) = ( glb ‘ 𝐾 ) ) |
6 |
5 2
|
eqtr4di |
⊢ ( 𝑝 = 𝐾 → ( glb ‘ 𝑝 ) = 𝐺 ) |
7 |
|
fveq2 |
⊢ ( 𝑝 = 𝐾 → ( Base ‘ 𝑝 ) = ( Base ‘ 𝐾 ) ) |
8 |
7 1
|
eqtr4di |
⊢ ( 𝑝 = 𝐾 → ( Base ‘ 𝑝 ) = 𝐵 ) |
9 |
6 8
|
fveq12d |
⊢ ( 𝑝 = 𝐾 → ( ( glb ‘ 𝑝 ) ‘ ( Base ‘ 𝑝 ) ) = ( 𝐺 ‘ 𝐵 ) ) |
10 |
|
df-p0 |
⊢ 0. = ( 𝑝 ∈ V ↦ ( ( glb ‘ 𝑝 ) ‘ ( Base ‘ 𝑝 ) ) ) |
11 |
|
fvex |
⊢ ( 𝐺 ‘ 𝐵 ) ∈ V |
12 |
9 10 11
|
fvmpt |
⊢ ( 𝐾 ∈ V → ( 0. ‘ 𝐾 ) = ( 𝐺 ‘ 𝐵 ) ) |
13 |
3 12
|
syl5eq |
⊢ ( 𝐾 ∈ V → 0 = ( 𝐺 ‘ 𝐵 ) ) |
14 |
4 13
|
syl |
⊢ ( 𝐾 ∈ 𝑉 → 0 = ( 𝐺 ‘ 𝐵 ) ) |