Step |
Hyp |
Ref |
Expression |
1 |
|
cvmlift2.b |
⊢ 𝐵 = ∪ 𝐶 |
2 |
|
cvmlift2.f |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) ) |
3 |
|
cvmlift2.g |
⊢ ( 𝜑 → 𝐺 ∈ ( ( II ×t II ) Cn 𝐽 ) ) |
4 |
|
cvmlift2.p |
⊢ ( 𝜑 → 𝑃 ∈ 𝐵 ) |
5 |
|
cvmlift2.i |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝑃 ) = ( 0 𝐺 0 ) ) |
6 |
|
cvmlift2.h |
⊢ 𝐻 = ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑧 𝐺 0 ) ) ∧ ( 𝑓 ‘ 0 ) = 𝑃 ) ) |
7 |
|
cvmlift2.k |
⊢ 𝐾 = ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑥 ) ) ) ‘ 𝑦 ) ) |
8 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 0 [,] 1 ) ) → 𝑋 ∈ ( 0 [,] 1 ) ) |
9 |
|
0elunit |
⊢ 0 ∈ ( 0 [,] 1 ) |
10 |
1 2 3 4 5 6 7
|
cvmlift2lem4 |
⊢ ( ( 𝑋 ∈ ( 0 [,] 1 ) ∧ 0 ∈ ( 0 [,] 1 ) ) → ( 𝑋 𝐾 0 ) = ( ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑋 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑋 ) ) ) ‘ 0 ) ) |
11 |
8 9 10
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 0 [,] 1 ) ) → ( 𝑋 𝐾 0 ) = ( ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑋 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑋 ) ) ) ‘ 0 ) ) |
12 |
|
eqid |
⊢ ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑋 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑋 ) ) ) = ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑋 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑋 ) ) ) |
13 |
1 2 3 4 5 6 12
|
cvmlift2lem3 |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 0 [,] 1 ) ) → ( ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑋 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑋 ) ) ) ∈ ( II Cn 𝐶 ) ∧ ( 𝐹 ∘ ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑋 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑋 ) ) ) ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑋 𝐺 𝑧 ) ) ∧ ( ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑋 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑋 ) ) ) ‘ 0 ) = ( 𝐻 ‘ 𝑋 ) ) ) |
14 |
13
|
simp3d |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 0 [,] 1 ) ) → ( ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑋 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑋 ) ) ) ‘ 0 ) = ( 𝐻 ‘ 𝑋 ) ) |
15 |
11 14
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 0 [,] 1 ) ) → ( 𝑋 𝐾 0 ) = ( 𝐻 ‘ 𝑋 ) ) |