Step |
Hyp |
Ref |
Expression |
1 |
|
om1bas.o |
⊢ 𝑂 = ( 𝐽 Ω1 𝑌 ) |
2 |
|
om1bas.j |
⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
3 |
|
om1bas.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝑋 ) |
4 |
|
om1bas.b |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝑂 ) ) |
5 |
|
om1addcl.h |
⊢ ( 𝜑 → 𝐻 ∈ 𝐵 ) |
6 |
|
om1addcl.k |
⊢ ( 𝜑 → 𝐾 ∈ 𝐵 ) |
7 |
1 2 3 4
|
om1elbas |
⊢ ( 𝜑 → ( 𝐻 ∈ 𝐵 ↔ ( 𝐻 ∈ ( II Cn 𝐽 ) ∧ ( 𝐻 ‘ 0 ) = 𝑌 ∧ ( 𝐻 ‘ 1 ) = 𝑌 ) ) ) |
8 |
5 7
|
mpbid |
⊢ ( 𝜑 → ( 𝐻 ∈ ( II Cn 𝐽 ) ∧ ( 𝐻 ‘ 0 ) = 𝑌 ∧ ( 𝐻 ‘ 1 ) = 𝑌 ) ) |
9 |
8
|
simp1d |
⊢ ( 𝜑 → 𝐻 ∈ ( II Cn 𝐽 ) ) |
10 |
1 2 3 4
|
om1elbas |
⊢ ( 𝜑 → ( 𝐾 ∈ 𝐵 ↔ ( 𝐾 ∈ ( II Cn 𝐽 ) ∧ ( 𝐾 ‘ 0 ) = 𝑌 ∧ ( 𝐾 ‘ 1 ) = 𝑌 ) ) ) |
11 |
6 10
|
mpbid |
⊢ ( 𝜑 → ( 𝐾 ∈ ( II Cn 𝐽 ) ∧ ( 𝐾 ‘ 0 ) = 𝑌 ∧ ( 𝐾 ‘ 1 ) = 𝑌 ) ) |
12 |
11
|
simp1d |
⊢ ( 𝜑 → 𝐾 ∈ ( II Cn 𝐽 ) ) |
13 |
8
|
simp3d |
⊢ ( 𝜑 → ( 𝐻 ‘ 1 ) = 𝑌 ) |
14 |
11
|
simp2d |
⊢ ( 𝜑 → ( 𝐾 ‘ 0 ) = 𝑌 ) |
15 |
13 14
|
eqtr4d |
⊢ ( 𝜑 → ( 𝐻 ‘ 1 ) = ( 𝐾 ‘ 0 ) ) |
16 |
9 12 15
|
pcocn |
⊢ ( 𝜑 → ( 𝐻 ( *𝑝 ‘ 𝐽 ) 𝐾 ) ∈ ( II Cn 𝐽 ) ) |
17 |
9 12
|
pco0 |
⊢ ( 𝜑 → ( ( 𝐻 ( *𝑝 ‘ 𝐽 ) 𝐾 ) ‘ 0 ) = ( 𝐻 ‘ 0 ) ) |
18 |
8
|
simp2d |
⊢ ( 𝜑 → ( 𝐻 ‘ 0 ) = 𝑌 ) |
19 |
17 18
|
eqtrd |
⊢ ( 𝜑 → ( ( 𝐻 ( *𝑝 ‘ 𝐽 ) 𝐾 ) ‘ 0 ) = 𝑌 ) |
20 |
9 12
|
pco1 |
⊢ ( 𝜑 → ( ( 𝐻 ( *𝑝 ‘ 𝐽 ) 𝐾 ) ‘ 1 ) = ( 𝐾 ‘ 1 ) ) |
21 |
11
|
simp3d |
⊢ ( 𝜑 → ( 𝐾 ‘ 1 ) = 𝑌 ) |
22 |
20 21
|
eqtrd |
⊢ ( 𝜑 → ( ( 𝐻 ( *𝑝 ‘ 𝐽 ) 𝐾 ) ‘ 1 ) = 𝑌 ) |
23 |
1 2 3 4
|
om1elbas |
⊢ ( 𝜑 → ( ( 𝐻 ( *𝑝 ‘ 𝐽 ) 𝐾 ) ∈ 𝐵 ↔ ( ( 𝐻 ( *𝑝 ‘ 𝐽 ) 𝐾 ) ∈ ( II Cn 𝐽 ) ∧ ( ( 𝐻 ( *𝑝 ‘ 𝐽 ) 𝐾 ) ‘ 0 ) = 𝑌 ∧ ( ( 𝐻 ( *𝑝 ‘ 𝐽 ) 𝐾 ) ‘ 1 ) = 𝑌 ) ) ) |
24 |
16 19 22 23
|
mpbir3and |
⊢ ( 𝜑 → ( 𝐻 ( *𝑝 ‘ 𝐽 ) 𝐾 ) ∈ 𝐵 ) |