Step |
Hyp |
Ref |
Expression |
1 |
|
o1compt.1 |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ ℂ ) |
2 |
|
o1compt.2 |
⊢ ( 𝜑 → 𝐹 ∈ 𝑂(1) ) |
3 |
|
o1compt.3 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → 𝐶 ∈ 𝐴 ) |
4 |
|
o1compt.4 |
⊢ ( 𝜑 → 𝐵 ⊆ ℝ ) |
5 |
|
o1compt.5 |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℝ ) → ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 → 𝑚 ≤ 𝐶 ) ) |
6 |
3
|
fmpttd |
⊢ ( 𝜑 → ( 𝑦 ∈ 𝐵 ↦ 𝐶 ) : 𝐵 ⟶ 𝐴 ) |
7 |
|
nfv |
⊢ Ⅎ 𝑦 𝑥 ≤ 𝑧 |
8 |
|
nfcv |
⊢ Ⅎ 𝑦 𝑚 |
9 |
|
nfcv |
⊢ Ⅎ 𝑦 ≤ |
10 |
|
nffvmpt1 |
⊢ Ⅎ 𝑦 ( ( 𝑦 ∈ 𝐵 ↦ 𝐶 ) ‘ 𝑧 ) |
11 |
8 9 10
|
nfbr |
⊢ Ⅎ 𝑦 𝑚 ≤ ( ( 𝑦 ∈ 𝐵 ↦ 𝐶 ) ‘ 𝑧 ) |
12 |
7 11
|
nfim |
⊢ Ⅎ 𝑦 ( 𝑥 ≤ 𝑧 → 𝑚 ≤ ( ( 𝑦 ∈ 𝐵 ↦ 𝐶 ) ‘ 𝑧 ) ) |
13 |
|
nfv |
⊢ Ⅎ 𝑧 ( 𝑥 ≤ 𝑦 → 𝑚 ≤ ( ( 𝑦 ∈ 𝐵 ↦ 𝐶 ) ‘ 𝑦 ) ) |
14 |
|
breq2 |
⊢ ( 𝑧 = 𝑦 → ( 𝑥 ≤ 𝑧 ↔ 𝑥 ≤ 𝑦 ) ) |
15 |
|
fveq2 |
⊢ ( 𝑧 = 𝑦 → ( ( 𝑦 ∈ 𝐵 ↦ 𝐶 ) ‘ 𝑧 ) = ( ( 𝑦 ∈ 𝐵 ↦ 𝐶 ) ‘ 𝑦 ) ) |
16 |
15
|
breq2d |
⊢ ( 𝑧 = 𝑦 → ( 𝑚 ≤ ( ( 𝑦 ∈ 𝐵 ↦ 𝐶 ) ‘ 𝑧 ) ↔ 𝑚 ≤ ( ( 𝑦 ∈ 𝐵 ↦ 𝐶 ) ‘ 𝑦 ) ) ) |
17 |
14 16
|
imbi12d |
⊢ ( 𝑧 = 𝑦 → ( ( 𝑥 ≤ 𝑧 → 𝑚 ≤ ( ( 𝑦 ∈ 𝐵 ↦ 𝐶 ) ‘ 𝑧 ) ) ↔ ( 𝑥 ≤ 𝑦 → 𝑚 ≤ ( ( 𝑦 ∈ 𝐵 ↦ 𝐶 ) ‘ 𝑦 ) ) ) ) |
18 |
12 13 17
|
cbvralw |
⊢ ( ∀ 𝑧 ∈ 𝐵 ( 𝑥 ≤ 𝑧 → 𝑚 ≤ ( ( 𝑦 ∈ 𝐵 ↦ 𝐶 ) ‘ 𝑧 ) ) ↔ ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 → 𝑚 ≤ ( ( 𝑦 ∈ 𝐵 ↦ 𝐶 ) ‘ 𝑦 ) ) ) |
19 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → 𝑦 ∈ 𝐵 ) |
20 |
|
eqid |
⊢ ( 𝑦 ∈ 𝐵 ↦ 𝐶 ) = ( 𝑦 ∈ 𝐵 ↦ 𝐶 ) |
21 |
20
|
fvmpt2 |
⊢ ( ( 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴 ) → ( ( 𝑦 ∈ 𝐵 ↦ 𝐶 ) ‘ 𝑦 ) = 𝐶 ) |
22 |
19 3 21
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝑦 ∈ 𝐵 ↦ 𝐶 ) ‘ 𝑦 ) = 𝐶 ) |
23 |
22
|
breq2d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑚 ≤ ( ( 𝑦 ∈ 𝐵 ↦ 𝐶 ) ‘ 𝑦 ) ↔ 𝑚 ≤ 𝐶 ) ) |
24 |
23
|
imbi2d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝑥 ≤ 𝑦 → 𝑚 ≤ ( ( 𝑦 ∈ 𝐵 ↦ 𝐶 ) ‘ 𝑦 ) ) ↔ ( 𝑥 ≤ 𝑦 → 𝑚 ≤ 𝐶 ) ) ) |
25 |
24
|
ralbidva |
⊢ ( 𝜑 → ( ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 → 𝑚 ≤ ( ( 𝑦 ∈ 𝐵 ↦ 𝐶 ) ‘ 𝑦 ) ) ↔ ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 → 𝑚 ≤ 𝐶 ) ) ) |
26 |
18 25
|
syl5bb |
⊢ ( 𝜑 → ( ∀ 𝑧 ∈ 𝐵 ( 𝑥 ≤ 𝑧 → 𝑚 ≤ ( ( 𝑦 ∈ 𝐵 ↦ 𝐶 ) ‘ 𝑧 ) ) ↔ ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 → 𝑚 ≤ 𝐶 ) ) ) |
27 |
26
|
rexbidv |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ ℝ ∀ 𝑧 ∈ 𝐵 ( 𝑥 ≤ 𝑧 → 𝑚 ≤ ( ( 𝑦 ∈ 𝐵 ↦ 𝐶 ) ‘ 𝑧 ) ) ↔ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 → 𝑚 ≤ 𝐶 ) ) ) |
28 |
27
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℝ ) → ( ∃ 𝑥 ∈ ℝ ∀ 𝑧 ∈ 𝐵 ( 𝑥 ≤ 𝑧 → 𝑚 ≤ ( ( 𝑦 ∈ 𝐵 ↦ 𝐶 ) ‘ 𝑧 ) ) ↔ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 → 𝑚 ≤ 𝐶 ) ) ) |
29 |
5 28
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ℝ ) → ∃ 𝑥 ∈ ℝ ∀ 𝑧 ∈ 𝐵 ( 𝑥 ≤ 𝑧 → 𝑚 ≤ ( ( 𝑦 ∈ 𝐵 ↦ 𝐶 ) ‘ 𝑧 ) ) ) |
30 |
1 2 6 4 29
|
o1co |
⊢ ( 𝜑 → ( 𝐹 ∘ ( 𝑦 ∈ 𝐵 ↦ 𝐶 ) ) ∈ 𝑂(1) ) |