Step |
Hyp |
Ref |
Expression |
1 |
|
o1add2.1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 ) |
2 |
|
o1add2.2 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐶 ∈ 𝑉 ) |
3 |
|
o1add2.3 |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝑂(1) ) |
4 |
|
o1add2.4 |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ∈ 𝑂(1) ) |
5 |
1
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 ) |
6 |
|
dmmptg |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = 𝐴 ) |
7 |
5 6
|
syl |
⊢ ( 𝜑 → dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = 𝐴 ) |
8 |
|
o1dm |
⊢ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝑂(1) → dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ⊆ ℝ ) |
9 |
3 8
|
syl |
⊢ ( 𝜑 → dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ⊆ ℝ ) |
10 |
7 9
|
eqsstrrd |
⊢ ( 𝜑 → 𝐴 ⊆ ℝ ) |
11 |
|
reex |
⊢ ℝ ∈ V |
12 |
11
|
ssex |
⊢ ( 𝐴 ⊆ ℝ → 𝐴 ∈ V ) |
13 |
10 12
|
syl |
⊢ ( 𝜑 → 𝐴 ∈ V ) |
14 |
|
eqidd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
15 |
|
eqidd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ) |
16 |
13 1 2 14 15
|
offval2 |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∘f · ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ) = ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 · 𝐶 ) ) ) |
17 |
|
o1mul |
⊢ ( ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝑂(1) ∧ ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ∈ 𝑂(1) ) → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∘f · ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ) ∈ 𝑂(1) ) |
18 |
3 4 17
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∘f · ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ) ∈ 𝑂(1) ) |
19 |
16 18
|
eqeltrrd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 · 𝐶 ) ) ∈ 𝑂(1) ) |