Step |
Hyp |
Ref |
Expression |
1 |
|
i1fposd.1 |
⊢ ( 𝜑 → ( 𝑥 ∈ ℝ ↦ 𝐴 ) ∈ dom ∫1 ) |
2 |
|
nfcv |
⊢ Ⅎ 𝑥 0 |
3 |
|
nfcv |
⊢ Ⅎ 𝑥 ≤ |
4 |
|
nffvmpt1 |
⊢ Ⅎ 𝑥 ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑦 ) |
5 |
2 3 4
|
nfbr |
⊢ Ⅎ 𝑥 0 ≤ ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑦 ) |
6 |
5 4 2
|
nfif |
⊢ Ⅎ 𝑥 if ( 0 ≤ ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑦 ) , ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑦 ) , 0 ) |
7 |
|
nfcv |
⊢ Ⅎ 𝑦 if ( 0 ≤ ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑥 ) , ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑥 ) , 0 ) |
8 |
|
fveq2 |
⊢ ( 𝑦 = 𝑥 → ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑦 ) = ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑥 ) ) |
9 |
8
|
breq2d |
⊢ ( 𝑦 = 𝑥 → ( 0 ≤ ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑦 ) ↔ 0 ≤ ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑥 ) ) ) |
10 |
9 8
|
ifbieq1d |
⊢ ( 𝑦 = 𝑥 → if ( 0 ≤ ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑦 ) , ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑦 ) , 0 ) = if ( 0 ≤ ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑥 ) , ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑥 ) , 0 ) ) |
11 |
6 7 10
|
cbvmpt |
⊢ ( 𝑦 ∈ ℝ ↦ if ( 0 ≤ ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑦 ) , ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑦 ) , 0 ) ) = ( 𝑥 ∈ ℝ ↦ if ( 0 ≤ ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑥 ) , ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑥 ) , 0 ) ) |
12 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → 𝑥 ∈ ℝ ) |
13 |
|
i1ff |
⊢ ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ∈ dom ∫1 → ( 𝑥 ∈ ℝ ↦ 𝐴 ) : ℝ ⟶ ℝ ) |
14 |
1 13
|
syl |
⊢ ( 𝜑 → ( 𝑥 ∈ ℝ ↦ 𝐴 ) : ℝ ⟶ ℝ ) |
15 |
14
|
fvmptelrn |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → 𝐴 ∈ ℝ ) |
16 |
|
eqid |
⊢ ( 𝑥 ∈ ℝ ↦ 𝐴 ) = ( 𝑥 ∈ ℝ ↦ 𝐴 ) |
17 |
16
|
fvmpt2 |
⊢ ( ( 𝑥 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑥 ) = 𝐴 ) |
18 |
12 15 17
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑥 ) = 𝐴 ) |
19 |
18
|
breq2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → ( 0 ≤ ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑥 ) ↔ 0 ≤ 𝐴 ) ) |
20 |
19 18
|
ifbieq1d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → if ( 0 ≤ ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑥 ) , ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑥 ) , 0 ) = if ( 0 ≤ 𝐴 , 𝐴 , 0 ) ) |
21 |
20
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑥 ∈ ℝ ↦ if ( 0 ≤ ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑥 ) , ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑥 ) , 0 ) ) = ( 𝑥 ∈ ℝ ↦ if ( 0 ≤ 𝐴 , 𝐴 , 0 ) ) ) |
22 |
11 21
|
syl5eq |
⊢ ( 𝜑 → ( 𝑦 ∈ ℝ ↦ if ( 0 ≤ ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑦 ) , ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑦 ) , 0 ) ) = ( 𝑥 ∈ ℝ ↦ if ( 0 ≤ 𝐴 , 𝐴 , 0 ) ) ) |
23 |
|
eqid |
⊢ ( 𝑦 ∈ ℝ ↦ if ( 0 ≤ ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑦 ) , ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑦 ) , 0 ) ) = ( 𝑦 ∈ ℝ ↦ if ( 0 ≤ ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑦 ) , ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑦 ) , 0 ) ) |
24 |
23
|
i1fpos |
⊢ ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ∈ dom ∫1 → ( 𝑦 ∈ ℝ ↦ if ( 0 ≤ ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑦 ) , ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑦 ) , 0 ) ) ∈ dom ∫1 ) |
25 |
1 24
|
syl |
⊢ ( 𝜑 → ( 𝑦 ∈ ℝ ↦ if ( 0 ≤ ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑦 ) , ( ( 𝑥 ∈ ℝ ↦ 𝐴 ) ‘ 𝑦 ) , 0 ) ) ∈ dom ∫1 ) |
26 |
22 25
|
eqeltrrd |
⊢ ( 𝜑 → ( 𝑥 ∈ ℝ ↦ if ( 0 ≤ 𝐴 , 𝐴 , 0 ) ) ∈ dom ∫1 ) |