Step | Hyp | Ref
| Expression |
1 | | nfcv 2909 |
. . . . . . . . 9
⊢
Ⅎ𝑥0 |
2 | | nfcv 2909 |
. . . . . . . . 9
⊢
Ⅎ𝑥
≤ |
3 | | nffvmpt1 6782 |
. . . . . . . . 9
⊢
Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) |
4 | 1, 2, 3 | nfbr 5126 |
. . . . . . . 8
⊢
Ⅎ𝑥0 ≤
((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) |
5 | 4, 3, 1 | nfif 4495 |
. . . . . . 7
⊢
Ⅎ𝑥if(0
≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0) |
6 | | nfcv 2909 |
. . . . . . 7
⊢
Ⅎ𝑦if(0
≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), 0) |
7 | | fveq2 6771 |
. . . . . . . . 9
⊢ (𝑦 = 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) |
8 | 7 | breq2d 5091 |
. . . . . . . 8
⊢ (𝑦 = 𝑥 → (0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) ↔ 0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))) |
9 | 8, 7 | ifbieq1d 4489 |
. . . . . . 7
⊢ (𝑦 = 𝑥 → if(0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0) = if(0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), 0)) |
10 | 5, 6, 9 | cbvmpt 5190 |
. . . . . 6
⊢ (𝑦 ∈ 𝐴 ↦ if(0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)) = (𝑥 ∈ 𝐴 ↦ if(0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), 0)) |
11 | | simpr 485 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) |
12 | | mbfpos.1 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
13 | | eqid 2740 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) |
14 | 13 | fvmpt2 6883 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵) |
15 | 11, 12, 14 | syl2anc 584 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵) |
16 | 15 | breq2d 5091 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ↔ 0 ≤ 𝐵)) |
17 | 16, 15 | ifbieq1d 4489 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), 0) = if(0 ≤ 𝐵, 𝐵, 0)) |
18 | 17 | mpteq2dva 5179 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), 0)) = (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))) |
19 | 10, 18 | eqtrid 2792 |
. . . . 5
⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ if(0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)) = (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))) |
20 | 19 | adantr 481 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑦 ∈ 𝐴 ↦ if(0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)) = (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))) |
21 | 12 | fmpttd 6986 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ) |
22 | 21 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ) |
23 | 22 | ffvelrnda 6958 |
. . . . 5
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) ∈ ℝ) |
24 | | nfcv 2909 |
. . . . . . . . 9
⊢
Ⅎ𝑦((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) |
25 | 3, 24, 7 | cbvmpt 5190 |
. . . . . . . 8
⊢ (𝑦 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)) = (𝑥 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) |
26 | 15 | mpteq2dva 5179 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) = (𝑥 ∈ 𝐴 ↦ 𝐵)) |
27 | 25, 26 | eqtrid 2792 |
. . . . . . 7
⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)) = (𝑥 ∈ 𝐴 ↦ 𝐵)) |
28 | 27 | eleq1d 2825 |
. . . . . 6
⊢ (𝜑 → ((𝑦 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)) ∈ MblFn ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)) |
29 | 28 | biimpar 478 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑦 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)) ∈ MblFn) |
30 | 23, 29 | mbfpos 24813 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑦 ∈ 𝐴 ↦ if(0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)) ∈ MblFn) |
31 | 20, 30 | eqeltrrd 2842 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn) |
32 | 3 | nfneg 11217 |
. . . . . . . . 9
⊢
Ⅎ𝑥-((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) |
33 | 1, 2, 32 | nfbr 5126 |
. . . . . . . 8
⊢
Ⅎ𝑥0 ≤
-((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) |
34 | 33, 32, 1 | nfif 4495 |
. . . . . . 7
⊢
Ⅎ𝑥if(0
≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0) |
35 | | nfcv 2909 |
. . . . . . 7
⊢
Ⅎ𝑦if(0
≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), 0) |
36 | 7 | negeqd 11215 |
. . . . . . . . 9
⊢ (𝑦 = 𝑥 → -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) = -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) |
37 | 36 | breq2d 5091 |
. . . . . . . 8
⊢ (𝑦 = 𝑥 → (0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) ↔ 0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))) |
38 | 37, 36 | ifbieq1d 4489 |
. . . . . . 7
⊢ (𝑦 = 𝑥 → if(0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0) = if(0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), 0)) |
39 | 34, 35, 38 | cbvmpt 5190 |
. . . . . 6
⊢ (𝑦 ∈ 𝐴 ↦ if(0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)) = (𝑥 ∈ 𝐴 ↦ if(0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), 0)) |
40 | 15 | negeqd 11215 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = -𝐵) |
41 | 40 | breq2d 5091 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ↔ 0 ≤ -𝐵)) |
42 | 41, 40 | ifbieq1d 4489 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), 0) = if(0 ≤ -𝐵, -𝐵, 0)) |
43 | 42 | mpteq2dva 5179 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), 0)) = (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))) |
44 | 39, 43 | eqtrid 2792 |
. . . . 5
⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ if(0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)) = (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))) |
45 | 44 | adantr 481 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑦 ∈ 𝐴 ↦ if(0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)) = (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))) |
46 | 23 | renegcld 11402 |
. . . . 5
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) ∧ 𝑦 ∈ 𝐴) → -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) ∈ ℝ) |
47 | 23, 29 | mbfneg 24812 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑦 ∈ 𝐴 ↦ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)) ∈ MblFn) |
48 | 46, 47 | mbfpos 24813 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑦 ∈ 𝐴 ↦ if(0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)) ∈ MblFn) |
49 | 45, 48 | eqeltrrd 2842 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn) |
50 | 31, 49 | jca 512 |
. 2
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) |
51 | 27 | adantr 481 |
. . 3
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑦 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)) = (𝑥 ∈ 𝐴 ↦ 𝐵)) |
52 | 21 | ffvelrnda 6958 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) ∈ ℝ) |
53 | 52 | adantlr 712 |
. . . 4
⊢ (((𝜑 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) ∈ ℝ) |
54 | 19 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑦 ∈ 𝐴 ↦ if(0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)) = (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))) |
55 | | simprl 768 |
. . . . 5
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn) |
56 | 54, 55 | eqeltrd 2841 |
. . . 4
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑦 ∈ 𝐴 ↦ if(0 ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)) ∈ MblFn) |
57 | 44 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑦 ∈ 𝐴 ↦ if(0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)) = (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))) |
58 | | simprr 770 |
. . . . 5
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn) |
59 | 57, 58 | eqeltrd 2841 |
. . . 4
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑦 ∈ 𝐴 ↦ if(0 ≤ -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), -((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), 0)) ∈ MblFn) |
60 | 53, 56, 59 | mbfposr 24814 |
. . 3
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑦 ∈ 𝐴 ↦ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)) ∈ MblFn) |
61 | 51, 60 | eqeltrrd 2842 |
. 2
⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) |
62 | 50, 61 | impbida 798 |
1
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ↔ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn))) |