| Step |
Hyp |
Ref |
Expression |
| 1 |
|
mbfmax.1 |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ ℝ ) |
| 2 |
|
mbfmax.2 |
⊢ ( 𝜑 → 𝐹 ∈ MblFn ) |
| 3 |
|
mbfmax.3 |
⊢ ( 𝜑 → 𝐺 : 𝐴 ⟶ ℝ ) |
| 4 |
|
mbfmax.4 |
⊢ ( 𝜑 → 𝐺 ∈ MblFn ) |
| 5 |
|
mbfmax.5 |
⊢ 𝐻 = ( 𝑥 ∈ 𝐴 ↦ if ( ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐺 ‘ 𝑥 ) , ( 𝐺 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑥 ) ) ) |
| 6 |
3
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑥 ) ∈ ℝ ) |
| 7 |
1
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) ∈ ℝ ) |
| 8 |
6 7
|
ifcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → if ( ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐺 ‘ 𝑥 ) , ( 𝐺 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑥 ) ) ∈ ℝ ) |
| 9 |
8 5
|
fmptd |
⊢ ( 𝜑 → 𝐻 : 𝐴 ⟶ ℝ ) |
| 10 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) → 𝐹 : 𝐴 ⟶ ℝ ) |
| 11 |
10
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑧 ) ∈ ℝ ) |
| 12 |
11
|
rexrd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑧 ) ∈ ℝ* ) |
| 13 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) → 𝐺 : 𝐴 ⟶ ℝ ) |
| 14 |
13
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑧 ) ∈ ℝ ) |
| 15 |
14
|
rexrd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑧 ) ∈ ℝ* ) |
| 16 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → 𝑦 ∈ ℝ* ) |
| 17 |
|
xrmaxle |
⊢ ( ( ( 𝐹 ‘ 𝑧 ) ∈ ℝ* ∧ ( 𝐺 ‘ 𝑧 ) ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) → ( if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ≤ 𝑦 ↔ ( ( 𝐹 ‘ 𝑧 ) ≤ 𝑦 ∧ ( 𝐺 ‘ 𝑧 ) ≤ 𝑦 ) ) ) |
| 18 |
12 15 16 17
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ≤ 𝑦 ↔ ( ( 𝐹 ‘ 𝑧 ) ≤ 𝑦 ∧ ( 𝐺 ‘ 𝑧 ) ≤ 𝑦 ) ) ) |
| 19 |
18
|
notbid |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( ¬ if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ≤ 𝑦 ↔ ¬ ( ( 𝐹 ‘ 𝑧 ) ≤ 𝑦 ∧ ( 𝐺 ‘ 𝑧 ) ≤ 𝑦 ) ) ) |
| 20 |
|
ianor |
⊢ ( ¬ ( ( 𝐹 ‘ 𝑧 ) ≤ 𝑦 ∧ ( 𝐺 ‘ 𝑧 ) ≤ 𝑦 ) ↔ ( ¬ ( 𝐹 ‘ 𝑧 ) ≤ 𝑦 ∨ ¬ ( 𝐺 ‘ 𝑧 ) ≤ 𝑦 ) ) |
| 21 |
19 20
|
bitrdi |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( ¬ if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ≤ 𝑦 ↔ ( ¬ ( 𝐹 ‘ 𝑧 ) ≤ 𝑦 ∨ ¬ ( 𝐺 ‘ 𝑧 ) ≤ 𝑦 ) ) ) |
| 22 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
| 23 |
|
elioo2 |
⊢ ( ( 𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ* ) → ( if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∈ ( 𝑦 (,) +∞ ) ↔ ( if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∈ ℝ ∧ 𝑦 < if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∧ if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) < +∞ ) ) ) |
| 24 |
16 22 23
|
sylancl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∈ ( 𝑦 (,) +∞ ) ↔ ( if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∈ ℝ ∧ 𝑦 < if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∧ if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) < +∞ ) ) ) |
| 25 |
|
3anan12 |
⊢ ( ( if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∈ ℝ ∧ 𝑦 < if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∧ if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) < +∞ ) ↔ ( 𝑦 < if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∧ ( if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∈ ℝ ∧ if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) < +∞ ) ) ) |
| 26 |
24 25
|
bitrdi |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∈ ( 𝑦 (,) +∞ ) ↔ ( 𝑦 < if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∧ ( if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∈ ℝ ∧ if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) < +∞ ) ) ) ) |
| 27 |
|
fveq2 |
⊢ ( 𝑥 = 𝑧 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑧 ) ) |
| 28 |
|
fveq2 |
⊢ ( 𝑥 = 𝑧 → ( 𝐺 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑧 ) ) |
| 29 |
27 28
|
breq12d |
⊢ ( 𝑥 = 𝑧 → ( ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐺 ‘ 𝑥 ) ↔ ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) ) ) |
| 30 |
29 28 27
|
ifbieq12d |
⊢ ( 𝑥 = 𝑧 → if ( ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐺 ‘ 𝑥 ) , ( 𝐺 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑥 ) ) = if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ) |
| 31 |
|
fvex |
⊢ ( 𝐺 ‘ 𝑧 ) ∈ V |
| 32 |
|
fvex |
⊢ ( 𝐹 ‘ 𝑧 ) ∈ V |
| 33 |
31 32
|
ifex |
⊢ if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∈ V |
| 34 |
30 5 33
|
fvmpt |
⊢ ( 𝑧 ∈ 𝐴 → ( 𝐻 ‘ 𝑧 ) = if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ) |
| 35 |
34
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( 𝐻 ‘ 𝑧 ) = if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ) |
| 36 |
35
|
eleq1d |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝐻 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ↔ if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∈ ( 𝑦 (,) +∞ ) ) ) |
| 37 |
14 11
|
ifcld |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∈ ℝ ) |
| 38 |
|
ltpnf |
⊢ ( if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∈ ℝ → if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) < +∞ ) |
| 39 |
37 38
|
jccir |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∈ ℝ ∧ if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) < +∞ ) ) |
| 40 |
39
|
biantrud |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( 𝑦 < if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ↔ ( 𝑦 < if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∧ ( if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∈ ℝ ∧ if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) < +∞ ) ) ) ) |
| 41 |
26 36 40
|
3bitr4d |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝐻 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ↔ 𝑦 < if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ) ) |
| 42 |
37
|
rexrd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∈ ℝ* ) |
| 43 |
|
xrltnle |
⊢ ( ( 𝑦 ∈ ℝ* ∧ if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∈ ℝ* ) → ( 𝑦 < if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ↔ ¬ if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ≤ 𝑦 ) ) |
| 44 |
16 42 43
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( 𝑦 < if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ↔ ¬ if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ≤ 𝑦 ) ) |
| 45 |
41 44
|
bitrd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝐻 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ↔ ¬ if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ≤ 𝑦 ) ) |
| 46 |
|
elioo2 |
⊢ ( ( 𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ* ) → ( ( 𝐹 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ↔ ( ( 𝐹 ‘ 𝑧 ) ∈ ℝ ∧ 𝑦 < ( 𝐹 ‘ 𝑧 ) ∧ ( 𝐹 ‘ 𝑧 ) < +∞ ) ) ) |
| 47 |
16 22 46
|
sylancl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ↔ ( ( 𝐹 ‘ 𝑧 ) ∈ ℝ ∧ 𝑦 < ( 𝐹 ‘ 𝑧 ) ∧ ( 𝐹 ‘ 𝑧 ) < +∞ ) ) ) |
| 48 |
|
3anan12 |
⊢ ( ( ( 𝐹 ‘ 𝑧 ) ∈ ℝ ∧ 𝑦 < ( 𝐹 ‘ 𝑧 ) ∧ ( 𝐹 ‘ 𝑧 ) < +∞ ) ↔ ( 𝑦 < ( 𝐹 ‘ 𝑧 ) ∧ ( ( 𝐹 ‘ 𝑧 ) ∈ ℝ ∧ ( 𝐹 ‘ 𝑧 ) < +∞ ) ) ) |
| 49 |
47 48
|
bitrdi |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ↔ ( 𝑦 < ( 𝐹 ‘ 𝑧 ) ∧ ( ( 𝐹 ‘ 𝑧 ) ∈ ℝ ∧ ( 𝐹 ‘ 𝑧 ) < +∞ ) ) ) ) |
| 50 |
|
ltpnf |
⊢ ( ( 𝐹 ‘ 𝑧 ) ∈ ℝ → ( 𝐹 ‘ 𝑧 ) < +∞ ) |
| 51 |
11 50
|
jccir |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝑧 ) ∈ ℝ ∧ ( 𝐹 ‘ 𝑧 ) < +∞ ) ) |
| 52 |
51
|
biantrud |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( 𝑦 < ( 𝐹 ‘ 𝑧 ) ↔ ( 𝑦 < ( 𝐹 ‘ 𝑧 ) ∧ ( ( 𝐹 ‘ 𝑧 ) ∈ ℝ ∧ ( 𝐹 ‘ 𝑧 ) < +∞ ) ) ) ) |
| 53 |
|
xrltnle |
⊢ ( ( 𝑦 ∈ ℝ* ∧ ( 𝐹 ‘ 𝑧 ) ∈ ℝ* ) → ( 𝑦 < ( 𝐹 ‘ 𝑧 ) ↔ ¬ ( 𝐹 ‘ 𝑧 ) ≤ 𝑦 ) ) |
| 54 |
16 12 53
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( 𝑦 < ( 𝐹 ‘ 𝑧 ) ↔ ¬ ( 𝐹 ‘ 𝑧 ) ≤ 𝑦 ) ) |
| 55 |
49 52 54
|
3bitr2d |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ↔ ¬ ( 𝐹 ‘ 𝑧 ) ≤ 𝑦 ) ) |
| 56 |
|
elioo2 |
⊢ ( ( 𝑦 ∈ ℝ* ∧ +∞ ∈ ℝ* ) → ( ( 𝐺 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ↔ ( ( 𝐺 ‘ 𝑧 ) ∈ ℝ ∧ 𝑦 < ( 𝐺 ‘ 𝑧 ) ∧ ( 𝐺 ‘ 𝑧 ) < +∞ ) ) ) |
| 57 |
16 22 56
|
sylancl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝐺 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ↔ ( ( 𝐺 ‘ 𝑧 ) ∈ ℝ ∧ 𝑦 < ( 𝐺 ‘ 𝑧 ) ∧ ( 𝐺 ‘ 𝑧 ) < +∞ ) ) ) |
| 58 |
|
3anan12 |
⊢ ( ( ( 𝐺 ‘ 𝑧 ) ∈ ℝ ∧ 𝑦 < ( 𝐺 ‘ 𝑧 ) ∧ ( 𝐺 ‘ 𝑧 ) < +∞ ) ↔ ( 𝑦 < ( 𝐺 ‘ 𝑧 ) ∧ ( ( 𝐺 ‘ 𝑧 ) ∈ ℝ ∧ ( 𝐺 ‘ 𝑧 ) < +∞ ) ) ) |
| 59 |
57 58
|
bitrdi |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝐺 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ↔ ( 𝑦 < ( 𝐺 ‘ 𝑧 ) ∧ ( ( 𝐺 ‘ 𝑧 ) ∈ ℝ ∧ ( 𝐺 ‘ 𝑧 ) < +∞ ) ) ) ) |
| 60 |
|
ltpnf |
⊢ ( ( 𝐺 ‘ 𝑧 ) ∈ ℝ → ( 𝐺 ‘ 𝑧 ) < +∞ ) |
| 61 |
14 60
|
jccir |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝐺 ‘ 𝑧 ) ∈ ℝ ∧ ( 𝐺 ‘ 𝑧 ) < +∞ ) ) |
| 62 |
61
|
biantrud |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( 𝑦 < ( 𝐺 ‘ 𝑧 ) ↔ ( 𝑦 < ( 𝐺 ‘ 𝑧 ) ∧ ( ( 𝐺 ‘ 𝑧 ) ∈ ℝ ∧ ( 𝐺 ‘ 𝑧 ) < +∞ ) ) ) ) |
| 63 |
|
xrltnle |
⊢ ( ( 𝑦 ∈ ℝ* ∧ ( 𝐺 ‘ 𝑧 ) ∈ ℝ* ) → ( 𝑦 < ( 𝐺 ‘ 𝑧 ) ↔ ¬ ( 𝐺 ‘ 𝑧 ) ≤ 𝑦 ) ) |
| 64 |
16 15 63
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( 𝑦 < ( 𝐺 ‘ 𝑧 ) ↔ ¬ ( 𝐺 ‘ 𝑧 ) ≤ 𝑦 ) ) |
| 65 |
59 62 64
|
3bitr2d |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝐺 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ↔ ¬ ( 𝐺 ‘ 𝑧 ) ≤ 𝑦 ) ) |
| 66 |
55 65
|
orbi12d |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( ( ( 𝐹 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ∨ ( 𝐺 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ) ↔ ( ¬ ( 𝐹 ‘ 𝑧 ) ≤ 𝑦 ∨ ¬ ( 𝐺 ‘ 𝑧 ) ≤ 𝑦 ) ) ) |
| 67 |
21 45 66
|
3bitr4d |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝐻 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ↔ ( ( 𝐹 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ∨ ( 𝐺 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ) ) ) |
| 68 |
67
|
pm5.32da |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) → ( ( 𝑧 ∈ 𝐴 ∧ ( 𝐻 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ) ↔ ( 𝑧 ∈ 𝐴 ∧ ( ( 𝐹 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ∨ ( 𝐺 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ) ) ) ) |
| 69 |
|
andi |
⊢ ( ( 𝑧 ∈ 𝐴 ∧ ( ( 𝐹 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ∨ ( 𝐺 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ) ) ↔ ( ( 𝑧 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ) ∨ ( 𝑧 ∈ 𝐴 ∧ ( 𝐺 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ) ) ) |
| 70 |
68 69
|
bitrdi |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) → ( ( 𝑧 ∈ 𝐴 ∧ ( 𝐻 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ) ↔ ( ( 𝑧 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ) ∨ ( 𝑧 ∈ 𝐴 ∧ ( 𝐺 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ) ) ) ) |
| 71 |
9
|
ffnd |
⊢ ( 𝜑 → 𝐻 Fn 𝐴 ) |
| 72 |
71
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) → 𝐻 Fn 𝐴 ) |
| 73 |
|
elpreima |
⊢ ( 𝐻 Fn 𝐴 → ( 𝑧 ∈ ( ◡ 𝐻 “ ( 𝑦 (,) +∞ ) ) ↔ ( 𝑧 ∈ 𝐴 ∧ ( 𝐻 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ) ) ) |
| 74 |
72 73
|
syl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) → ( 𝑧 ∈ ( ◡ 𝐻 “ ( 𝑦 (,) +∞ ) ) ↔ ( 𝑧 ∈ 𝐴 ∧ ( 𝐻 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ) ) ) |
| 75 |
10
|
ffnd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) → 𝐹 Fn 𝐴 ) |
| 76 |
|
elpreima |
⊢ ( 𝐹 Fn 𝐴 → ( 𝑧 ∈ ( ◡ 𝐹 “ ( 𝑦 (,) +∞ ) ) ↔ ( 𝑧 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ) ) ) |
| 77 |
75 76
|
syl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) → ( 𝑧 ∈ ( ◡ 𝐹 “ ( 𝑦 (,) +∞ ) ) ↔ ( 𝑧 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ) ) ) |
| 78 |
13
|
ffnd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) → 𝐺 Fn 𝐴 ) |
| 79 |
|
elpreima |
⊢ ( 𝐺 Fn 𝐴 → ( 𝑧 ∈ ( ◡ 𝐺 “ ( 𝑦 (,) +∞ ) ) ↔ ( 𝑧 ∈ 𝐴 ∧ ( 𝐺 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ) ) ) |
| 80 |
78 79
|
syl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) → ( 𝑧 ∈ ( ◡ 𝐺 “ ( 𝑦 (,) +∞ ) ) ↔ ( 𝑧 ∈ 𝐴 ∧ ( 𝐺 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ) ) ) |
| 81 |
77 80
|
orbi12d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) → ( ( 𝑧 ∈ ( ◡ 𝐹 “ ( 𝑦 (,) +∞ ) ) ∨ 𝑧 ∈ ( ◡ 𝐺 “ ( 𝑦 (,) +∞ ) ) ) ↔ ( ( 𝑧 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ) ∨ ( 𝑧 ∈ 𝐴 ∧ ( 𝐺 ‘ 𝑧 ) ∈ ( 𝑦 (,) +∞ ) ) ) ) ) |
| 82 |
70 74 81
|
3bitr4d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) → ( 𝑧 ∈ ( ◡ 𝐻 “ ( 𝑦 (,) +∞ ) ) ↔ ( 𝑧 ∈ ( ◡ 𝐹 “ ( 𝑦 (,) +∞ ) ) ∨ 𝑧 ∈ ( ◡ 𝐺 “ ( 𝑦 (,) +∞ ) ) ) ) ) |
| 83 |
|
elun |
⊢ ( 𝑧 ∈ ( ( ◡ 𝐹 “ ( 𝑦 (,) +∞ ) ) ∪ ( ◡ 𝐺 “ ( 𝑦 (,) +∞ ) ) ) ↔ ( 𝑧 ∈ ( ◡ 𝐹 “ ( 𝑦 (,) +∞ ) ) ∨ 𝑧 ∈ ( ◡ 𝐺 “ ( 𝑦 (,) +∞ ) ) ) ) |
| 84 |
82 83
|
bitr4di |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) → ( 𝑧 ∈ ( ◡ 𝐻 “ ( 𝑦 (,) +∞ ) ) ↔ 𝑧 ∈ ( ( ◡ 𝐹 “ ( 𝑦 (,) +∞ ) ) ∪ ( ◡ 𝐺 “ ( 𝑦 (,) +∞ ) ) ) ) ) |
| 85 |
84
|
eqrdv |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) → ( ◡ 𝐻 “ ( 𝑦 (,) +∞ ) ) = ( ( ◡ 𝐹 “ ( 𝑦 (,) +∞ ) ) ∪ ( ◡ 𝐺 “ ( 𝑦 (,) +∞ ) ) ) ) |
| 86 |
|
mbfima |
⊢ ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ ℝ ) → ( ◡ 𝐹 “ ( 𝑦 (,) +∞ ) ) ∈ dom vol ) |
| 87 |
2 1 86
|
syl2anc |
⊢ ( 𝜑 → ( ◡ 𝐹 “ ( 𝑦 (,) +∞ ) ) ∈ dom vol ) |
| 88 |
|
mbfima |
⊢ ( ( 𝐺 ∈ MblFn ∧ 𝐺 : 𝐴 ⟶ ℝ ) → ( ◡ 𝐺 “ ( 𝑦 (,) +∞ ) ) ∈ dom vol ) |
| 89 |
4 3 88
|
syl2anc |
⊢ ( 𝜑 → ( ◡ 𝐺 “ ( 𝑦 (,) +∞ ) ) ∈ dom vol ) |
| 90 |
|
unmbl |
⊢ ( ( ( ◡ 𝐹 “ ( 𝑦 (,) +∞ ) ) ∈ dom vol ∧ ( ◡ 𝐺 “ ( 𝑦 (,) +∞ ) ) ∈ dom vol ) → ( ( ◡ 𝐹 “ ( 𝑦 (,) +∞ ) ) ∪ ( ◡ 𝐺 “ ( 𝑦 (,) +∞ ) ) ) ∈ dom vol ) |
| 91 |
87 89 90
|
syl2anc |
⊢ ( 𝜑 → ( ( ◡ 𝐹 “ ( 𝑦 (,) +∞ ) ) ∪ ( ◡ 𝐺 “ ( 𝑦 (,) +∞ ) ) ) ∈ dom vol ) |
| 92 |
91
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) → ( ( ◡ 𝐹 “ ( 𝑦 (,) +∞ ) ) ∪ ( ◡ 𝐺 “ ( 𝑦 (,) +∞ ) ) ) ∈ dom vol ) |
| 93 |
85 92
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) → ( ◡ 𝐻 “ ( 𝑦 (,) +∞ ) ) ∈ dom vol ) |
| 94 |
|
xrmaxlt |
⊢ ( ( ( 𝐹 ‘ 𝑧 ) ∈ ℝ* ∧ ( 𝐺 ‘ 𝑧 ) ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) → ( if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) < 𝑦 ↔ ( ( 𝐹 ‘ 𝑧 ) < 𝑦 ∧ ( 𝐺 ‘ 𝑧 ) < 𝑦 ) ) ) |
| 95 |
12 15 16 94
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) < 𝑦 ↔ ( ( 𝐹 ‘ 𝑧 ) < 𝑦 ∧ ( 𝐺 ‘ 𝑧 ) < 𝑦 ) ) ) |
| 96 |
|
mnfxr |
⊢ -∞ ∈ ℝ* |
| 97 |
|
elioo2 |
⊢ ( ( -∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) → ( if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∈ ( -∞ (,) 𝑦 ) ↔ ( if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∈ ℝ ∧ -∞ < if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∧ if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) < 𝑦 ) ) ) |
| 98 |
96 16 97
|
sylancr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∈ ( -∞ (,) 𝑦 ) ↔ ( if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∈ ℝ ∧ -∞ < if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∧ if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) < 𝑦 ) ) ) |
| 99 |
|
df-3an |
⊢ ( ( if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∈ ℝ ∧ -∞ < if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∧ if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) < 𝑦 ) ↔ ( ( if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∈ ℝ ∧ -∞ < if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ) ∧ if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) < 𝑦 ) ) |
| 100 |
98 99
|
bitrdi |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∈ ( -∞ (,) 𝑦 ) ↔ ( ( if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∈ ℝ ∧ -∞ < if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ) ∧ if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) < 𝑦 ) ) ) |
| 101 |
35
|
eleq1d |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝐻 ‘ 𝑧 ) ∈ ( -∞ (,) 𝑦 ) ↔ if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∈ ( -∞ (,) 𝑦 ) ) ) |
| 102 |
|
mnflt |
⊢ ( if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∈ ℝ → -∞ < if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ) |
| 103 |
37 102
|
jccir |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∈ ℝ ∧ -∞ < if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ) ) |
| 104 |
103
|
biantrurd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) < 𝑦 ↔ ( ( if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ∈ ℝ ∧ -∞ < if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) ) ∧ if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) < 𝑦 ) ) ) |
| 105 |
100 101 104
|
3bitr4d |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝐻 ‘ 𝑧 ) ∈ ( -∞ (,) 𝑦 ) ↔ if ( ( 𝐹 ‘ 𝑧 ) ≤ ( 𝐺 ‘ 𝑧 ) , ( 𝐺 ‘ 𝑧 ) , ( 𝐹 ‘ 𝑧 ) ) < 𝑦 ) ) |
| 106 |
|
mnflt |
⊢ ( ( 𝐹 ‘ 𝑧 ) ∈ ℝ → -∞ < ( 𝐹 ‘ 𝑧 ) ) |
| 107 |
11 106
|
jccir |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝑧 ) ∈ ℝ ∧ -∞ < ( 𝐹 ‘ 𝑧 ) ) ) |
| 108 |
|
elioo2 |
⊢ ( ( -∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) → ( ( 𝐹 ‘ 𝑧 ) ∈ ( -∞ (,) 𝑦 ) ↔ ( ( 𝐹 ‘ 𝑧 ) ∈ ℝ ∧ -∞ < ( 𝐹 ‘ 𝑧 ) ∧ ( 𝐹 ‘ 𝑧 ) < 𝑦 ) ) ) |
| 109 |
96 16 108
|
sylancr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝑧 ) ∈ ( -∞ (,) 𝑦 ) ↔ ( ( 𝐹 ‘ 𝑧 ) ∈ ℝ ∧ -∞ < ( 𝐹 ‘ 𝑧 ) ∧ ( 𝐹 ‘ 𝑧 ) < 𝑦 ) ) ) |
| 110 |
|
df-3an |
⊢ ( ( ( 𝐹 ‘ 𝑧 ) ∈ ℝ ∧ -∞ < ( 𝐹 ‘ 𝑧 ) ∧ ( 𝐹 ‘ 𝑧 ) < 𝑦 ) ↔ ( ( ( 𝐹 ‘ 𝑧 ) ∈ ℝ ∧ -∞ < ( 𝐹 ‘ 𝑧 ) ) ∧ ( 𝐹 ‘ 𝑧 ) < 𝑦 ) ) |
| 111 |
109 110
|
bitrdi |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝑧 ) ∈ ( -∞ (,) 𝑦 ) ↔ ( ( ( 𝐹 ‘ 𝑧 ) ∈ ℝ ∧ -∞ < ( 𝐹 ‘ 𝑧 ) ) ∧ ( 𝐹 ‘ 𝑧 ) < 𝑦 ) ) ) |
| 112 |
107 111
|
mpbirand |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝑧 ) ∈ ( -∞ (,) 𝑦 ) ↔ ( 𝐹 ‘ 𝑧 ) < 𝑦 ) ) |
| 113 |
|
mnflt |
⊢ ( ( 𝐺 ‘ 𝑧 ) ∈ ℝ → -∞ < ( 𝐺 ‘ 𝑧 ) ) |
| 114 |
14 113
|
jccir |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝐺 ‘ 𝑧 ) ∈ ℝ ∧ -∞ < ( 𝐺 ‘ 𝑧 ) ) ) |
| 115 |
|
elioo2 |
⊢ ( ( -∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) → ( ( 𝐺 ‘ 𝑧 ) ∈ ( -∞ (,) 𝑦 ) ↔ ( ( 𝐺 ‘ 𝑧 ) ∈ ℝ ∧ -∞ < ( 𝐺 ‘ 𝑧 ) ∧ ( 𝐺 ‘ 𝑧 ) < 𝑦 ) ) ) |
| 116 |
96 16 115
|
sylancr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝐺 ‘ 𝑧 ) ∈ ( -∞ (,) 𝑦 ) ↔ ( ( 𝐺 ‘ 𝑧 ) ∈ ℝ ∧ -∞ < ( 𝐺 ‘ 𝑧 ) ∧ ( 𝐺 ‘ 𝑧 ) < 𝑦 ) ) ) |
| 117 |
|
df-3an |
⊢ ( ( ( 𝐺 ‘ 𝑧 ) ∈ ℝ ∧ -∞ < ( 𝐺 ‘ 𝑧 ) ∧ ( 𝐺 ‘ 𝑧 ) < 𝑦 ) ↔ ( ( ( 𝐺 ‘ 𝑧 ) ∈ ℝ ∧ -∞ < ( 𝐺 ‘ 𝑧 ) ) ∧ ( 𝐺 ‘ 𝑧 ) < 𝑦 ) ) |
| 118 |
116 117
|
bitrdi |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝐺 ‘ 𝑧 ) ∈ ( -∞ (,) 𝑦 ) ↔ ( ( ( 𝐺 ‘ 𝑧 ) ∈ ℝ ∧ -∞ < ( 𝐺 ‘ 𝑧 ) ) ∧ ( 𝐺 ‘ 𝑧 ) < 𝑦 ) ) ) |
| 119 |
114 118
|
mpbirand |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝐺 ‘ 𝑧 ) ∈ ( -∞ (,) 𝑦 ) ↔ ( 𝐺 ‘ 𝑧 ) < 𝑦 ) ) |
| 120 |
112 119
|
anbi12d |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( ( ( 𝐹 ‘ 𝑧 ) ∈ ( -∞ (,) 𝑦 ) ∧ ( 𝐺 ‘ 𝑧 ) ∈ ( -∞ (,) 𝑦 ) ) ↔ ( ( 𝐹 ‘ 𝑧 ) < 𝑦 ∧ ( 𝐺 ‘ 𝑧 ) < 𝑦 ) ) ) |
| 121 |
95 105 120
|
3bitr4d |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝐻 ‘ 𝑧 ) ∈ ( -∞ (,) 𝑦 ) ↔ ( ( 𝐹 ‘ 𝑧 ) ∈ ( -∞ (,) 𝑦 ) ∧ ( 𝐺 ‘ 𝑧 ) ∈ ( -∞ (,) 𝑦 ) ) ) ) |
| 122 |
121
|
pm5.32da |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) → ( ( 𝑧 ∈ 𝐴 ∧ ( 𝐻 ‘ 𝑧 ) ∈ ( -∞ (,) 𝑦 ) ) ↔ ( 𝑧 ∈ 𝐴 ∧ ( ( 𝐹 ‘ 𝑧 ) ∈ ( -∞ (,) 𝑦 ) ∧ ( 𝐺 ‘ 𝑧 ) ∈ ( -∞ (,) 𝑦 ) ) ) ) ) |
| 123 |
|
anandi |
⊢ ( ( 𝑧 ∈ 𝐴 ∧ ( ( 𝐹 ‘ 𝑧 ) ∈ ( -∞ (,) 𝑦 ) ∧ ( 𝐺 ‘ 𝑧 ) ∈ ( -∞ (,) 𝑦 ) ) ) ↔ ( ( 𝑧 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑧 ) ∈ ( -∞ (,) 𝑦 ) ) ∧ ( 𝑧 ∈ 𝐴 ∧ ( 𝐺 ‘ 𝑧 ) ∈ ( -∞ (,) 𝑦 ) ) ) ) |
| 124 |
122 123
|
bitrdi |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) → ( ( 𝑧 ∈ 𝐴 ∧ ( 𝐻 ‘ 𝑧 ) ∈ ( -∞ (,) 𝑦 ) ) ↔ ( ( 𝑧 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑧 ) ∈ ( -∞ (,) 𝑦 ) ) ∧ ( 𝑧 ∈ 𝐴 ∧ ( 𝐺 ‘ 𝑧 ) ∈ ( -∞ (,) 𝑦 ) ) ) ) ) |
| 125 |
|
elpreima |
⊢ ( 𝐻 Fn 𝐴 → ( 𝑧 ∈ ( ◡ 𝐻 “ ( -∞ (,) 𝑦 ) ) ↔ ( 𝑧 ∈ 𝐴 ∧ ( 𝐻 ‘ 𝑧 ) ∈ ( -∞ (,) 𝑦 ) ) ) ) |
| 126 |
72 125
|
syl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) → ( 𝑧 ∈ ( ◡ 𝐻 “ ( -∞ (,) 𝑦 ) ) ↔ ( 𝑧 ∈ 𝐴 ∧ ( 𝐻 ‘ 𝑧 ) ∈ ( -∞ (,) 𝑦 ) ) ) ) |
| 127 |
|
elpreima |
⊢ ( 𝐹 Fn 𝐴 → ( 𝑧 ∈ ( ◡ 𝐹 “ ( -∞ (,) 𝑦 ) ) ↔ ( 𝑧 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑧 ) ∈ ( -∞ (,) 𝑦 ) ) ) ) |
| 128 |
75 127
|
syl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) → ( 𝑧 ∈ ( ◡ 𝐹 “ ( -∞ (,) 𝑦 ) ) ↔ ( 𝑧 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑧 ) ∈ ( -∞ (,) 𝑦 ) ) ) ) |
| 129 |
|
elpreima |
⊢ ( 𝐺 Fn 𝐴 → ( 𝑧 ∈ ( ◡ 𝐺 “ ( -∞ (,) 𝑦 ) ) ↔ ( 𝑧 ∈ 𝐴 ∧ ( 𝐺 ‘ 𝑧 ) ∈ ( -∞ (,) 𝑦 ) ) ) ) |
| 130 |
78 129
|
syl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) → ( 𝑧 ∈ ( ◡ 𝐺 “ ( -∞ (,) 𝑦 ) ) ↔ ( 𝑧 ∈ 𝐴 ∧ ( 𝐺 ‘ 𝑧 ) ∈ ( -∞ (,) 𝑦 ) ) ) ) |
| 131 |
128 130
|
anbi12d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) → ( ( 𝑧 ∈ ( ◡ 𝐹 “ ( -∞ (,) 𝑦 ) ) ∧ 𝑧 ∈ ( ◡ 𝐺 “ ( -∞ (,) 𝑦 ) ) ) ↔ ( ( 𝑧 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑧 ) ∈ ( -∞ (,) 𝑦 ) ) ∧ ( 𝑧 ∈ 𝐴 ∧ ( 𝐺 ‘ 𝑧 ) ∈ ( -∞ (,) 𝑦 ) ) ) ) ) |
| 132 |
124 126 131
|
3bitr4d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) → ( 𝑧 ∈ ( ◡ 𝐻 “ ( -∞ (,) 𝑦 ) ) ↔ ( 𝑧 ∈ ( ◡ 𝐹 “ ( -∞ (,) 𝑦 ) ) ∧ 𝑧 ∈ ( ◡ 𝐺 “ ( -∞ (,) 𝑦 ) ) ) ) ) |
| 133 |
|
elin |
⊢ ( 𝑧 ∈ ( ( ◡ 𝐹 “ ( -∞ (,) 𝑦 ) ) ∩ ( ◡ 𝐺 “ ( -∞ (,) 𝑦 ) ) ) ↔ ( 𝑧 ∈ ( ◡ 𝐹 “ ( -∞ (,) 𝑦 ) ) ∧ 𝑧 ∈ ( ◡ 𝐺 “ ( -∞ (,) 𝑦 ) ) ) ) |
| 134 |
132 133
|
bitr4di |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) → ( 𝑧 ∈ ( ◡ 𝐻 “ ( -∞ (,) 𝑦 ) ) ↔ 𝑧 ∈ ( ( ◡ 𝐹 “ ( -∞ (,) 𝑦 ) ) ∩ ( ◡ 𝐺 “ ( -∞ (,) 𝑦 ) ) ) ) ) |
| 135 |
134
|
eqrdv |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) → ( ◡ 𝐻 “ ( -∞ (,) 𝑦 ) ) = ( ( ◡ 𝐹 “ ( -∞ (,) 𝑦 ) ) ∩ ( ◡ 𝐺 “ ( -∞ (,) 𝑦 ) ) ) ) |
| 136 |
|
mbfima |
⊢ ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴 ⟶ ℝ ) → ( ◡ 𝐹 “ ( -∞ (,) 𝑦 ) ) ∈ dom vol ) |
| 137 |
2 1 136
|
syl2anc |
⊢ ( 𝜑 → ( ◡ 𝐹 “ ( -∞ (,) 𝑦 ) ) ∈ dom vol ) |
| 138 |
|
mbfima |
⊢ ( ( 𝐺 ∈ MblFn ∧ 𝐺 : 𝐴 ⟶ ℝ ) → ( ◡ 𝐺 “ ( -∞ (,) 𝑦 ) ) ∈ dom vol ) |
| 139 |
4 3 138
|
syl2anc |
⊢ ( 𝜑 → ( ◡ 𝐺 “ ( -∞ (,) 𝑦 ) ) ∈ dom vol ) |
| 140 |
|
inmbl |
⊢ ( ( ( ◡ 𝐹 “ ( -∞ (,) 𝑦 ) ) ∈ dom vol ∧ ( ◡ 𝐺 “ ( -∞ (,) 𝑦 ) ) ∈ dom vol ) → ( ( ◡ 𝐹 “ ( -∞ (,) 𝑦 ) ) ∩ ( ◡ 𝐺 “ ( -∞ (,) 𝑦 ) ) ) ∈ dom vol ) |
| 141 |
137 139 140
|
syl2anc |
⊢ ( 𝜑 → ( ( ◡ 𝐹 “ ( -∞ (,) 𝑦 ) ) ∩ ( ◡ 𝐺 “ ( -∞ (,) 𝑦 ) ) ) ∈ dom vol ) |
| 142 |
141
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) → ( ( ◡ 𝐹 “ ( -∞ (,) 𝑦 ) ) ∩ ( ◡ 𝐺 “ ( -∞ (,) 𝑦 ) ) ) ∈ dom vol ) |
| 143 |
135 142
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ* ) → ( ◡ 𝐻 “ ( -∞ (,) 𝑦 ) ) ∈ dom vol ) |
| 144 |
9 93 143
|
ismbfd |
⊢ ( 𝜑 → 𝐻 ∈ MblFn ) |