Proof of Theorem hsphoif
Step | Hyp | Ref
| Expression |
1 | | hsphoif.b |
. . . . 5
⊢ (𝜑 → 𝐵:𝑋⟶ℝ) |
2 | 1 | ffvelrnda 6359 |
. . . 4
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (𝐵‘𝑗) ∈ ℝ) |
3 | | hsphoif.a |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ ℝ) |
4 | 3 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → 𝐴 ∈ ℝ) |
5 | 2, 4 | ifcld 4131 |
. . . 4
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴) ∈ ℝ) |
6 | 2, 5 | ifcld 4131 |
. . 3
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴)) ∈ ℝ) |
7 | | eqid 2622 |
. . 3
⊢ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))) = (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))) |
8 | 6, 7 | fmptd 6385 |
. 2
⊢ (𝜑 → (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))):𝑋⟶ℝ) |
9 | | hsphoif.h |
. . . . . 6
⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑𝑚
𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝑥, (𝑎‘𝑗), 𝑥))))) |
10 | 9 | a1i 11 |
. . . . 5
⊢ (𝜑 → 𝐻 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑𝑚
𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝑥, (𝑎‘𝑗), 𝑥)))))) |
11 | | breq2 4657 |
. . . . . . . . . 10
⊢ (𝑥 = 𝐴 → ((𝑎‘𝑗) ≤ 𝑥 ↔ (𝑎‘𝑗) ≤ 𝐴)) |
12 | | id 22 |
. . . . . . . . . 10
⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) |
13 | 11, 12 | ifbieq2d 4111 |
. . . . . . . . 9
⊢ (𝑥 = 𝐴 → if((𝑎‘𝑗) ≤ 𝑥, (𝑎‘𝑗), 𝑥) = if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴)) |
14 | 13 | ifeq2d 4105 |
. . . . . . . 8
⊢ (𝑥 = 𝐴 → if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝑥, (𝑎‘𝑗), 𝑥)) = if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴))) |
15 | 14 | mpteq2dv 4745 |
. . . . . . 7
⊢ (𝑥 = 𝐴 → (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝑥, (𝑎‘𝑗), 𝑥))) = (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴)))) |
16 | 15 | mpteq2dv 4745 |
. . . . . 6
⊢ (𝑥 = 𝐴 → (𝑎 ∈ (ℝ ↑𝑚
𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝑥, (𝑎‘𝑗), 𝑥)))) = (𝑎 ∈ (ℝ ↑𝑚
𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴))))) |
17 | 16 | adantl 482 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝑎 ∈ (ℝ ↑𝑚
𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝑥, (𝑎‘𝑗), 𝑥)))) = (𝑎 ∈ (ℝ ↑𝑚
𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴))))) |
18 | | ovex 6678 |
. . . . . . 7
⊢ (ℝ
↑𝑚 𝑋) ∈ V |
19 | 18 | mptex 6486 |
. . . . . 6
⊢ (𝑎 ∈ (ℝ
↑𝑚 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴)))) ∈ V |
20 | 19 | a1i 11 |
. . . . 5
⊢ (𝜑 → (𝑎 ∈ (ℝ ↑𝑚
𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴)))) ∈ V) |
21 | 10, 17, 3, 20 | fvmptd 6288 |
. . . 4
⊢ (𝜑 → (𝐻‘𝐴) = (𝑎 ∈ (ℝ ↑𝑚
𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴))))) |
22 | | fveq1 6190 |
. . . . . . 7
⊢ (𝑎 = 𝐵 → (𝑎‘𝑗) = (𝐵‘𝑗)) |
23 | 22 | breq1d 4663 |
. . . . . . . 8
⊢ (𝑎 = 𝐵 → ((𝑎‘𝑗) ≤ 𝐴 ↔ (𝐵‘𝑗) ≤ 𝐴)) |
24 | 23, 22 | ifbieq1d 4109 |
. . . . . . 7
⊢ (𝑎 = 𝐵 → if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴) = if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴)) |
25 | 22, 24 | ifeq12d 4106 |
. . . . . 6
⊢ (𝑎 = 𝐵 → if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴)) = if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))) |
26 | 25 | mpteq2dv 4745 |
. . . . 5
⊢ (𝑎 = 𝐵 → (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴))) = (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴)))) |
27 | 26 | adantl 482 |
. . . 4
⊢ ((𝜑 ∧ 𝑎 = 𝐵) → (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴))) = (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴)))) |
28 | | reex 10027 |
. . . . . . . 8
⊢ ℝ
∈ V |
29 | 28 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → ℝ ∈
V) |
30 | | hsphoif.x |
. . . . . . 7
⊢ (𝜑 → 𝑋 ∈ 𝑉) |
31 | 29, 30 | jca 554 |
. . . . . 6
⊢ (𝜑 → (ℝ ∈ V ∧
𝑋 ∈ 𝑉)) |
32 | | elmapg 7870 |
. . . . . 6
⊢ ((ℝ
∈ V ∧ 𝑋 ∈
𝑉) → (𝐵 ∈ (ℝ
↑𝑚 𝑋) ↔ 𝐵:𝑋⟶ℝ)) |
33 | 31, 32 | syl 17 |
. . . . 5
⊢ (𝜑 → (𝐵 ∈ (ℝ ↑𝑚
𝑋) ↔ 𝐵:𝑋⟶ℝ)) |
34 | 1, 33 | mpbird 247 |
. . . 4
⊢ (𝜑 → 𝐵 ∈ (ℝ ↑𝑚
𝑋)) |
35 | | mptexg 6484 |
. . . . 5
⊢ (𝑋 ∈ 𝑉 → (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))) ∈ V) |
36 | 30, 35 | syl 17 |
. . . 4
⊢ (𝜑 → (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))) ∈ V) |
37 | 21, 27, 34, 36 | fvmptd 6288 |
. . 3
⊢ (𝜑 → ((𝐻‘𝐴)‘𝐵) = (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴)))) |
38 | 37 | feq1d 6030 |
. 2
⊢ (𝜑 → (((𝐻‘𝐴)‘𝐵):𝑋⟶ℝ ↔ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))):𝑋⟶ℝ)) |
39 | 8, 38 | mpbird 247 |
1
⊢ (𝜑 → ((𝐻‘𝐴)‘𝐵):𝑋⟶ℝ) |