Proof of Theorem hsphoif
Step | Hyp | Ref
| Expression |
1 | | hsphoif.b |
. . . . 5
⊢ (𝜑 → 𝐵:𝑋⟶ℝ) |
2 | 1 | ffvelrnda 6954 |
. . . 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 4506 |
. . . 4
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴) ∈ ℝ) |
6 | 2, 5 | ifcld 4506 |
. . 3
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴)) ∈ ℝ) |
7 | | eqid 2738 |
. . 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 6981 |
. 2
⊢ (𝜑 → (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))):𝑋⟶ℝ) |
9 | | hsphoif.h |
. . . . 5
⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝑥, (𝑎‘𝑗), 𝑥))))) |
10 | | breq2 5078 |
. . . . . . . . 9
⊢ (𝑥 = 𝐴 → ((𝑎‘𝑗) ≤ 𝑥 ↔ (𝑎‘𝑗) ≤ 𝐴)) |
11 | | id 22 |
. . . . . . . . 9
⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) |
12 | 10, 11 | ifbieq2d 4486 |
. . . . . . . 8
⊢ (𝑥 = 𝐴 → if((𝑎‘𝑗) ≤ 𝑥, (𝑎‘𝑗), 𝑥) = if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴)) |
13 | 12 | ifeq2d 4480 |
. . . . . . 7
⊢ (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))) |
14 | 13 | mpteq2dv 5176 |
. . . . . 6
⊢ (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)))) |
15 | 14 | mpteq2dv 5176 |
. . . . 5
⊢ (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))))) |
16 | | ovex 7301 |
. . . . . . 7
⊢ (ℝ
↑m 𝑋)
∈ V |
17 | 16 | mptex 7092 |
. . . . . 6
⊢ (𝑎 ∈ (ℝ
↑m 𝑋)
↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴)))) ∈ V |
18 | 17 | a1i 11 |
. . . . 5
⊢ (𝜑 → (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴)))) ∈ V) |
19 | 9, 15, 3, 18 | fvmptd3 6891 |
. . . 4
⊢ (𝜑 → (𝐻‘𝐴) = (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴))))) |
20 | | fveq1 6766 |
. . . . . . 7
⊢ (𝑎 = 𝐵 → (𝑎‘𝑗) = (𝐵‘𝑗)) |
21 | 20 | breq1d 5084 |
. . . . . . . 8
⊢ (𝑎 = 𝐵 → ((𝑎‘𝑗) ≤ 𝐴 ↔ (𝐵‘𝑗) ≤ 𝐴)) |
22 | 21, 20 | ifbieq1d 4484 |
. . . . . . 7
⊢ (𝑎 = 𝐵 → if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴) = if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴)) |
23 | 20, 22 | ifeq12d 4481 |
. . . . . 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))) |
24 | 23 | mpteq2dv 5176 |
. . . . 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)))) |
25 | 24 | 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)))) |
26 | | reex 10950 |
. . . . . . . 8
⊢ ℝ
∈ V |
27 | 26 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → ℝ ∈
V) |
28 | | hsphoif.x |
. . . . . . 7
⊢ (𝜑 → 𝑋 ∈ 𝑉) |
29 | 27, 28 | jca 512 |
. . . . . 6
⊢ (𝜑 → (ℝ ∈ V ∧
𝑋 ∈ 𝑉)) |
30 | | elmapg 8616 |
. . . . . 6
⊢ ((ℝ
∈ V ∧ 𝑋 ∈
𝑉) → (𝐵 ∈ (ℝ
↑m 𝑋)
↔ 𝐵:𝑋⟶ℝ)) |
31 | 29, 30 | syl 17 |
. . . . 5
⊢ (𝜑 → (𝐵 ∈ (ℝ ↑m 𝑋) ↔ 𝐵:𝑋⟶ℝ)) |
32 | 1, 31 | mpbird 256 |
. . . 4
⊢ (𝜑 → 𝐵 ∈ (ℝ ↑m 𝑋)) |
33 | | mptexg 7090 |
. . . . 5
⊢ (𝑋 ∈ 𝑉 → (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))) ∈ V) |
34 | 28, 33 | syl 17 |
. . . 4
⊢ (𝜑 → (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))) ∈ V) |
35 | 19, 25, 32, 34 | fvmptd 6875 |
. . 3
⊢ (𝜑 → ((𝐻‘𝐴)‘𝐵) = (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴)))) |
36 | 35 | feq1d 6578 |
. 2
⊢ (𝜑 → (((𝐻‘𝐴)‘𝐵):𝑋⟶ℝ ↔ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))):𝑋⟶ℝ)) |
37 | 8, 36 | mpbird 256 |
1
⊢ (𝜑 → ((𝐻‘𝐴)‘𝐵):𝑋⟶ℝ) |