Proof of Theorem hsphoival
Step | Hyp | Ref
| Expression |
1 | | hsphoival.h |
. . . . 5
⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑𝑚
𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝑥, (𝑎‘𝑗), 𝑥))))) |
2 | 1 | a1i 11 |
. . . 4
⊢ (𝜑 → 𝐻 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑𝑚
𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝑥, (𝑎‘𝑗), 𝑥)))))) |
3 | | breq2 4587 |
. . . . . . . . 9
⊢ (𝑥 = 𝐴 → ((𝑎‘𝑗) ≤ 𝑥 ↔ (𝑎‘𝑗) ≤ 𝐴)) |
4 | | id 22 |
. . . . . . . . 9
⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) |
5 | 3, 4 | ifbieq2d 4061 |
. . . . . . . 8
⊢ (𝑥 = 𝐴 → if((𝑎‘𝑗) ≤ 𝑥, (𝑎‘𝑗), 𝑥) = if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴)) |
6 | 5 | ifeq2d 4055 |
. . . . . . 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))) |
7 | 6 | mpteq2dv 4673 |
. . . . . 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)))) |
8 | 7 | mpteq2dv 4673 |
. . . . 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))))) |
9 | 8 | adantl 481 |
. . . 4
⊢ ((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))))) |
10 | | hsphoival.a |
. . . 4
⊢ (𝜑 → 𝐴 ∈ ℝ) |
11 | | ovex 6577 |
. . . . . 6
⊢ (ℝ
↑𝑚 𝑋) ∈ V |
12 | 11 | mptex 6390 |
. . . . 5
⊢ (𝑎 ∈ (ℝ
↑𝑚 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴)))) ∈ V |
13 | 12 | a1i 11 |
. . . 4
⊢ (𝜑 → (𝑎 ∈ (ℝ ↑𝑚
𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴)))) ∈ V) |
14 | 2, 9, 10, 13 | fvmptd 6197 |
. . 3
⊢ (𝜑 → (𝐻‘𝐴) = (𝑎 ∈ (ℝ ↑𝑚
𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴))))) |
15 | | fveq1 6102 |
. . . . . 6
⊢ (𝑎 = 𝐵 → (𝑎‘𝑗) = (𝐵‘𝑗)) |
16 | 15 | breq1d 4593 |
. . . . . . 7
⊢ (𝑎 = 𝐵 → ((𝑎‘𝑗) ≤ 𝐴 ↔ (𝐵‘𝑗) ≤ 𝐴)) |
17 | 16, 15 | ifbieq1d 4059 |
. . . . . 6
⊢ (𝑎 = 𝐵 → if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴) = if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴)) |
18 | 15, 17 | ifeq12d 4056 |
. . . . 5
⊢ (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))) |
19 | 18 | mpteq2dv 4673 |
. . . 4
⊢ (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)))) |
20 | 19 | adantl 481 |
. . 3
⊢ ((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)))) |
21 | | hsphoival.b |
. . . 4
⊢ (𝜑 → 𝐵:𝑋⟶ℝ) |
22 | | reex 9906 |
. . . . . . 7
⊢ ℝ
∈ V |
23 | 22 | a1i 11 |
. . . . . 6
⊢ (𝜑 → ℝ ∈
V) |
24 | | hsphoival.x |
. . . . . 6
⊢ (𝜑 → 𝑋 ∈ 𝑉) |
25 | 23, 24 | jca 553 |
. . . . 5
⊢ (𝜑 → (ℝ ∈ V ∧
𝑋 ∈ 𝑉)) |
26 | | elmapg 7757 |
. . . . 5
⊢ ((ℝ
∈ V ∧ 𝑋 ∈
𝑉) → (𝐵 ∈ (ℝ
↑𝑚 𝑋) ↔ 𝐵:𝑋⟶ℝ)) |
27 | 25, 26 | syl 17 |
. . . 4
⊢ (𝜑 → (𝐵 ∈ (ℝ ↑𝑚
𝑋) ↔ 𝐵:𝑋⟶ℝ)) |
28 | 21, 27 | mpbird 246 |
. . 3
⊢ (𝜑 → 𝐵 ∈ (ℝ ↑𝑚
𝑋)) |
29 | | mptexg 6389 |
. . . 4
⊢ (𝑋 ∈ 𝑉 → (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))) ∈ V) |
30 | 24, 29 | syl 17 |
. . 3
⊢ (𝜑 → (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴))) ∈ V) |
31 | 14, 20, 28, 30 | fvmptd 6197 |
. 2
⊢ (𝜑 → ((𝐻‘𝐴)‘𝐵) = (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴)))) |
32 | | eleq1 2676 |
. . . 4
⊢ (𝑗 = 𝐾 → (𝑗 ∈ 𝑌 ↔ 𝐾 ∈ 𝑌)) |
33 | | fveq2 6103 |
. . . 4
⊢ (𝑗 = 𝐾 → (𝐵‘𝑗) = (𝐵‘𝐾)) |
34 | 33 | breq1d 4593 |
. . . . 5
⊢ (𝑗 = 𝐾 → ((𝐵‘𝑗) ≤ 𝐴 ↔ (𝐵‘𝐾) ≤ 𝐴)) |
35 | 34, 33 | ifbieq1d 4059 |
. . . 4
⊢ (𝑗 = 𝐾 → if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴) = if((𝐵‘𝐾) ≤ 𝐴, (𝐵‘𝐾), 𝐴)) |
36 | 32, 33, 35 | ifbieq12d 4063 |
. . 3
⊢ (𝑗 = 𝐾 → if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴)) = if(𝐾 ∈ 𝑌, (𝐵‘𝐾), if((𝐵‘𝐾) ≤ 𝐴, (𝐵‘𝐾), 𝐴))) |
37 | 36 | adantl 481 |
. 2
⊢ ((𝜑 ∧ 𝑗 = 𝐾) → if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴)) = if(𝐾 ∈ 𝑌, (𝐵‘𝐾), if((𝐵‘𝐾) ≤ 𝐴, (𝐵‘𝐾), 𝐴))) |
38 | | hsphoival.k |
. 2
⊢ (𝜑 → 𝐾 ∈ 𝑋) |
39 | 21, 38 | ffvelrnd 6268 |
. . 3
⊢ (𝜑 → (𝐵‘𝐾) ∈ ℝ) |
40 | 39, 10 | ifcld 4081 |
. . 3
⊢ (𝜑 → if((𝐵‘𝐾) ≤ 𝐴, (𝐵‘𝐾), 𝐴) ∈ ℝ) |
41 | | ifexg 4107 |
. . 3
⊢ (((𝐵‘𝐾) ∈ ℝ ∧ if((𝐵‘𝐾) ≤ 𝐴, (𝐵‘𝐾), 𝐴) ∈ ℝ) → if(𝐾 ∈ 𝑌, (𝐵‘𝐾), if((𝐵‘𝐾) ≤ 𝐴, (𝐵‘𝐾), 𝐴)) ∈ V) |
42 | 39, 40, 41 | syl2anc 691 |
. 2
⊢ (𝜑 → if(𝐾 ∈ 𝑌, (𝐵‘𝐾), if((𝐵‘𝐾) ≤ 𝐴, (𝐵‘𝐾), 𝐴)) ∈ V) |
43 | 31, 37, 38, 42 | fvmptd 6197 |
1
⊢ (𝜑 → (((𝐻‘𝐴)‘𝐵)‘𝐾) = if(𝐾 ∈ 𝑌, (𝐵‘𝐾), if((𝐵‘𝐾) ≤ 𝐴, (𝐵‘𝐾), 𝐴))) |