Proof of Theorem hsphoif
| Step | Hyp | Ref
| Expression |
| 1 | | hsphoif.b |
. . . . 5
⊢ (𝜑 → 𝐵:𝑋⟶ℝ) |
| 2 | 1 | ffvelcdmda 7104 |
. . . 4
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (𝐵‘𝑗) ∈ ℝ) |
| 3 | | hsphoif.a |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ ℝ) |
| 4 | 3 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → 𝐴 ∈ ℝ) |
| 5 | 2, 4 | ifcld 4572 |
. . . 4
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴) ∈ ℝ) |
| 6 | 2, 5 | ifcld 4572 |
. . 3
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴)) ∈ ℝ) |
| 7 | | eqid 2737 |
. . 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 7134 |
. 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 5147 |
. . . . . . . . 9
⊢ (𝑥 = 𝐴 → ((𝑎‘𝑗) ≤ 𝑥 ↔ (𝑎‘𝑗) ≤ 𝐴)) |
| 11 | | id 22 |
. . . . . . . . 9
⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) |
| 12 | 10, 11 | ifbieq2d 4552 |
. . . . . . . 8
⊢ (𝑥 = 𝐴 → if((𝑎‘𝑗) ≤ 𝑥, (𝑎‘𝑗), 𝑥) = if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴)) |
| 13 | 12 | ifeq2d 4546 |
. . . . . . 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 5244 |
. . . . . 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 5244 |
. . . . 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 7464 |
. . . . . . 7
⊢ (ℝ
↑m 𝑋)
∈ V |
| 17 | 16 | mptex 7243 |
. . . . . 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 7039 |
. . . 4
⊢ (𝜑 → (𝐻‘𝐴) = (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝑎‘𝑗), if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴))))) |
| 20 | | fveq1 6905 |
. . . . . . 7
⊢ (𝑎 = 𝐵 → (𝑎‘𝑗) = (𝐵‘𝑗)) |
| 21 | 20 | breq1d 5153 |
. . . . . . . 8
⊢ (𝑎 = 𝐵 → ((𝑎‘𝑗) ≤ 𝐴 ↔ (𝐵‘𝑗) ≤ 𝐴)) |
| 22 | 21, 20 | ifbieq1d 4550 |
. . . . . . 7
⊢ (𝑎 = 𝐵 → if((𝑎‘𝑗) ≤ 𝐴, (𝑎‘𝑗), 𝐴) = if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴)) |
| 23 | 20, 22 | ifeq12d 4547 |
. . . . . 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 5244 |
. . . . 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 481 |
. . . 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 11246 |
. . . . . . . 8
⊢ ℝ
∈ V |
| 27 | 26 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → ℝ ∈
V) |
| 28 | | hsphoif.x |
. . . . . . 7
⊢ (𝜑 → 𝑋 ∈ 𝑉) |
| 29 | 27, 28 | jca 511 |
. . . . . 6
⊢ (𝜑 → (ℝ ∈ V ∧
𝑋 ∈ 𝑉)) |
| 30 | | elmapg 8879 |
. . . . . 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 257 |
. . . 4
⊢ (𝜑 → 𝐵 ∈ (ℝ ↑m 𝑋)) |
| 33 | | mptexg 7241 |
. . . . 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 7023 |
. . 3
⊢ (𝜑 → ((𝐻‘𝐴)‘𝐵) = (𝑗 ∈ 𝑋 ↦ if(𝑗 ∈ 𝑌, (𝐵‘𝑗), if((𝐵‘𝑗) ≤ 𝐴, (𝐵‘𝑗), 𝐴)))) |
| 36 | 35 | feq1d 6720 |
. 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 257 |
1
⊢ (𝜑 → ((𝐻‘𝐴)‘𝐵):𝑋⟶ℝ) |