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| Mirrors > Home > MPE Home > Th. List > Mathboxes > hoidifhspval2 | Structured version Visualization version GIF version | ||
| Description: 𝐷 is a function that returns the representation of the left side of the difference of a half-open interval and a half-space. Used in Lemma 115F of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
| Ref | Expression |
|---|---|
| hoidifhspval2.d | ⊢ 𝐷 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑥), (𝑎‘𝑘))))) |
| hoidifhspval2.y | ⊢ (𝜑 → 𝑌 ∈ ℝ) |
| hoidifhspval2.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
| hoidifhspval2.a | ⊢ (𝜑 → 𝐴:𝑋⟶ℝ) |
| Ref | Expression |
|---|---|
| hoidifhspval2 | ⊢ (𝜑 → ((𝐷‘𝑌)‘𝐴) = (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hoidifhspval2.d | . . 3 ⊢ 𝐷 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑥), (𝑎‘𝑘))))) | |
| 2 | hoidifhspval2.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ ℝ) | |
| 3 | 1, 2 | hoidifhspval 46623 | . 2 ⊢ (𝜑 → (𝐷‘𝑌) = (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌), (𝑎‘𝑘))))) |
| 4 | fveq1 6905 | . . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎‘𝑘) = (𝐴‘𝑘)) | |
| 5 | 4 | breq2d 5155 | . . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑌 ≤ (𝑎‘𝑘) ↔ 𝑌 ≤ (𝐴‘𝑘))) |
| 6 | 5, 4 | ifbieq1d 4550 | . . . . 5 ⊢ (𝑎 = 𝐴 → if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌) = if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌)) |
| 7 | 6, 4 | ifeq12d 4547 | . . . 4 ⊢ (𝑎 = 𝐴 → if(𝑘 = 𝐾, if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌), (𝑎‘𝑘)) = if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘))) |
| 8 | 7 | mpteq2dv 5244 | . . 3 ⊢ (𝑎 = 𝐴 → (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌), (𝑎‘𝑘))) = (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘)))) |
| 9 | 8 | adantl 481 | . 2 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌), (𝑎‘𝑘))) = (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘)))) |
| 10 | hoidifhspval2.a | . . 3 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ) | |
| 11 | reex 11246 | . . . . . 6 ⊢ ℝ ∈ V | |
| 12 | 11 | a1i 11 | . . . . 5 ⊢ (𝜑 → ℝ ∈ V) |
| 13 | hoidifhspval2.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
| 14 | 12, 13 | jca 511 | . . . 4 ⊢ (𝜑 → (ℝ ∈ V ∧ 𝑋 ∈ 𝑉)) |
| 15 | elmapg 8879 | . . . 4 ⊢ ((ℝ ∈ V ∧ 𝑋 ∈ 𝑉) → (𝐴 ∈ (ℝ ↑m 𝑋) ↔ 𝐴:𝑋⟶ℝ)) | |
| 16 | 14, 15 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 ∈ (ℝ ↑m 𝑋) ↔ 𝐴:𝑋⟶ℝ)) |
| 17 | 10, 16 | mpbird 257 | . 2 ⊢ (𝜑 → 𝐴 ∈ (ℝ ↑m 𝑋)) |
| 18 | mptexg 7241 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘))) ∈ V) | |
| 19 | 13, 18 | syl 17 | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘))) ∈ V) |
| 20 | 3, 9, 17, 19 | fvmptd 7023 | 1 ⊢ (𝜑 → ((𝐷‘𝑌)‘𝐴) = (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘)))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ifcif 4525 class class class wbr 5143 ↦ cmpt 5225 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 ℝcr 11154 ≤ cle 11296 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 |
| This theorem is referenced by: hoidifhspf 46633 hoidifhspval3 46634 |
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