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Mathbox for Glauco Siliprandi |
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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 46564 | . 2 ⊢ (𝜑 → (𝐷‘𝑌) = (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌), (𝑎‘𝑘))))) |
4 | fveq1 6906 | . . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎‘𝑘) = (𝐴‘𝑘)) | |
5 | 4 | breq2d 5160 | . . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑌 ≤ (𝑎‘𝑘) ↔ 𝑌 ≤ (𝐴‘𝑘))) |
6 | 5, 4 | ifbieq1d 4555 | . . . . 5 ⊢ (𝑎 = 𝐴 → if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌) = if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌)) |
7 | 6, 4 | ifeq12d 4552 | . . . 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 5250 | . . 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 11244 | . . . . . 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 8878 | . . . 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 1537 ∈ wcel 2106 Vcvv 3478 ifcif 4531 class class class wbr 5148 ↦ cmpt 5231 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 ℝcr 11152 ≤ cle 11294 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8867 |
This theorem is referenced by: hoidifhspf 46574 hoidifhspval3 46575 |
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