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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hoidifhspval3 | 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 |
---|---|
hoidifhspval3.d | ⊢ 𝐷 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑥), (𝑎‘𝑘))))) |
hoidifhspval3.y | ⊢ (𝜑 → 𝑌 ∈ ℝ) |
hoidifhspval3.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
hoidifhspval3.a | ⊢ (𝜑 → 𝐴:𝑋⟶ℝ) |
hoidifhspval3.j | ⊢ (𝜑 → 𝐽 ∈ 𝑋) |
Ref | Expression |
---|---|
hoidifhspval3 | ⊢ (𝜑 → (((𝐷‘𝑌)‘𝐴)‘𝐽) = if(𝐽 = 𝐾, if(𝑌 ≤ (𝐴‘𝐽), (𝐴‘𝐽), 𝑌), (𝐴‘𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hoidifhspval3.d | . . 3 ⊢ 𝐷 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑥), (𝑎‘𝑘))))) | |
2 | hoidifhspval3.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ ℝ) | |
3 | hoidifhspval3.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
4 | hoidifhspval3.a | . . 3 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ) | |
5 | 1, 2, 3, 4 | hoidifhspval2 41335 | . 2 ⊢ (𝜑 → ((𝐷‘𝑌)‘𝐴) = (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘)))) |
6 | eqeq1 2764 | . . . 4 ⊢ (𝑘 = 𝐽 → (𝑘 = 𝐾 ↔ 𝐽 = 𝐾)) | |
7 | fveq2 6352 | . . . . . 6 ⊢ (𝑘 = 𝐽 → (𝐴‘𝑘) = (𝐴‘𝐽)) | |
8 | 7 | breq2d 4816 | . . . . 5 ⊢ (𝑘 = 𝐽 → (𝑌 ≤ (𝐴‘𝑘) ↔ 𝑌 ≤ (𝐴‘𝐽))) |
9 | 8, 7 | ifbieq1d 4253 | . . . 4 ⊢ (𝑘 = 𝐽 → if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌) = if(𝑌 ≤ (𝐴‘𝐽), (𝐴‘𝐽), 𝑌)) |
10 | 6, 9, 7 | ifbieq12d 4257 | . . 3 ⊢ (𝑘 = 𝐽 → if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘)) = if(𝐽 = 𝐾, if(𝑌 ≤ (𝐴‘𝐽), (𝐴‘𝐽), 𝑌), (𝐴‘𝐽))) |
11 | 10 | adantl 473 | . 2 ⊢ ((𝜑 ∧ 𝑘 = 𝐽) → if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘)) = if(𝐽 = 𝐾, if(𝑌 ≤ (𝐴‘𝐽), (𝐴‘𝐽), 𝑌), (𝐴‘𝐽))) |
12 | hoidifhspval3.j | . 2 ⊢ (𝜑 → 𝐽 ∈ 𝑋) | |
13 | fvexd 6364 | . . . 4 ⊢ (𝜑 → (𝐴‘𝐽) ∈ V) | |
14 | 2 | elexd 3354 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ V) |
15 | 13, 14 | ifcld 4275 | . . 3 ⊢ (𝜑 → if(𝑌 ≤ (𝐴‘𝐽), (𝐴‘𝐽), 𝑌) ∈ V) |
16 | 15, 13 | ifcld 4275 | . 2 ⊢ (𝜑 → if(𝐽 = 𝐾, if(𝑌 ≤ (𝐴‘𝐽), (𝐴‘𝐽), 𝑌), (𝐴‘𝐽)) ∈ V) |
17 | 5, 11, 12, 16 | fvmptd 6450 | 1 ⊢ (𝜑 → (((𝐷‘𝑌)‘𝐴)‘𝐽) = if(𝐽 = 𝐾, if(𝑌 ≤ (𝐴‘𝐽), (𝐴‘𝐽), 𝑌), (𝐴‘𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1632 ∈ wcel 2139 Vcvv 3340 ifcif 4230 class class class wbr 4804 ↦ cmpt 4881 ⟶wf 6045 ‘cfv 6049 (class class class)co 6813 ↑𝑚 cmap 8023 ℝcr 10127 ≤ cle 10267 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-map 8025 |
This theorem is referenced by: hoidifhspdmvle 41340 hspmbllem1 41346 hspmbllem2 41347 |
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