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 42896 | . 2 ⊢ (𝜑 → ((𝐷‘𝑌)‘𝐴) = (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘)))) |
6 | eqeq1 2825 | . . . 4 ⊢ (𝑘 = 𝐽 → (𝑘 = 𝐾 ↔ 𝐽 = 𝐾)) | |
7 | fveq2 6669 | . . . . . 6 ⊢ (𝑘 = 𝐽 → (𝐴‘𝑘) = (𝐴‘𝐽)) | |
8 | 7 | breq2d 5077 | . . . . 5 ⊢ (𝑘 = 𝐽 → (𝑌 ≤ (𝐴‘𝑘) ↔ 𝑌 ≤ (𝐴‘𝐽))) |
9 | 8, 7 | ifbieq1d 4489 | . . . 4 ⊢ (𝑘 = 𝐽 → if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌) = if(𝑌 ≤ (𝐴‘𝐽), (𝐴‘𝐽), 𝑌)) |
10 | 6, 9, 7 | ifbieq12d 4493 | . . 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 484 | . 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 6684 | . . . 4 ⊢ (𝜑 → (𝐴‘𝐽) ∈ V) | |
14 | 2 | elexd 3514 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ V) |
15 | 13, 14 | ifcld 4511 | . . 3 ⊢ (𝜑 → if(𝑌 ≤ (𝐴‘𝐽), (𝐴‘𝐽), 𝑌) ∈ V) |
16 | 15, 13 | ifcld 4511 | . 2 ⊢ (𝜑 → if(𝐽 = 𝐾, if(𝑌 ≤ (𝐴‘𝐽), (𝐴‘𝐽), 𝑌), (𝐴‘𝐽)) ∈ V) |
17 | 5, 11, 12, 16 | fvmptd 6774 | 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 1533 ∈ wcel 2110 Vcvv 3494 ifcif 4466 class class class wbr 5065 ↦ cmpt 5145 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 ↑m cmap 8405 ℝcr 10535 ≤ cle 10675 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-map 8407 |
This theorem is referenced by: hoidifhspdmvle 42901 hspmbllem1 42907 hspmbllem2 42908 |
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