Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hoidifhspval | 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 |
---|---|
hoidifhspval.d | ⊢ 𝐷 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑥), (𝑎‘𝑘))))) |
hoidifhspval.y | ⊢ (𝜑 → 𝑌 ∈ ℝ) |
Ref | Expression |
---|---|
hoidifhspval | ⊢ (𝜑 → (𝐷‘𝑌) = (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌), (𝑎‘𝑘))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hoidifhspval.d | . 2 ⊢ 𝐷 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑥), (𝑎‘𝑘))))) | |
2 | breq1 5071 | . . . . . 6 ⊢ (𝑥 = 𝑌 → (𝑥 ≤ (𝑎‘𝑘) ↔ 𝑌 ≤ (𝑎‘𝑘))) | |
3 | id 22 | . . . . . 6 ⊢ (𝑥 = 𝑌 → 𝑥 = 𝑌) | |
4 | 2, 3 | ifbieq2d 4494 | . . . . 5 ⊢ (𝑥 = 𝑌 → if(𝑥 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑥) = if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌)) |
5 | 4 | ifeq1d 4487 | . . . 4 ⊢ (𝑥 = 𝑌 → if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑥), (𝑎‘𝑘)) = if(𝑘 = 𝐾, if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌), (𝑎‘𝑘))) |
6 | 5 | mpteq2dv 5164 | . . 3 ⊢ (𝑥 = 𝑌 → (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑥), (𝑎‘𝑘))) = (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌), (𝑎‘𝑘)))) |
7 | 6 | mpteq2dv 5164 | . 2 ⊢ (𝑥 = 𝑌 → (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑥), (𝑎‘𝑘)))) = (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌), (𝑎‘𝑘))))) |
8 | hoidifhspval.y | . 2 ⊢ (𝜑 → 𝑌 ∈ ℝ) | |
9 | ovex 7191 | . . . 4 ⊢ (ℝ ↑m 𝑋) ∈ V | |
10 | 9 | mptex 6988 | . . 3 ⊢ (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌), (𝑎‘𝑘)))) ∈ V |
11 | 10 | a1i 11 | . 2 ⊢ (𝜑 → (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌), (𝑎‘𝑘)))) ∈ V) |
12 | 1, 7, 8, 11 | fvmptd3 6793 | 1 ⊢ (𝜑 → (𝐷‘𝑌) = (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌), (𝑎‘𝑘))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ifcif 4469 class class class wbr 5068 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 ↑m cmap 8408 ℝcr 10538 ≤ cle 10678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 |
This theorem is referenced by: hoidifhspval2 42904 |
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