Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hoidifhspf | 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 |
---|---|
hoidifhspf.d | ⊢ 𝐷 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑥), (𝑎‘𝑘))))) |
hoidifhspf.y | ⊢ (𝜑 → 𝑌 ∈ ℝ) |
hoidifhspf.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
hoidifhspf.a | ⊢ (𝜑 → 𝐴:𝑋⟶ℝ) |
Ref | Expression |
---|---|
hoidifhspf | ⊢ (𝜑 → ((𝐷‘𝑌)‘𝐴):𝑋⟶ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hoidifhspf.a | . . . . . 6 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ) | |
2 | 1 | ffvelrnda 6359 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ) |
3 | hoidifhspf.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ ℝ) | |
4 | 3 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑌 ∈ ℝ) |
5 | 2, 4 | ifcld 4131 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌) ∈ ℝ) |
6 | 5, 2 | ifcld 4131 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘)) ∈ ℝ) |
7 | eqid 2622 | . . 3 ⊢ (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))) | |
8 | 6, 7 | fmptd 6385 | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘))):𝑋⟶ℝ) |
9 | hoidifhspf.d | . . . 4 ⊢ 𝐷 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑥), (𝑎‘𝑘))))) | |
10 | hoidifhspf.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
11 | 9, 3, 10, 1 | hoidifhspval2 40829 | . . 3 ⊢ (𝜑 → ((𝐷‘𝑌)‘𝐴) = (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘)))) |
12 | 11 | feq1d 6030 | . 2 ⊢ (𝜑 → (((𝐷‘𝑌)‘𝐴):𝑋⟶ℝ ↔ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘))):𝑋⟶ℝ)) |
13 | 8, 12 | mpbird 247 | 1 ⊢ (𝜑 → ((𝐷‘𝑌)‘𝐴):𝑋⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ifcif 4086 class class class wbr 4653 ↦ cmpt 4729 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↑𝑚 cmap 7857 ℝcr 9935 ≤ cle 10075 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-map 7859 |
This theorem is referenced by: hoidifhspdmvle 40834 hspmbllem1 40840 hspmbllem2 40841 |
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