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Theorem hoidifhspf 39302
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.)
Hypotheses
Ref Expression
hoidifhspf.d 𝐷 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎𝑘), (𝑎𝑘), 𝑥), (𝑎𝑘)))))
hoidifhspf.y (𝜑𝑌 ∈ ℝ)
hoidifhspf.x (𝜑𝑋𝑉)
hoidifhspf.a (𝜑𝐴:𝑋⟶ℝ)
Assertion
Ref Expression
hoidifhspf (𝜑 → ((𝐷𝑌)‘𝐴):𝑋⟶ℝ)
Distinct variable groups:   𝐴,𝑎,𝑘   𝐾,𝑎,𝑥   𝑋,𝑎,𝑘,𝑥   𝑌,𝑎,𝑘,𝑥   𝜑,𝑎,𝑘,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐷(𝑥,𝑘,𝑎)   𝐾(𝑘)   𝑉(𝑥,𝑘,𝑎)

Proof of Theorem hoidifhspf
StepHypRef Expression
1 hoidifhspf.a . . . . . 6 (𝜑𝐴:𝑋⟶ℝ)
21ffvelrnda 6252 . . . . 5 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
3 hoidifhspf.y . . . . . 6 (𝜑𝑌 ∈ ℝ)
43adantr 480 . . . . 5 ((𝜑𝑘𝑋) → 𝑌 ∈ ℝ)
52, 4ifcld 4081 . . . 4 ((𝜑𝑘𝑋) → if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌) ∈ ℝ)
65, 2ifcld 4081 . . 3 ((𝜑𝑘𝑋) → if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌), (𝐴𝑘)) ∈ ℝ)
7 eqid 2610 . . 3 (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌), (𝐴𝑘))) = (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌), (𝐴𝑘)))
86, 7fmptd 6277 . 2 (𝜑 → (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌), (𝐴𝑘))):𝑋⟶ℝ)
9 hoidifhspf.d . . . 4 𝐷 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎𝑘), (𝑎𝑘), 𝑥), (𝑎𝑘)))))
10 hoidifhspf.x . . . 4 (𝜑𝑋𝑉)
119, 3, 10, 1hoidifhspval2 39299 . . 3 (𝜑 → ((𝐷𝑌)‘𝐴) = (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌), (𝐴𝑘))))
1211feq1d 5929 . 2 (𝜑 → (((𝐷𝑌)‘𝐴):𝑋⟶ℝ ↔ (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌), (𝐴𝑘))):𝑋⟶ℝ))
138, 12mpbird 246 1 (𝜑 → ((𝐷𝑌)‘𝐴):𝑋⟶ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  ifcif 4036   class class class wbr 4578  cmpt 4638  wf 5786  cfv 5790  (class class class)co 6527  𝑚 cmap 7722  cr 9792  cle 9932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-cnex 9849  ax-resscn 9850
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-map 7724
This theorem is referenced by:  hoidifhspdmvle  39304  hspmbllem1  39310  hspmbllem2  39311
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