Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hoidifhspf Structured version   Visualization version   GIF version

Theorem hoidifhspf 46539
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)
21ffvelcdmda 7118 . . . . 5 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
3 hoidifhspf.y . . . . . 6 (𝜑𝑌 ∈ ℝ)
43adantr 480 . . . . 5 ((𝜑𝑘𝑋) → 𝑌 ∈ ℝ)
52, 4ifcld 4594 . . . 4 ((𝜑𝑘𝑋) → if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌) ∈ ℝ)
65, 2ifcld 4594 . . 3 ((𝜑𝑘𝑋) → if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌), (𝐴𝑘)) ∈ ℝ)
76fmpttd 7149 . 2 (𝜑 → (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌), (𝐴𝑘))):𝑋⟶ℝ)
8 hoidifhspf.d . . . 4 𝐷 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎𝑘), (𝑎𝑘), 𝑥), (𝑎𝑘)))))
9 hoidifhspf.x . . . 4 (𝜑𝑋𝑉)
108, 3, 9, 1hoidifhspval2 46536 . . 3 (𝜑 → ((𝐷𝑌)‘𝐴) = (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌), (𝐴𝑘))))
1110feq1d 6732 . 2 (𝜑 → (((𝐷𝑌)‘𝐴):𝑋⟶ℝ ↔ (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌), (𝐴𝑘))):𝑋⟶ℝ))
127, 11mpbird 257 1 (𝜑 → ((𝐷𝑌)‘𝐴):𝑋⟶ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2108  ifcif 4548   class class class wbr 5166  cmpt 5249  wf 6569  cfv 6573  (class class class)co 7448  m cmap 8884  cr 11183  cle 11325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-map 8886
This theorem is referenced by:  hoidifhspdmvle  46541  hspmbllem1  46547  hspmbllem2  46548
  Copyright terms: Public domain W3C validator