Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hoidifhspval3 Structured version   Visualization version   GIF version

Theorem hoidifhspval3 39509
 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
hoidifhspval3.d 𝐷 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎𝑘), (𝑎𝑘), 𝑥), (𝑎𝑘)))))
hoidifhspval3.y (𝜑𝑌 ∈ ℝ)
hoidifhspval3.x (𝜑𝑋𝑉)
hoidifhspval3.a (𝜑𝐴:𝑋⟶ℝ)
hoidifhspval3.j (𝜑𝐽𝑋)
Assertion
Ref Expression
hoidifhspval3 (𝜑 → (((𝐷𝑌)‘𝐴)‘𝐽) = if(𝐽 = 𝐾, if(𝑌 ≤ (𝐴𝐽), (𝐴𝐽), 𝑌), (𝐴𝐽)))
Distinct variable groups:   𝐴,𝑎,𝑘   𝑘,𝐽   𝐾,𝑎,𝑘,𝑥   𝑋,𝑎,𝑘,𝑥   𝑌,𝑎,𝑘,𝑥   𝜑,𝑎,𝑘,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐷(𝑥,𝑘,𝑎)   𝐽(𝑥,𝑎)   𝑉(𝑥,𝑘,𝑎)

Proof of Theorem hoidifhspval3
StepHypRef Expression
1 hoidifhspval3.d . . 3 𝐷 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎𝑘), (𝑎𝑘), 𝑥), (𝑎𝑘)))))
2 hoidifhspval3.y . . 3 (𝜑𝑌 ∈ ℝ)
3 hoidifhspval3.x . . 3 (𝜑𝑋𝑉)
4 hoidifhspval3.a . . 3 (𝜑𝐴:𝑋⟶ℝ)
51, 2, 3, 4hoidifhspval2 39505 . 2 (𝜑 → ((𝐷𝑌)‘𝐴) = (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌), (𝐴𝑘))))
6 eqeq1 2614 . . . 4 (𝑘 = 𝐽 → (𝑘 = 𝐾𝐽 = 𝐾))
7 fveq2 6103 . . . . . 6 (𝑘 = 𝐽 → (𝐴𝑘) = (𝐴𝐽))
87breq2d 4595 . . . . 5 (𝑘 = 𝐽 → (𝑌 ≤ (𝐴𝑘) ↔ 𝑌 ≤ (𝐴𝐽)))
98, 7ifbieq1d 4059 . . . 4 (𝑘 = 𝐽 → if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌) = if(𝑌 ≤ (𝐴𝐽), (𝐴𝐽), 𝑌))
106, 9, 7ifbieq12d 4063 . . 3 (𝑘 = 𝐽 → if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌), (𝐴𝑘)) = if(𝐽 = 𝐾, if(𝑌 ≤ (𝐴𝐽), (𝐴𝐽), 𝑌), (𝐴𝐽)))
1110adantl 481 . 2 ((𝜑𝑘 = 𝐽) → if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌), (𝐴𝑘)) = if(𝐽 = 𝐾, if(𝑌 ≤ (𝐴𝐽), (𝐴𝐽), 𝑌), (𝐴𝐽)))
12 hoidifhspval3.j . 2 (𝜑𝐽𝑋)
13 fvex 6113 . . . . 5 (𝐴𝐽) ∈ V
1413a1i 11 . . . 4 (𝜑 → (𝐴𝐽) ∈ V)
152elexd 3187 . . . 4 (𝜑𝑌 ∈ V)
1614, 15ifcld 4081 . . 3 (𝜑 → if(𝑌 ≤ (𝐴𝐽), (𝐴𝐽), 𝑌) ∈ V)
1716, 14ifcld 4081 . 2 (𝜑 → if(𝐽 = 𝐾, if(𝑌 ≤ (𝐴𝐽), (𝐴𝐽), 𝑌), (𝐴𝐽)) ∈ V)
185, 11, 12, 17fvmptd 6197 1 (𝜑 → (((𝐷𝑌)‘𝐴)‘𝐽) = if(𝐽 = 𝐾, if(𝑌 ≤ (𝐴𝐽), (𝐴𝐽), 𝑌), (𝐴𝐽)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ifcif 4036   class class class wbr 4583   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  ℝcr 9814   ≤ cle 9954 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 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872 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 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746 This theorem is referenced by:  hoidifhspdmvle  39510  hspmbllem1  39516  hspmbllem2  39517
 Copyright terms: Public domain W3C validator