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Theorem hoidifhspf 44849
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 7035 . . . . 5 ((𝜑𝑘𝑋) → (𝐴𝑘) ∈ ℝ)
3 hoidifhspf.y . . . . . 6 (𝜑𝑌 ∈ ℝ)
43adantr 481 . . . . 5 ((𝜑𝑘𝑋) → 𝑌 ∈ ℝ)
52, 4ifcld 4532 . . . 4 ((𝜑𝑘𝑋) → if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌) ∈ ℝ)
65, 2ifcld 4532 . . 3 ((𝜑𝑘𝑋) → if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌), (𝐴𝑘)) ∈ ℝ)
76fmpttd 7063 . 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 44846 . . 3 (𝜑 → ((𝐷𝑌)‘𝐴) = (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌), (𝐴𝑘))))
1110feq1d 6653 . 2 (𝜑 → (((𝐷𝑌)‘𝐴):𝑋⟶ℝ ↔ (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌), (𝐴𝑘))):𝑋⟶ℝ))
127, 11mpbird 256 1 (𝜑 → ((𝐷𝑌)‘𝐴):𝑋⟶ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  ifcif 4486   class class class wbr 5105  cmpt 5188  wf 6492  cfv 6496  (class class class)co 7357  m cmap 8765  cr 11050  cle 11190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-map 8767
This theorem is referenced by:  hoidifhspdmvle  44851  hspmbllem1  44857  hspmbllem2  44858
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