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Theorem hoidifhspval3 44157
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 44153 . 2 (𝜑 → ((𝐷𝑌)‘𝐴) = (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌), (𝐴𝑘))))
6 eqeq1 2742 . . . 4 (𝑘 = 𝐽 → (𝑘 = 𝐾𝐽 = 𝐾))
7 fveq2 6774 . . . . . 6 (𝑘 = 𝐽 → (𝐴𝑘) = (𝐴𝐽))
87breq2d 5086 . . . . 5 (𝑘 = 𝐽 → (𝑌 ≤ (𝐴𝑘) ↔ 𝑌 ≤ (𝐴𝐽)))
98, 7ifbieq1d 4483 . . . 4 (𝑘 = 𝐽 → if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌) = if(𝑌 ≤ (𝐴𝐽), (𝐴𝐽), 𝑌))
106, 9, 7ifbieq12d 4487 . . 3 (𝑘 = 𝐽 → if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌), (𝐴𝑘)) = if(𝐽 = 𝐾, if(𝑌 ≤ (𝐴𝐽), (𝐴𝐽), 𝑌), (𝐴𝐽)))
1110adantl 482 . 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 fvexd 6789 . . . 4 (𝜑 → (𝐴𝐽) ∈ V)
142elexd 3452 . . . 4 (𝜑𝑌 ∈ V)
1513, 14ifcld 4505 . . 3 (𝜑 → if(𝑌 ≤ (𝐴𝐽), (𝐴𝐽), 𝑌) ∈ V)
1615, 13ifcld 4505 . 2 (𝜑 → if(𝐽 = 𝐾, if(𝑌 ≤ (𝐴𝐽), (𝐴𝐽), 𝑌), (𝐴𝐽)) ∈ V)
175, 11, 12, 16fvmptd 6882 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 1539  wcel 2106  Vcvv 3432  ifcif 4459   class class class wbr 5074  cmpt 5157  wf 6429  cfv 6433  (class class class)co 7275  m cmap 8615  cr 10870  cle 11010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617
This theorem is referenced by:  hoidifhspdmvle  44158  hspmbllem1  44164  hspmbllem2  44165
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