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Theorem hoidifhspval3 41339
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 41335 . 2 (𝜑 → ((𝐷𝑌)‘𝐴) = (𝑘𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌), (𝐴𝑘))))
6 eqeq1 2764 . . . 4 (𝑘 = 𝐽 → (𝑘 = 𝐾𝐽 = 𝐾))
7 fveq2 6352 . . . . . 6 (𝑘 = 𝐽 → (𝐴𝑘) = (𝐴𝐽))
87breq2d 4816 . . . . 5 (𝑘 = 𝐽 → (𝑌 ≤ (𝐴𝑘) ↔ 𝑌 ≤ (𝐴𝐽)))
98, 7ifbieq1d 4253 . . . 4 (𝑘 = 𝐽 → if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌) = if(𝑌 ≤ (𝐴𝐽), (𝐴𝐽), 𝑌))
106, 9, 7ifbieq12d 4257 . . 3 (𝑘 = 𝐽 → if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴𝑘), (𝐴𝑘), 𝑌), (𝐴𝑘)) = if(𝐽 = 𝐾, if(𝑌 ≤ (𝐴𝐽), (𝐴𝐽), 𝑌), (𝐴𝐽)))
1110adantl 473 . 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 6364 . . . 4 (𝜑 → (𝐴𝐽) ∈ V)
142elexd 3354 . . . 4 (𝜑𝑌 ∈ V)
1513, 14ifcld 4275 . . 3 (𝜑 → if(𝑌 ≤ (𝐴𝐽), (𝐴𝐽), 𝑌) ∈ V)
1615, 13ifcld 4275 . 2 (𝜑 → if(𝐽 = 𝐾, if(𝑌 ≤ (𝐴𝐽), (𝐴𝐽), 𝑌), (𝐴𝐽)) ∈ V)
175, 11, 12, 16fvmptd 6450 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 1632  wcel 2139  Vcvv 3340  ifcif 4230   class class class wbr 4804  cmpt 4881  wf 6045  cfv 6049  (class class class)co 6813  𝑚 cmap 8023  cr 10127  cle 10267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-map 8025
This theorem is referenced by:  hoidifhspdmvle  41340  hspmbllem1  41346  hspmbllem2  41347
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