Theorem hoidifhspval 47891

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
hoidifhspval.d	⊢ 𝐷 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑥), (𝑎‘𝑘)))))
hoidifhspval.y	⊢ (𝜑 → 𝑌 ∈ ℝ)

Assertion

Ref	Expression
hoidifhspval	⊢ (𝜑 → (𝐷‘𝑌) = (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌), (𝑎‘𝑘)))))

Distinct variable groups:   𝑥,𝐾   𝑋,𝑎,𝑥   𝑌,𝑎,𝑥   𝑘,𝑌,𝑥   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑘, 𝑎)   𝐷(𝑥, 𝑘, 𝑎)   𝐾(𝑘, 𝑎)   𝑋(𝑘)

Proof of Theorem hoidifhspval

Step	Hyp	Ref	Expression
1		hoidifhspval.d	. 2 ⊢ 𝐷 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑥), (𝑎‘𝑘)))))
2		breq1 5111	. . . . . 6 ⊢ (𝑥 = 𝑌 → (𝑥 ≤ (𝑎‘𝑘) ↔ 𝑌 ≤ (𝑎‘𝑘)))
3		id 23	. . . . . 6 ⊢ (𝑥 = 𝑌 → 𝑥 = 𝑌)
4	2, 3	ifbieq2d 4512	. . . . 5 ⊢ (𝑥 = 𝑌 → if(𝑥 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑥) = if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌))
5	4	ifeq1d 4505	. . . 4 ⊢ (𝑥 = 𝑌 → if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑥), (𝑎‘𝑘)) = if(𝑘 = 𝐾, if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌), (𝑎‘𝑘)))
6	5	mpteq2dv 5205	. . 3 ⊢ (𝑥 = 𝑌 → (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑥), (𝑎‘𝑘))) = (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌), (𝑎‘𝑘))))
7	6	mpteq2dv 5205	. 2 ⊢ (𝑥 = 𝑌 → (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑥), (𝑎‘𝑘)))) = (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌), (𝑎‘𝑘)))))
8		hoidifhspval.y	. 2 ⊢ (𝜑 → 𝑌 ∈ ℝ)
9		ovex 7484	. . . 4 ⊢ (ℝ ↑m 𝑋) ∈ V
10	9	mptex 7260	. . 3 ⊢ (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌), (𝑎‘𝑘)))) ∈ V
11	10	a1i 11	. 2 ⊢ (𝜑 → (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌), (𝑎‘𝑘)))) ∈ V)
12	1, 7, 8, 11	fvmptd3 7048	1 ⊢ (𝜑 → (𝐷‘𝑌) = (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌), (𝑎‘𝑘)))))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ifcif 4485   class class class wbr 5108   ↦ cmpt 5192  ‘cfv 6566  (class class class)co 7451   ↑_m cmap 8885  ℝcr 11238   ≤ cle 11383

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454

This theorem is used by:  hoidifhspval2  47898