Theorem hoidifhspf 44046

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) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑥), (𝑎‘𝑘)))))
hoidifhspf.y	⊢ (𝜑 → 𝑌 ∈ ℝ)
hoidifhspf.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
hoidifhspf.a	⊢ (𝜑 → 𝐴:𝑋⟶ℝ)

Assertion

Ref	Expression
hoidifhspf	⊢ (𝜑 → ((𝐷‘𝑌)‘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

Step	Hyp	Ref	Expression
1		hoidifhspf.a	. . . . . 6 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
2	1	ffvelrnda 6943	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
3		hoidifhspf.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ ℝ)
4	3	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑌 ∈ ℝ)
5	2, 4	ifcld 4502	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌) ∈ ℝ)
6	5, 2	ifcld 4502	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘)) ∈ ℝ)
7	6	fmpttd 6971	. 2 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘))):𝑋⟶ℝ)
8		hoidifhspf.d	. . . 4 ⊢ 𝐷 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑥), (𝑎‘𝑘)))))
9		hoidifhspf.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
10	8, 3, 9, 1	hoidifhspval2 44043	. . 3 ⊢ (𝜑 → ((𝐷‘𝑌)‘𝐴) = (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘))))
11	10	feq1d 6569	. 2 ⊢ (𝜑 → (((𝐷‘𝑌)‘𝐴):𝑋⟶ℝ ↔ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘))):𝑋⟶ℝ))
12	7, 11	mpbird 256	1 ⊢ (𝜑 → ((𝐷‘𝑌)‘𝐴):𝑋⟶ℝ)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ifcif 4456   class class class wbr 5070   ↦ cmpt 5153  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ↑_m cmap 8573  ℝcr 10801   ≤ cle 10941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-map 8575

This theorem is referenced by:  hoidifhspdmvle  44048  hspmbllem1  44054  hspmbllem2  44055