Theorem hoidifhspval 45809

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 5141	. . . . . 6 ⊢ (𝑥 = 𝑌 → (𝑥 ≤ (𝑎‘𝑘) ↔ 𝑌 ≤ (𝑎‘𝑘)))
3		id 22	. . . . . 6 ⊢ (𝑥 = 𝑌 → 𝑥 = 𝑌)
4	2, 3	ifbieq2d 4546	. . . . 5 ⊢ (𝑥 = 𝑌 → if(𝑥 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑥) = if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌))
5	4	ifeq1d 4539	. . . 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 5240	. . 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 5240	. 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 7434	. . . 4 ⊢ (ℝ ↑m 𝑋) ∈ V
10	9	mptex 7216	. . 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 7011	1 ⊢ (𝜑 → (𝐷‘𝑌) = (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌), (𝑎‘𝑘)))))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  Vcvv 3466  ifcif 4520   class class class wbr 5138   ↦ cmpt 5221  ‘cfv 6533  (class class class)co 7401   ↑_m cmap 8816  ℝcr 11105   ≤ cle 11246

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404

This theorem is referenced by:  hoidifhspval2  45816