Theorem hoidifhspval2 43254

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
hoidifhspval2.d	⊢ 𝐷 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑥), (𝑎‘𝑘)))))
hoidifhspval2.y	⊢ (𝜑 → 𝑌 ∈ ℝ)
hoidifhspval2.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
hoidifhspval2.a	⊢ (𝜑 → 𝐴:𝑋⟶ℝ)

Assertion

Ref	Expression
hoidifhspval2	⊢ (𝜑 → ((𝐷‘𝑌)‘𝐴) = (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘))))

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

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

Proof of Theorem hoidifhspval2

Step	Hyp	Ref	Expression
1		hoidifhspval2.d	. . 3 ⊢ 𝐷 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑥), (𝑎‘𝑘)))))
2		hoidifhspval2.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ ℝ)
3	1, 2	hoidifhspval 43247	. 2 ⊢ (𝜑 → (𝐷‘𝑌) = (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌), (𝑎‘𝑘)))))
4		fveq1 6644	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎‘𝑘) = (𝐴‘𝑘))
5	4	breq2d 5042	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑌 ≤ (𝑎‘𝑘) ↔ 𝑌 ≤ (𝐴‘𝑘)))
6	5, 4	ifbieq1d 4448	. . . . 5 ⊢ (𝑎 = 𝐴 → if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌) = if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌))
7	6, 4	ifeq12d 4445	. . . 4 ⊢ (𝑎 = 𝐴 → if(𝑘 = 𝐾, if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌), (𝑎‘𝑘)) = if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘)))
8	7	mpteq2dv 5126	. . 3 ⊢ (𝑎 = 𝐴 → (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌), (𝑎‘𝑘))) = (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘))))
9	8	adantl 485	. 2 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌), (𝑎‘𝑘))) = (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘))))
10		hoidifhspval2.a	. . 3 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
11		reex 10617	. . . . . 6 ⊢ ℝ ∈ V
12	11	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ V)
13		hoidifhspval2.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
14	12, 13	jca 515	. . . 4 ⊢ (𝜑 → (ℝ ∈ V ∧ 𝑋 ∈ 𝑉))
15		elmapg 8402	. . . 4 ⊢ ((ℝ ∈ V ∧ 𝑋 ∈ 𝑉) → (𝐴 ∈ (ℝ ↑m 𝑋) ↔ 𝐴:𝑋⟶ℝ))
16	14, 15	syl 17	. . 3 ⊢ (𝜑 → (𝐴 ∈ (ℝ ↑m 𝑋) ↔ 𝐴:𝑋⟶ℝ))
17	10, 16	mpbird 260	. 2 ⊢ (𝜑 → 𝐴 ∈ (ℝ ↑m 𝑋))
18		mptexg 6961	. . 3 ⊢ (𝑋 ∈ 𝑉 → (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘))) ∈ V)
19	13, 18	syl 17	. 2 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘))) ∈ V)
20	3, 9, 17, 19	fvmptd 6752	1 ⊢ (𝜑 → ((𝐷‘𝑌)‘𝐴) = (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3441  ifcif 4425   class class class wbr 5030   ↦ cmpt 5110  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ↑_m cmap 8389  ℝcr 10525   ≤ cle 10665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-map 8391

This theorem is referenced by:  hoidifhspf  43257  hoidifhspval3  43258