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) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑥), (𝑎‘𝑘)))))
hoidifhspval3.y	⊢ (𝜑 → 𝑌 ∈ ℝ)
hoidifhspval3.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
hoidifhspval3.a	⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
hoidifhspval3.j	⊢ (𝜑 → 𝐽 ∈ 𝑋)

Assertion

Ref	Expression
hoidifhspval3	⊢ (𝜑 → (((𝐷‘𝑌)‘𝐴)‘𝐽) = if(𝐽 = 𝐾, if(𝑌 ≤ (𝐴‘𝐽), (𝐴‘𝐽), 𝑌), (𝐴‘𝐽)))

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

Step	Hyp	Ref	Expression
1		hoidifhspval3.d	. . 3 ⊢ 𝐷 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑥), (𝑎‘𝑘)))))
2		hoidifhspval3.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ ℝ)
3		hoidifhspval3.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
4		hoidifhspval3.a	. . 3 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
5	1, 2, 3, 4	hoidifhspval2 41335	. 2 ⊢ (𝜑 → ((𝐷‘𝑌)‘𝐴) = (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘))))
6		eqeq1 2764	. . . 4 ⊢ (𝑘 = 𝐽 → (𝑘 = 𝐾 ↔ 𝐽 = 𝐾))
7		fveq2 6352	. . . . . 6 ⊢ (𝑘 = 𝐽 → (𝐴‘𝑘) = (𝐴‘𝐽))
8	7	breq2d 4816	. . . . 5 ⊢ (𝑘 = 𝐽 → (𝑌 ≤ (𝐴‘𝑘) ↔ 𝑌 ≤ (𝐴‘𝐽)))
9	8, 7	ifbieq1d 4253	. . . 4 ⊢ (𝑘 = 𝐽 → if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌) = if(𝑌 ≤ (𝐴‘𝐽), (𝐴‘𝐽), 𝑌))
10	6, 9, 7	ifbieq12d 4257	. . 3 ⊢ (𝑘 = 𝐽 → if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘)) = if(𝐽 = 𝐾, if(𝑌 ≤ (𝐴‘𝐽), (𝐴‘𝐽), 𝑌), (𝐴‘𝐽)))
11	10	adantl 473	. 2 ⊢ ((𝜑 ∧ 𝑘 = 𝐽) → if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘)) = if(𝐽 = 𝐾, if(𝑌 ≤ (𝐴‘𝐽), (𝐴‘𝐽), 𝑌), (𝐴‘𝐽)))
12		hoidifhspval3.j	. 2 ⊢ (𝜑 → 𝐽 ∈ 𝑋)
13		fvexd 6364	. . . 4 ⊢ (𝜑 → (𝐴‘𝐽) ∈ V)
14	2	elexd 3354	. . . 4 ⊢ (𝜑 → 𝑌 ∈ V)
15	13, 14	ifcld 4275	. . 3 ⊢ (𝜑 → if(𝑌 ≤ (𝐴‘𝐽), (𝐴‘𝐽), 𝑌) ∈ V)
16	15, 13	ifcld 4275	. 2 ⊢ (𝜑 → if(𝐽 = 𝐾, if(𝑌 ≤ (𝐴‘𝐽), (𝐴‘𝐽), 𝑌), (𝐴‘𝐽)) ∈ V)
17	5, 11, 12, 16	fvmptd 6450	1 ⊢ (𝜑 → (((𝐷‘𝑌)‘𝐴)‘𝐽) = if(𝐽 = 𝐾, if(𝑌 ≤ (𝐴‘𝐽), (𝐴‘𝐽), 𝑌), (𝐴‘𝐽)))

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  ℝcr 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