Theorem hoidifhspval3 44157

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 44153	. 2 ⊢ (𝜑 → ((𝐷‘𝑌)‘𝐴) = (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘))))
6		eqeq1 2742	. . . 4 ⊢ (𝑘 = 𝐽 → (𝑘 = 𝐾 ↔ 𝐽 = 𝐾))
7		fveq2 6774	. . . . . 6 ⊢ (𝑘 = 𝐽 → (𝐴‘𝑘) = (𝐴‘𝐽))
8	7	breq2d 5086	. . . . 5 ⊢ (𝑘 = 𝐽 → (𝑌 ≤ (𝐴‘𝑘) ↔ 𝑌 ≤ (𝐴‘𝐽)))
9	8, 7	ifbieq1d 4483	. . . 4 ⊢ (𝑘 = 𝐽 → if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌) = if(𝑌 ≤ (𝐴‘𝐽), (𝐴‘𝐽), 𝑌))
10	6, 9, 7	ifbieq12d 4487	. . 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 482	. 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 6789	. . . 4 ⊢ (𝜑 → (𝐴‘𝐽) ∈ V)
14	2	elexd 3452	. . . 4 ⊢ (𝜑 → 𝑌 ∈ V)
15	13, 14	ifcld 4505	. . 3 ⊢ (𝜑 → if(𝑌 ≤ (𝐴‘𝐽), (𝐴‘𝐽), 𝑌) ∈ V)
16	15, 13	ifcld 4505	. 2 ⊢ (𝜑 → if(𝐽 = 𝐾, if(𝑌 ≤ (𝐴‘𝐽), (𝐴‘𝐽), 𝑌), (𝐴‘𝐽)) ∈ V)
17	5, 11, 12, 16	fvmptd 6882	1 ⊢ (𝜑 → (((𝐷‘𝑌)‘𝐴)‘𝐽) = if(𝐽 = 𝐾, if(𝑌 ≤ (𝐴‘𝐽), (𝐴‘𝐽), 𝑌), (𝐴‘𝐽)))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ifcif 4459   class class class wbr 5074   ↦ cmpt 5157  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ↑_m cmap 8615  ℝcr 10870   ≤ cle 11010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617

This theorem is referenced by:  hoidifhspdmvle  44158  hspmbllem1  44164  hspmbllem2  44165