Theorem hoidifhspval3 43699

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 43695	. 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 6674	. . . . . 6 ⊢ (𝑘 = 𝐽 → (𝐴‘𝑘) = (𝐴‘𝐽))
8	7	breq2d 5042	. . . . 5 ⊢ (𝑘 = 𝐽 → (𝑌 ≤ (𝐴‘𝑘) ↔ 𝑌 ≤ (𝐴‘𝐽)))
9	8, 7	ifbieq1d 4438	. . . 4 ⊢ (𝑘 = 𝐽 → if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌) = if(𝑌 ≤ (𝐴‘𝐽), (𝐴‘𝐽), 𝑌))
10	6, 9, 7	ifbieq12d 4442	. . 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 485	. 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 6689	. . . 4 ⊢ (𝜑 → (𝐴‘𝐽) ∈ V)
14	2	elexd 3418	. . . 4 ⊢ (𝜑 → 𝑌 ∈ V)
15	13, 14	ifcld 4460	. . 3 ⊢ (𝜑 → if(𝑌 ≤ (𝐴‘𝐽), (𝐴‘𝐽), 𝑌) ∈ V)
16	15, 13	ifcld 4460	. 2 ⊢ (𝜑 → if(𝐽 = 𝐾, if(𝑌 ≤ (𝐴‘𝐽), (𝐴‘𝐽), 𝑌), (𝐴‘𝐽)) ∈ V)
17	5, 11, 12, 16	fvmptd 6782	1 ⊢ (𝜑 → (((𝐷‘𝑌)‘𝐴)‘𝐽) = if(𝐽 = 𝐾, if(𝑌 ≤ (𝐴‘𝐽), (𝐴‘𝐽), 𝑌), (𝐴‘𝐽)))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3398  ifcif 4414   class class class wbr 5030   ↦ cmpt 5110  ⟶wf 6335  ‘cfv 6339  (class class class)co 7170   ↑_m cmap 8437  ℝcr 10614   ≤ cle 10754

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-cnex 10671  ax-resscn 10672

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-map 8439

This theorem is referenced by:  hoidifhspdmvle  43700  hspmbllem1  43706  hspmbllem2  43707