Theorem hoidifhspval2 46727

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 46720	. 2 ⊢ (𝜑 → (𝐷‘𝑌) = (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌), (𝑎‘𝑘)))))
4		fveq1 6830	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎‘𝑘) = (𝐴‘𝑘))
5	4	breq2d 5107	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑌 ≤ (𝑎‘𝑘) ↔ 𝑌 ≤ (𝐴‘𝑘)))
6	5, 4	ifbieq1d 4501	. . . . 5 ⊢ (𝑎 = 𝐴 → if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌) = if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌))
7	6, 4	ifeq12d 4498	. . . 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 5189	. . 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 481	. 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 11107	. . . . . 6 ⊢ ℝ ∈ V
12	11	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ V)
13		hoidifhspval2.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
14	12, 13	jca 511	. . . 4 ⊢ (𝜑 → (ℝ ∈ V ∧ 𝑋 ∈ 𝑉))
15		elmapg 8772	. . . 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 257	. 2 ⊢ (𝜑 → 𝐴 ∈ (ℝ ↑m 𝑋))
18		mptexg 7164	. . 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 6945	1 ⊢ (𝜑 → ((𝐷‘𝑌)‘𝐴) = (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3438  ifcif 4476   class class class wbr 5095   ↦ cmpt 5176  ⟶wf 6485  ‘cfv 6489  (class class class)co 7355   ↑_m cmap 8759  ℝcr 11015   ≤ cle 11157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-cnex 11072  ax-resscn 11073

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-map 8761

This theorem is referenced by:  hoidifhspf  46730  hoidifhspval3  46731