Theorem hoidifhspval2 46141

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 46134	. 2 ⊢ (𝜑 → (𝐷‘𝑌) = (𝑎 ∈ (ℝ ↑m 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌), (𝑎‘𝑘)))))
4		fveq1 6895	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎‘𝑘) = (𝐴‘𝑘))
5	4	breq2d 5161	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑌 ≤ (𝑎‘𝑘) ↔ 𝑌 ≤ (𝐴‘𝑘)))
6	5, 4	ifbieq1d 4554	. . . . 5 ⊢ (𝑎 = 𝐴 → if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌) = if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌))
7	6, 4	ifeq12d 4551	. . . 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 5251	. . 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 480	. 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 11231	. . . . . 6 ⊢ ℝ ∈ V
12	11	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ V)
13		hoidifhspval2.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
14	12, 13	jca 510	. . . 4 ⊢ (𝜑 → (ℝ ∈ V ∧ 𝑋 ∈ 𝑉))
15		elmapg 8858	. . . 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 256	. 2 ⊢ (𝜑 → 𝐴 ∈ (ℝ ↑m 𝑋))
18		mptexg 7233	. . 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 7011	1 ⊢ (𝜑 → ((𝐷‘𝑌)‘𝐴) = (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3461  ifcif 4530   class class class wbr 5149   ↦ cmpt 5232  ⟶wf 6545  ‘cfv 6549  (class class class)co 7419   ↑_m cmap 8845  ℝcr 11139   ≤ cle 11281

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-map 8847

This theorem is referenced by:  hoidifhspf  46144  hoidifhspval3  46145