Theorem hspval 47445

Description: The value of the half-space of n-dimensional Real numbers. (Contributed by Glauco Siliprandi, 24-Dec-2020.)

Hypotheses

Ref	Expression
hspval.h	⊢ 𝐻 = (𝑥 ∈ Fin ↦ (𝑖 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑥 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)))
hspval.x	⊢ (𝜑 → 𝑋 ∈ Fin)
hspval.i	⊢ (𝜑 → 𝐼 ∈ 𝑋)
hspval.y	⊢ (𝜑 → 𝑌 ∈ ℝ)

Assertion

Ref	Expression
hspval	⊢ (𝜑 → (𝐼(𝐻‘𝑋)𝑌) = X𝑘 ∈ 𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ))

Distinct variable groups:   𝑖,𝐼,𝑘,𝑦   𝑖,𝑋,𝑘,𝑥,𝑦   𝑖,𝑌,𝑘,𝑦   𝜑,𝑖,𝑘,𝑥,𝑦

Allowed substitution hints:   𝐻(𝑥, 𝑦, 𝑖, 𝑘)   𝐼(𝑥)   𝑌(𝑥)

Proof of Theorem hspval

Step	Hyp	Ref	Expression
1		hspval.h	. . 3 ⊢ 𝐻 = (𝑥 ∈ Fin ↦ (𝑖 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑥 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)))
2		id 23	. . . 4 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
3		eqidd 2763	. . . 4 ⊢ (𝑥 = 𝑋 → ℝ = ℝ)
4		ixpeq1 8919	. . . 4 ⊢ (𝑥 = 𝑋 → X𝑘 ∈ 𝑥 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ) = X𝑘 ∈ 𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ))
5	2, 3, 4	mpoeq123dv 7492	. . 3 ⊢ (𝑥 = 𝑋 → (𝑖 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑥 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)) = (𝑖 ∈ 𝑋, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)))
6		hspval.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ Fin)
7		reex 11219	. . . . 5 ⊢ ℝ ∈ V
8	7	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ∈ V)
9		eqid 2762	. . . . 5 ⊢ (𝑖 ∈ 𝑋, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)) = (𝑖 ∈ 𝑋, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ))
10	9	mpoexg 8079	. . . 4 ⊢ ((𝑋 ∈ Fin ∧ ℝ ∈ V) → (𝑖 ∈ 𝑋, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)) ∈ V)
11	6, 8, 10	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑖 ∈ 𝑋, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)) ∈ V)
12	1, 5, 6, 11	fvmptd3 7014	. 2 ⊢ (𝜑 → (𝐻‘𝑋) = (𝑖 ∈ 𝑋, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)))
13		simpl 488	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑦 = 𝑌) → 𝑖 = 𝐼)
14	13	eqeq2d 2773	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑦 = 𝑌) → (𝑘 = 𝑖 ↔ 𝑘 = 𝐼))
15		simpr 490	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
16	15	oveq2d 7433	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑦 = 𝑌) → (-∞(,)𝑦) = (-∞(,)𝑌))
17	14, 16	ifbieq1d 4510	. . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑦 = 𝑌) → if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ) = if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ))
18	17	ixpeq2dv 8924	. . 3 ⊢ ((𝑖 = 𝐼 ∧ 𝑦 = 𝑌) → X𝑘 ∈ 𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ) = X𝑘 ∈ 𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ))
19	18	adantl 487	. 2 ⊢ ((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑦 = 𝑌)) → X𝑘 ∈ 𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ) = X𝑘 ∈ 𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ))
20		hspval.i	. 2 ⊢ (𝜑 → 𝐼 ∈ 𝑋)
21		hspval.y	. 2 ⊢ (𝜑 → 𝑌 ∈ ℝ)
22		ovex 7450	. . . . . 6 ⊢ (-∞(,)𝑌) ∈ V
23	22, 7	ifcli 4533	. . . . 5 ⊢ if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V
24	23	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V)
25	24	ralrimiva 3156	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V)
26		ixpexg 8933	. . 3 ⊢ (∀𝑘 ∈ 𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V → X𝑘 ∈ 𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V)
27	25, 26	syl 18	. 2 ⊢ (𝜑 → X𝑘 ∈ 𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V)
28	12, 19, 20, 21, 27	ovmpod 7569	1 ⊢ (𝜑 → (𝐼(𝐻‘𝑋)𝑌) = X𝑘 ∈ 𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  Vcvv 3453  ifcif 4485   ↦ cmpt 5190  ‘cfv 6537  (class class class)co 7417   ∈ cmpo 7419  Xcixp 8908  Fincfn 8956  ℝcr 11127  -∞cmnf 11269  (,)cioo 13402

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-1st 7990  df-2nd 7991  df-ixp 8909

This theorem is used by:  hspdifhsp  47452  hspmbllem2  47463  hspmbl  47465