Theorem hspval 47055

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 22	. . . 4 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
3		eqidd 2738	. . . 4 ⊢ (𝑥 = 𝑋 → ℝ = ℝ)
4		ixpeq1 8849	. . . 4 ⊢ (𝑥 = 𝑋 → X𝑘 ∈ 𝑥 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ) = X𝑘 ∈ 𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ))
5	2, 3, 4	mpoeq123dv 7435	. . 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 11120	. . . . 5 ⊢ ℝ ∈ V
8	7	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ∈ V)
9		eqid 2737	. . . . 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 8022	. . . 4 ⊢ ((𝑋 ∈ Fin ∧ ℝ ∈ V) → (𝑖 ∈ 𝑋, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)) ∈ V)
11	6, 8, 10	syl2anc 585	. . 3 ⊢ (𝜑 → (𝑖 ∈ 𝑋, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)) ∈ V)
12	1, 5, 6, 11	fvmptd3 6965	. 2 ⊢ (𝜑 → (𝐻‘𝑋) = (𝑖 ∈ 𝑋, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)))
13		simpl 482	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑦 = 𝑌) → 𝑖 = 𝐼)
14	13	eqeq2d 2748	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑦 = 𝑌) → (𝑘 = 𝑖 ↔ 𝑘 = 𝐼))
15		simpr 484	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
16	15	oveq2d 7376	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑦 = 𝑌) → (-∞(,)𝑦) = (-∞(,)𝑌))
17	14, 16	ifbieq1d 4492	. . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑦 = 𝑌) → if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ) = if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ))
18	17	ixpeq2dv 8854	. . 3 ⊢ ((𝑖 = 𝐼 ∧ 𝑦 = 𝑌) → X𝑘 ∈ 𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ) = X𝑘 ∈ 𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ))
19	18	adantl 481	. 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 7393	. . . . . 6 ⊢ (-∞(,)𝑌) ∈ V
23	22, 7	ifcli 4515	. . . . 5 ⊢ if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V
24	23	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V)
25	24	ralrimiva 3130	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V)
26		ixpexg 8863	. . 3 ⊢ (∀𝑘 ∈ 𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V → X𝑘 ∈ 𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V)
27	25, 26	syl 17	. 2 ⊢ (𝜑 → X𝑘 ∈ 𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V)
28	12, 19, 20, 21, 27	ovmpod 7512	1 ⊢ (𝜑 → (𝐼(𝐻‘𝑋)𝑌) = X𝑘 ∈ 𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3430  ifcif 4467   ↦ cmpt 5167  ‘cfv 6492  (class class class)co 7360   ∈ cmpo 7362  Xcixp 8838  Fincfn 8886  ℝcr 11028  -∞cmnf 11168  (,)cioo 13289

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-ixp 8839

This theorem is referenced by:  hspdifhsp  47062  hspmbllem2  47073  hspmbl  47075