Theorem hspval 44840

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 2737	. . . 4 ⊢ (𝑥 = 𝑋 → ℝ = ℝ)
4		ixpeq1 8846	. . . 4 ⊢ (𝑥 = 𝑋 → X𝑘 ∈ 𝑥 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ) = X𝑘 ∈ 𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ))
5	2, 3, 4	mpoeq123dv 7432	. . 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 11142	. . . . 5 ⊢ ℝ ∈ V
8	7	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ∈ V)
9		eqid 2736	. . . . 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 8009	. . . 4 ⊢ ((𝑋 ∈ Fin ∧ ℝ ∈ V) → (𝑖 ∈ 𝑋, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)) ∈ V)
11	6, 8, 10	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑖 ∈ 𝑋, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)) ∈ V)
12	1, 5, 6, 11	fvmptd3 6971	. 2 ⊢ (𝜑 → (𝐻‘𝑋) = (𝑖 ∈ 𝑋, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)))
13		simpl 483	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑦 = 𝑌) → 𝑖 = 𝐼)
14	13	eqeq2d 2747	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑦 = 𝑌) → (𝑘 = 𝑖 ↔ 𝑘 = 𝐼))
15		simpr 485	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
16	15	oveq2d 7373	. . . . 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 8851	. . 3 ⊢ ((𝑖 = 𝐼 ∧ 𝑦 = 𝑌) → X𝑘 ∈ 𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ) = X𝑘 ∈ 𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ))
19	18	adantl 482	. 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 7390	. . . . . 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 3143	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V)
26		ixpexg 8860	. . 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 7507	1 ⊢ (𝜑 → (𝐼(𝐻‘𝑋)𝑌) = X𝑘 ∈ 𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3064  Vcvv 3445  ifcif 4486   ↦ cmpt 5188  ‘cfv 6496  (class class class)co 7357   ∈ cmpo 7359  Xcixp 8835  Fincfn 8883  ℝcr 11050  -∞cmnf 11187  (,)cioo 13264

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-ixp 8836

This theorem is referenced by:  hspdifhsp  44847  hspmbllem2  44858  hspmbl  44860