Theorem hspval 42898

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 2824	. . . 4 ⊢ (𝑥 = 𝑋 → ℝ = ℝ)
4		ixpeq1 8474	. . . 4 ⊢ (𝑥 = 𝑋 → X𝑘 ∈ 𝑥 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ) = X𝑘 ∈ 𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ))
5	2, 3, 4	mpoeq123dv 7231	. . 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 10630	. . . . 5 ⊢ ℝ ∈ V
8	7	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ∈ V)
9		eqid 2823	. . . . 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 7776	. . . 4 ⊢ ((𝑋 ∈ Fin ∧ ℝ ∈ V) → (𝑖 ∈ 𝑋, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)) ∈ V)
11	6, 8, 10	syl2anc 586	. . 3 ⊢ (𝜑 → (𝑖 ∈ 𝑋, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)) ∈ V)
12	1, 5, 6, 11	fvmptd3 6793	. 2 ⊢ (𝜑 → (𝐻‘𝑋) = (𝑖 ∈ 𝑋, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)))
13		simpl 485	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑦 = 𝑌) → 𝑖 = 𝐼)
14	13	eqeq2d 2834	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑦 = 𝑌) → (𝑘 = 𝑖 ↔ 𝑘 = 𝐼))
15		simpr 487	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
16	15	oveq2d 7174	. . . . 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 8479	. . 3 ⊢ ((𝑖 = 𝐼 ∧ 𝑦 = 𝑌) → X𝑘 ∈ 𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ) = X𝑘 ∈ 𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ))
19	18	adantl 484	. 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 7191	. . . . . 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 3184	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V)
26		ixpexg 8488	. . 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 7304	1 ⊢ (𝜑 → (𝐼(𝐻‘𝑋)𝑌) = X𝑘 ∈ 𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  Vcvv 3496  ifcif 4469   ↦ cmpt 5148  ‘cfv 6357  (class class class)co 7158   ∈ cmpo 7160  Xcixp 8463  Fincfn 8511  ℝcr 10538  -∞cmnf 10675  (,)cioo 12741

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-1st 7691  df-2nd 7692  df-ixp 8464

This theorem is referenced by:  hspdifhsp  42905  hspmbllem2  42916  hspmbl  42918