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Theorem hspval 45311
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 ↦ (𝑖𝑥, 𝑦 ∈ ℝ ↦ X𝑘𝑥 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)))
hspval.x (𝜑𝑋 ∈ Fin)
hspval.i (𝜑𝐼𝑋)
hspval.y (𝜑𝑌 ∈ ℝ)
Assertion
Ref Expression
hspval (𝜑 → (𝐼(𝐻𝑋)𝑌) = X𝑘𝑋 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
StepHypRef Expression
1 hspval.h . . 3 𝐻 = (𝑥 ∈ Fin ↦ (𝑖𝑥, 𝑦 ∈ ℝ ↦ X𝑘𝑥 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)))
2 id 22 . . . 4 (𝑥 = 𝑋𝑥 = 𝑋)
3 eqidd 2733 . . . 4 (𝑥 = 𝑋 → ℝ = ℝ)
4 ixpeq1 8898 . . . 4 (𝑥 = 𝑋X𝑘𝑥 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ) = X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ))
52, 3, 4mpoeq123dv 7480 . . 3 (𝑥 = 𝑋 → (𝑖𝑥, 𝑦 ∈ ℝ ↦ X𝑘𝑥 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)) = (𝑖𝑋, 𝑦 ∈ ℝ ↦ X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)))
6 hspval.x . . 3 (𝜑𝑋 ∈ Fin)
7 reex 11197 . . . . 5 ℝ ∈ V
87a1i 11 . . . 4 (𝜑 → ℝ ∈ V)
9 eqid 2732 . . . . 5 (𝑖𝑋, 𝑦 ∈ ℝ ↦ X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)) = (𝑖𝑋, 𝑦 ∈ ℝ ↦ X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ))
109mpoexg 8059 . . . 4 ((𝑋 ∈ Fin ∧ ℝ ∈ V) → (𝑖𝑋, 𝑦 ∈ ℝ ↦ X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)) ∈ V)
116, 8, 10syl2anc 584 . . 3 (𝜑 → (𝑖𝑋, 𝑦 ∈ ℝ ↦ X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)) ∈ V)
121, 5, 6, 11fvmptd3 7018 . 2 (𝜑 → (𝐻𝑋) = (𝑖𝑋, 𝑦 ∈ ℝ ↦ X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)))
13 simpl 483 . . . . . 6 ((𝑖 = 𝐼𝑦 = 𝑌) → 𝑖 = 𝐼)
1413eqeq2d 2743 . . . . 5 ((𝑖 = 𝐼𝑦 = 𝑌) → (𝑘 = 𝑖𝑘 = 𝐼))
15 simpr 485 . . . . . 6 ((𝑖 = 𝐼𝑦 = 𝑌) → 𝑦 = 𝑌)
1615oveq2d 7421 . . . . 5 ((𝑖 = 𝐼𝑦 = 𝑌) → (-∞(,)𝑦) = (-∞(,)𝑌))
1714, 16ifbieq1d 4551 . . . 4 ((𝑖 = 𝐼𝑦 = 𝑌) → if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ) = if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ))
1817ixpeq2dv 8903 . . 3 ((𝑖 = 𝐼𝑦 = 𝑌) → X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ) = X𝑘𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ))
1918adantl 482 . 2 ((𝜑 ∧ (𝑖 = 𝐼𝑦 = 𝑌)) → X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ) = X𝑘𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ))
20 hspval.i . 2 (𝜑𝐼𝑋)
21 hspval.y . 2 (𝜑𝑌 ∈ ℝ)
22 ovex 7438 . . . . . 6 (-∞(,)𝑌) ∈ V
2322, 7ifcli 4574 . . . . 5 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V
2423a1i 11 . . . 4 ((𝜑𝑘𝑋) → if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V)
2524ralrimiva 3146 . . 3 (𝜑 → ∀𝑘𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V)
26 ixpexg 8912 . . 3 (∀𝑘𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V → X𝑘𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V)
2725, 26syl 17 . 2 (𝜑X𝑘𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V)
2812, 19, 20, 21, 27ovmpod 7556 1 (𝜑 → (𝐼(𝐻𝑋)𝑌) = X𝑘𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3061  Vcvv 3474  ifcif 4527  cmpt 5230  cfv 6540  (class class class)co 7405  cmpo 7407  Xcixp 8887  Fincfn 8935  cr 11105  -∞cmnf 11242  (,)cioo 13320
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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-ixp 8888
This theorem is referenced by:  hspdifhsp  45318  hspmbllem2  45329  hspmbl  45331
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