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Theorem hspval 46565
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 2736 . . . 4 (𝑥 = 𝑋 → ℝ = ℝ)
4 ixpeq1 8947 . . . 4 (𝑥 = 𝑋X𝑘𝑥 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ) = X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ))
52, 3, 4mpoeq123dv 7508 . . 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 11244 . . . . 5 ℝ ∈ V
87a1i 11 . . . 4 (𝜑 → ℝ ∈ V)
9 eqid 2735 . . . . 5 (𝑖𝑋, 𝑦 ∈ ℝ ↦ X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)) = (𝑖𝑋, 𝑦 ∈ ℝ ↦ X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ))
109mpoexg 8100 . . . 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 7039 . 2 (𝜑 → (𝐻𝑋) = (𝑖𝑋, 𝑦 ∈ ℝ ↦ X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)))
13 simpl 482 . . . . . 6 ((𝑖 = 𝐼𝑦 = 𝑌) → 𝑖 = 𝐼)
1413eqeq2d 2746 . . . . 5 ((𝑖 = 𝐼𝑦 = 𝑌) → (𝑘 = 𝑖𝑘 = 𝐼))
15 simpr 484 . . . . . 6 ((𝑖 = 𝐼𝑦 = 𝑌) → 𝑦 = 𝑌)
1615oveq2d 7447 . . . . 5 ((𝑖 = 𝐼𝑦 = 𝑌) → (-∞(,)𝑦) = (-∞(,)𝑌))
1714, 16ifbieq1d 4555 . . . 4 ((𝑖 = 𝐼𝑦 = 𝑌) → if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ) = if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ))
1817ixpeq2dv 8952 . . 3 ((𝑖 = 𝐼𝑦 = 𝑌) → X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ) = X𝑘𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ))
1918adantl 481 . 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 7464 . . . . . 6 (-∞(,)𝑌) ∈ V
2322, 7ifcli 4578 . . . . 5 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V
2423a1i 11 . . . 4 ((𝜑𝑘𝑋) → if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V)
2524ralrimiva 3144 . . 3 (𝜑 → ∀𝑘𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V)
26 ixpexg 8961 . . 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 7585 1 (𝜑 → (𝐼(𝐻𝑋)𝑌) = X𝑘𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  ifcif 4531  cmpt 5231  cfv 6563  (class class class)co 7431  cmpo 7433  Xcixp 8936  Fincfn 8984  cr 11152  -∞cmnf 11291  (,)cioo 13384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-ixp 8937
This theorem is referenced by:  hspdifhsp  46572  hspmbllem2  46583  hspmbl  46585
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