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Theorem hspval 43179
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 2825 . . . 4 (𝑥 = 𝑋 → ℝ = ℝ)
4 ixpeq1 8468 . . . 4 (𝑥 = 𝑋X𝑘𝑥 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ) = X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ))
52, 3, 4mpoeq123dv 7222 . . 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 10626 . . . . 5 ℝ ∈ V
87a1i 11 . . . 4 (𝜑 → ℝ ∈ V)
9 eqid 2824 . . . . 5 (𝑖𝑋, 𝑦 ∈ ℝ ↦ X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)) = (𝑖𝑋, 𝑦 ∈ ℝ ↦ X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ))
109mpoexg 7770 . . . 4 ((𝑋 ∈ Fin ∧ ℝ ∈ V) → (𝑖𝑋, 𝑦 ∈ ℝ ↦ X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)) ∈ V)
116, 8, 10syl2anc 587 . . 3 (𝜑 → (𝑖𝑋, 𝑦 ∈ ℝ ↦ X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)) ∈ V)
121, 5, 6, 11fvmptd3 6782 . 2 (𝜑 → (𝐻𝑋) = (𝑖𝑋, 𝑦 ∈ ℝ ↦ X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)))
13 simpl 486 . . . . . 6 ((𝑖 = 𝐼𝑦 = 𝑌) → 𝑖 = 𝐼)
1413eqeq2d 2835 . . . . 5 ((𝑖 = 𝐼𝑦 = 𝑌) → (𝑘 = 𝑖𝑘 = 𝐼))
15 simpr 488 . . . . . 6 ((𝑖 = 𝐼𝑦 = 𝑌) → 𝑦 = 𝑌)
1615oveq2d 7165 . . . . 5 ((𝑖 = 𝐼𝑦 = 𝑌) → (-∞(,)𝑦) = (-∞(,)𝑌))
1714, 16ifbieq1d 4473 . . . 4 ((𝑖 = 𝐼𝑦 = 𝑌) → if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ) = if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ))
1817ixpeq2dv 8473 . . 3 ((𝑖 = 𝐼𝑦 = 𝑌) → X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ) = X𝑘𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ))
1918adantl 485 . 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 7182 . . . . . 6 (-∞(,)𝑌) ∈ V
2322, 7ifcli 4496 . . . . 5 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V
2423a1i 11 . . . 4 ((𝜑𝑘𝑋) → if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V)
2524ralrimiva 3177 . . 3 (𝜑 → ∀𝑘𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V)
26 ixpexg 8482 . . 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 7295 1 (𝜑 → (𝐼(𝐻𝑋)𝑌) = X𝑘𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wral 3133  Vcvv 3480  ifcif 4450  cmpt 5132  cfv 6343  (class class class)co 7149  cmpo 7151  Xcixp 8457  Fincfn 8505  cr 10534  -∞cmnf 10671  (,)cioo 12735
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-ixp 8458
This theorem is referenced by:  hspdifhsp  43186  hspmbllem2  43197  hspmbl  43199
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