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Theorem hspval 41750
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 2779 . . . 4 (𝑥 = 𝑋 → ℝ = ℝ)
4 ixpeq1 8205 . . . 4 (𝑥 = 𝑋X𝑘𝑥 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ) = X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ))
52, 3, 4mpt2eq123dv 6994 . . 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 10363 . . . . 5 ℝ ∈ V
87a1i 11 . . . 4 (𝜑 → ℝ ∈ V)
9 eqid 2778 . . . . 5 (𝑖𝑋, 𝑦 ∈ ℝ ↦ X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)) = (𝑖𝑋, 𝑦 ∈ ℝ ↦ X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ))
109mpt2exg 7525 . . . 4 ((𝑋 ∈ Fin ∧ ℝ ∈ V) → (𝑖𝑋, 𝑦 ∈ ℝ ↦ X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)) ∈ V)
116, 8, 10syl2anc 579 . . 3 (𝜑 → (𝑖𝑋, 𝑦 ∈ ℝ ↦ X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)) ∈ V)
121, 5, 6, 11fvmptd3 6564 . 2 (𝜑 → (𝐻𝑋) = (𝑖𝑋, 𝑦 ∈ ℝ ↦ X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)))
13 simpl 476 . . . . . 6 ((𝑖 = 𝐼𝑦 = 𝑌) → 𝑖 = 𝐼)
1413eqeq2d 2788 . . . . 5 ((𝑖 = 𝐼𝑦 = 𝑌) → (𝑘 = 𝑖𝑘 = 𝐼))
15 simpr 479 . . . . . 6 ((𝑖 = 𝐼𝑦 = 𝑌) → 𝑦 = 𝑌)
1615oveq2d 6938 . . . . 5 ((𝑖 = 𝐼𝑦 = 𝑌) → (-∞(,)𝑦) = (-∞(,)𝑌))
1714, 16ifbieq1d 4330 . . . 4 ((𝑖 = 𝐼𝑦 = 𝑌) → if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ) = if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ))
1817ixpeq2dv 8210 . . 3 ((𝑖 = 𝐼𝑦 = 𝑌) → X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ) = X𝑘𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ))
1918adantl 475 . 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 6954 . . . . . 6 (-∞(,)𝑌) ∈ V
2322, 7ifcli 4353 . . . . 5 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V
2423a1i 11 . . . 4 ((𝜑𝑘𝑋) → if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V)
2524ralrimiva 3148 . . 3 (𝜑 → ∀𝑘𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V)
26 ixpexg 8218 . . 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, 27ovmpt2d 7065 1 (𝜑 → (𝐼(𝐻𝑋)𝑌) = X𝑘𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2107  wral 3090  Vcvv 3398  ifcif 4307  cmpt 4965  cfv 6135  (class class class)co 6922  cmpt2 6924  Xcixp 8194  Fincfn 8241  cr 10271  -∞cmnf 10409  (,)cioo 12487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446  df-ixp 8195
This theorem is referenced by:  hspdifhsp  41757  hspmbllem2  41768  hspmbl  41770
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