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Theorem hspval 40586
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:   𝑥,𝑘   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 . . . 4 𝐻 = (𝑥 ∈ Fin ↦ (𝑖𝑥, 𝑦 ∈ ℝ ↦ X𝑘𝑥 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)))
21a1i 11 . . 3 (𝜑𝐻 = (𝑥 ∈ Fin ↦ (𝑖𝑥, 𝑦 ∈ ℝ ↦ X𝑘𝑥 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ))))
3 id 22 . . . . 5 (𝑥 = 𝑋𝑥 = 𝑋)
4 eqidd 2621 . . . . 5 (𝑥 = 𝑋 → ℝ = ℝ)
5 ixpeq1 7904 . . . . 5 (𝑥 = 𝑋X𝑘𝑥 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ) = X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ))
63, 4, 5mpt2eq123dv 6702 . . . 4 (𝑥 = 𝑋 → (𝑖𝑥, 𝑦 ∈ ℝ ↦ X𝑘𝑥 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)) = (𝑖𝑋, 𝑦 ∈ ℝ ↦ X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)))
76adantl 482 . . 3 ((𝜑𝑥 = 𝑋) → (𝑖𝑥, 𝑦 ∈ ℝ ↦ X𝑘𝑥 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)) = (𝑖𝑋, 𝑦 ∈ ℝ ↦ X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)))
8 hspval.x . . 3 (𝜑𝑋 ∈ Fin)
9 reex 10012 . . . . 5 ℝ ∈ V
109a1i 11 . . . 4 (𝜑 → ℝ ∈ V)
11 eqid 2620 . . . . 5 (𝑖𝑋, 𝑦 ∈ ℝ ↦ X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)) = (𝑖𝑋, 𝑦 ∈ ℝ ↦ X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ))
1211mpt2exg 7230 . . . 4 ((𝑋 ∈ Fin ∧ ℝ ∈ V) → (𝑖𝑋, 𝑦 ∈ ℝ ↦ X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)) ∈ V)
138, 10, 12syl2anc 692 . . 3 (𝜑 → (𝑖𝑋, 𝑦 ∈ ℝ ↦ X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)) ∈ V)
142, 7, 8, 13fvmptd 6275 . 2 (𝜑 → (𝐻𝑋) = (𝑖𝑋, 𝑦 ∈ ℝ ↦ X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)))
15 simpl 473 . . . . . 6 ((𝑖 = 𝐼𝑦 = 𝑌) → 𝑖 = 𝐼)
1615eqeq2d 2630 . . . . 5 ((𝑖 = 𝐼𝑦 = 𝑌) → (𝑘 = 𝑖𝑘 = 𝐼))
17 simpr 477 . . . . . 6 ((𝑖 = 𝐼𝑦 = 𝑌) → 𝑦 = 𝑌)
1817oveq2d 6651 . . . . 5 ((𝑖 = 𝐼𝑦 = 𝑌) → (-∞(,)𝑦) = (-∞(,)𝑌))
1916, 18ifbieq1d 4100 . . . 4 ((𝑖 = 𝐼𝑦 = 𝑌) → if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ) = if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ))
2019ixpeq2dv 7909 . . 3 ((𝑖 = 𝐼𝑦 = 𝑌) → X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ) = X𝑘𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ))
2120adantl 482 . 2 ((𝜑 ∧ (𝑖 = 𝐼𝑦 = 𝑌)) → X𝑘𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ) = X𝑘𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ))
22 hspval.i . 2 (𝜑𝐼𝑋)
23 hspval.y . 2 (𝜑𝑌 ∈ ℝ)
24 ovex 6663 . . . . . 6 (-∞(,)𝑌) ∈ V
2524, 9keepel 4146 . . . . 5 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V
2625a1i 11 . . . 4 ((𝜑𝑘𝑋) → if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V)
2726ralrimiva 2963 . . 3 (𝜑 → ∀𝑘𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V)
28 ixpexg 7917 . . 3 (∀𝑘𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V → X𝑘𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V)
2927, 28syl 17 . 2 (𝜑X𝑘𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V)
3014, 21, 22, 23, 29ovmpt2d 6773 1 (𝜑 → (𝐼(𝐻𝑋)𝑌) = X𝑘𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  wral 2909  Vcvv 3195  ifcif 4077  cmpt 4720  cfv 5876  (class class class)co 6635  cmpt2 6637  Xcixp 7893  Fincfn 7940  cr 9920  -∞cmnf 10057  (,)cioo 12160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-ixp 7894
This theorem is referenced by:  hspdifhsp  40593  hspmbllem2  40604  hspmbl  40606
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