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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hspval | Structured version Visualization version GIF version |
Description: The value of the half-space of n-dimensional Real numbers. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
hspval.h | ⊢ 𝐻 = (𝑥 ∈ Fin ↦ (𝑖 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑥 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ))) |
hspval.x | ⊢ (𝜑 → 𝑋 ∈ Fin) |
hspval.i | ⊢ (𝜑 → 𝐼 ∈ 𝑋) |
hspval.y | ⊢ (𝜑 → 𝑌 ∈ ℝ) |
Ref | Expression |
---|---|
hspval | ⊢ (𝜑 → (𝐼(𝐻‘𝑋)𝑌) = X𝑘 ∈ 𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hspval.h | . . 3 ⊢ 𝐻 = (𝑥 ∈ Fin ↦ (𝑖 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑥 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ))) | |
2 | id 22 | . . . 4 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋) | |
3 | eqidd 2727 | . . . 4 ⊢ (𝑥 = 𝑋 → ℝ = ℝ) | |
4 | ixpeq1 8901 | . . . 4 ⊢ (𝑥 = 𝑋 → X𝑘 ∈ 𝑥 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ) = X𝑘 ∈ 𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)) | |
5 | 2, 3, 4 | mpoeq123dv 7479 | . . 3 ⊢ (𝑥 = 𝑋 → (𝑖 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑥 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)) = (𝑖 ∈ 𝑋, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ))) |
6 | hspval.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
7 | reex 11200 | . . . . 5 ⊢ ℝ ∈ V | |
8 | 7 | a1i 11 | . . . 4 ⊢ (𝜑 → ℝ ∈ V) |
9 | eqid 2726 | . . . . 5 ⊢ (𝑖 ∈ 𝑋, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)) = (𝑖 ∈ 𝑋, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)) | |
10 | 9 | mpoexg 8059 | . . . 4 ⊢ ((𝑋 ∈ Fin ∧ ℝ ∈ V) → (𝑖 ∈ 𝑋, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)) ∈ V) |
11 | 6, 8, 10 | syl2anc 583 | . . 3 ⊢ (𝜑 → (𝑖 ∈ 𝑋, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)) ∈ V) |
12 | 1, 5, 6, 11 | fvmptd3 7014 | . 2 ⊢ (𝜑 → (𝐻‘𝑋) = (𝑖 ∈ 𝑋, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ))) |
13 | simpl 482 | . . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑦 = 𝑌) → 𝑖 = 𝐼) | |
14 | 13 | eqeq2d 2737 | . . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑦 = 𝑌) → (𝑘 = 𝑖 ↔ 𝑘 = 𝐼)) |
15 | simpr 484 | . . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌) | |
16 | 15 | oveq2d 7420 | . . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑦 = 𝑌) → (-∞(,)𝑦) = (-∞(,)𝑌)) |
17 | 14, 16 | ifbieq1d 4547 | . . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑦 = 𝑌) → if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ) = if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ)) |
18 | 17 | ixpeq2dv 8906 | . . 3 ⊢ ((𝑖 = 𝐼 ∧ 𝑦 = 𝑌) → X𝑘 ∈ 𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ) = X𝑘 ∈ 𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ)) |
19 | 18 | adantl 481 | . 2 ⊢ ((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑦 = 𝑌)) → X𝑘 ∈ 𝑋 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ) = X𝑘 ∈ 𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ)) |
20 | hspval.i | . 2 ⊢ (𝜑 → 𝐼 ∈ 𝑋) | |
21 | hspval.y | . 2 ⊢ (𝜑 → 𝑌 ∈ ℝ) | |
22 | ovex 7437 | . . . . . 6 ⊢ (-∞(,)𝑌) ∈ V | |
23 | 22, 7 | ifcli 4570 | . . . . 5 ⊢ if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V |
24 | 23 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V) |
25 | 24 | ralrimiva 3140 | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V) |
26 | ixpexg 8915 | . . 3 ⊢ (∀𝑘 ∈ 𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V → X𝑘 ∈ 𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V) | |
27 | 25, 26 | syl 17 | . 2 ⊢ (𝜑 → X𝑘 ∈ 𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ) ∈ V) |
28 | 12, 19, 20, 21, 27 | ovmpod 7555 | 1 ⊢ (𝜑 → (𝐼(𝐻‘𝑋)𝑌) = X𝑘 ∈ 𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3055 Vcvv 3468 ifcif 4523 ↦ cmpt 5224 ‘cfv 6536 (class class class)co 7404 ∈ cmpo 7406 Xcixp 8890 Fincfn 8938 ℝcr 11108 -∞cmnf 11247 (,)cioo 13327 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-ixp 8891 |
This theorem is referenced by: hspdifhsp 45885 hspmbllem2 45896 hspmbl 45898 |
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