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Mirrors > Home > MPE Home > Th. List > Mathboxes > hoicoto2 | Structured version Visualization version GIF version |
Description: The half-open interval expressed using a composition of a function into (ℝ × ℝ) and using two distinct real-valued functions for the borders. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
hoicoto2.i | ⊢ (𝜑 → 𝐼:𝑋⟶(ℝ × ℝ)) |
hoicoto2.a | ⊢ 𝐴 = (𝑘 ∈ 𝑋 ↦ (1st ‘(𝐼‘𝑘))) |
hoicoto2.b | ⊢ 𝐵 = (𝑘 ∈ 𝑋 ↦ (2nd ‘(𝐼‘𝑘))) |
Ref | Expression |
---|---|
hoicoto2 | ⊢ (𝜑 → X𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hoicoto2.i | . . . . 5 ⊢ (𝜑 → 𝐼:𝑋⟶(ℝ × ℝ)) | |
2 | 1 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐼:𝑋⟶(ℝ × ℝ)) |
3 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋) | |
4 | 2, 3 | fvovco 41447 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) = ((1st ‘(𝐼‘𝑘))[,)(2nd ‘(𝐼‘𝑘)))) |
5 | 1 | ffvelrnda 6846 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐼‘𝑘) ∈ (ℝ × ℝ)) |
6 | xp1st 7715 | . . . . . . . 8 ⊢ ((𝐼‘𝑘) ∈ (ℝ × ℝ) → (1st ‘(𝐼‘𝑘)) ∈ ℝ) | |
7 | 5, 6 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1st ‘(𝐼‘𝑘)) ∈ ℝ) |
8 | 7 | elexd 3515 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1st ‘(𝐼‘𝑘)) ∈ V) |
9 | hoicoto2.a | . . . . . . 7 ⊢ 𝐴 = (𝑘 ∈ 𝑋 ↦ (1st ‘(𝐼‘𝑘))) | |
10 | 9 | fvmpt2 6774 | . . . . . 6 ⊢ ((𝑘 ∈ 𝑋 ∧ (1st ‘(𝐼‘𝑘)) ∈ V) → (𝐴‘𝑘) = (1st ‘(𝐼‘𝑘))) |
11 | 3, 8, 10 | syl2anc 586 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) = (1st ‘(𝐼‘𝑘))) |
12 | 11 | eqcomd 2827 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1st ‘(𝐼‘𝑘)) = (𝐴‘𝑘)) |
13 | xp2nd 7716 | . . . . . . . 8 ⊢ ((𝐼‘𝑘) ∈ (ℝ × ℝ) → (2nd ‘(𝐼‘𝑘)) ∈ ℝ) | |
14 | 5, 13 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2nd ‘(𝐼‘𝑘)) ∈ ℝ) |
15 | 14 | elexd 3515 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2nd ‘(𝐼‘𝑘)) ∈ V) |
16 | hoicoto2.b | . . . . . . 7 ⊢ 𝐵 = (𝑘 ∈ 𝑋 ↦ (2nd ‘(𝐼‘𝑘))) | |
17 | 16 | fvmpt2 6774 | . . . . . 6 ⊢ ((𝑘 ∈ 𝑋 ∧ (2nd ‘(𝐼‘𝑘)) ∈ V) → (𝐵‘𝑘) = (2nd ‘(𝐼‘𝑘))) |
18 | 3, 15, 17 | syl2anc 586 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) = (2nd ‘(𝐼‘𝑘))) |
19 | 18 | eqcomd 2827 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2nd ‘(𝐼‘𝑘)) = (𝐵‘𝑘)) |
20 | 12, 19 | oveq12d 7168 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((1st ‘(𝐼‘𝑘))[,)(2nd ‘(𝐼‘𝑘))) = ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
21 | 4, 20 | eqtrd 2856 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) = ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
22 | 21 | ixpeq2dva 8470 | 1 ⊢ (𝜑 → X𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3495 ↦ cmpt 5139 × cxp 5548 ∘ ccom 5554 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 1st c1st 7681 2nd c2nd 7682 Xcixp 8455 ℝcr 10530 [,)cico 12734 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-fv 6358 df-ov 7153 df-1st 7683 df-2nd 7684 df-ixp 8456 |
This theorem is referenced by: opnvonmbllem2 42908 |
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