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Mirrors > Home > MPE Home > Th. List > elicore | Structured version Visualization version GIF version |
Description: A member of a left-closed right-open interval of reals is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
elicore | ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ico 13279 | . . . . . . 7 ⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | |
2 | 1 | elixx3g 13286 | . . . . . 6 ⊢ (𝐶 ∈ (𝐴[,)𝐵) ↔ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))) |
3 | 2 | biimpi 215 | . . . . 5 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))) |
4 | 3 | simpld 496 | . . . 4 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*)) |
5 | 4 | simp3d 1145 | . . 3 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐶 ∈ ℝ*) |
6 | 5 | adantl 483 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 ∈ ℝ*) |
7 | simpl 484 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐴 ∈ ℝ) | |
8 | 3 | simprd 497 | . . . 4 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → (𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) |
9 | 8 | simpld 496 | . . 3 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐴 ≤ 𝐶) |
10 | 9 | adantl 483 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐴 ≤ 𝐶) |
11 | 4 | simp2d 1144 | . . . 4 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐵 ∈ ℝ*) |
12 | 11 | adantl 483 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐵 ∈ ℝ*) |
13 | pnfxr 11217 | . . . 4 ⊢ +∞ ∈ ℝ* | |
14 | 13 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → +∞ ∈ ℝ*) |
15 | 8 | simprd 497 | . . . 4 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐶 < 𝐵) |
16 | 15 | adantl 483 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 < 𝐵) |
17 | pnfge 13059 | . . . . 5 ⊢ (𝐵 ∈ ℝ* → 𝐵 ≤ +∞) | |
18 | 11, 17 | syl 17 | . . . 4 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐵 ≤ +∞) |
19 | 18 | adantl 483 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐵 ≤ +∞) |
20 | 6, 12, 14, 16, 19 | xrltletrd 13089 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 < +∞) |
21 | xrre3 13099 | . 2 ⊢ (((𝐶 ∈ ℝ* ∧ 𝐴 ∈ ℝ) ∧ (𝐴 ≤ 𝐶 ∧ 𝐶 < +∞)) → 𝐶 ∈ ℝ) | |
22 | 6, 7, 10, 20, 21 | syl22anc 838 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 ∈ wcel 2107 class class class wbr 5109 (class class class)co 7361 ℝcr 11058 +∞cpnf 11194 ℝ*cxr 11196 < clt 11197 ≤ cle 11198 [,)cico 13275 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-pre-lttri 11133 ax-pre-lttrn 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-po 5549 df-so 5550 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-1st 7925 df-2nd 7926 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-ico 13279 |
This theorem is referenced by: relowlpssretop 35885 limsupresico 44031 liminfresico 44102 fourierdlem43 44481 |
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