Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 12738 | . . . . . . 7 ⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | |
2 | 1 | elixx3g 12745 | . . . . . 6 ⊢ (𝐶 ∈ (𝐴[,)𝐵) ↔ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))) |
3 | 2 | biimpi 218 | . . . . 5 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))) |
4 | 3 | simpld 497 | . . . 4 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*)) |
5 | 4 | simp3d 1140 | . . 3 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐶 ∈ ℝ*) |
6 | 5 | adantl 484 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 ∈ ℝ*) |
7 | simpl 485 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐴 ∈ ℝ) | |
8 | 3 | simprd 498 | . . . 4 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → (𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) |
9 | 8 | simpld 497 | . . 3 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐴 ≤ 𝐶) |
10 | 9 | adantl 484 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐴 ≤ 𝐶) |
11 | 4 | simp2d 1139 | . . . 4 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐵 ∈ ℝ*) |
12 | 11 | adantl 484 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐵 ∈ ℝ*) |
13 | pnfxr 10689 | . . . 4 ⊢ +∞ ∈ ℝ* | |
14 | 13 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → +∞ ∈ ℝ*) |
15 | 8 | simprd 498 | . . . 4 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐶 < 𝐵) |
16 | 15 | adantl 484 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 < 𝐵) |
17 | pnfge 12519 | . . . . 5 ⊢ (𝐵 ∈ ℝ* → 𝐵 ≤ +∞) | |
18 | 11, 17 | syl 17 | . . . 4 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐵 ≤ +∞) |
19 | 18 | adantl 484 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐵 ≤ +∞) |
20 | 6, 12, 14, 16, 19 | xrltletrd 12548 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 < +∞) |
21 | xrre3 12558 | . 2 ⊢ (((𝐶 ∈ ℝ* ∧ 𝐴 ∈ ℝ) ∧ (𝐴 ≤ 𝐶 ∧ 𝐶 < +∞)) → 𝐶 ∈ ℝ) | |
22 | 6, 7, 10, 20, 21 | syl22anc 836 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2110 class class class wbr 5058 (class class class)co 7150 ℝcr 10530 +∞cpnf 10666 ℝ*cxr 10668 < clt 10669 ≤ cle 10670 [,)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 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-pre-lttri 10605 ax-pre-lttrn 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 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-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-ico 12738 |
This theorem is referenced by: relowlpssretop 34639 limsupresico 41974 liminfresico 42045 fourierdlem43 42429 |
Copyright terms: Public domain | W3C validator |