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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 12052 | . . . . . . 7 ⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | |
2 | 1 | elixx3g 12059 | . . . . . 6 ⊢ (𝐶 ∈ (𝐴[,)𝐵) ↔ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))) |
3 | 2 | biimpi 205 | . . . . 5 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))) |
4 | 3 | simpld 474 | . . . 4 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*)) |
5 | 4 | simp3d 1068 | . . 3 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐶 ∈ ℝ*) |
6 | 5 | adantl 481 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 ∈ ℝ*) |
7 | simpl 472 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐴 ∈ ℝ) | |
8 | 3 | simprd 478 | . . . 4 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → (𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) |
9 | 8 | simpld 474 | . . 3 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐴 ≤ 𝐶) |
10 | 9 | adantl 481 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐴 ≤ 𝐶) |
11 | 4 | simp2d 1067 | . . . 4 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐵 ∈ ℝ*) |
12 | 11 | adantl 481 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐵 ∈ ℝ*) |
13 | pnfxr 9971 | . . . 4 ⊢ +∞ ∈ ℝ* | |
14 | 13 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → +∞ ∈ ℝ*) |
15 | 8 | simprd 478 | . . . 4 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐶 < 𝐵) |
16 | 15 | adantl 481 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 < 𝐵) |
17 | pnfge 11840 | . . . . 5 ⊢ (𝐵 ∈ ℝ* → 𝐵 ≤ +∞) | |
18 | 11, 17 | syl 17 | . . . 4 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐵 ≤ +∞) |
19 | 18 | adantl 481 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐵 ≤ +∞) |
20 | 6, 12, 14, 16, 19 | xrltletrd 11868 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 < +∞) |
21 | xrre3 11876 | . 2 ⊢ (((𝐶 ∈ ℝ* ∧ 𝐴 ∈ ℝ) ∧ (𝐴 ≤ 𝐶 ∧ 𝐶 < +∞)) → 𝐶 ∈ ℝ) | |
22 | 6, 7, 10, 20, 21 | syl22anc 1319 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 ∈ wcel 1977 class class class wbr 4583 (class class class)co 6549 ℝcr 9814 +∞cpnf 9950 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 [,)cico 12048 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-ico 12052 |
This theorem is referenced by: relowlpssretop 32388 fourierdlem43 39043 |
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