Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > eliooshift | Structured version Visualization version GIF version |
Description: Element of an open interval shifted by a displacement. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
eliooshift.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
eliooshift.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
eliooshift.c | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
eliooshift.d | ⊢ (𝜑 → 𝐷 ∈ ℝ) |
Ref | Expression |
---|---|
eliooshift | ⊢ (𝜑 → (𝐴 ∈ (𝐵(,)𝐶) ↔ (𝐴 + 𝐷) ∈ ((𝐵 + 𝐷)(,)(𝐶 + 𝐷)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliooshift.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | eliooshift.d | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℝ) | |
3 | 1, 2 | readdcld 10751 | . . . 4 ⊢ (𝜑 → (𝐴 + 𝐷) ∈ ℝ) |
4 | 3, 1 | 2thd 268 | . . 3 ⊢ (𝜑 → ((𝐴 + 𝐷) ∈ ℝ ↔ 𝐴 ∈ ℝ)) |
5 | eliooshift.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
6 | 5, 1, 2 | ltadd1d 11314 | . . . 4 ⊢ (𝜑 → (𝐵 < 𝐴 ↔ (𝐵 + 𝐷) < (𝐴 + 𝐷))) |
7 | 6 | bicomd 226 | . . 3 ⊢ (𝜑 → ((𝐵 + 𝐷) < (𝐴 + 𝐷) ↔ 𝐵 < 𝐴)) |
8 | eliooshift.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
9 | 1, 8, 2 | ltadd1d 11314 | . . . 4 ⊢ (𝜑 → (𝐴 < 𝐶 ↔ (𝐴 + 𝐷) < (𝐶 + 𝐷))) |
10 | 9 | bicomd 226 | . . 3 ⊢ (𝜑 → ((𝐴 + 𝐷) < (𝐶 + 𝐷) ↔ 𝐴 < 𝐶)) |
11 | 4, 7, 10 | 3anbi123d 1437 | . 2 ⊢ (𝜑 → (((𝐴 + 𝐷) ∈ ℝ ∧ (𝐵 + 𝐷) < (𝐴 + 𝐷) ∧ (𝐴 + 𝐷) < (𝐶 + 𝐷)) ↔ (𝐴 ∈ ℝ ∧ 𝐵 < 𝐴 ∧ 𝐴 < 𝐶))) |
12 | 5, 2 | readdcld 10751 | . . . 4 ⊢ (𝜑 → (𝐵 + 𝐷) ∈ ℝ) |
13 | 12 | rexrd 10772 | . . 3 ⊢ (𝜑 → (𝐵 + 𝐷) ∈ ℝ*) |
14 | 8, 2 | readdcld 10751 | . . . 4 ⊢ (𝜑 → (𝐶 + 𝐷) ∈ ℝ) |
15 | 14 | rexrd 10772 | . . 3 ⊢ (𝜑 → (𝐶 + 𝐷) ∈ ℝ*) |
16 | elioo2 12865 | . . 3 ⊢ (((𝐵 + 𝐷) ∈ ℝ* ∧ (𝐶 + 𝐷) ∈ ℝ*) → ((𝐴 + 𝐷) ∈ ((𝐵 + 𝐷)(,)(𝐶 + 𝐷)) ↔ ((𝐴 + 𝐷) ∈ ℝ ∧ (𝐵 + 𝐷) < (𝐴 + 𝐷) ∧ (𝐴 + 𝐷) < (𝐶 + 𝐷)))) | |
17 | 13, 15, 16 | syl2anc 587 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐷) ∈ ((𝐵 + 𝐷)(,)(𝐶 + 𝐷)) ↔ ((𝐴 + 𝐷) ∈ ℝ ∧ (𝐵 + 𝐷) < (𝐴 + 𝐷) ∧ (𝐴 + 𝐷) < (𝐶 + 𝐷)))) |
18 | 5 | rexrd 10772 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
19 | 8 | rexrd 10772 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ*) |
20 | elioo2 12865 | . . 3 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ∈ (𝐵(,)𝐶) ↔ (𝐴 ∈ ℝ ∧ 𝐵 < 𝐴 ∧ 𝐴 < 𝐶))) | |
21 | 18, 19, 20 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝐴 ∈ (𝐵(,)𝐶) ↔ (𝐴 ∈ ℝ ∧ 𝐵 < 𝐴 ∧ 𝐴 < 𝐶))) |
22 | 11, 17, 21 | 3bitr4rd 315 | 1 ⊢ (𝜑 → (𝐴 ∈ (𝐵(,)𝐶) ↔ (𝐴 + 𝐷) ∈ ((𝐵 + 𝐷)(,)(𝐶 + 𝐷)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ w3a 1088 ∈ wcel 2114 class class class wbr 5031 (class class class)co 7173 ℝcr 10617 + caddc 10621 ℝ*cxr 10755 < clt 10756 (,)cioo 12824 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-id 5430 df-po 5443 df-so 5444 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-ov 7176 df-oprab 7177 df-mpo 7178 df-1st 7717 df-2nd 7718 df-er 8323 df-en 8559 df-dom 8560 df-sdom 8561 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-ioo 12828 |
This theorem is referenced by: fourierdlem88 43300 |
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