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 10670 | . . . 4 ⊢ (𝜑 → (𝐴 + 𝐷) ∈ ℝ) |
4 | 3, 1 | 2thd 267 | . . 3 ⊢ (𝜑 → ((𝐴 + 𝐷) ∈ ℝ ↔ 𝐴 ∈ ℝ)) |
5 | eliooshift.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
6 | 5, 1, 2 | ltadd1d 11233 | . . . 4 ⊢ (𝜑 → (𝐵 < 𝐴 ↔ (𝐵 + 𝐷) < (𝐴 + 𝐷))) |
7 | 6 | bicomd 225 | . . 3 ⊢ (𝜑 → ((𝐵 + 𝐷) < (𝐴 + 𝐷) ↔ 𝐵 < 𝐴)) |
8 | eliooshift.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
9 | 1, 8, 2 | ltadd1d 11233 | . . . 4 ⊢ (𝜑 → (𝐴 < 𝐶 ↔ (𝐴 + 𝐷) < (𝐶 + 𝐷))) |
10 | 9 | bicomd 225 | . . 3 ⊢ (𝜑 → ((𝐴 + 𝐷) < (𝐶 + 𝐷) ↔ 𝐴 < 𝐶)) |
11 | 4, 7, 10 | 3anbi123d 1432 | . 2 ⊢ (𝜑 → (((𝐴 + 𝐷) ∈ ℝ ∧ (𝐵 + 𝐷) < (𝐴 + 𝐷) ∧ (𝐴 + 𝐷) < (𝐶 + 𝐷)) ↔ (𝐴 ∈ ℝ ∧ 𝐵 < 𝐴 ∧ 𝐴 < 𝐶))) |
12 | 5, 2 | readdcld 10670 | . . . 4 ⊢ (𝜑 → (𝐵 + 𝐷) ∈ ℝ) |
13 | 12 | rexrd 10691 | . . 3 ⊢ (𝜑 → (𝐵 + 𝐷) ∈ ℝ*) |
14 | 8, 2 | readdcld 10670 | . . . 4 ⊢ (𝜑 → (𝐶 + 𝐷) ∈ ℝ) |
15 | 14 | rexrd 10691 | . . 3 ⊢ (𝜑 → (𝐶 + 𝐷) ∈ ℝ*) |
16 | elioo2 12780 | . . 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 586 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐷) ∈ ((𝐵 + 𝐷)(,)(𝐶 + 𝐷)) ↔ ((𝐴 + 𝐷) ∈ ℝ ∧ (𝐵 + 𝐷) < (𝐴 + 𝐷) ∧ (𝐴 + 𝐷) < (𝐶 + 𝐷)))) |
18 | 5 | rexrd 10691 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
19 | 8 | rexrd 10691 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ*) |
20 | elioo2 12780 | . . 3 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ∈ (𝐵(,)𝐶) ↔ (𝐴 ∈ ℝ ∧ 𝐵 < 𝐴 ∧ 𝐴 < 𝐶))) | |
21 | 18, 19, 20 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝐴 ∈ (𝐵(,)𝐶) ↔ (𝐴 ∈ ℝ ∧ 𝐵 < 𝐴 ∧ 𝐴 < 𝐶))) |
22 | 11, 17, 21 | 3bitr4rd 314 | 1 ⊢ (𝜑 → (𝐴 ∈ (𝐵(,)𝐶) ↔ (𝐴 + 𝐷) ∈ ((𝐵 + 𝐷)(,)(𝐶 + 𝐷)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 ∈ wcel 2114 class class class wbr 5066 (class class class)co 7156 ℝcr 10536 + caddc 10540 ℝ*cxr 10674 < clt 10675 (,)cioo 12739 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-ioo 12743 |
This theorem is referenced by: fourierdlem88 42499 |
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