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 10069 | . . . 4 ⊢ (𝜑 → (𝐴 + 𝐷) ∈ ℝ) |
4 | 3, 1 | 2thd 255 | . . 3 ⊢ (𝜑 → ((𝐴 + 𝐷) ∈ ℝ ↔ 𝐴 ∈ ℝ)) |
5 | eliooshift.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
6 | 5, 1, 2 | ltadd1d 10620 | . . . 4 ⊢ (𝜑 → (𝐵 < 𝐴 ↔ (𝐵 + 𝐷) < (𝐴 + 𝐷))) |
7 | 6 | bicomd 213 | . . 3 ⊢ (𝜑 → ((𝐵 + 𝐷) < (𝐴 + 𝐷) ↔ 𝐵 < 𝐴)) |
8 | eliooshift.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
9 | 1, 8, 2 | ltadd1d 10620 | . . . 4 ⊢ (𝜑 → (𝐴 < 𝐶 ↔ (𝐴 + 𝐷) < (𝐶 + 𝐷))) |
10 | 9 | bicomd 213 | . . 3 ⊢ (𝜑 → ((𝐴 + 𝐷) < (𝐶 + 𝐷) ↔ 𝐴 < 𝐶)) |
11 | 4, 7, 10 | 3anbi123d 1399 | . 2 ⊢ (𝜑 → (((𝐴 + 𝐷) ∈ ℝ ∧ (𝐵 + 𝐷) < (𝐴 + 𝐷) ∧ (𝐴 + 𝐷) < (𝐶 + 𝐷)) ↔ (𝐴 ∈ ℝ ∧ 𝐵 < 𝐴 ∧ 𝐴 < 𝐶))) |
12 | 5, 2 | readdcld 10069 | . . . 4 ⊢ (𝜑 → (𝐵 + 𝐷) ∈ ℝ) |
13 | 12 | rexrd 10089 | . . 3 ⊢ (𝜑 → (𝐵 + 𝐷) ∈ ℝ*) |
14 | 8, 2 | readdcld 10069 | . . . 4 ⊢ (𝜑 → (𝐶 + 𝐷) ∈ ℝ) |
15 | 14 | rexrd 10089 | . . 3 ⊢ (𝜑 → (𝐶 + 𝐷) ∈ ℝ*) |
16 | elioo2 12216 | . . 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 693 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐷) ∈ ((𝐵 + 𝐷)(,)(𝐶 + 𝐷)) ↔ ((𝐴 + 𝐷) ∈ ℝ ∧ (𝐵 + 𝐷) < (𝐴 + 𝐷) ∧ (𝐴 + 𝐷) < (𝐶 + 𝐷)))) |
18 | 5 | rexrd 10089 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
19 | 8 | rexrd 10089 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ*) |
20 | elioo2 12216 | . . 3 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ∈ (𝐵(,)𝐶) ↔ (𝐴 ∈ ℝ ∧ 𝐵 < 𝐴 ∧ 𝐴 < 𝐶))) | |
21 | 18, 19, 20 | syl2anc 693 | . 2 ⊢ (𝜑 → (𝐴 ∈ (𝐵(,)𝐶) ↔ (𝐴 ∈ ℝ ∧ 𝐵 < 𝐴 ∧ 𝐴 < 𝐶))) |
22 | 11, 17, 21 | 3bitr4rd 301 | 1 ⊢ (𝜑 → (𝐴 ∈ (𝐵(,)𝐶) ↔ (𝐴 + 𝐷) ∈ ((𝐵 + 𝐷)(,)(𝐶 + 𝐷)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ w3a 1037 ∈ wcel 1990 class class class wbr 4653 (class class class)co 6650 ℝcr 9935 + caddc 9939 ℝ*cxr 10073 < clt 10074 (,)cioo 12175 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-ioo 12179 |
This theorem is referenced by: fourierdlem88 40411 |
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