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 11004 | . . . 4 ⊢ (𝜑 → (𝐴 + 𝐷) ∈ ℝ) |
4 | 3, 1 | 2thd 264 | . . 3 ⊢ (𝜑 → ((𝐴 + 𝐷) ∈ ℝ ↔ 𝐴 ∈ ℝ)) |
5 | eliooshift.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
6 | 5, 1, 2 | ltadd1d 11568 | . . . 4 ⊢ (𝜑 → (𝐵 < 𝐴 ↔ (𝐵 + 𝐷) < (𝐴 + 𝐷))) |
7 | 6 | bicomd 222 | . . 3 ⊢ (𝜑 → ((𝐵 + 𝐷) < (𝐴 + 𝐷) ↔ 𝐵 < 𝐴)) |
8 | eliooshift.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
9 | 1, 8, 2 | ltadd1d 11568 | . . . 4 ⊢ (𝜑 → (𝐴 < 𝐶 ↔ (𝐴 + 𝐷) < (𝐶 + 𝐷))) |
10 | 9 | bicomd 222 | . . 3 ⊢ (𝜑 → ((𝐴 + 𝐷) < (𝐶 + 𝐷) ↔ 𝐴 < 𝐶)) |
11 | 4, 7, 10 | 3anbi123d 1435 | . 2 ⊢ (𝜑 → (((𝐴 + 𝐷) ∈ ℝ ∧ (𝐵 + 𝐷) < (𝐴 + 𝐷) ∧ (𝐴 + 𝐷) < (𝐶 + 𝐷)) ↔ (𝐴 ∈ ℝ ∧ 𝐵 < 𝐴 ∧ 𝐴 < 𝐶))) |
12 | 5, 2 | readdcld 11004 | . . . 4 ⊢ (𝜑 → (𝐵 + 𝐷) ∈ ℝ) |
13 | 12 | rexrd 11025 | . . 3 ⊢ (𝜑 → (𝐵 + 𝐷) ∈ ℝ*) |
14 | 8, 2 | readdcld 11004 | . . . 4 ⊢ (𝜑 → (𝐶 + 𝐷) ∈ ℝ) |
15 | 14 | rexrd 11025 | . . 3 ⊢ (𝜑 → (𝐶 + 𝐷) ∈ ℝ*) |
16 | elioo2 13120 | . . 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 584 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐷) ∈ ((𝐵 + 𝐷)(,)(𝐶 + 𝐷)) ↔ ((𝐴 + 𝐷) ∈ ℝ ∧ (𝐵 + 𝐷) < (𝐴 + 𝐷) ∧ (𝐴 + 𝐷) < (𝐶 + 𝐷)))) |
18 | 5 | rexrd 11025 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
19 | 8 | rexrd 11025 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ*) |
20 | elioo2 13120 | . . 3 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ∈ (𝐵(,)𝐶) ↔ (𝐴 ∈ ℝ ∧ 𝐵 < 𝐴 ∧ 𝐴 < 𝐶))) | |
21 | 18, 19, 20 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐴 ∈ (𝐵(,)𝐶) ↔ (𝐴 ∈ ℝ ∧ 𝐵 < 𝐴 ∧ 𝐴 < 𝐶))) |
22 | 11, 17, 21 | 3bitr4rd 312 | 1 ⊢ (𝜑 → (𝐴 ∈ (𝐵(,)𝐶) ↔ (𝐴 + 𝐷) ∈ ((𝐵 + 𝐷)(,)(𝐶 + 𝐷)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1086 ∈ wcel 2106 class class class wbr 5074 (class class class)co 7275 ℝcr 10870 + caddc 10874 ℝ*cxr 11008 < clt 11009 (,)cioo 13079 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-ioo 13083 |
This theorem is referenced by: fourierdlem88 43735 |
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