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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 11185 | . . . 4 ⊢ (𝜑 → (𝐴 + 𝐷) ∈ ℝ) |
4 | 3, 1 | 2thd 265 | . . 3 ⊢ (𝜑 → ((𝐴 + 𝐷) ∈ ℝ ↔ 𝐴 ∈ ℝ)) |
5 | eliooshift.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
6 | 5, 1, 2 | ltadd1d 11749 | . . . 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 11749 | . . . 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 1437 | . 2 ⊢ (𝜑 → (((𝐴 + 𝐷) ∈ ℝ ∧ (𝐵 + 𝐷) < (𝐴 + 𝐷) ∧ (𝐴 + 𝐷) < (𝐶 + 𝐷)) ↔ (𝐴 ∈ ℝ ∧ 𝐵 < 𝐴 ∧ 𝐴 < 𝐶))) |
12 | 5, 2 | readdcld 11185 | . . . 4 ⊢ (𝜑 → (𝐵 + 𝐷) ∈ ℝ) |
13 | 12 | rexrd 11206 | . . 3 ⊢ (𝜑 → (𝐵 + 𝐷) ∈ ℝ*) |
14 | 8, 2 | readdcld 11185 | . . . 4 ⊢ (𝜑 → (𝐶 + 𝐷) ∈ ℝ) |
15 | 14 | rexrd 11206 | . . 3 ⊢ (𝜑 → (𝐶 + 𝐷) ∈ ℝ*) |
16 | elioo2 13306 | . . 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 585 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐷) ∈ ((𝐵 + 𝐷)(,)(𝐶 + 𝐷)) ↔ ((𝐴 + 𝐷) ∈ ℝ ∧ (𝐵 + 𝐷) < (𝐴 + 𝐷) ∧ (𝐴 + 𝐷) < (𝐶 + 𝐷)))) |
18 | 5 | rexrd 11206 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
19 | 8 | rexrd 11206 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ*) |
20 | elioo2 13306 | . . 3 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ∈ (𝐵(,)𝐶) ↔ (𝐴 ∈ ℝ ∧ 𝐵 < 𝐴 ∧ 𝐴 < 𝐶))) | |
21 | 18, 19, 20 | syl2anc 585 | . 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 1088 ∈ wcel 2107 class class class wbr 5106 (class class class)co 7358 ℝcr 11051 + caddc 11055 ℝ*cxr 11189 < clt 11190 (,)cioo 13265 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-ioo 13269 |
This theorem is referenced by: fourierdlem88 44442 |
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