eliooshift - Mathbox for Glauco Siliprandi

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Theorem eliooshift 45504

Description: Element of an open interval shifted by a displacement. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
eliooshift.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
eliooshift.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
eliooshift.c	⊢ (𝜑 → 𝐶 ∈ ℝ)
eliooshift.d	⊢ (𝜑 → 𝐷 ∈ ℝ)

**Assertion**
Ref	Expression
eliooshift	⊢ (𝜑 → (𝐴 ∈ (𝐵(,)𝐶) ↔ (𝐴 + 𝐷) ∈ ((𝐵 + 𝐷)(,)(𝐶 + 𝐷))))

**Proof of Theorem eliooshift**
Step	Hyp	Ref	Expression
1		eliooshift.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2		eliooshift.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℝ)
3	1, 2	readdcld 11203	. . . 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 11771	. . . 4 ⊢ (𝜑 → (𝐵 < 𝐴 ↔ (𝐵 + 𝐷) < (𝐴 + 𝐷)))
7	6	bicomd 223	. . 3 ⊢ (𝜑 → ((𝐵 + 𝐷) < (𝐴 + 𝐷) ↔ 𝐵 < 𝐴))
8		eliooshift.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ)
9	1, 8, 2	ltadd1d 11771	. . . 4 ⊢ (𝜑 → (𝐴 < 𝐶 ↔ (𝐴 + 𝐷) < (𝐶 + 𝐷)))
10	9	bicomd 223	. . 3 ⊢ (𝜑 → ((𝐴 + 𝐷) < (𝐶 + 𝐷) ↔ 𝐴 < 𝐶))
11	4, 7, 10	3anbi123d 1438	. 2 ⊢ (𝜑 → (((𝐴 + 𝐷) ∈ ℝ ∧ (𝐵 + 𝐷) < (𝐴 + 𝐷) ∧ (𝐴 + 𝐷) < (𝐶 + 𝐷)) ↔ (𝐴 ∈ ℝ ∧ 𝐵 < 𝐴 ∧ 𝐴 < 𝐶)))
12	5, 2	readdcld 11203	. . . 4 ⊢ (𝜑 → (𝐵 + 𝐷) ∈ ℝ)
13	12	rexrd 11224	. . 3 ⊢ (𝜑 → (𝐵 + 𝐷) ∈ ℝ^*)
14	8, 2	readdcld 11203	. . . 4 ⊢ (𝜑 → (𝐶 + 𝐷) ∈ ℝ)
15	14	rexrd 11224	. . 3 ⊢ (𝜑 → (𝐶 + 𝐷) ∈ ℝ^*)
16		elioo2 13347	. . 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 11224	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
19	8	rexrd 11224	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
20		elioo2 13347	. . 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 206 ∧ w3a 1086 ∈ wcel 2109 class class class wbr 5107 (class class class)co 7387 ℝcr 11067 + caddc 11071 ℝ^*cxr 11207 < clt 11208 (,)cioo 13306

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-po 5546 df-so 5547 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-ov 7390 df-oprab 7391 df-mpo 7392 df-1st 7968 df-2nd 7969 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-ioo 13310

This theorem is referenced by: fourierdlem88 46192

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