eliooshift - Mathbox for Glauco Siliprandi

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Theorem eliooshift 45483

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 11262	. . . 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 11828	. . . 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 11828	. . . 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 11262	. . . 4 ⊢ (𝜑 → (𝐵 + 𝐷) ∈ ℝ)
13	12	rexrd 11283	. . 3 ⊢ (𝜑 → (𝐵 + 𝐷) ∈ ℝ^*)
14	8, 2	readdcld 11262	. . . 4 ⊢ (𝜑 → (𝐶 + 𝐷) ∈ ℝ)
15	14	rexrd 11283	. . 3 ⊢ (𝜑 → (𝐶 + 𝐷) ∈ ℝ^*)
16		elioo2 13401	. . 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 11283	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
19	8	rexrd 11283	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
20		elioo2 13401	. . 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 2108 class class class wbr 5119 (class class class)co 7403 ℝcr 11126 + caddc 11130 ℝ^*cxr 11266 < clt 11267 (,)cioo 13360

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-po 5561 df-so 5562 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-ov 7406 df-oprab 7407 df-mpo 7408 df-1st 7986 df-2nd 7987 df-er 8717 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-ioo 13364

This theorem is referenced by: fourierdlem88 46171

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