eliooshift - Mathbox for Glauco Siliprandi

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Theorem eliooshift 45954

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 11165	. . . 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 11734	. . . 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 11734	. . . 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 1439	. 2 ⊢ (𝜑 → (((𝐴 + 𝐷) ∈ ℝ ∧ (𝐵 + 𝐷) < (𝐴 + 𝐷) ∧ (𝐴 + 𝐷) < (𝐶 + 𝐷)) ↔ (𝐴 ∈ ℝ ∧ 𝐵 < 𝐴 ∧ 𝐴 < 𝐶)))
12	5, 2	readdcld 11165	. . . 4 ⊢ (𝜑 → (𝐵 + 𝐷) ∈ ℝ)
13	12	rexrd 11186	. . 3 ⊢ (𝜑 → (𝐵 + 𝐷) ∈ ℝ^*)
14	8, 2	readdcld 11165	. . . 4 ⊢ (𝜑 → (𝐶 + 𝐷) ∈ ℝ)
15	14	rexrd 11186	. . 3 ⊢ (𝜑 → (𝐶 + 𝐷) ∈ ℝ^*)
16		elioo2 13330	. . 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 11186	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
19	8	rexrd 11186	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
20		elioo2 13330	. . 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 206 ∧ w3a 1087 ∈ wcel 2114 class class class wbr 5086 (class class class)co 7360 ℝcr 11028 + caddc 11032 ℝ^*cxr 11169 < clt 11170 (,)cioo 13289

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5519 df-po 5532 df-so 5533 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7363 df-oprab 7364 df-mpo 7365 df-1st 7935 df-2nd 7936 df-er 8636 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-ioo 13293

This theorem is referenced by: fourierdlem88 46640

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