iccshift - Mathbox for Glauco Siliprandi

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Theorem iccshift 45436

Description: A closed interval shifted by a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
iccshift.1	⊢ (𝜑 → 𝐴 ∈ ℝ)
iccshift.2	⊢ (𝜑 → 𝐵 ∈ ℝ)
iccshift.3	⊢ (𝜑 → 𝑇 ∈ ℝ)

**Assertion**
Ref	Expression
iccshift	⊢ (𝜑 → ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)})

Distinct variable groups: 𝑤,𝐴,𝑧 𝑤,𝐵,𝑧 𝑤,𝑇,𝑧 𝜑,𝑧 Allowed substitution hint: 𝜑(𝑤)

Proof of Theorem iccshift Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2744	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (𝑤 = (𝑧 + 𝑇) ↔ 𝑥 = (𝑧 + 𝑇)))
2	1	rexbidv 3185	. . . . . 6 ⊢ (𝑤 = 𝑥 → (∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇) ↔ ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇)))
3	2	elrab 3708	. . . . 5 ⊢ (𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)} ↔ (𝑥 ∈ ℂ ∧ ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇)))
4		simprr 772	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇))) → ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇))
5		nfv 1913	. . . . . . . 8 ⊢ Ⅎ𝑧𝜑
6		nfv 1913	. . . . . . . . 9 ⊢ Ⅎ𝑧 𝑥 ∈ ℂ
7		nfre1 3291	. . . . . . . . 9 ⊢ Ⅎ𝑧∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇)
8	6, 7	nfan 1898	. . . . . . . 8 ⊢ Ⅎ𝑧(𝑥 ∈ ℂ ∧ ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇))
9	5, 8	nfan 1898	. . . . . . 7 ⊢ Ⅎ𝑧(𝜑 ∧ (𝑥 ∈ ℂ ∧ ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇)))
10		nfv 1913	. . . . . . 7 ⊢ Ⅎ𝑧 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))
11		simp3 1138	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵) ∧ 𝑥 = (𝑧 + 𝑇)) → 𝑥 = (𝑧 + 𝑇))
12		iccshift.1	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ ℝ)
13		iccshift.2	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 ∈ ℝ)
14	12, 13	iccssred 13494	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
15	14	sselda 4008	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → 𝑧 ∈ ℝ)
16		iccshift.3	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑇 ∈ ℝ)
17	16	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → 𝑇 ∈ ℝ)
18	15, 17	readdcld 11319	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → (𝑧 + 𝑇) ∈ ℝ)
19	12	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ)
20		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → 𝑧 ∈ (𝐴[,]𝐵))
21	13	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
22		elicc2 13472	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑧 ∈ (𝐴[,]𝐵) ↔ (𝑧 ∈ ℝ ∧ 𝐴 ≤ 𝑧 ∧ 𝑧 ≤ 𝐵)))
23	19, 21, 22	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → (𝑧 ∈ (𝐴[,]𝐵) ↔ (𝑧 ∈ ℝ ∧ 𝐴 ≤ 𝑧 ∧ 𝑧 ≤ 𝐵)))
24	20, 23	mpbid 232	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → (𝑧 ∈ ℝ ∧ 𝐴 ≤ 𝑧 ∧ 𝑧 ≤ 𝐵))
25	24	simp2d 1143	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝑧)
26	19, 15, 17, 25	leadd1dd 11904	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → (𝐴 + 𝑇) ≤ (𝑧 + 𝑇))
27	24	simp3d 1144	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → 𝑧 ≤ 𝐵)
28	15, 21, 17, 27	leadd1dd 11904	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → (𝑧 + 𝑇) ≤ (𝐵 + 𝑇))
29	18, 26, 28	3jca 1128	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵)) → ((𝑧 + 𝑇) ∈ ℝ ∧ (𝐴 + 𝑇) ≤ (𝑧 + 𝑇) ∧ (𝑧 + 𝑇) ≤ (𝐵 + 𝑇)))
30	29	3adant3 1132	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵) ∧ 𝑥 = (𝑧 + 𝑇)) → ((𝑧 + 𝑇) ∈ ℝ ∧ (𝐴 + 𝑇) ≤ (𝑧 + 𝑇) ∧ (𝑧 + 𝑇) ≤ (𝐵 + 𝑇)))
31	12, 16	readdcld 11319	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 + 𝑇) ∈ ℝ)
32	31	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵) ∧ 𝑥 = (𝑧 + 𝑇)) → (𝐴 + 𝑇) ∈ ℝ)
33	13, 16	readdcld 11319	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 + 𝑇) ∈ ℝ)
34	33	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵) ∧ 𝑥 = (𝑧 + 𝑇)) → (𝐵 + 𝑇) ∈ ℝ)
35		elicc2 13472	. . . . . . . . . . . 12 ⊢ (((𝐴 + 𝑇) ∈ ℝ ∧ (𝐵 + 𝑇) ∈ ℝ) → ((𝑧 + 𝑇) ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↔ ((𝑧 + 𝑇) ∈ ℝ ∧ (𝐴 + 𝑇) ≤ (𝑧 + 𝑇) ∧ (𝑧 + 𝑇) ≤ (𝐵 + 𝑇))))
36	32, 34, 35	syl2anc 583	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵) ∧ 𝑥 = (𝑧 + 𝑇)) → ((𝑧 + 𝑇) ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↔ ((𝑧 + 𝑇) ∈ ℝ ∧ (𝐴 + 𝑇) ≤ (𝑧 + 𝑇) ∧ (𝑧 + 𝑇) ≤ (𝐵 + 𝑇))))
37	30, 36	mpbird 257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵) ∧ 𝑥 = (𝑧 + 𝑇)) → (𝑧 + 𝑇) ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))
38	11, 37	eqeltrd 2844	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴[,]𝐵) ∧ 𝑥 = (𝑧 + 𝑇)) → 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))
39	38	3exp 1119	. . . . . . . 8 ⊢ (𝜑 → (𝑧 ∈ (𝐴[,]𝐵) → (𝑥 = (𝑧 + 𝑇) → 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))))
40	39	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇))) → (𝑧 ∈ (𝐴[,]𝐵) → (𝑥 = (𝑧 + 𝑇) → 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))))
41	9, 10, 40	rexlimd 3272	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇))) → (∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇) → 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))))
42	4, 41	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇))) → 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))
43	3, 42	sylan2b 593	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)}) → 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))
44	31	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐴 + 𝑇) ∈ ℝ)
45	33	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐵 + 𝑇) ∈ ℝ)
46		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))
47		eliccre 45423	. . . . . . 7 ⊢ (((𝐴 + 𝑇) ∈ ℝ ∧ (𝐵 + 𝑇) ∈ ℝ ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ∈ ℝ)
48	44, 45, 46, 47	syl3anc 1371	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ∈ ℝ)
49	48	recnd 11318	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ∈ ℂ)
50	12	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐴 ∈ ℝ)
51	13	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐵 ∈ ℝ)
52	16	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑇 ∈ ℝ)
53	48, 52	resubcld 11718	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 − 𝑇) ∈ ℝ)
54	12	recnd 11318	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℂ)
55	16	recnd 11318	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ∈ ℂ)
56	54, 55	pncand 11648	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 + 𝑇) − 𝑇) = 𝐴)
57	56	eqcomd 2746	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 = ((𝐴 + 𝑇) − 𝑇))
58	57	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐴 = ((𝐴 + 𝑇) − 𝑇))
59		elicc2 13472	. . . . . . . . . . . 12 ⊢ (((𝐴 + 𝑇) ∈ ℝ ∧ (𝐵 + 𝑇) ∈ ℝ) → (𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↔ (𝑥 ∈ ℝ ∧ (𝐴 + 𝑇) ≤ 𝑥 ∧ 𝑥 ≤ (𝐵 + 𝑇))))
60	44, 45, 59	syl2anc 583	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↔ (𝑥 ∈ ℝ ∧ (𝐴 + 𝑇) ≤ 𝑥 ∧ 𝑥 ≤ (𝐵 + 𝑇))))
61	46, 60	mpbid 232	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 ∈ ℝ ∧ (𝐴 + 𝑇) ≤ 𝑥 ∧ 𝑥 ≤ (𝐵 + 𝑇)))
62	61	simp2d 1143	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝐴 + 𝑇) ≤ 𝑥)
63	44, 48, 52, 62	lesub1dd 11906	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ((𝐴 + 𝑇) − 𝑇) ≤ (𝑥 − 𝑇))
64	58, 63	eqbrtrd 5188	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝐴 ≤ (𝑥 − 𝑇))
65	61	simp3d 1144	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ≤ (𝐵 + 𝑇))
66	48, 45, 52, 65	lesub1dd 11906	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 − 𝑇) ≤ ((𝐵 + 𝑇) − 𝑇))
67	13	recnd 11318	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℂ)
68	67, 55	pncand 11648	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 + 𝑇) − 𝑇) = 𝐵)
69	68	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ((𝐵 + 𝑇) − 𝑇) = 𝐵)
70	66, 69	breqtrd 5192	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 − 𝑇) ≤ 𝐵)
71	50, 51, 53, 64, 70	eliccd 45422	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → (𝑥 − 𝑇) ∈ (𝐴[,]𝐵))
72	55	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑇 ∈ ℂ)
73	49, 72	npcand 11651	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ((𝑥 − 𝑇) + 𝑇) = 𝑥)
74	73	eqcomd 2746	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 = ((𝑥 − 𝑇) + 𝑇))
75		oveq1 7455	. . . . . . 7 ⊢ (𝑧 = (𝑥 − 𝑇) → (𝑧 + 𝑇) = ((𝑥 − 𝑇) + 𝑇))
76	75	rspceeqv 3658	. . . . . 6 ⊢ (((𝑥 − 𝑇) ∈ (𝐴[,]𝐵) ∧ 𝑥 = ((𝑥 − 𝑇) + 𝑇)) → ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇))
77	71, 74, 76	syl2anc 583	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → ∃𝑧 ∈ (𝐴[,]𝐵)𝑥 = (𝑧 + 𝑇))
78	49, 77, 3	sylanbrc 582	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))) → 𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)})
79	43, 78	impbida 800	. . 3 ⊢ (𝜑 → (𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)} ↔ 𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇))))
80	79	eqrdv 2738	. 2 ⊢ (𝜑 → {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)} = ((𝐴 + 𝑇)[,](𝐵 + 𝑇)))
81	80	eqcomd 2746	1 ⊢ (𝜑 → ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ∃wrex 3076 {crab 3443 class class class wbr 5166 (class class class)co 7448 ℂcc 11182 ℝcr 11183 + caddc 11187 ≤ cle 11325 − cmin 11520 [,]cicc 13410

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-po 5607 df-so 5608 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-icc 13414

This theorem is referenced by: itgiccshift 45901 itgperiod 45902

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