2elfz2melfz - Mathbox for Alexander van der Vekens

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Theorem 2elfz2melfz 47323

Description: If the sum of two integers of a 0-based finite set of sequential integers is greater than the upper bound, the difference between one of the integers and the difference between the upper bound and the other integer is in the 0-based finite set of sequential integers with the first integer as upper bound. (Contributed by Alexander van der Vekens, 7-Apr-2018.) (Revised by Alexander van der Vekens, 31-May-2018.)

**Assertion**
Ref	Expression
2elfz2melfz	⊢ ((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → (𝑁 < (𝐴 + 𝐵) → (𝐵 − (𝑁 − 𝐴)) ∈ (0...𝐴)))

**Proof of Theorem 2elfz2melfz**
Step	Hyp	Ref	Expression
1		elfzelz 13492	. . . . 5 ⊢ (𝐴 ∈ (0...𝑁) → 𝐴 ∈ ℤ)
2		elfzel2 13490	. . . . . 6 ⊢ (𝐵 ∈ (0...𝑁) → 𝑁 ∈ ℤ)
3		elfzelz 13492	. . . . . 6 ⊢ (𝐵 ∈ (0...𝑁) → 𝐵 ∈ ℤ)
4		simplr 768	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) → 𝐵 ∈ ℤ)
5		zsubcl 12582	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝑁 − 𝐴) ∈ ℤ)
6	5	adantlr 715	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) → (𝑁 − 𝐴) ∈ ℤ)
7	4, 6	zsubcld 12650	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) → (𝐵 − (𝑁 − 𝐴)) ∈ ℤ)
8	7	adantr 480	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) ∧ 𝑁 < (𝐴 + 𝐵)) → (𝐵 − (𝑁 − 𝐴)) ∈ ℤ)
9		zre 12540	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
10	9	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) → 𝑁 ∈ ℝ)
11		zaddcl 12580	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 + 𝐵) ∈ ℤ)
12	11	zred 12645	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 + 𝐵) ∈ ℝ)
13	12	expcom 413	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℤ → (𝐴 ∈ ℤ → (𝐴 + 𝐵) ∈ ℝ))
14	13	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ∈ ℤ → (𝐴 + 𝐵) ∈ ℝ))
15	14	imp 406	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) → (𝐴 + 𝐵) ∈ ℝ)
16	10, 15, 10	ltsub1d 11794	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) → (𝑁 < (𝐴 + 𝐵) ↔ (𝑁 − 𝑁) < ((𝐴 + 𝐵) − 𝑁)))
17		zre 12540	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ)
18	9, 17	anim12i 613	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝑁 ∈ ℝ ∧ 𝐵 ∈ ℝ))
19		zre 12540	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
20	18, 19	anim12i 613	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) → ((𝑁 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ))
21		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℝ → 𝑁 ∈ ℝ)
22	21, 21	resubcld 11613	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℝ → (𝑁 − 𝑁) ∈ ℝ)
23	22	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → (𝑁 − 𝑁) ∈ ℝ)
24		readdcl 11158	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
25	24	expcom 413	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℝ → (𝐴 ∈ ℝ → (𝐴 + 𝐵) ∈ ℝ))
26	25	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ∈ ℝ → (𝐴 + 𝐵) ∈ ℝ))
27	26	imp 406	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
28		simpll 766	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → 𝑁 ∈ ℝ)
29	27, 28	resubcld 11613	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → ((𝐴 + 𝐵) − 𝑁) ∈ ℝ)
30	23, 29	jca 511	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → ((𝑁 − 𝑁) ∈ ℝ ∧ ((𝐴 + 𝐵) − 𝑁) ∈ ℝ))
31		ltle 11269	. . . . . . . . . . . 12 ⊢ (((𝑁 − 𝑁) ∈ ℝ ∧ ((𝐴 + 𝐵) − 𝑁) ∈ ℝ) → ((𝑁 − 𝑁) < ((𝐴 + 𝐵) − 𝑁) → (𝑁 − 𝑁) ≤ ((𝐴 + 𝐵) − 𝑁)))
32	20, 30, 31	3syl 18	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) → ((𝑁 − 𝑁) < ((𝐴 + 𝐵) − 𝑁) → (𝑁 − 𝑁) ≤ ((𝐴 + 𝐵) − 𝑁)))
33		zcn 12541	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
34	33	subidd 11528	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℤ → (𝑁 − 𝑁) = 0)
35	34	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) → (𝑁 − 𝑁) = 0)
36		zcn 12541	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℂ)
37	36	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℂ)
38	37	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) → 𝐵 ∈ ℂ)
39	33	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) → 𝑁 ∈ ℂ)
40		zcn 12541	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ)
41	40	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) → 𝐴 ∈ ℂ)
42		simp3 1138	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ ℂ ∧ 𝑁 ∈ ℂ ∧ 𝐴 ∈ ℂ) → 𝐴 ∈ ℂ)
43		simp1 1136	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ ℂ ∧ 𝑁 ∈ ℂ ∧ 𝐴 ∈ ℂ) → 𝐵 ∈ ℂ)
44	42, 43	addcomd 11383	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ ℂ ∧ 𝑁 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
45	44	oveq1d 7405	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℂ ∧ 𝑁 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((𝐴 + 𝐵) − 𝑁) = ((𝐵 + 𝐴) − 𝑁))
46		subsub3 11461	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℂ ∧ 𝑁 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝐵 − (𝑁 − 𝐴)) = ((𝐵 + 𝐴) − 𝑁))
47	45, 46	eqtr4d 2768	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ℂ ∧ 𝑁 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((𝐴 + 𝐵) − 𝑁) = (𝐵 − (𝑁 − 𝐴)))
48	38, 39, 41, 47	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) → ((𝐴 + 𝐵) − 𝑁) = (𝐵 − (𝑁 − 𝐴)))
49	35, 48	breq12d 5123	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) → ((𝑁 − 𝑁) ≤ ((𝐴 + 𝐵) − 𝑁) ↔ 0 ≤ (𝐵 − (𝑁 − 𝐴))))
50	32, 49	sylibd 239	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) → ((𝑁 − 𝑁) < ((𝐴 + 𝐵) − 𝑁) → 0 ≤ (𝐵 − (𝑁 − 𝐴))))
51	16, 50	sylbid 240	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) → (𝑁 < (𝐴 + 𝐵) → 0 ≤ (𝐵 − (𝑁 − 𝐴))))
52	51	imp 406	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) ∧ 𝑁 < (𝐴 + 𝐵)) → 0 ≤ (𝐵 − (𝑁 − 𝐴)))
53		elnn0z 12549	. . . . . . . 8 ⊢ ((𝐵 − (𝑁 − 𝐴)) ∈ ℕ₀ ↔ ((𝐵 − (𝑁 − 𝐴)) ∈ ℤ ∧ 0 ≤ (𝐵 − (𝑁 − 𝐴))))
54	8, 52, 53	sylanbrc 583	. . . . . . 7 ⊢ ((((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) ∧ 𝑁 < (𝐴 + 𝐵)) → (𝐵 − (𝑁 − 𝐴)) ∈ ℕ₀)
55	54	exp31 419	. . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ∈ ℤ → (𝑁 < (𝐴 + 𝐵) → (𝐵 − (𝑁 − 𝐴)) ∈ ℕ₀)))
56	2, 3, 55	syl2anc 584	. . . . 5 ⊢ (𝐵 ∈ (0...𝑁) → (𝐴 ∈ ℤ → (𝑁 < (𝐴 + 𝐵) → (𝐵 − (𝑁 − 𝐴)) ∈ ℕ₀)))
57	1, 56	mpan9 506	. . . 4 ⊢ ((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → (𝑁 < (𝐴 + 𝐵) → (𝐵 − (𝑁 − 𝐴)) ∈ ℕ₀))
58	57	imp 406	. . 3 ⊢ (((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) ∧ 𝑁 < (𝐴 + 𝐵)) → (𝐵 − (𝑁 − 𝐴)) ∈ ℕ₀)
59		elfznn0 13588	. . . 4 ⊢ (𝐴 ∈ (0...𝑁) → 𝐴 ∈ ℕ₀)
60	59	ad2antrr 726	. . 3 ⊢ (((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) ∧ 𝑁 < (𝐴 + 𝐵)) → 𝐴 ∈ ℕ₀)
61		elfzle2 13496	. . . . . . 7 ⊢ (𝐵 ∈ (0...𝑁) → 𝐵 ≤ 𝑁)
62	61	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → 𝐵 ≤ 𝑁)
63		elfzel2 13490	. . . . . . . . . 10 ⊢ (𝐴 ∈ (0...𝑁) → 𝑁 ∈ ℤ)
64	63	zcnd 12646	. . . . . . . . 9 ⊢ (𝐴 ∈ (0...𝑁) → 𝑁 ∈ ℂ)
65	1	zcnd 12646	. . . . . . . . 9 ⊢ (𝐴 ∈ (0...𝑁) → 𝐴 ∈ ℂ)
66	64, 65	jca 511	. . . . . . . 8 ⊢ (𝐴 ∈ (0...𝑁) → (𝑁 ∈ ℂ ∧ 𝐴 ∈ ℂ))
67	66	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → (𝑁 ∈ ℂ ∧ 𝐴 ∈ ℂ))
68		npcan 11437	. . . . . . 7 ⊢ ((𝑁 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((𝑁 − 𝐴) + 𝐴) = 𝑁)
69	67, 68	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → ((𝑁 − 𝐴) + 𝐴) = 𝑁)
70	62, 69	breqtrrd 5138	. . . . 5 ⊢ ((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → 𝐵 ≤ ((𝑁 − 𝐴) + 𝐴))
71	3	zred 12645	. . . . . . 7 ⊢ (𝐵 ∈ (0...𝑁) → 𝐵 ∈ ℝ)
72	71	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → 𝐵 ∈ ℝ)
73	63	zred 12645	. . . . . . . 8 ⊢ (𝐴 ∈ (0...𝑁) → 𝑁 ∈ ℝ)
74	1	zred 12645	. . . . . . . 8 ⊢ (𝐴 ∈ (0...𝑁) → 𝐴 ∈ ℝ)
75	73, 74	resubcld 11613	. . . . . . 7 ⊢ (𝐴 ∈ (0...𝑁) → (𝑁 − 𝐴) ∈ ℝ)
76	75	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → (𝑁 − 𝐴) ∈ ℝ)
77	74	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → 𝐴 ∈ ℝ)
78	72, 76, 77	lesubadd2d 11784	. . . . 5 ⊢ ((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → ((𝐵 − (𝑁 − 𝐴)) ≤ 𝐴 ↔ 𝐵 ≤ ((𝑁 − 𝐴) + 𝐴)))
79	70, 78	mpbird 257	. . . 4 ⊢ ((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → (𝐵 − (𝑁 − 𝐴)) ≤ 𝐴)
80	79	adantr 480	. . 3 ⊢ (((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) ∧ 𝑁 < (𝐴 + 𝐵)) → (𝐵 − (𝑁 − 𝐴)) ≤ 𝐴)
81		elfz2nn0 13586	. . 3 ⊢ ((𝐵 − (𝑁 − 𝐴)) ∈ (0...𝐴) ↔ ((𝐵 − (𝑁 − 𝐴)) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ (𝐵 − (𝑁 − 𝐴)) ≤ 𝐴))
82	58, 60, 80, 81	syl3anbrc 1344	. 2 ⊢ (((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) ∧ 𝑁 < (𝐴 + 𝐵)) → (𝐵 − (𝑁 − 𝐴)) ∈ (0...𝐴))
83	82	ex 412	1 ⊢ ((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → (𝑁 < (𝐴 + 𝐵) → (𝐵 − (𝑁 − 𝐴)) ∈ (0...𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 class class class wbr 5110 (class class class)co 7390 ℂcc 11073 ℝcr 11074 0cc0 11075 + caddc 11078 < clt 11215 ≤ cle 11216 − cmin 11412 ℕ₀cn0 12449 ℤcz 12536 ...cfz 13475

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 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-n0 12450 df-z 12537 df-uz 12801 df-fz 13476

This theorem is referenced by: (None)

W3C validator