2elfz2melfz - Mathbox for Alexander van der Vekens

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Theorem 2elfz2melfz 43525

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 12911	. . . . 5 ⊢ (𝐴 ∈ (0...𝑁) → 𝐴 ∈ ℤ)
2		elfzel2 12909	. . . . . 6 ⊢ (𝐵 ∈ (0...𝑁) → 𝑁 ∈ ℤ)
3		elfzelz 12911	. . . . . 6 ⊢ (𝐵 ∈ (0...𝑁) → 𝐵 ∈ ℤ)
4		simplr 767	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) → 𝐵 ∈ ℤ)
5		zsubcl 12027	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝑁 − 𝐴) ∈ ℤ)
6	5	adantlr 713	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) → (𝑁 − 𝐴) ∈ ℤ)
7	4, 6	zsubcld 12095	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) → (𝐵 − (𝑁 − 𝐴)) ∈ ℤ)
8	7	adantr 483	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) ∧ 𝑁 < (𝐴 + 𝐵)) → (𝐵 − (𝑁 − 𝐴)) ∈ ℤ)
9		zre 11988	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
10	9	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) → 𝑁 ∈ ℝ)
11		zaddcl 12025	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 + 𝐵) ∈ ℤ)
12	11	zred 12090	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 + 𝐵) ∈ ℝ)
13	12	expcom 416	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℤ → (𝐴 ∈ ℤ → (𝐴 + 𝐵) ∈ ℝ))
14	13	adantl 484	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ∈ ℤ → (𝐴 + 𝐵) ∈ ℝ))
15	14	imp 409	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) → (𝐴 + 𝐵) ∈ ℝ)
16	10, 15, 10	ltsub1d 11251	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) → (𝑁 < (𝐴 + 𝐵) ↔ (𝑁 − 𝑁) < ((𝐴 + 𝐵) − 𝑁)))
17		zre 11988	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ)
18	9, 17	anim12i 614	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝑁 ∈ ℝ ∧ 𝐵 ∈ ℝ))
19		zre 11988	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
20	18, 19	anim12i 614	. . . . . . . . . . . 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 11070	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℝ → (𝑁 − 𝑁) ∈ ℝ)
23	22	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → (𝑁 − 𝑁) ∈ ℝ)
24		readdcl 10622	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
25	24	expcom 416	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℝ → (𝐴 ∈ ℝ → (𝐴 + 𝐵) ∈ ℝ))
26	25	adantl 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ∈ ℝ → (𝐴 + 𝐵) ∈ ℝ))
27	26	imp 409	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
28		simpll 765	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → 𝑁 ∈ ℝ)
29	27, 28	resubcld 11070	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → ((𝐴 + 𝐵) − 𝑁) ∈ ℝ)
30	23, 29	jca 514	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ∈ ℝ) → ((𝑁 − 𝑁) ∈ ℝ ∧ ((𝐴 + 𝐵) − 𝑁) ∈ ℝ))
31		ltle 10731	. . . . . . . . . . . 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 11989	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
34	33	subidd 10987	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℤ → (𝑁 − 𝑁) = 0)
35	34	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) → (𝑁 − 𝑁) = 0)
36		zcn 11989	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℂ)
37	36	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℂ)
38	37	adantr 483	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) → 𝐵 ∈ ℂ)
39	33	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) → 𝑁 ∈ ℂ)
40		zcn 11989	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ)
41	40	adantl 484	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) → 𝐴 ∈ ℂ)
42		simp3 1134	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ ℂ ∧ 𝑁 ∈ ℂ ∧ 𝐴 ∈ ℂ) → 𝐴 ∈ ℂ)
43		simp1 1132	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ ℂ ∧ 𝑁 ∈ ℂ ∧ 𝐴 ∈ ℂ) → 𝐵 ∈ ℂ)
44	42, 43	addcomd 10844	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ ℂ ∧ 𝑁 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
45	44	oveq1d 7173	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℂ ∧ 𝑁 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((𝐴 + 𝐵) − 𝑁) = ((𝐵 + 𝐴) − 𝑁))
46		subsub3 10920	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℂ ∧ 𝑁 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝐵 − (𝑁 − 𝐴)) = ((𝐵 + 𝐴) − 𝑁))
47	45, 46	eqtr4d 2861	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ℂ ∧ 𝑁 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((𝐴 + 𝐵) − 𝑁) = (𝐵 − (𝑁 − 𝐴)))
48	38, 39, 41, 47	syl3anc 1367	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) → ((𝐴 + 𝐵) − 𝑁) = (𝐵 − (𝑁 − 𝐴)))
49	35, 48	breq12d 5081	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) → ((𝑁 − 𝑁) ≤ ((𝐴 + 𝐵) − 𝑁) ↔ 0 ≤ (𝐵 − (𝑁 − 𝐴))))
50	32, 49	sylibd 241	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) → ((𝑁 − 𝑁) < ((𝐴 + 𝐵) − 𝑁) → 0 ≤ (𝐵 − (𝑁 − 𝐴))))
51	16, 50	sylbid 242	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) → (𝑁 < (𝐴 + 𝐵) → 0 ≤ (𝐵 − (𝑁 − 𝐴))))
52	51	imp 409	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) ∧ 𝑁 < (𝐴 + 𝐵)) → 0 ≤ (𝐵 − (𝑁 − 𝐴)))
53		elnn0z 11997	. . . . . . . 8 ⊢ ((𝐵 − (𝑁 − 𝐴)) ∈ ℕ₀ ↔ ((𝐵 − (𝑁 − 𝐴)) ∈ ℤ ∧ 0 ≤ (𝐵 − (𝑁 − 𝐴))))
54	8, 52, 53	sylanbrc 585	. . . . . . 7 ⊢ ((((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐴 ∈ ℤ) ∧ 𝑁 < (𝐴 + 𝐵)) → (𝐵 − (𝑁 − 𝐴)) ∈ ℕ₀)
55	54	exp31 422	. . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ∈ ℤ → (𝑁 < (𝐴 + 𝐵) → (𝐵 − (𝑁 − 𝐴)) ∈ ℕ₀)))
56	2, 3, 55	syl2anc 586	. . . . 5 ⊢ (𝐵 ∈ (0...𝑁) → (𝐴 ∈ ℤ → (𝑁 < (𝐴 + 𝐵) → (𝐵 − (𝑁 − 𝐴)) ∈ ℕ₀)))
57	1, 56	mpan9 509	. . . 4 ⊢ ((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → (𝑁 < (𝐴 + 𝐵) → (𝐵 − (𝑁 − 𝐴)) ∈ ℕ₀))
58	57	imp 409	. . 3 ⊢ (((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) ∧ 𝑁 < (𝐴 + 𝐵)) → (𝐵 − (𝑁 − 𝐴)) ∈ ℕ₀)
59		elfznn0 13003	. . . 4 ⊢ (𝐴 ∈ (0...𝑁) → 𝐴 ∈ ℕ₀)
60	59	ad2antrr 724	. . 3 ⊢ (((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) ∧ 𝑁 < (𝐴 + 𝐵)) → 𝐴 ∈ ℕ₀)
61		elfzle2 12914	. . . . . . 7 ⊢ (𝐵 ∈ (0...𝑁) → 𝐵 ≤ 𝑁)
62	61	adantl 484	. . . . . 6 ⊢ ((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → 𝐵 ≤ 𝑁)
63		elfzel2 12909	. . . . . . . . . 10 ⊢ (𝐴 ∈ (0...𝑁) → 𝑁 ∈ ℤ)
64	63	zcnd 12091	. . . . . . . . 9 ⊢ (𝐴 ∈ (0...𝑁) → 𝑁 ∈ ℂ)
65	1	zcnd 12091	. . . . . . . . 9 ⊢ (𝐴 ∈ (0...𝑁) → 𝐴 ∈ ℂ)
66	64, 65	jca 514	. . . . . . . 8 ⊢ (𝐴 ∈ (0...𝑁) → (𝑁 ∈ ℂ ∧ 𝐴 ∈ ℂ))
67	66	adantr 483	. . . . . . 7 ⊢ ((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → (𝑁 ∈ ℂ ∧ 𝐴 ∈ ℂ))
68		npcan 10897	. . . . . . 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 5096	. . . . 5 ⊢ ((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → 𝐵 ≤ ((𝑁 − 𝐴) + 𝐴))
71	3	zred 12090	. . . . . . 7 ⊢ (𝐵 ∈ (0...𝑁) → 𝐵 ∈ ℝ)
72	71	adantl 484	. . . . . 6 ⊢ ((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → 𝐵 ∈ ℝ)
73	63	zred 12090	. . . . . . . 8 ⊢ (𝐴 ∈ (0...𝑁) → 𝑁 ∈ ℝ)
74	1	zred 12090	. . . . . . . 8 ⊢ (𝐴 ∈ (0...𝑁) → 𝐴 ∈ ℝ)
75	73, 74	resubcld 11070	. . . . . . 7 ⊢ (𝐴 ∈ (0...𝑁) → (𝑁 − 𝐴) ∈ ℝ)
76	75	adantr 483	. . . . . 6 ⊢ ((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → (𝑁 − 𝐴) ∈ ℝ)
77	74	adantr 483	. . . . . 6 ⊢ ((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → 𝐴 ∈ ℝ)
78	72, 76, 77	lesubadd2d 11241	. . . . 5 ⊢ ((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → ((𝐵 − (𝑁 − 𝐴)) ≤ 𝐴 ↔ 𝐵 ≤ ((𝑁 − 𝐴) + 𝐴)))
79	70, 78	mpbird 259	. . . 4 ⊢ ((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → (𝐵 − (𝑁 − 𝐴)) ≤ 𝐴)
80	79	adantr 483	. . 3 ⊢ (((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) ∧ 𝑁 < (𝐴 + 𝐵)) → (𝐵 − (𝑁 − 𝐴)) ≤ 𝐴)
81		elfz2nn0 13001	. . 3 ⊢ ((𝐵 − (𝑁 − 𝐴)) ∈ (0...𝐴) ↔ ((𝐵 − (𝑁 − 𝐴)) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀ ∧ (𝐵 − (𝑁 − 𝐴)) ≤ 𝐴))
82	58, 60, 80, 81	syl3anbrc 1339	. 2 ⊢ (((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) ∧ 𝑁 < (𝐴 + 𝐵)) → (𝐵 − (𝑁 − 𝐴)) ∈ (0...𝐴))
83	82	ex 415	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 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 (class class class)co 7158 ℂcc 10537 ℝcr 10538 0cc0 10539 + caddc 10542 < clt 10677 ≤ cle 10678 − cmin 10872 ℕ₀cn0 11900 ℤcz 11984 ...cfz 12895

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896

This theorem is referenced by: (None)

W3C validator