modaddmodup - Metamath Proof Explorer

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Theorem modaddmodup 13582

Description: The sum of an integer modulo a positive integer and another integer minus the positive integer equals the sum of the two integers modulo the positive integer if the other integer is in the upper part of the range between 0 and the positive integer. (Contributed by AV, 30-Oct-2018.)

**Assertion**
Ref	Expression
modaddmodup	⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) → (𝐵 ∈ ((𝑀 − (𝐴 mod 𝑀))..^𝑀) → ((𝐵 + (𝐴 mod 𝑀)) − 𝑀) = ((𝐵 + 𝐴) mod 𝑀)))

**Proof of Theorem modaddmodup**
Step	Hyp	Ref	Expression
1		elfzoelz 13316	. . . . . . . 8 ⊢ (𝐵 ∈ ((𝑀 − (𝐴 mod 𝑀))..^𝑀) → 𝐵 ∈ ℤ)
2	1	zred 12355	. . . . . . 7 ⊢ (𝐵 ∈ ((𝑀 − (𝐴 mod 𝑀))..^𝑀) → 𝐵 ∈ ℝ)
3	2	adantr 480	. . . . . 6 ⊢ ((𝐵 ∈ ((𝑀 − (𝐴 mod 𝑀))..^𝑀) ∧ (𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ)) → 𝐵 ∈ ℝ)
4		zmodcl 13539	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) → (𝐴 mod 𝑀) ∈ ℕ₀)
5	4	nn0red 12224	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) → (𝐴 mod 𝑀) ∈ ℝ)
6	5	adantl 481	. . . . . 6 ⊢ ((𝐵 ∈ ((𝑀 − (𝐴 mod 𝑀))..^𝑀) ∧ (𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ)) → (𝐴 mod 𝑀) ∈ ℝ)
7	3, 6	readdcld 10935	. . . . 5 ⊢ ((𝐵 ∈ ((𝑀 − (𝐴 mod 𝑀))..^𝑀) ∧ (𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ)) → (𝐵 + (𝐴 mod 𝑀)) ∈ ℝ)
8	7	ancoms 458	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) ∧ 𝐵 ∈ ((𝑀 − (𝐴 mod 𝑀))..^𝑀)) → (𝐵 + (𝐴 mod 𝑀)) ∈ ℝ)
9		nnrp 12670	. . . . 5 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ⁺)
10	9	ad2antlr 723	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) ∧ 𝐵 ∈ ((𝑀 − (𝐴 mod 𝑀))..^𝑀)) → 𝑀 ∈ ℝ⁺)
11		elfzo2 13319	. . . . . 6 ⊢ (𝐵 ∈ ((𝑀 − (𝐴 mod 𝑀))..^𝑀) ↔ (𝐵 ∈ (ℤ_≥‘(𝑀 − (𝐴 mod 𝑀))) ∧ 𝑀 ∈ ℤ ∧ 𝐵 < 𝑀))
12		eluz2 12517	. . . . . . . 8 ⊢ (𝐵 ∈ (ℤ_≥‘(𝑀 − (𝐴 mod 𝑀))) ↔ ((𝑀 − (𝐴 mod 𝑀)) ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ (𝑀 − (𝐴 mod 𝑀)) ≤ 𝐵))
13		nnre 11910	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ)
14	13	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) → 𝑀 ∈ ℝ)
15	14	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℤ ∧ (𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ)) → 𝑀 ∈ ℝ)
16	5	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℤ ∧ (𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ)) → (𝐴 mod 𝑀) ∈ ℝ)
17		zre 12253	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ)
18	17	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℤ ∧ (𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ)) → 𝐵 ∈ ℝ)
19	15, 16, 18	lesubaddd 11502	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℤ ∧ (𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ)) → ((𝑀 − (𝐴 mod 𝑀)) ≤ 𝐵 ↔ 𝑀 ≤ (𝐵 + (𝐴 mod 𝑀))))
20	19	biimpd 228	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℤ ∧ (𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ)) → ((𝑀 − (𝐴 mod 𝑀)) ≤ 𝐵 → 𝑀 ≤ (𝐵 + (𝐴 mod 𝑀))))
21	20	impancom 451	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℤ ∧ (𝑀 − (𝐴 mod 𝑀)) ≤ 𝐵) → ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) → 𝑀 ≤ (𝐵 + (𝐴 mod 𝑀))))
22	21	3adant1 1128	. . . . . . . 8 ⊢ (((𝑀 − (𝐴 mod 𝑀)) ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ (𝑀 − (𝐴 mod 𝑀)) ≤ 𝐵) → ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) → 𝑀 ≤ (𝐵 + (𝐴 mod 𝑀))))
23	12, 22	sylbi 216	. . . . . . 7 ⊢ (𝐵 ∈ (ℤ_≥‘(𝑀 − (𝐴 mod 𝑀))) → ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) → 𝑀 ≤ (𝐵 + (𝐴 mod 𝑀))))
24	23	3ad2ant1 1131	. . . . . 6 ⊢ ((𝐵 ∈ (ℤ_≥‘(𝑀 − (𝐴 mod 𝑀))) ∧ 𝑀 ∈ ℤ ∧ 𝐵 < 𝑀) → ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) → 𝑀 ≤ (𝐵 + (𝐴 mod 𝑀))))
25	11, 24	sylbi 216	. . . . 5 ⊢ (𝐵 ∈ ((𝑀 − (𝐴 mod 𝑀))..^𝑀) → ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) → 𝑀 ≤ (𝐵 + (𝐴 mod 𝑀))))
26	25	impcom 407	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) ∧ 𝐵 ∈ ((𝑀 − (𝐴 mod 𝑀))..^𝑀)) → 𝑀 ≤ (𝐵 + (𝐴 mod 𝑀)))
27		eluzelz 12521	. . . . . . . . 9 ⊢ (𝐵 ∈ (ℤ_≥‘(𝑀 − (𝐴 mod 𝑀))) → 𝐵 ∈ ℤ)
28	17, 5	anim12i 612	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ ℤ ∧ (𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ)) → (𝐵 ∈ ℝ ∧ (𝐴 mod 𝑀) ∈ ℝ))
29	13, 13	jca 511	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℕ → (𝑀 ∈ ℝ ∧ 𝑀 ∈ ℝ))
30	29	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) → (𝑀 ∈ ℝ ∧ 𝑀 ∈ ℝ))
31	30	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ ℤ ∧ (𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ)) → (𝑀 ∈ ℝ ∧ 𝑀 ∈ ℝ))
32	28, 31	jca 511	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℤ ∧ (𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ)) → ((𝐵 ∈ ℝ ∧ (𝐴 mod 𝑀) ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑀 ∈ ℝ)))
33	32	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ ℤ ∧ (𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ)) ∧ 𝐵 < 𝑀) → ((𝐵 ∈ ℝ ∧ (𝐴 mod 𝑀) ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑀 ∈ ℝ)))
34		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ ℤ ∧ (𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ)) ∧ 𝐵 < 𝑀) → 𝐵 < 𝑀)
35		zre 12253	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
36		modlt 13528	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℝ⁺) → (𝐴 mod 𝑀) < 𝑀)
37	35, 9, 36	syl2an 595	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) → (𝐴 mod 𝑀) < 𝑀)
38	5, 14, 37	ltled 11053	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) → (𝐴 mod 𝑀) ≤ 𝑀)
39	38	ad2antlr 723	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ ℤ ∧ (𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ)) ∧ 𝐵 < 𝑀) → (𝐴 mod 𝑀) ≤ 𝑀)
40	34, 39	jca 511	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ ℤ ∧ (𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ)) ∧ 𝐵 < 𝑀) → (𝐵 < 𝑀 ∧ (𝐴 mod 𝑀) ≤ 𝑀))
41		ltleadd 11388	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ ℝ ∧ (𝐴 mod 𝑀) ∈ ℝ) ∧ (𝑀 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → ((𝐵 < 𝑀 ∧ (𝐴 mod 𝑀) ≤ 𝑀) → (𝐵 + (𝐴 mod 𝑀)) < (𝑀 + 𝑀)))
42	33, 40, 41	sylc 65	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ ℤ ∧ (𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ)) ∧ 𝐵 < 𝑀) → (𝐵 + (𝐴 mod 𝑀)) < (𝑀 + 𝑀))
43		nncn 11911	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℂ)
44	43	2timesd 12146	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ → (2 · 𝑀) = (𝑀 + 𝑀))
45	44	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) → (2 · 𝑀) = (𝑀 + 𝑀))
46	45	ad2antlr 723	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ ℤ ∧ (𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ)) ∧ 𝐵 < 𝑀) → (2 · 𝑀) = (𝑀 + 𝑀))
47	42, 46	breqtrrd 5098	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ ℤ ∧ (𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ)) ∧ 𝐵 < 𝑀) → (𝐵 + (𝐴 mod 𝑀)) < (2 · 𝑀))
48	47	exp31 419	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℤ → ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) → (𝐵 < 𝑀 → (𝐵 + (𝐴 mod 𝑀)) < (2 · 𝑀))))
49	48	com23 86	. . . . . . . . 9 ⊢ (𝐵 ∈ ℤ → (𝐵 < 𝑀 → ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) → (𝐵 + (𝐴 mod 𝑀)) < (2 · 𝑀))))
50	27, 49	syl 17	. . . . . . . 8 ⊢ (𝐵 ∈ (ℤ_≥‘(𝑀 − (𝐴 mod 𝑀))) → (𝐵 < 𝑀 → ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) → (𝐵 + (𝐴 mod 𝑀)) < (2 · 𝑀))))
51	50	imp 406	. . . . . . 7 ⊢ ((𝐵 ∈ (ℤ_≥‘(𝑀 − (𝐴 mod 𝑀))) ∧ 𝐵 < 𝑀) → ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) → (𝐵 + (𝐴 mod 𝑀)) < (2 · 𝑀)))
52	51	3adant2 1129	. . . . . 6 ⊢ ((𝐵 ∈ (ℤ_≥‘(𝑀 − (𝐴 mod 𝑀))) ∧ 𝑀 ∈ ℤ ∧ 𝐵 < 𝑀) → ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) → (𝐵 + (𝐴 mod 𝑀)) < (2 · 𝑀)))
53	11, 52	sylbi 216	. . . . 5 ⊢ (𝐵 ∈ ((𝑀 − (𝐴 mod 𝑀))..^𝑀) → ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) → (𝐵 + (𝐴 mod 𝑀)) < (2 · 𝑀)))
54	53	impcom 407	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) ∧ 𝐵 ∈ ((𝑀 − (𝐴 mod 𝑀))..^𝑀)) → (𝐵 + (𝐴 mod 𝑀)) < (2 · 𝑀))
55		2submod 13580	. . . . 5 ⊢ ((((𝐵 + (𝐴 mod 𝑀)) ∈ ℝ ∧ 𝑀 ∈ ℝ⁺) ∧ (𝑀 ≤ (𝐵 + (𝐴 mod 𝑀)) ∧ (𝐵 + (𝐴 mod 𝑀)) < (2 · 𝑀))) → ((𝐵 + (𝐴 mod 𝑀)) mod 𝑀) = ((𝐵 + (𝐴 mod 𝑀)) − 𝑀))
56	55	eqcomd 2744	. . . 4 ⊢ ((((𝐵 + (𝐴 mod 𝑀)) ∈ ℝ ∧ 𝑀 ∈ ℝ⁺) ∧ (𝑀 ≤ (𝐵 + (𝐴 mod 𝑀)) ∧ (𝐵 + (𝐴 mod 𝑀)) < (2 · 𝑀))) → ((𝐵 + (𝐴 mod 𝑀)) − 𝑀) = ((𝐵 + (𝐴 mod 𝑀)) mod 𝑀))
57	8, 10, 26, 54, 56	syl22anc 835	. . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) ∧ 𝐵 ∈ ((𝑀 − (𝐴 mod 𝑀))..^𝑀)) → ((𝐵 + (𝐴 mod 𝑀)) − 𝑀) = ((𝐵 + (𝐴 mod 𝑀)) mod 𝑀))
58	35	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) → 𝐴 ∈ ℝ)
59	58	adantr 480	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) ∧ 𝐵 ∈ ((𝑀 − (𝐴 mod 𝑀))..^𝑀)) → 𝐴 ∈ ℝ)
60	2	adantl 481	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) ∧ 𝐵 ∈ ((𝑀 − (𝐴 mod 𝑀))..^𝑀)) → 𝐵 ∈ ℝ)
61		modadd2mod 13569	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ⁺) → ((𝐵 + (𝐴 mod 𝑀)) mod 𝑀) = ((𝐵 + 𝐴) mod 𝑀))
62	59, 60, 10, 61	syl3anc 1369	. . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) ∧ 𝐵 ∈ ((𝑀 − (𝐴 mod 𝑀))..^𝑀)) → ((𝐵 + (𝐴 mod 𝑀)) mod 𝑀) = ((𝐵 + 𝐴) mod 𝑀))
63	57, 62	eqtrd 2778	. 2 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) ∧ 𝐵 ∈ ((𝑀 − (𝐴 mod 𝑀))..^𝑀)) → ((𝐵 + (𝐴 mod 𝑀)) − 𝑀) = ((𝐵 + 𝐴) mod 𝑀))
64	63	ex 412	1 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) → (𝐵 ∈ ((𝑀 − (𝐴 mod 𝑀))..^𝑀) → ((𝐵 + (𝐴 mod 𝑀)) − 𝑀) = ((𝐵 + 𝐴) mod 𝑀)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 class class class wbr 5070 ‘cfv 6418 (class class class)co 7255 ℝcr 10801 + caddc 10805 · cmul 10807 < clt 10940 ≤ cle 10941 − cmin 11135 ℕcn 11903 2c2 11958 ℤcz 12249 ℤ_≥cuz 12511 ℝ⁺crp 12659 ..^cfzo 13311 mod cmo 13517

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-sup 9131 df-inf 9132 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-n0 12164 df-z 12250 df-uz 12512 df-rp 12660 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518

This theorem is referenced by: cshwidxmod 14444

W3C validator