Metamath Proof Explorer

Theorem mod42tp1mod8

Description: If a number is 3 modulo 4 , twice the number plus 1 is 7 modulo 8 . (Contributed by AV, 19-Aug-2021)

Ref Expression
Assertion mod42tp1mod8 ⊢ N ∈ ℤ ∧ N mod 4 = 3 → 2 ⋅ N + 1 mod 8 = 7

Proof

Step Hyp Ref Expression
1 4nn ⊢ 4 ∈ ℕ
2 1 a1i ⊢ N ∈ ℤ → 4 ∈ ℕ
3 3nn0 ⊢ 3 ∈ ℕ 0
4 3 a1i ⊢ N ∈ ℤ → 3 ∈ ℕ 0
5 3lt4 ⊢ 3 < 4
6 4 5 jctir ⊢ N ∈ ℤ → 3 ∈ ℕ 0 ∧ 3 < 4
7 modremain ⊢ N ∈ ℤ ∧ 4 ∈ ℕ ∧ 3 ∈ ℕ 0 ∧ 3 < 4 → N mod 4 = 3 ↔ ∃ z ∈ ℤ z ⋅ 4 + 3 = N
8 2 6 7 mpd3an23 ⊢ N ∈ ℤ → N mod 4 = 3 ↔ ∃ z ∈ ℤ z ⋅ 4 + 3 = N
9 2cnd ⊢ N ∈ ℤ ∧ z ∈ ℤ → 2 ∈ ℂ
10 simpr ⊢ N ∈ ℤ ∧ z ∈ ℤ → z ∈ ℤ
11 4z ⊢ 4 ∈ ℤ
12 11 a1i ⊢ N ∈ ℤ ∧ z ∈ ℤ → 4 ∈ ℤ
13 10 12 zmulcld ⊢ N ∈ ℤ ∧ z ∈ ℤ → z ⋅ 4 ∈ ℤ
14 13 zcnd ⊢ N ∈ ℤ ∧ z ∈ ℤ → z ⋅ 4 ∈ ℂ
15 3cn ⊢ 3 ∈ ℂ
16 15 a1i ⊢ N ∈ ℤ ∧ z ∈ ℤ → 3 ∈ ℂ
17 9 14 16 adddid ⊢ N ∈ ℤ ∧ z ∈ ℤ → 2 ⁢ z ⋅ 4 + 3 = 2 ⁢ z ⋅ 4 + 2 ⋅ 3
18 10 zcnd ⊢ N ∈ ℤ ∧ z ∈ ℤ → z ∈ ℂ
19 4cn ⊢ 4 ∈ ℂ
20 19 a1i ⊢ N ∈ ℤ ∧ z ∈ ℤ → 4 ∈ ℂ
21 9 18 20 mul12d ⊢ N ∈ ℤ ∧ z ∈ ℤ → 2 ⁢ z ⋅ 4 = z ⁢ 2 ⋅ 4
22 2t4e8 ⊢ 2 ⋅ 4 = 8
23 22 oveq2i ⊢ z ⁢ 2 ⋅ 4 = z ⋅ 8
24 21 23 eqtrdi ⊢ N ∈ ℤ ∧ z ∈ ℤ → 2 ⁢ z ⋅ 4 = z ⋅ 8
25 2t3e6 ⊢ 2 ⋅ 3 = 6
26 25 a1i ⊢ N ∈ ℤ ∧ z ∈ ℤ → 2 ⋅ 3 = 6
27 24 26 oveq12d ⊢ N ∈ ℤ ∧ z ∈ ℤ → 2 ⁢ z ⋅ 4 + 2 ⋅ 3 = z ⋅ 8 + 6
28 17 27 eqtrd ⊢ N ∈ ℤ ∧ z ∈ ℤ → 2 ⁢ z ⋅ 4 + 3 = z ⋅ 8 + 6
29 28 oveq1d ⊢ N ∈ ℤ ∧ z ∈ ℤ → 2 ⁢ z ⋅ 4 + 3 + 1 = z ⋅ 8 + 6 + 1
30 id ⊢ z ∈ ℤ → z ∈ ℤ
31 8nn ⊢ 8 ∈ ℕ
32 31 nnzi ⊢ 8 ∈ ℤ
33 32 a1i ⊢ z ∈ ℤ → 8 ∈ ℤ
34 30 33 zmulcld ⊢ z ∈ ℤ → z ⋅ 8 ∈ ℤ
35 34 zcnd ⊢ z ∈ ℤ → z ⋅ 8 ∈ ℂ
36 6cn ⊢ 6 ∈ ℂ
37 36 a1i ⊢ z ∈ ℤ → 6 ∈ ℂ
38 1cnd ⊢ z ∈ ℤ → 1 ∈ ℂ
39 35 37 38 addassd ⊢ z ∈ ℤ → z ⋅ 8 + 6 + 1 = z ⋅ 8 + 6 + 1
40 6p1e7 ⊢ 6 + 1 = 7
41 40 a1i ⊢ z ∈ ℤ → 6 + 1 = 7
42 41 oveq2d ⊢ z ∈ ℤ → z ⋅ 8 + 6 + 1 = z ⋅ 8 + 7
43 39 42 eqtrd ⊢ z ∈ ℤ → z ⋅ 8 + 6 + 1 = z ⋅ 8 + 7
44 43 adantl ⊢ N ∈ ℤ ∧ z ∈ ℤ → z ⋅ 8 + 6 + 1 = z ⋅ 8 + 7
45 29 44 eqtrd ⊢ N ∈ ℤ ∧ z ∈ ℤ → 2 ⁢ z ⋅ 4 + 3 + 1 = z ⋅ 8 + 7
46 45 oveq1d ⊢ N ∈ ℤ ∧ z ∈ ℤ → 2 ⁢ z ⋅ 4 + 3 + 1 mod 8 = z ⋅ 8 + 7 mod 8
47 nnrp ⊢ 8 ∈ ℕ → 8 ∈ ℝ +
48 31 47 mp1i ⊢ z ∈ ℤ → 8 ∈ ℝ +
49 0xr ⊢ 0 ∈ ℝ *
50 49 a1i ⊢ z ∈ ℤ → 0 ∈ ℝ *
51 8re ⊢ 8 ∈ ℝ
52 51 rexri ⊢ 8 ∈ ℝ *
53 52 a1i ⊢ z ∈ ℤ → 8 ∈ ℝ *
54 7re ⊢ 7 ∈ ℝ
55 54 rexri ⊢ 7 ∈ ℝ *
56 55 a1i ⊢ z ∈ ℤ → 7 ∈ ℝ *
57 0re ⊢ 0 ∈ ℝ
58 7pos ⊢ 0 < 7
59 57 54 58 ltleii ⊢ 0 ≤ 7
60 59 a1i ⊢ z ∈ ℤ → 0 ≤ 7
61 7lt8 ⊢ 7 < 8
62 61 a1i ⊢ z ∈ ℤ → 7 < 8
63 50 53 56 60 62 elicod ⊢ z ∈ ℤ → 7 ∈ 0 8
64 muladdmodid ⊢ z ∈ ℤ ∧ 8 ∈ ℝ + ∧ 7 ∈ 0 8 → z ⋅ 8 + 7 mod 8 = 7
65 48 63 64 mpd3an23 ⊢ z ∈ ℤ → z ⋅ 8 + 7 mod 8 = 7
66 65 adantl ⊢ N ∈ ℤ ∧ z ∈ ℤ → z ⋅ 8 + 7 mod 8 = 7
67 46 66 eqtrd ⊢ N ∈ ℤ ∧ z ∈ ℤ → 2 ⁢ z ⋅ 4 + 3 + 1 mod 8 = 7
68 oveq2 ⊢ z ⋅ 4 + 3 = N → 2 ⁢ z ⋅ 4 + 3 = 2 ⋅ N
69 68 oveq1d ⊢ z ⋅ 4 + 3 = N → 2 ⁢ z ⋅ 4 + 3 + 1 = 2 ⋅ N + 1
70 69 oveq1d ⊢ z ⋅ 4 + 3 = N → 2 ⁢ z ⋅ 4 + 3 + 1 mod 8 = 2 ⋅ N + 1 mod 8
71 70 eqeq1d ⊢ z ⋅ 4 + 3 = N → 2 ⁢ z ⋅ 4 + 3 + 1 mod 8 = 7 ↔ 2 ⋅ N + 1 mod 8 = 7
72 67 71 syl5ibcom ⊢ N ∈ ℤ ∧ z ∈ ℤ → z ⋅ 4 + 3 = N → 2 ⋅ N + 1 mod 8 = 7
73 72 rexlimdva ⊢ N ∈ ℤ → ∃ z ∈ ℤ z ⋅ 4 + 3 = N → 2 ⋅ N + 1 mod 8 = 7
74 8 73 sylbid ⊢ N ∈ ℤ → N mod 4 = 3 → 2 ⋅ N + 1 mod 8 = 7
75 74 imp ⊢ N ∈ ℤ ∧ N mod 4 = 3 → 2 ⋅ N + 1 mod 8 = 7