Step |
Hyp |
Ref |
Expression |
1 |
|
eqeq2 |
⊢ ( 𝑗 = 0 → ( ( ( 2 · 𝑛 ) + 1 ) = 𝑗 ↔ ( ( 2 · 𝑛 ) + 1 ) = 0 ) ) |
2 |
1
|
rexbidv |
⊢ ( 𝑗 = 0 → ( ∃ 𝑛 ∈ ℤ ( ( 2 · 𝑛 ) + 1 ) = 𝑗 ↔ ∃ 𝑛 ∈ ℤ ( ( 2 · 𝑛 ) + 1 ) = 0 ) ) |
3 |
|
eqeq2 |
⊢ ( 𝑗 = 0 → ( ( 𝑘 · 2 ) = 𝑗 ↔ ( 𝑘 · 2 ) = 0 ) ) |
4 |
3
|
rexbidv |
⊢ ( 𝑗 = 0 → ( ∃ 𝑘 ∈ ℤ ( 𝑘 · 2 ) = 𝑗 ↔ ∃ 𝑘 ∈ ℤ ( 𝑘 · 2 ) = 0 ) ) |
5 |
2 4
|
orbi12d |
⊢ ( 𝑗 = 0 → ( ( ∃ 𝑛 ∈ ℤ ( ( 2 · 𝑛 ) + 1 ) = 𝑗 ∨ ∃ 𝑘 ∈ ℤ ( 𝑘 · 2 ) = 𝑗 ) ↔ ( ∃ 𝑛 ∈ ℤ ( ( 2 · 𝑛 ) + 1 ) = 0 ∨ ∃ 𝑘 ∈ ℤ ( 𝑘 · 2 ) = 0 ) ) ) |
6 |
|
eqeq2 |
⊢ ( 𝑗 = 𝑚 → ( ( ( 2 · 𝑛 ) + 1 ) = 𝑗 ↔ ( ( 2 · 𝑛 ) + 1 ) = 𝑚 ) ) |
7 |
6
|
rexbidv |
⊢ ( 𝑗 = 𝑚 → ( ∃ 𝑛 ∈ ℤ ( ( 2 · 𝑛 ) + 1 ) = 𝑗 ↔ ∃ 𝑛 ∈ ℤ ( ( 2 · 𝑛 ) + 1 ) = 𝑚 ) ) |
8 |
|
oveq2 |
⊢ ( 𝑛 = 𝑥 → ( 2 · 𝑛 ) = ( 2 · 𝑥 ) ) |
9 |
8
|
oveq1d |
⊢ ( 𝑛 = 𝑥 → ( ( 2 · 𝑛 ) + 1 ) = ( ( 2 · 𝑥 ) + 1 ) ) |
10 |
9
|
eqeq1d |
⊢ ( 𝑛 = 𝑥 → ( ( ( 2 · 𝑛 ) + 1 ) = 𝑚 ↔ ( ( 2 · 𝑥 ) + 1 ) = 𝑚 ) ) |
11 |
10
|
cbvrexvw |
⊢ ( ∃ 𝑛 ∈ ℤ ( ( 2 · 𝑛 ) + 1 ) = 𝑚 ↔ ∃ 𝑥 ∈ ℤ ( ( 2 · 𝑥 ) + 1 ) = 𝑚 ) |
12 |
7 11
|
bitrdi |
⊢ ( 𝑗 = 𝑚 → ( ∃ 𝑛 ∈ ℤ ( ( 2 · 𝑛 ) + 1 ) = 𝑗 ↔ ∃ 𝑥 ∈ ℤ ( ( 2 · 𝑥 ) + 1 ) = 𝑚 ) ) |
13 |
|
eqeq2 |
⊢ ( 𝑗 = 𝑚 → ( ( 𝑘 · 2 ) = 𝑗 ↔ ( 𝑘 · 2 ) = 𝑚 ) ) |
14 |
13
|
rexbidv |
⊢ ( 𝑗 = 𝑚 → ( ∃ 𝑘 ∈ ℤ ( 𝑘 · 2 ) = 𝑗 ↔ ∃ 𝑘 ∈ ℤ ( 𝑘 · 2 ) = 𝑚 ) ) |
15 |
|
oveq1 |
⊢ ( 𝑘 = 𝑦 → ( 𝑘 · 2 ) = ( 𝑦 · 2 ) ) |
16 |
15
|
eqeq1d |
⊢ ( 𝑘 = 𝑦 → ( ( 𝑘 · 2 ) = 𝑚 ↔ ( 𝑦 · 2 ) = 𝑚 ) ) |
17 |
16
|
cbvrexvw |
⊢ ( ∃ 𝑘 ∈ ℤ ( 𝑘 · 2 ) = 𝑚 ↔ ∃ 𝑦 ∈ ℤ ( 𝑦 · 2 ) = 𝑚 ) |
18 |
14 17
|
bitrdi |
⊢ ( 𝑗 = 𝑚 → ( ∃ 𝑘 ∈ ℤ ( 𝑘 · 2 ) = 𝑗 ↔ ∃ 𝑦 ∈ ℤ ( 𝑦 · 2 ) = 𝑚 ) ) |
19 |
12 18
|
orbi12d |
⊢ ( 𝑗 = 𝑚 → ( ( ∃ 𝑛 ∈ ℤ ( ( 2 · 𝑛 ) + 1 ) = 𝑗 ∨ ∃ 𝑘 ∈ ℤ ( 𝑘 · 2 ) = 𝑗 ) ↔ ( ∃ 𝑥 ∈ ℤ ( ( 2 · 𝑥 ) + 1 ) = 𝑚 ∨ ∃ 𝑦 ∈ ℤ ( 𝑦 · 2 ) = 𝑚 ) ) ) |
20 |
|
eqeq2 |
⊢ ( 𝑗 = ( 𝑚 + 1 ) → ( ( ( 2 · 𝑛 ) + 1 ) = 𝑗 ↔ ( ( 2 · 𝑛 ) + 1 ) = ( 𝑚 + 1 ) ) ) |
21 |
20
|
rexbidv |
⊢ ( 𝑗 = ( 𝑚 + 1 ) → ( ∃ 𝑛 ∈ ℤ ( ( 2 · 𝑛 ) + 1 ) = 𝑗 ↔ ∃ 𝑛 ∈ ℤ ( ( 2 · 𝑛 ) + 1 ) = ( 𝑚 + 1 ) ) ) |
22 |
|
eqeq2 |
⊢ ( 𝑗 = ( 𝑚 + 1 ) → ( ( 𝑘 · 2 ) = 𝑗 ↔ ( 𝑘 · 2 ) = ( 𝑚 + 1 ) ) ) |
23 |
22
|
rexbidv |
⊢ ( 𝑗 = ( 𝑚 + 1 ) → ( ∃ 𝑘 ∈ ℤ ( 𝑘 · 2 ) = 𝑗 ↔ ∃ 𝑘 ∈ ℤ ( 𝑘 · 2 ) = ( 𝑚 + 1 ) ) ) |
24 |
21 23
|
orbi12d |
⊢ ( 𝑗 = ( 𝑚 + 1 ) → ( ( ∃ 𝑛 ∈ ℤ ( ( 2 · 𝑛 ) + 1 ) = 𝑗 ∨ ∃ 𝑘 ∈ ℤ ( 𝑘 · 2 ) = 𝑗 ) ↔ ( ∃ 𝑛 ∈ ℤ ( ( 2 · 𝑛 ) + 1 ) = ( 𝑚 + 1 ) ∨ ∃ 𝑘 ∈ ℤ ( 𝑘 · 2 ) = ( 𝑚 + 1 ) ) ) ) |
25 |
|
eqeq2 |
⊢ ( 𝑗 = 𝑁 → ( ( ( 2 · 𝑛 ) + 1 ) = 𝑗 ↔ ( ( 2 · 𝑛 ) + 1 ) = 𝑁 ) ) |
26 |
25
|
rexbidv |
⊢ ( 𝑗 = 𝑁 → ( ∃ 𝑛 ∈ ℤ ( ( 2 · 𝑛 ) + 1 ) = 𝑗 ↔ ∃ 𝑛 ∈ ℤ ( ( 2 · 𝑛 ) + 1 ) = 𝑁 ) ) |
27 |
|
eqeq2 |
⊢ ( 𝑗 = 𝑁 → ( ( 𝑘 · 2 ) = 𝑗 ↔ ( 𝑘 · 2 ) = 𝑁 ) ) |
28 |
27
|
rexbidv |
⊢ ( 𝑗 = 𝑁 → ( ∃ 𝑘 ∈ ℤ ( 𝑘 · 2 ) = 𝑗 ↔ ∃ 𝑘 ∈ ℤ ( 𝑘 · 2 ) = 𝑁 ) ) |
29 |
26 28
|
orbi12d |
⊢ ( 𝑗 = 𝑁 → ( ( ∃ 𝑛 ∈ ℤ ( ( 2 · 𝑛 ) + 1 ) = 𝑗 ∨ ∃ 𝑘 ∈ ℤ ( 𝑘 · 2 ) = 𝑗 ) ↔ ( ∃ 𝑛 ∈ ℤ ( ( 2 · 𝑛 ) + 1 ) = 𝑁 ∨ ∃ 𝑘 ∈ ℤ ( 𝑘 · 2 ) = 𝑁 ) ) ) |
30 |
|
0z |
⊢ 0 ∈ ℤ |
31 |
|
2cn |
⊢ 2 ∈ ℂ |
32 |
31
|
mul02i |
⊢ ( 0 · 2 ) = 0 |
33 |
|
oveq1 |
⊢ ( 𝑘 = 0 → ( 𝑘 · 2 ) = ( 0 · 2 ) ) |
34 |
33
|
eqeq1d |
⊢ ( 𝑘 = 0 → ( ( 𝑘 · 2 ) = 0 ↔ ( 0 · 2 ) = 0 ) ) |
35 |
34
|
rspcev |
⊢ ( ( 0 ∈ ℤ ∧ ( 0 · 2 ) = 0 ) → ∃ 𝑘 ∈ ℤ ( 𝑘 · 2 ) = 0 ) |
36 |
30 32 35
|
mp2an |
⊢ ∃ 𝑘 ∈ ℤ ( 𝑘 · 2 ) = 0 |
37 |
36
|
olci |
⊢ ( ∃ 𝑛 ∈ ℤ ( ( 2 · 𝑛 ) + 1 ) = 0 ∨ ∃ 𝑘 ∈ ℤ ( 𝑘 · 2 ) = 0 ) |
38 |
|
orcom |
⊢ ( ( ∃ 𝑥 ∈ ℤ ( ( 2 · 𝑥 ) + 1 ) = 𝑚 ∨ ∃ 𝑦 ∈ ℤ ( 𝑦 · 2 ) = 𝑚 ) ↔ ( ∃ 𝑦 ∈ ℤ ( 𝑦 · 2 ) = 𝑚 ∨ ∃ 𝑥 ∈ ℤ ( ( 2 · 𝑥 ) + 1 ) = 𝑚 ) ) |
39 |
|
zcn |
⊢ ( 𝑦 ∈ ℤ → 𝑦 ∈ ℂ ) |
40 |
|
mulcom |
⊢ ( ( 𝑦 ∈ ℂ ∧ 2 ∈ ℂ ) → ( 𝑦 · 2 ) = ( 2 · 𝑦 ) ) |
41 |
39 31 40
|
sylancl |
⊢ ( 𝑦 ∈ ℤ → ( 𝑦 · 2 ) = ( 2 · 𝑦 ) ) |
42 |
41
|
adantl |
⊢ ( ( 𝑚 ∈ ℕ0 ∧ 𝑦 ∈ ℤ ) → ( 𝑦 · 2 ) = ( 2 · 𝑦 ) ) |
43 |
42
|
eqeq1d |
⊢ ( ( 𝑚 ∈ ℕ0 ∧ 𝑦 ∈ ℤ ) → ( ( 𝑦 · 2 ) = 𝑚 ↔ ( 2 · 𝑦 ) = 𝑚 ) ) |
44 |
|
eqid |
⊢ ( ( 2 · 𝑦 ) + 1 ) = ( ( 2 · 𝑦 ) + 1 ) |
45 |
|
oveq2 |
⊢ ( 𝑛 = 𝑦 → ( 2 · 𝑛 ) = ( 2 · 𝑦 ) ) |
46 |
45
|
oveq1d |
⊢ ( 𝑛 = 𝑦 → ( ( 2 · 𝑛 ) + 1 ) = ( ( 2 · 𝑦 ) + 1 ) ) |
47 |
46
|
eqeq1d |
⊢ ( 𝑛 = 𝑦 → ( ( ( 2 · 𝑛 ) + 1 ) = ( ( 2 · 𝑦 ) + 1 ) ↔ ( ( 2 · 𝑦 ) + 1 ) = ( ( 2 · 𝑦 ) + 1 ) ) ) |
48 |
47
|
rspcev |
⊢ ( ( 𝑦 ∈ ℤ ∧ ( ( 2 · 𝑦 ) + 1 ) = ( ( 2 · 𝑦 ) + 1 ) ) → ∃ 𝑛 ∈ ℤ ( ( 2 · 𝑛 ) + 1 ) = ( ( 2 · 𝑦 ) + 1 ) ) |
49 |
44 48
|
mpan2 |
⊢ ( 𝑦 ∈ ℤ → ∃ 𝑛 ∈ ℤ ( ( 2 · 𝑛 ) + 1 ) = ( ( 2 · 𝑦 ) + 1 ) ) |
50 |
|
oveq1 |
⊢ ( ( 2 · 𝑦 ) = 𝑚 → ( ( 2 · 𝑦 ) + 1 ) = ( 𝑚 + 1 ) ) |
51 |
50
|
eqeq2d |
⊢ ( ( 2 · 𝑦 ) = 𝑚 → ( ( ( 2 · 𝑛 ) + 1 ) = ( ( 2 · 𝑦 ) + 1 ) ↔ ( ( 2 · 𝑛 ) + 1 ) = ( 𝑚 + 1 ) ) ) |
52 |
51
|
rexbidv |
⊢ ( ( 2 · 𝑦 ) = 𝑚 → ( ∃ 𝑛 ∈ ℤ ( ( 2 · 𝑛 ) + 1 ) = ( ( 2 · 𝑦 ) + 1 ) ↔ ∃ 𝑛 ∈ ℤ ( ( 2 · 𝑛 ) + 1 ) = ( 𝑚 + 1 ) ) ) |
53 |
49 52
|
syl5ibcom |
⊢ ( 𝑦 ∈ ℤ → ( ( 2 · 𝑦 ) = 𝑚 → ∃ 𝑛 ∈ ℤ ( ( 2 · 𝑛 ) + 1 ) = ( 𝑚 + 1 ) ) ) |
54 |
53
|
adantl |
⊢ ( ( 𝑚 ∈ ℕ0 ∧ 𝑦 ∈ ℤ ) → ( ( 2 · 𝑦 ) = 𝑚 → ∃ 𝑛 ∈ ℤ ( ( 2 · 𝑛 ) + 1 ) = ( 𝑚 + 1 ) ) ) |
55 |
43 54
|
sylbid |
⊢ ( ( 𝑚 ∈ ℕ0 ∧ 𝑦 ∈ ℤ ) → ( ( 𝑦 · 2 ) = 𝑚 → ∃ 𝑛 ∈ ℤ ( ( 2 · 𝑛 ) + 1 ) = ( 𝑚 + 1 ) ) ) |
56 |
55
|
rexlimdva |
⊢ ( 𝑚 ∈ ℕ0 → ( ∃ 𝑦 ∈ ℤ ( 𝑦 · 2 ) = 𝑚 → ∃ 𝑛 ∈ ℤ ( ( 2 · 𝑛 ) + 1 ) = ( 𝑚 + 1 ) ) ) |
57 |
|
peano2z |
⊢ ( 𝑥 ∈ ℤ → ( 𝑥 + 1 ) ∈ ℤ ) |
58 |
|
zcn |
⊢ ( 𝑥 ∈ ℤ → 𝑥 ∈ ℂ ) |
59 |
|
mulcom |
⊢ ( ( 𝑥 ∈ ℂ ∧ 2 ∈ ℂ ) → ( 𝑥 · 2 ) = ( 2 · 𝑥 ) ) |
60 |
31 59
|
mpan2 |
⊢ ( 𝑥 ∈ ℂ → ( 𝑥 · 2 ) = ( 2 · 𝑥 ) ) |
61 |
31
|
mulid2i |
⊢ ( 1 · 2 ) = 2 |
62 |
61
|
a1i |
⊢ ( 𝑥 ∈ ℂ → ( 1 · 2 ) = 2 ) |
63 |
60 62
|
oveq12d |
⊢ ( 𝑥 ∈ ℂ → ( ( 𝑥 · 2 ) + ( 1 · 2 ) ) = ( ( 2 · 𝑥 ) + 2 ) ) |
64 |
|
df-2 |
⊢ 2 = ( 1 + 1 ) |
65 |
64
|
oveq2i |
⊢ ( ( 2 · 𝑥 ) + 2 ) = ( ( 2 · 𝑥 ) + ( 1 + 1 ) ) |
66 |
63 65
|
eqtrdi |
⊢ ( 𝑥 ∈ ℂ → ( ( 𝑥 · 2 ) + ( 1 · 2 ) ) = ( ( 2 · 𝑥 ) + ( 1 + 1 ) ) ) |
67 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
68 |
|
adddir |
⊢ ( ( 𝑥 ∈ ℂ ∧ 1 ∈ ℂ ∧ 2 ∈ ℂ ) → ( ( 𝑥 + 1 ) · 2 ) = ( ( 𝑥 · 2 ) + ( 1 · 2 ) ) ) |
69 |
67 31 68
|
mp3an23 |
⊢ ( 𝑥 ∈ ℂ → ( ( 𝑥 + 1 ) · 2 ) = ( ( 𝑥 · 2 ) + ( 1 · 2 ) ) ) |
70 |
|
mulcl |
⊢ ( ( 2 ∈ ℂ ∧ 𝑥 ∈ ℂ ) → ( 2 · 𝑥 ) ∈ ℂ ) |
71 |
31 70
|
mpan |
⊢ ( 𝑥 ∈ ℂ → ( 2 · 𝑥 ) ∈ ℂ ) |
72 |
|
addass |
⊢ ( ( ( 2 · 𝑥 ) ∈ ℂ ∧ 1 ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( ( 2 · 𝑥 ) + 1 ) + 1 ) = ( ( 2 · 𝑥 ) + ( 1 + 1 ) ) ) |
73 |
67 67 72
|
mp3an23 |
⊢ ( ( 2 · 𝑥 ) ∈ ℂ → ( ( ( 2 · 𝑥 ) + 1 ) + 1 ) = ( ( 2 · 𝑥 ) + ( 1 + 1 ) ) ) |
74 |
71 73
|
syl |
⊢ ( 𝑥 ∈ ℂ → ( ( ( 2 · 𝑥 ) + 1 ) + 1 ) = ( ( 2 · 𝑥 ) + ( 1 + 1 ) ) ) |
75 |
66 69 74
|
3eqtr4d |
⊢ ( 𝑥 ∈ ℂ → ( ( 𝑥 + 1 ) · 2 ) = ( ( ( 2 · 𝑥 ) + 1 ) + 1 ) ) |
76 |
58 75
|
syl |
⊢ ( 𝑥 ∈ ℤ → ( ( 𝑥 + 1 ) · 2 ) = ( ( ( 2 · 𝑥 ) + 1 ) + 1 ) ) |
77 |
76
|
adantl |
⊢ ( ( 𝑚 ∈ ℕ0 ∧ 𝑥 ∈ ℤ ) → ( ( 𝑥 + 1 ) · 2 ) = ( ( ( 2 · 𝑥 ) + 1 ) + 1 ) ) |
78 |
|
oveq1 |
⊢ ( 𝑘 = ( 𝑥 + 1 ) → ( 𝑘 · 2 ) = ( ( 𝑥 + 1 ) · 2 ) ) |
79 |
78
|
eqeq1d |
⊢ ( 𝑘 = ( 𝑥 + 1 ) → ( ( 𝑘 · 2 ) = ( ( ( 2 · 𝑥 ) + 1 ) + 1 ) ↔ ( ( 𝑥 + 1 ) · 2 ) = ( ( ( 2 · 𝑥 ) + 1 ) + 1 ) ) ) |
80 |
79
|
rspcev |
⊢ ( ( ( 𝑥 + 1 ) ∈ ℤ ∧ ( ( 𝑥 + 1 ) · 2 ) = ( ( ( 2 · 𝑥 ) + 1 ) + 1 ) ) → ∃ 𝑘 ∈ ℤ ( 𝑘 · 2 ) = ( ( ( 2 · 𝑥 ) + 1 ) + 1 ) ) |
81 |
57 77 80
|
syl2an2 |
⊢ ( ( 𝑚 ∈ ℕ0 ∧ 𝑥 ∈ ℤ ) → ∃ 𝑘 ∈ ℤ ( 𝑘 · 2 ) = ( ( ( 2 · 𝑥 ) + 1 ) + 1 ) ) |
82 |
|
oveq1 |
⊢ ( ( ( 2 · 𝑥 ) + 1 ) = 𝑚 → ( ( ( 2 · 𝑥 ) + 1 ) + 1 ) = ( 𝑚 + 1 ) ) |
83 |
82
|
eqeq2d |
⊢ ( ( ( 2 · 𝑥 ) + 1 ) = 𝑚 → ( ( 𝑘 · 2 ) = ( ( ( 2 · 𝑥 ) + 1 ) + 1 ) ↔ ( 𝑘 · 2 ) = ( 𝑚 + 1 ) ) ) |
84 |
83
|
rexbidv |
⊢ ( ( ( 2 · 𝑥 ) + 1 ) = 𝑚 → ( ∃ 𝑘 ∈ ℤ ( 𝑘 · 2 ) = ( ( ( 2 · 𝑥 ) + 1 ) + 1 ) ↔ ∃ 𝑘 ∈ ℤ ( 𝑘 · 2 ) = ( 𝑚 + 1 ) ) ) |
85 |
81 84
|
syl5ibcom |
⊢ ( ( 𝑚 ∈ ℕ0 ∧ 𝑥 ∈ ℤ ) → ( ( ( 2 · 𝑥 ) + 1 ) = 𝑚 → ∃ 𝑘 ∈ ℤ ( 𝑘 · 2 ) = ( 𝑚 + 1 ) ) ) |
86 |
85
|
rexlimdva |
⊢ ( 𝑚 ∈ ℕ0 → ( ∃ 𝑥 ∈ ℤ ( ( 2 · 𝑥 ) + 1 ) = 𝑚 → ∃ 𝑘 ∈ ℤ ( 𝑘 · 2 ) = ( 𝑚 + 1 ) ) ) |
87 |
56 86
|
orim12d |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ∃ 𝑦 ∈ ℤ ( 𝑦 · 2 ) = 𝑚 ∨ ∃ 𝑥 ∈ ℤ ( ( 2 · 𝑥 ) + 1 ) = 𝑚 ) → ( ∃ 𝑛 ∈ ℤ ( ( 2 · 𝑛 ) + 1 ) = ( 𝑚 + 1 ) ∨ ∃ 𝑘 ∈ ℤ ( 𝑘 · 2 ) = ( 𝑚 + 1 ) ) ) ) |
88 |
38 87
|
syl5bi |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ∃ 𝑥 ∈ ℤ ( ( 2 · 𝑥 ) + 1 ) = 𝑚 ∨ ∃ 𝑦 ∈ ℤ ( 𝑦 · 2 ) = 𝑚 ) → ( ∃ 𝑛 ∈ ℤ ( ( 2 · 𝑛 ) + 1 ) = ( 𝑚 + 1 ) ∨ ∃ 𝑘 ∈ ℤ ( 𝑘 · 2 ) = ( 𝑚 + 1 ) ) ) ) |
89 |
5 19 24 29 37 88
|
nn0ind |
⊢ ( 𝑁 ∈ ℕ0 → ( ∃ 𝑛 ∈ ℤ ( ( 2 · 𝑛 ) + 1 ) = 𝑁 ∨ ∃ 𝑘 ∈ ℤ ( 𝑘 · 2 ) = 𝑁 ) ) |