Step |
Hyp |
Ref |
Expression |
1 |
|
3odd |
⊢ 3 ∈ Odd |
2 |
1
|
a1i |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ ; 1 2 ) → 3 ∈ Odd ) |
3 |
2
|
anim1i |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ ; 1 2 ) ∧ 𝑁 ∈ Even ) → ( 3 ∈ Odd ∧ 𝑁 ∈ Even ) ) |
4 |
3
|
ancomd |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ ; 1 2 ) ∧ 𝑁 ∈ Even ) → ( 𝑁 ∈ Even ∧ 3 ∈ Odd ) ) |
5 |
|
emoo |
⊢ ( ( 𝑁 ∈ Even ∧ 3 ∈ Odd ) → ( 𝑁 − 3 ) ∈ Odd ) |
6 |
4 5
|
syl |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ ; 1 2 ) ∧ 𝑁 ∈ Even ) → ( 𝑁 − 3 ) ∈ Odd ) |
7 |
|
breq2 |
⊢ ( 𝑚 = ( 𝑁 − 3 ) → ( 7 < 𝑚 ↔ 7 < ( 𝑁 − 3 ) ) ) |
8 |
|
eleq1 |
⊢ ( 𝑚 = ( 𝑁 − 3 ) → ( 𝑚 ∈ GoldbachOdd ↔ ( 𝑁 − 3 ) ∈ GoldbachOdd ) ) |
9 |
7 8
|
imbi12d |
⊢ ( 𝑚 = ( 𝑁 − 3 ) → ( ( 7 < 𝑚 → 𝑚 ∈ GoldbachOdd ) ↔ ( 7 < ( 𝑁 − 3 ) → ( 𝑁 − 3 ) ∈ GoldbachOdd ) ) ) |
10 |
9
|
adantl |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ ; 1 2 ) ∧ 𝑁 ∈ Even ) ∧ 𝑚 = ( 𝑁 − 3 ) ) → ( ( 7 < 𝑚 → 𝑚 ∈ GoldbachOdd ) ↔ ( 7 < ( 𝑁 − 3 ) → ( 𝑁 − 3 ) ∈ GoldbachOdd ) ) ) |
11 |
6 10
|
rspcdv |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ ; 1 2 ) ∧ 𝑁 ∈ Even ) → ( ∀ 𝑚 ∈ Odd ( 7 < 𝑚 → 𝑚 ∈ GoldbachOdd ) → ( 7 < ( 𝑁 − 3 ) → ( 𝑁 − 3 ) ∈ GoldbachOdd ) ) ) |
12 |
|
eluz2 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ ; 1 2 ) ↔ ( ; 1 2 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ; 1 2 ≤ 𝑁 ) ) |
13 |
|
7p3e10 |
⊢ ( 7 + 3 ) = ; 1 0 |
14 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
15 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
16 |
|
2nn |
⊢ 2 ∈ ℕ |
17 |
|
2pos |
⊢ 0 < 2 |
18 |
14 15 16 17
|
declt |
⊢ ; 1 0 < ; 1 2 |
19 |
13 18
|
eqbrtri |
⊢ ( 7 + 3 ) < ; 1 2 |
20 |
|
7re |
⊢ 7 ∈ ℝ |
21 |
|
3re |
⊢ 3 ∈ ℝ |
22 |
20 21
|
readdcli |
⊢ ( 7 + 3 ) ∈ ℝ |
23 |
|
2nn0 |
⊢ 2 ∈ ℕ0 |
24 |
14 23
|
deccl |
⊢ ; 1 2 ∈ ℕ0 |
25 |
24
|
nn0rei |
⊢ ; 1 2 ∈ ℝ |
26 |
|
zre |
⊢ ( 𝑁 ∈ ℤ → 𝑁 ∈ ℝ ) |
27 |
|
ltletr |
⊢ ( ( ( 7 + 3 ) ∈ ℝ ∧ ; 1 2 ∈ ℝ ∧ 𝑁 ∈ ℝ ) → ( ( ( 7 + 3 ) < ; 1 2 ∧ ; 1 2 ≤ 𝑁 ) → ( 7 + 3 ) < 𝑁 ) ) |
28 |
22 25 26 27
|
mp3an12i |
⊢ ( 𝑁 ∈ ℤ → ( ( ( 7 + 3 ) < ; 1 2 ∧ ; 1 2 ≤ 𝑁 ) → ( 7 + 3 ) < 𝑁 ) ) |
29 |
19 28
|
mpani |
⊢ ( 𝑁 ∈ ℤ → ( ; 1 2 ≤ 𝑁 → ( 7 + 3 ) < 𝑁 ) ) |
30 |
29
|
imp |
⊢ ( ( 𝑁 ∈ ℤ ∧ ; 1 2 ≤ 𝑁 ) → ( 7 + 3 ) < 𝑁 ) |
31 |
30
|
3adant1 |
⊢ ( ( ; 1 2 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ; 1 2 ≤ 𝑁 ) → ( 7 + 3 ) < 𝑁 ) |
32 |
20
|
a1i |
⊢ ( ( ; 1 2 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ; 1 2 ≤ 𝑁 ) → 7 ∈ ℝ ) |
33 |
21
|
a1i |
⊢ ( ( ; 1 2 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ; 1 2 ≤ 𝑁 ) → 3 ∈ ℝ ) |
34 |
26
|
3ad2ant2 |
⊢ ( ( ; 1 2 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ; 1 2 ≤ 𝑁 ) → 𝑁 ∈ ℝ ) |
35 |
32 33 34
|
ltaddsubd |
⊢ ( ( ; 1 2 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ; 1 2 ≤ 𝑁 ) → ( ( 7 + 3 ) < 𝑁 ↔ 7 < ( 𝑁 − 3 ) ) ) |
36 |
31 35
|
mpbid |
⊢ ( ( ; 1 2 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ ; 1 2 ≤ 𝑁 ) → 7 < ( 𝑁 − 3 ) ) |
37 |
12 36
|
sylbi |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ ; 1 2 ) → 7 < ( 𝑁 − 3 ) ) |
38 |
37
|
adantr |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ ; 1 2 ) ∧ 𝑁 ∈ Even ) → 7 < ( 𝑁 − 3 ) ) |
39 |
|
simpr |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ ; 1 2 ) ∧ 𝑁 ∈ Even ) ∧ ( 𝑁 − 3 ) ∈ GoldbachOdd ) → ( 𝑁 − 3 ) ∈ GoldbachOdd ) |
40 |
|
oveq1 |
⊢ ( 𝑜 = ( 𝑁 − 3 ) → ( 𝑜 + 3 ) = ( ( 𝑁 − 3 ) + 3 ) ) |
41 |
40
|
eqeq2d |
⊢ ( 𝑜 = ( 𝑁 − 3 ) → ( 𝑁 = ( 𝑜 + 3 ) ↔ 𝑁 = ( ( 𝑁 − 3 ) + 3 ) ) ) |
42 |
41
|
adantl |
⊢ ( ( ( ( 𝑁 ∈ ( ℤ≥ ‘ ; 1 2 ) ∧ 𝑁 ∈ Even ) ∧ ( 𝑁 − 3 ) ∈ GoldbachOdd ) ∧ 𝑜 = ( 𝑁 − 3 ) ) → ( 𝑁 = ( 𝑜 + 3 ) ↔ 𝑁 = ( ( 𝑁 − 3 ) + 3 ) ) ) |
43 |
|
eluzelcn |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ ; 1 2 ) → 𝑁 ∈ ℂ ) |
44 |
|
3cn |
⊢ 3 ∈ ℂ |
45 |
43 44
|
jctir |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ ; 1 2 ) → ( 𝑁 ∈ ℂ ∧ 3 ∈ ℂ ) ) |
46 |
45
|
adantr |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ ; 1 2 ) ∧ 𝑁 ∈ Even ) → ( 𝑁 ∈ ℂ ∧ 3 ∈ ℂ ) ) |
47 |
46
|
adantr |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ ; 1 2 ) ∧ 𝑁 ∈ Even ) ∧ ( 𝑁 − 3 ) ∈ GoldbachOdd ) → ( 𝑁 ∈ ℂ ∧ 3 ∈ ℂ ) ) |
48 |
|
npcan |
⊢ ( ( 𝑁 ∈ ℂ ∧ 3 ∈ ℂ ) → ( ( 𝑁 − 3 ) + 3 ) = 𝑁 ) |
49 |
48
|
eqcomd |
⊢ ( ( 𝑁 ∈ ℂ ∧ 3 ∈ ℂ ) → 𝑁 = ( ( 𝑁 − 3 ) + 3 ) ) |
50 |
47 49
|
syl |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ ; 1 2 ) ∧ 𝑁 ∈ Even ) ∧ ( 𝑁 − 3 ) ∈ GoldbachOdd ) → 𝑁 = ( ( 𝑁 − 3 ) + 3 ) ) |
51 |
39 42 50
|
rspcedvd |
⊢ ( ( ( 𝑁 ∈ ( ℤ≥ ‘ ; 1 2 ) ∧ 𝑁 ∈ Even ) ∧ ( 𝑁 − 3 ) ∈ GoldbachOdd ) → ∃ 𝑜 ∈ GoldbachOdd 𝑁 = ( 𝑜 + 3 ) ) |
52 |
51
|
ex |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ ; 1 2 ) ∧ 𝑁 ∈ Even ) → ( ( 𝑁 − 3 ) ∈ GoldbachOdd → ∃ 𝑜 ∈ GoldbachOdd 𝑁 = ( 𝑜 + 3 ) ) ) |
53 |
38 52
|
embantd |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ ; 1 2 ) ∧ 𝑁 ∈ Even ) → ( ( 7 < ( 𝑁 − 3 ) → ( 𝑁 − 3 ) ∈ GoldbachOdd ) → ∃ 𝑜 ∈ GoldbachOdd 𝑁 = ( 𝑜 + 3 ) ) ) |
54 |
11 53
|
syldc |
⊢ ( ∀ 𝑚 ∈ Odd ( 7 < 𝑚 → 𝑚 ∈ GoldbachOdd ) → ( ( 𝑁 ∈ ( ℤ≥ ‘ ; 1 2 ) ∧ 𝑁 ∈ Even ) → ∃ 𝑜 ∈ GoldbachOdd 𝑁 = ( 𝑜 + 3 ) ) ) |