Step |
Hyp |
Ref |
Expression |
1 |
|
isgbe |
⊢ ( 𝑍 ∈ GoldbachEven ↔ ( 𝑍 ∈ Even ∧ ∃ 𝑝 ∈ ℙ ∃ 𝑞 ∈ ℙ ( 𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = ( 𝑝 + 𝑞 ) ) ) ) |
2 |
|
oddprmuzge3 |
⊢ ( ( 𝑝 ∈ ℙ ∧ 𝑝 ∈ Odd ) → 𝑝 ∈ ( ℤ≥ ‘ 3 ) ) |
3 |
2
|
ancoms |
⊢ ( ( 𝑝 ∈ Odd ∧ 𝑝 ∈ ℙ ) → 𝑝 ∈ ( ℤ≥ ‘ 3 ) ) |
4 |
|
oddprmuzge3 |
⊢ ( ( 𝑞 ∈ ℙ ∧ 𝑞 ∈ Odd ) → 𝑞 ∈ ( ℤ≥ ‘ 3 ) ) |
5 |
4
|
ancoms |
⊢ ( ( 𝑞 ∈ Odd ∧ 𝑞 ∈ ℙ ) → 𝑞 ∈ ( ℤ≥ ‘ 3 ) ) |
6 |
|
eluz2 |
⊢ ( 𝑝 ∈ ( ℤ≥ ‘ 3 ) ↔ ( 3 ∈ ℤ ∧ 𝑝 ∈ ℤ ∧ 3 ≤ 𝑝 ) ) |
7 |
|
eluz2 |
⊢ ( 𝑞 ∈ ( ℤ≥ ‘ 3 ) ↔ ( 3 ∈ ℤ ∧ 𝑞 ∈ ℤ ∧ 3 ≤ 𝑞 ) ) |
8 |
|
zre |
⊢ ( 𝑞 ∈ ℤ → 𝑞 ∈ ℝ ) |
9 |
|
zre |
⊢ ( 𝑝 ∈ ℤ → 𝑝 ∈ ℝ ) |
10 |
|
3re |
⊢ 3 ∈ ℝ |
11 |
10 10
|
pm3.2i |
⊢ ( 3 ∈ ℝ ∧ 3 ∈ ℝ ) |
12 |
|
pm3.22 |
⊢ ( ( 𝑞 ∈ ℝ ∧ 𝑝 ∈ ℝ ) → ( 𝑝 ∈ ℝ ∧ 𝑞 ∈ ℝ ) ) |
13 |
|
le2add |
⊢ ( ( ( 3 ∈ ℝ ∧ 3 ∈ ℝ ) ∧ ( 𝑝 ∈ ℝ ∧ 𝑞 ∈ ℝ ) ) → ( ( 3 ≤ 𝑝 ∧ 3 ≤ 𝑞 ) → ( 3 + 3 ) ≤ ( 𝑝 + 𝑞 ) ) ) |
14 |
11 12 13
|
sylancr |
⊢ ( ( 𝑞 ∈ ℝ ∧ 𝑝 ∈ ℝ ) → ( ( 3 ≤ 𝑝 ∧ 3 ≤ 𝑞 ) → ( 3 + 3 ) ≤ ( 𝑝 + 𝑞 ) ) ) |
15 |
14
|
ancomsd |
⊢ ( ( 𝑞 ∈ ℝ ∧ 𝑝 ∈ ℝ ) → ( ( 3 ≤ 𝑞 ∧ 3 ≤ 𝑝 ) → ( 3 + 3 ) ≤ ( 𝑝 + 𝑞 ) ) ) |
16 |
|
3p3e6 |
⊢ ( 3 + 3 ) = 6 |
17 |
16
|
breq1i |
⊢ ( ( 3 + 3 ) ≤ ( 𝑝 + 𝑞 ) ↔ 6 ≤ ( 𝑝 + 𝑞 ) ) |
18 |
|
5lt6 |
⊢ 5 < 6 |
19 |
|
5re |
⊢ 5 ∈ ℝ |
20 |
19
|
a1i |
⊢ ( ( 𝑞 ∈ ℝ ∧ 𝑝 ∈ ℝ ) → 5 ∈ ℝ ) |
21 |
|
6re |
⊢ 6 ∈ ℝ |
22 |
21
|
a1i |
⊢ ( ( 𝑞 ∈ ℝ ∧ 𝑝 ∈ ℝ ) → 6 ∈ ℝ ) |
23 |
|
readdcl |
⊢ ( ( 𝑝 ∈ ℝ ∧ 𝑞 ∈ ℝ ) → ( 𝑝 + 𝑞 ) ∈ ℝ ) |
24 |
23
|
ancoms |
⊢ ( ( 𝑞 ∈ ℝ ∧ 𝑝 ∈ ℝ ) → ( 𝑝 + 𝑞 ) ∈ ℝ ) |
25 |
|
ltletr |
⊢ ( ( 5 ∈ ℝ ∧ 6 ∈ ℝ ∧ ( 𝑝 + 𝑞 ) ∈ ℝ ) → ( ( 5 < 6 ∧ 6 ≤ ( 𝑝 + 𝑞 ) ) → 5 < ( 𝑝 + 𝑞 ) ) ) |
26 |
20 22 24 25
|
syl3anc |
⊢ ( ( 𝑞 ∈ ℝ ∧ 𝑝 ∈ ℝ ) → ( ( 5 < 6 ∧ 6 ≤ ( 𝑝 + 𝑞 ) ) → 5 < ( 𝑝 + 𝑞 ) ) ) |
27 |
18 26
|
mpani |
⊢ ( ( 𝑞 ∈ ℝ ∧ 𝑝 ∈ ℝ ) → ( 6 ≤ ( 𝑝 + 𝑞 ) → 5 < ( 𝑝 + 𝑞 ) ) ) |
28 |
17 27
|
syl5bi |
⊢ ( ( 𝑞 ∈ ℝ ∧ 𝑝 ∈ ℝ ) → ( ( 3 + 3 ) ≤ ( 𝑝 + 𝑞 ) → 5 < ( 𝑝 + 𝑞 ) ) ) |
29 |
15 28
|
syld |
⊢ ( ( 𝑞 ∈ ℝ ∧ 𝑝 ∈ ℝ ) → ( ( 3 ≤ 𝑞 ∧ 3 ≤ 𝑝 ) → 5 < ( 𝑝 + 𝑞 ) ) ) |
30 |
8 9 29
|
syl2an |
⊢ ( ( 𝑞 ∈ ℤ ∧ 𝑝 ∈ ℤ ) → ( ( 3 ≤ 𝑞 ∧ 3 ≤ 𝑝 ) → 5 < ( 𝑝 + 𝑞 ) ) ) |
31 |
30
|
ex |
⊢ ( 𝑞 ∈ ℤ → ( 𝑝 ∈ ℤ → ( ( 3 ≤ 𝑞 ∧ 3 ≤ 𝑝 ) → 5 < ( 𝑝 + 𝑞 ) ) ) ) |
32 |
31
|
adantl |
⊢ ( ( 3 ∈ ℤ ∧ 𝑞 ∈ ℤ ) → ( 𝑝 ∈ ℤ → ( ( 3 ≤ 𝑞 ∧ 3 ≤ 𝑝 ) → 5 < ( 𝑝 + 𝑞 ) ) ) ) |
33 |
32
|
com23 |
⊢ ( ( 3 ∈ ℤ ∧ 𝑞 ∈ ℤ ) → ( ( 3 ≤ 𝑞 ∧ 3 ≤ 𝑝 ) → ( 𝑝 ∈ ℤ → 5 < ( 𝑝 + 𝑞 ) ) ) ) |
34 |
33
|
exp4b |
⊢ ( 3 ∈ ℤ → ( 𝑞 ∈ ℤ → ( 3 ≤ 𝑞 → ( 3 ≤ 𝑝 → ( 𝑝 ∈ ℤ → 5 < ( 𝑝 + 𝑞 ) ) ) ) ) ) |
35 |
34
|
3imp |
⊢ ( ( 3 ∈ ℤ ∧ 𝑞 ∈ ℤ ∧ 3 ≤ 𝑞 ) → ( 3 ≤ 𝑝 → ( 𝑝 ∈ ℤ → 5 < ( 𝑝 + 𝑞 ) ) ) ) |
36 |
35
|
com13 |
⊢ ( 𝑝 ∈ ℤ → ( 3 ≤ 𝑝 → ( ( 3 ∈ ℤ ∧ 𝑞 ∈ ℤ ∧ 3 ≤ 𝑞 ) → 5 < ( 𝑝 + 𝑞 ) ) ) ) |
37 |
36
|
imp |
⊢ ( ( 𝑝 ∈ ℤ ∧ 3 ≤ 𝑝 ) → ( ( 3 ∈ ℤ ∧ 𝑞 ∈ ℤ ∧ 3 ≤ 𝑞 ) → 5 < ( 𝑝 + 𝑞 ) ) ) |
38 |
37
|
3adant1 |
⊢ ( ( 3 ∈ ℤ ∧ 𝑝 ∈ ℤ ∧ 3 ≤ 𝑝 ) → ( ( 3 ∈ ℤ ∧ 𝑞 ∈ ℤ ∧ 3 ≤ 𝑞 ) → 5 < ( 𝑝 + 𝑞 ) ) ) |
39 |
7 38
|
syl5bi |
⊢ ( ( 3 ∈ ℤ ∧ 𝑝 ∈ ℤ ∧ 3 ≤ 𝑝 ) → ( 𝑞 ∈ ( ℤ≥ ‘ 3 ) → 5 < ( 𝑝 + 𝑞 ) ) ) |
40 |
6 39
|
sylbi |
⊢ ( 𝑝 ∈ ( ℤ≥ ‘ 3 ) → ( 𝑞 ∈ ( ℤ≥ ‘ 3 ) → 5 < ( 𝑝 + 𝑞 ) ) ) |
41 |
40
|
imp |
⊢ ( ( 𝑝 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑞 ∈ ( ℤ≥ ‘ 3 ) ) → 5 < ( 𝑝 + 𝑞 ) ) |
42 |
3 5 41
|
syl2an |
⊢ ( ( ( 𝑝 ∈ Odd ∧ 𝑝 ∈ ℙ ) ∧ ( 𝑞 ∈ Odd ∧ 𝑞 ∈ ℙ ) ) → 5 < ( 𝑝 + 𝑞 ) ) |
43 |
42
|
an4s |
⊢ ( ( ( 𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ) ∧ ( 𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ ) ) → 5 < ( 𝑝 + 𝑞 ) ) |
44 |
43
|
ex |
⊢ ( ( 𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ) → ( ( 𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ ) → 5 < ( 𝑝 + 𝑞 ) ) ) |
45 |
44
|
3adant3 |
⊢ ( ( 𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = ( 𝑝 + 𝑞 ) ) → ( ( 𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ ) → 5 < ( 𝑝 + 𝑞 ) ) ) |
46 |
45
|
impcom |
⊢ ( ( ( 𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ ) ∧ ( 𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = ( 𝑝 + 𝑞 ) ) ) → 5 < ( 𝑝 + 𝑞 ) ) |
47 |
|
breq2 |
⊢ ( 𝑍 = ( 𝑝 + 𝑞 ) → ( 5 < 𝑍 ↔ 5 < ( 𝑝 + 𝑞 ) ) ) |
48 |
47
|
3ad2ant3 |
⊢ ( ( 𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = ( 𝑝 + 𝑞 ) ) → ( 5 < 𝑍 ↔ 5 < ( 𝑝 + 𝑞 ) ) ) |
49 |
48
|
adantl |
⊢ ( ( ( 𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ ) ∧ ( 𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = ( 𝑝 + 𝑞 ) ) ) → ( 5 < 𝑍 ↔ 5 < ( 𝑝 + 𝑞 ) ) ) |
50 |
46 49
|
mpbird |
⊢ ( ( ( 𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ ) ∧ ( 𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = ( 𝑝 + 𝑞 ) ) ) → 5 < 𝑍 ) |
51 |
50
|
ex |
⊢ ( ( 𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ ) → ( ( 𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = ( 𝑝 + 𝑞 ) ) → 5 < 𝑍 ) ) |
52 |
51
|
a1i |
⊢ ( 𝑍 ∈ Even → ( ( 𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ ) → ( ( 𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = ( 𝑝 + 𝑞 ) ) → 5 < 𝑍 ) ) ) |
53 |
52
|
rexlimdvv |
⊢ ( 𝑍 ∈ Even → ( ∃ 𝑝 ∈ ℙ ∃ 𝑞 ∈ ℙ ( 𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = ( 𝑝 + 𝑞 ) ) → 5 < 𝑍 ) ) |
54 |
53
|
imp |
⊢ ( ( 𝑍 ∈ Even ∧ ∃ 𝑝 ∈ ℙ ∃ 𝑞 ∈ ℙ ( 𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = ( 𝑝 + 𝑞 ) ) ) → 5 < 𝑍 ) |
55 |
1 54
|
sylbi |
⊢ ( 𝑍 ∈ GoldbachEven → 5 < 𝑍 ) |