Step |
Hyp |
Ref |
Expression |
1 |
|
olc |
⊢ ( 𝑅 = 2 → ( ( 𝑃 = 2 ∨ 𝑄 = 2 ) ∨ 𝑅 = 2 ) ) |
2 |
1
|
a1d |
⊢ ( 𝑅 = 2 → ( ( 𝑁 ∈ Even ∧ ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ ) ∧ 𝑁 = ( ( 𝑃 + 𝑄 ) + 𝑅 ) ) → ( ( 𝑃 = 2 ∨ 𝑄 = 2 ) ∨ 𝑅 = 2 ) ) ) |
3 |
|
df-ne |
⊢ ( 𝑅 ≠ 2 ↔ ¬ 𝑅 = 2 ) |
4 |
|
eldifsn |
⊢ ( 𝑅 ∈ ( ℙ ∖ { 2 } ) ↔ ( 𝑅 ∈ ℙ ∧ 𝑅 ≠ 2 ) ) |
5 |
|
oddprmALTV |
⊢ ( 𝑅 ∈ ( ℙ ∖ { 2 } ) → 𝑅 ∈ Odd ) |
6 |
|
emoo |
⊢ ( ( 𝑁 ∈ Even ∧ 𝑅 ∈ Odd ) → ( 𝑁 − 𝑅 ) ∈ Odd ) |
7 |
6
|
expcom |
⊢ ( 𝑅 ∈ Odd → ( 𝑁 ∈ Even → ( 𝑁 − 𝑅 ) ∈ Odd ) ) |
8 |
5 7
|
syl |
⊢ ( 𝑅 ∈ ( ℙ ∖ { 2 } ) → ( 𝑁 ∈ Even → ( 𝑁 − 𝑅 ) ∈ Odd ) ) |
9 |
4 8
|
sylbir |
⊢ ( ( 𝑅 ∈ ℙ ∧ 𝑅 ≠ 2 ) → ( 𝑁 ∈ Even → ( 𝑁 − 𝑅 ) ∈ Odd ) ) |
10 |
9
|
ex |
⊢ ( 𝑅 ∈ ℙ → ( 𝑅 ≠ 2 → ( 𝑁 ∈ Even → ( 𝑁 − 𝑅 ) ∈ Odd ) ) ) |
11 |
3 10
|
syl5bir |
⊢ ( 𝑅 ∈ ℙ → ( ¬ 𝑅 = 2 → ( 𝑁 ∈ Even → ( 𝑁 − 𝑅 ) ∈ Odd ) ) ) |
12 |
11
|
com23 |
⊢ ( 𝑅 ∈ ℙ → ( 𝑁 ∈ Even → ( ¬ 𝑅 = 2 → ( 𝑁 − 𝑅 ) ∈ Odd ) ) ) |
13 |
12
|
3ad2ant3 |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ ) → ( 𝑁 ∈ Even → ( ¬ 𝑅 = 2 → ( 𝑁 − 𝑅 ) ∈ Odd ) ) ) |
14 |
13
|
impcom |
⊢ ( ( 𝑁 ∈ Even ∧ ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ ) ) → ( ¬ 𝑅 = 2 → ( 𝑁 − 𝑅 ) ∈ Odd ) ) |
15 |
14
|
3adant3 |
⊢ ( ( 𝑁 ∈ Even ∧ ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ ) ∧ 𝑁 = ( ( 𝑃 + 𝑄 ) + 𝑅 ) ) → ( ¬ 𝑅 = 2 → ( 𝑁 − 𝑅 ) ∈ Odd ) ) |
16 |
15
|
impcom |
⊢ ( ( ¬ 𝑅 = 2 ∧ ( 𝑁 ∈ Even ∧ ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ ) ∧ 𝑁 = ( ( 𝑃 + 𝑄 ) + 𝑅 ) ) ) → ( 𝑁 − 𝑅 ) ∈ Odd ) |
17 |
|
3simpa |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ ) → ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ) |
18 |
17
|
3ad2ant2 |
⊢ ( ( 𝑁 ∈ Even ∧ ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ ) ∧ 𝑁 = ( ( 𝑃 + 𝑄 ) + 𝑅 ) ) → ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ) |
19 |
18
|
adantl |
⊢ ( ( ¬ 𝑅 = 2 ∧ ( 𝑁 ∈ Even ∧ ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ ) ∧ 𝑁 = ( ( 𝑃 + 𝑄 ) + 𝑅 ) ) ) → ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ) |
20 |
|
eqcom |
⊢ ( 𝑁 = ( ( 𝑃 + 𝑄 ) + 𝑅 ) ↔ ( ( 𝑃 + 𝑄 ) + 𝑅 ) = 𝑁 ) |
21 |
|
evenz |
⊢ ( 𝑁 ∈ Even → 𝑁 ∈ ℤ ) |
22 |
21
|
zcnd |
⊢ ( 𝑁 ∈ Even → 𝑁 ∈ ℂ ) |
23 |
22
|
adantr |
⊢ ( ( 𝑁 ∈ Even ∧ ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ ) ) → 𝑁 ∈ ℂ ) |
24 |
|
prmz |
⊢ ( 𝑅 ∈ ℙ → 𝑅 ∈ ℤ ) |
25 |
24
|
zcnd |
⊢ ( 𝑅 ∈ ℙ → 𝑅 ∈ ℂ ) |
26 |
25
|
3ad2ant3 |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ ) → 𝑅 ∈ ℂ ) |
27 |
26
|
adantl |
⊢ ( ( 𝑁 ∈ Even ∧ ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ ) ) → 𝑅 ∈ ℂ ) |
28 |
|
prmz |
⊢ ( 𝑃 ∈ ℙ → 𝑃 ∈ ℤ ) |
29 |
|
prmz |
⊢ ( 𝑄 ∈ ℙ → 𝑄 ∈ ℤ ) |
30 |
|
zaddcl |
⊢ ( ( 𝑃 ∈ ℤ ∧ 𝑄 ∈ ℤ ) → ( 𝑃 + 𝑄 ) ∈ ℤ ) |
31 |
28 29 30
|
syl2an |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) → ( 𝑃 + 𝑄 ) ∈ ℤ ) |
32 |
31
|
zcnd |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) → ( 𝑃 + 𝑄 ) ∈ ℂ ) |
33 |
32
|
3adant3 |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ ) → ( 𝑃 + 𝑄 ) ∈ ℂ ) |
34 |
33
|
adantl |
⊢ ( ( 𝑁 ∈ Even ∧ ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ ) ) → ( 𝑃 + 𝑄 ) ∈ ℂ ) |
35 |
23 27 34
|
subadd2d |
⊢ ( ( 𝑁 ∈ Even ∧ ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ ) ) → ( ( 𝑁 − 𝑅 ) = ( 𝑃 + 𝑄 ) ↔ ( ( 𝑃 + 𝑄 ) + 𝑅 ) = 𝑁 ) ) |
36 |
35
|
biimprd |
⊢ ( ( 𝑁 ∈ Even ∧ ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ ) ) → ( ( ( 𝑃 + 𝑄 ) + 𝑅 ) = 𝑁 → ( 𝑁 − 𝑅 ) = ( 𝑃 + 𝑄 ) ) ) |
37 |
20 36
|
syl5bi |
⊢ ( ( 𝑁 ∈ Even ∧ ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ ) ) → ( 𝑁 = ( ( 𝑃 + 𝑄 ) + 𝑅 ) → ( 𝑁 − 𝑅 ) = ( 𝑃 + 𝑄 ) ) ) |
38 |
37
|
3impia |
⊢ ( ( 𝑁 ∈ Even ∧ ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ ) ∧ 𝑁 = ( ( 𝑃 + 𝑄 ) + 𝑅 ) ) → ( 𝑁 − 𝑅 ) = ( 𝑃 + 𝑄 ) ) |
39 |
38
|
adantl |
⊢ ( ( ¬ 𝑅 = 2 ∧ ( 𝑁 ∈ Even ∧ ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ ) ∧ 𝑁 = ( ( 𝑃 + 𝑄 ) + 𝑅 ) ) ) → ( 𝑁 − 𝑅 ) = ( 𝑃 + 𝑄 ) ) |
40 |
|
odd2prm2 |
⊢ ( ( ( 𝑁 − 𝑅 ) ∈ Odd ∧ ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ ( 𝑁 − 𝑅 ) = ( 𝑃 + 𝑄 ) ) → ( 𝑃 = 2 ∨ 𝑄 = 2 ) ) |
41 |
16 19 39 40
|
syl3anc |
⊢ ( ( ¬ 𝑅 = 2 ∧ ( 𝑁 ∈ Even ∧ ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ ) ∧ 𝑁 = ( ( 𝑃 + 𝑄 ) + 𝑅 ) ) ) → ( 𝑃 = 2 ∨ 𝑄 = 2 ) ) |
42 |
41
|
orcd |
⊢ ( ( ¬ 𝑅 = 2 ∧ ( 𝑁 ∈ Even ∧ ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ ) ∧ 𝑁 = ( ( 𝑃 + 𝑄 ) + 𝑅 ) ) ) → ( ( 𝑃 = 2 ∨ 𝑄 = 2 ) ∨ 𝑅 = 2 ) ) |
43 |
42
|
ex |
⊢ ( ¬ 𝑅 = 2 → ( ( 𝑁 ∈ Even ∧ ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ ) ∧ 𝑁 = ( ( 𝑃 + 𝑄 ) + 𝑅 ) ) → ( ( 𝑃 = 2 ∨ 𝑄 = 2 ) ∨ 𝑅 = 2 ) ) ) |
44 |
2 43
|
pm2.61i |
⊢ ( ( 𝑁 ∈ Even ∧ ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ ) ∧ 𝑁 = ( ( 𝑃 + 𝑄 ) + 𝑅 ) ) → ( ( 𝑃 = 2 ∨ 𝑄 = 2 ) ∨ 𝑅 = 2 ) ) |
45 |
|
df-3or |
⊢ ( ( 𝑃 = 2 ∨ 𝑄 = 2 ∨ 𝑅 = 2 ) ↔ ( ( 𝑃 = 2 ∨ 𝑄 = 2 ) ∨ 𝑅 = 2 ) ) |
46 |
44 45
|
sylibr |
⊢ ( ( 𝑁 ∈ Even ∧ ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ ) ∧ 𝑁 = ( ( 𝑃 + 𝑄 ) + 𝑅 ) ) → ( 𝑃 = 2 ∨ 𝑄 = 2 ∨ 𝑅 = 2 ) ) |