Theorem odd2prm2 44236

Description: If an odd number is the sum of two prime numbers, one of the prime numbers must be 2. (Contributed by AV, 26-Dec-2021.)

Assertion

Ref	Expression
odd2prm2	⊢ ((𝑁 ∈ Odd ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ 𝑁 = (𝑃 + 𝑄)) → (𝑃 = 2 ∨ 𝑄 = 2))

Proof of Theorem odd2prm2

Step	Hyp	Ref	Expression
1		eleq1 2877	. . . . . 6 ⊢ (𝑁 = (𝑃 + 𝑄) → (𝑁 ∈ Odd ↔ (𝑃 + 𝑄) ∈ Odd ))
2		evennodd 44161	. . . . . . . . 9 ⊢ ((𝑃 + 𝑄) ∈ Even → ¬ (𝑃 + 𝑄) ∈ Odd )
3	2	pm2.21d 121	. . . . . . . 8 ⊢ ((𝑃 + 𝑄) ∈ Even → ((𝑃 + 𝑄) ∈ Odd → (𝑃 = 2 ∨ 𝑄 = 2)))
4		df-ne 2988	. . . . . . . . . . . 12 ⊢ (𝑃 ≠ 2 ↔ ¬ 𝑃 = 2)
5		eldifsn 4680	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ (ℙ ∖ {2}) ↔ (𝑃 ∈ ℙ ∧ 𝑃 ≠ 2))
6		oddprmALTV 44205	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ Odd )
7	5, 6	sylbir 238	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ ℙ ∧ 𝑃 ≠ 2) → 𝑃 ∈ Odd )
8	7	ex 416	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → (𝑃 ≠ 2 → 𝑃 ∈ Odd ))
9	4, 8	syl5bir 246	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℙ → (¬ 𝑃 = 2 → 𝑃 ∈ Odd ))
10		df-ne 2988	. . . . . . . . . . . 12 ⊢ (𝑄 ≠ 2 ↔ ¬ 𝑄 = 2)
11		eldifsn 4680	. . . . . . . . . . . . . 14 ⊢ (𝑄 ∈ (ℙ ∖ {2}) ↔ (𝑄 ∈ ℙ ∧ 𝑄 ≠ 2))
12		oddprmALTV 44205	. . . . . . . . . . . . . 14 ⊢ (𝑄 ∈ (ℙ ∖ {2}) → 𝑄 ∈ Odd )
13	11, 12	sylbir 238	. . . . . . . . . . . . 13 ⊢ ((𝑄 ∈ ℙ ∧ 𝑄 ≠ 2) → 𝑄 ∈ Odd )
14	13	ex 416	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ ℙ → (𝑄 ≠ 2 → 𝑄 ∈ Odd ))
15	10, 14	syl5bir 246	. . . . . . . . . . 11 ⊢ (𝑄 ∈ ℙ → (¬ 𝑄 = 2 → 𝑄 ∈ Odd ))
16	9, 15	im2anan9 622	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → ((¬ 𝑃 = 2 ∧ ¬ 𝑄 = 2) → (𝑃 ∈ Odd ∧ 𝑄 ∈ Odd )))
17	16	imp 410	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (¬ 𝑃 = 2 ∧ ¬ 𝑄 = 2)) → (𝑃 ∈ Odd ∧ 𝑄 ∈ Odd ))
18		opoeALTV 44201	. . . . . . . . 9 ⊢ ((𝑃 ∈ Odd ∧ 𝑄 ∈ Odd ) → (𝑃 + 𝑄) ∈ Even )
19	17, 18	syl 17	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (¬ 𝑃 = 2 ∧ ¬ 𝑄 = 2)) → (𝑃 + 𝑄) ∈ Even )
20	3, 19	syl11 33	. . . . . . 7 ⊢ ((𝑃 + 𝑄) ∈ Odd → (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (¬ 𝑃 = 2 ∧ ¬ 𝑄 = 2)) → (𝑃 = 2 ∨ 𝑄 = 2)))
21	20	expd 419	. . . . . 6 ⊢ ((𝑃 + 𝑄) ∈ Odd → ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → ((¬ 𝑃 = 2 ∧ ¬ 𝑄 = 2) → (𝑃 = 2 ∨ 𝑄 = 2))))
22	1, 21	syl6bi 256	. . . . 5 ⊢ (𝑁 = (𝑃 + 𝑄) → (𝑁 ∈ Odd → ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → ((¬ 𝑃 = 2 ∧ ¬ 𝑄 = 2) → (𝑃 = 2 ∨ 𝑄 = 2)))))
23	22	3imp231 1110	. . . 4 ⊢ ((𝑁 ∈ Odd ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ 𝑁 = (𝑃 + 𝑄)) → ((¬ 𝑃 = 2 ∧ ¬ 𝑄 = 2) → (𝑃 = 2 ∨ 𝑄 = 2)))
24	23	com12 32	. . 3 ⊢ ((¬ 𝑃 = 2 ∧ ¬ 𝑄 = 2) → ((𝑁 ∈ Odd ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ 𝑁 = (𝑃 + 𝑄)) → (𝑃 = 2 ∨ 𝑄 = 2)))
25	24	ex 416	. 2 ⊢ (¬ 𝑃 = 2 → (¬ 𝑄 = 2 → ((𝑁 ∈ Odd ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ 𝑁 = (𝑃 + 𝑄)) → (𝑃 = 2 ∨ 𝑄 = 2))))
26		orc 864	. . 3 ⊢ (𝑃 = 2 → (𝑃 = 2 ∨ 𝑄 = 2))
27	26	a1d 25	. 2 ⊢ (𝑃 = 2 → ((𝑁 ∈ Odd ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ 𝑁 = (𝑃 + 𝑄)) → (𝑃 = 2 ∨ 𝑄 = 2)))
28		olc 865	. . 3 ⊢ (𝑄 = 2 → (𝑃 = 2 ∨ 𝑄 = 2))
29	28	a1d 25	. 2 ⊢ (𝑄 = 2 → ((𝑁 ∈ Odd ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ 𝑁 = (𝑃 + 𝑄)) → (𝑃 = 2 ∨ 𝑄 = 2)))
30	25, 27, 29	pm2.61ii 186	1 ⊢ ((𝑁 ∈ Odd ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ 𝑁 = (𝑃 + 𝑄)) → (𝑃 = 2 ∨ 𝑄 = 2))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987   ∖ cdif 3878  {csn 4525  (class class class)co 7135   + caddc 10529  2c2 11680  ℙcprime 16005   Even ceven 44142   Odd codd 44143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-sup 8890  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-seq 13365  df-exp 13426  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-dvds 15600  df-prm 16006  df-even 44144  df-odd 44145

This theorem is referenced by:  even3prm2  44237