Theorem odd2prm2 46940

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 2815	. . . . . 6 ⊢ (𝑁 = (𝑃 + 𝑄) → (𝑁 ∈ Odd ↔ (𝑃 + 𝑄) ∈ Odd ))
2		evennodd 46865	. . . . . . . . 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 2935	. . . . . . . . . . . 12 ⊢ (𝑃 ≠ 2 ↔ ¬ 𝑃 = 2)
5		eldifsn 4785	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ (ℙ ∖ {2}) ↔ (𝑃 ∈ ℙ ∧ 𝑃 ≠ 2))
6		oddprmALTV 46909	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ Odd )
7	5, 6	sylbir 234	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ ℙ ∧ 𝑃 ≠ 2) → 𝑃 ∈ Odd )
8	7	ex 412	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → (𝑃 ≠ 2 → 𝑃 ∈ Odd ))
9	4, 8	biimtrrid 242	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℙ → (¬ 𝑃 = 2 → 𝑃 ∈ Odd ))
10		df-ne 2935	. . . . . . . . . . . 12 ⊢ (𝑄 ≠ 2 ↔ ¬ 𝑄 = 2)
11		eldifsn 4785	. . . . . . . . . . . . . 14 ⊢ (𝑄 ∈ (ℙ ∖ {2}) ↔ (𝑄 ∈ ℙ ∧ 𝑄 ≠ 2))
12		oddprmALTV 46909	. . . . . . . . . . . . . 14 ⊢ (𝑄 ∈ (ℙ ∖ {2}) → 𝑄 ∈ Odd )
13	11, 12	sylbir 234	. . . . . . . . . . . . 13 ⊢ ((𝑄 ∈ ℙ ∧ 𝑄 ≠ 2) → 𝑄 ∈ Odd )
14	13	ex 412	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ ℙ → (𝑄 ≠ 2 → 𝑄 ∈ Odd ))
15	10, 14	biimtrrid 242	. . . . . . . . . . 11 ⊢ (𝑄 ∈ ℙ → (¬ 𝑄 = 2 → 𝑄 ∈ Odd ))
16	9, 15	im2anan9 619	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → ((¬ 𝑃 = 2 ∧ ¬ 𝑄 = 2) → (𝑃 ∈ Odd ∧ 𝑄 ∈ Odd )))
17	16	imp 406	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (¬ 𝑃 = 2 ∧ ¬ 𝑄 = 2)) → (𝑃 ∈ Odd ∧ 𝑄 ∈ Odd ))
18		opoeALTV 46905	. . . . . . . . 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 415	. . . . . 6 ⊢ ((𝑃 + 𝑄) ∈ Odd → ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → ((¬ 𝑃 = 2 ∧ ¬ 𝑄 = 2) → (𝑃 = 2 ∨ 𝑄 = 2))))
22	1, 21	syl6bi 253	. . . . 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 412	. 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 183	1 ⊢ ((𝑁 ∈ Odd ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ 𝑁 = (𝑃 + 𝑄)) → (𝑃 = 2 ∨ 𝑄 = 2))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 844   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2934   ∖ cdif 3940  {csn 4623  (class class class)co 7404   + caddc 11112  2c2 12268  ℙcprime 16612   Even ceven 46846   Odd codd 46847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186  ax-pre-sup 11187

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-1o 8464  df-2o 8465  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-sup 9436  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-div 11873  df-nn 12214  df-2 12276  df-3 12277  df-n0 12474  df-z 12560  df-uz 12824  df-rp 12978  df-seq 13970  df-exp 14030  df-cj 15049  df-re 15050  df-im 15051  df-sqrt 15185  df-abs 15186  df-dvds 16202  df-prm 16613  df-even 46848  df-odd 46849

This theorem is referenced by:  even3prm2  46941