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Theorem odd2prm2 47087
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
StepHypRef Expression
1 eleq1 2817 . . . . . 6 (𝑁 = (𝑃 + 𝑄) → (𝑁 ∈ Odd ↔ (𝑃 + 𝑄) ∈ Odd ))
2 evennodd 47012 . . . . . . . . 9 ((𝑃 + 𝑄) ∈ Even → ¬ (𝑃 + 𝑄) ∈ Odd )
32pm2.21d 121 . . . . . . . 8 ((𝑃 + 𝑄) ∈ Even → ((𝑃 + 𝑄) ∈ Odd → (𝑃 = 2 ∨ 𝑄 = 2)))
4 df-ne 2938 . . . . . . . . . . . 12 (𝑃 ≠ 2 ↔ ¬ 𝑃 = 2)
5 eldifsn 4795 . . . . . . . . . . . . . 14 (𝑃 ∈ (ℙ ∖ {2}) ↔ (𝑃 ∈ ℙ ∧ 𝑃 ≠ 2))
6 oddprmALTV 47056 . . . . . . . . . . . . . 14 (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ Odd )
75, 6sylbir 234 . . . . . . . . . . . . 13 ((𝑃 ∈ ℙ ∧ 𝑃 ≠ 2) → 𝑃 ∈ Odd )
87ex 411 . . . . . . . . . . . 12 (𝑃 ∈ ℙ → (𝑃 ≠ 2 → 𝑃 ∈ Odd ))
94, 8biimtrrid 242 . . . . . . . . . . 11 (𝑃 ∈ ℙ → (¬ 𝑃 = 2 → 𝑃 ∈ Odd ))
10 df-ne 2938 . . . . . . . . . . . 12 (𝑄 ≠ 2 ↔ ¬ 𝑄 = 2)
11 eldifsn 4795 . . . . . . . . . . . . . 14 (𝑄 ∈ (ℙ ∖ {2}) ↔ (𝑄 ∈ ℙ ∧ 𝑄 ≠ 2))
12 oddprmALTV 47056 . . . . . . . . . . . . . 14 (𝑄 ∈ (ℙ ∖ {2}) → 𝑄 ∈ Odd )
1311, 12sylbir 234 . . . . . . . . . . . . 13 ((𝑄 ∈ ℙ ∧ 𝑄 ≠ 2) → 𝑄 ∈ Odd )
1413ex 411 . . . . . . . . . . . 12 (𝑄 ∈ ℙ → (𝑄 ≠ 2 → 𝑄 ∈ Odd ))
1510, 14biimtrrid 242 . . . . . . . . . . 11 (𝑄 ∈ ℙ → (¬ 𝑄 = 2 → 𝑄 ∈ Odd ))
169, 15im2anan9 618 . . . . . . . . . 10 ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → ((¬ 𝑃 = 2 ∧ ¬ 𝑄 = 2) → (𝑃 ∈ Odd ∧ 𝑄 ∈ Odd )))
1716imp 405 . . . . . . . . 9 (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (¬ 𝑃 = 2 ∧ ¬ 𝑄 = 2)) → (𝑃 ∈ Odd ∧ 𝑄 ∈ Odd ))
18 opoeALTV 47052 . . . . . . . . 9 ((𝑃 ∈ Odd ∧ 𝑄 ∈ Odd ) → (𝑃 + 𝑄) ∈ Even )
1917, 18syl 17 . . . . . . . 8 (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (¬ 𝑃 = 2 ∧ ¬ 𝑄 = 2)) → (𝑃 + 𝑄) ∈ Even )
203, 19syl11 33 . . . . . . 7 ((𝑃 + 𝑄) ∈ Odd → (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (¬ 𝑃 = 2 ∧ ¬ 𝑄 = 2)) → (𝑃 = 2 ∨ 𝑄 = 2)))
2120expd 414 . . . . . 6 ((𝑃 + 𝑄) ∈ Odd → ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → ((¬ 𝑃 = 2 ∧ ¬ 𝑄 = 2) → (𝑃 = 2 ∨ 𝑄 = 2))))
221, 21biimtrdi 252 . . . . 5 (𝑁 = (𝑃 + 𝑄) → (𝑁 ∈ Odd → ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → ((¬ 𝑃 = 2 ∧ ¬ 𝑄 = 2) → (𝑃 = 2 ∨ 𝑄 = 2)))))
23223imp231 1110 . . . 4 ((𝑁 ∈ Odd ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ 𝑁 = (𝑃 + 𝑄)) → ((¬ 𝑃 = 2 ∧ ¬ 𝑄 = 2) → (𝑃 = 2 ∨ 𝑄 = 2)))
2423com12 32 . . 3 ((¬ 𝑃 = 2 ∧ ¬ 𝑄 = 2) → ((𝑁 ∈ Odd ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ 𝑁 = (𝑃 + 𝑄)) → (𝑃 = 2 ∨ 𝑄 = 2)))
2524ex 411 . 2 𝑃 = 2 → (¬ 𝑄 = 2 → ((𝑁 ∈ Odd ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ 𝑁 = (𝑃 + 𝑄)) → (𝑃 = 2 ∨ 𝑄 = 2))))
26 orc 865 . . 3 (𝑃 = 2 → (𝑃 = 2 ∨ 𝑄 = 2))
2726a1d 25 . 2 (𝑃 = 2 → ((𝑁 ∈ Odd ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ 𝑁 = (𝑃 + 𝑄)) → (𝑃 = 2 ∨ 𝑄 = 2)))
28 olc 866 . . 3 (𝑄 = 2 → (𝑃 = 2 ∨ 𝑄 = 2))
2928a1d 25 . 2 (𝑄 = 2 → ((𝑁 ∈ Odd ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ 𝑁 = (𝑃 + 𝑄)) → (𝑃 = 2 ∨ 𝑄 = 2)))
3025, 27, 29pm2.61ii 183 1 ((𝑁 ∈ Odd ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ 𝑁 = (𝑃 + 𝑄)) → (𝑃 = 2 ∨ 𝑄 = 2))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  wo 845  w3a 1084   = wceq 1533  wcel 2098  wne 2937  cdif 3946  {csn 4632  (class class class)co 7426   + caddc 11149  2c2 12305  cprime 16649   Even ceven 46993   Odd codd 46994
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223  ax-pre-sup 11224
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-2o 8494  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-sup 9473  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-div 11910  df-nn 12251  df-2 12313  df-3 12314  df-n0 12511  df-z 12597  df-uz 12861  df-rp 13015  df-seq 14007  df-exp 14067  df-cj 15086  df-re 15087  df-im 15088  df-sqrt 15222  df-abs 15223  df-dvds 16239  df-prm 16650  df-even 46995  df-odd 46996
This theorem is referenced by:  even3prm2  47088
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