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Theorem even3prm2 44176
Description: If an even number is the sum of three prime numbers, one of the prime numbers must be 2. (Contributed by AV, 25-Dec-2021.)
Assertion
Ref Expression
even3prm2 ((𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) ∧ 𝑁 = ((𝑃 + 𝑄) + 𝑅)) → (𝑃 = 2 ∨ 𝑄 = 2 ∨ 𝑅 = 2))

Proof of Theorem even3prm2
StepHypRef Expression
1 olc 865 . . . 4 (𝑅 = 2 → ((𝑃 = 2 ∨ 𝑄 = 2) ∨ 𝑅 = 2))
21a1d 25 . . 3 (𝑅 = 2 → ((𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) ∧ 𝑁 = ((𝑃 + 𝑄) + 𝑅)) → ((𝑃 = 2 ∨ 𝑄 = 2) ∨ 𝑅 = 2)))
3 df-ne 3012 . . . . . . . . . . . 12 (𝑅 ≠ 2 ↔ ¬ 𝑅 = 2)
4 eldifsn 4693 . . . . . . . . . . . . . 14 (𝑅 ∈ (ℙ ∖ {2}) ↔ (𝑅 ∈ ℙ ∧ 𝑅 ≠ 2))
5 oddprmALTV 44144 . . . . . . . . . . . . . . 15 (𝑅 ∈ (ℙ ∖ {2}) → 𝑅 ∈ Odd )
6 emoo 44161 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ Even ∧ 𝑅 ∈ Odd ) → (𝑁𝑅) ∈ Odd )
76expcom 417 . . . . . . . . . . . . . . 15 (𝑅 ∈ Odd → (𝑁 ∈ Even → (𝑁𝑅) ∈ Odd ))
85, 7syl 17 . . . . . . . . . . . . . 14 (𝑅 ∈ (ℙ ∖ {2}) → (𝑁 ∈ Even → (𝑁𝑅) ∈ Odd ))
94, 8sylbir 238 . . . . . . . . . . . . 13 ((𝑅 ∈ ℙ ∧ 𝑅 ≠ 2) → (𝑁 ∈ Even → (𝑁𝑅) ∈ Odd ))
109ex 416 . . . . . . . . . . . 12 (𝑅 ∈ ℙ → (𝑅 ≠ 2 → (𝑁 ∈ Even → (𝑁𝑅) ∈ Odd )))
113, 10syl5bir 246 . . . . . . . . . . 11 (𝑅 ∈ ℙ → (¬ 𝑅 = 2 → (𝑁 ∈ Even → (𝑁𝑅) ∈ Odd )))
1211com23 86 . . . . . . . . . 10 (𝑅 ∈ ℙ → (𝑁 ∈ Even → (¬ 𝑅 = 2 → (𝑁𝑅) ∈ Odd )))
13123ad2ant3 1132 . . . . . . . . 9 ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) → (𝑁 ∈ Even → (¬ 𝑅 = 2 → (𝑁𝑅) ∈ Odd )))
1413impcom 411 . . . . . . . 8 ((𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ)) → (¬ 𝑅 = 2 → (𝑁𝑅) ∈ Odd ))
15143adant3 1129 . . . . . . 7 ((𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) ∧ 𝑁 = ((𝑃 + 𝑄) + 𝑅)) → (¬ 𝑅 = 2 → (𝑁𝑅) ∈ Odd ))
1615impcom 411 . . . . . 6 ((¬ 𝑅 = 2 ∧ (𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) ∧ 𝑁 = ((𝑃 + 𝑄) + 𝑅))) → (𝑁𝑅) ∈ Odd )
17 3simpa 1145 . . . . . . . 8 ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) → (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ))
18173ad2ant2 1131 . . . . . . 7 ((𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) ∧ 𝑁 = ((𝑃 + 𝑄) + 𝑅)) → (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ))
1918adantl 485 . . . . . 6 ((¬ 𝑅 = 2 ∧ (𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) ∧ 𝑁 = ((𝑃 + 𝑄) + 𝑅))) → (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ))
20 eqcom 2829 . . . . . . . . 9 (𝑁 = ((𝑃 + 𝑄) + 𝑅) ↔ ((𝑃 + 𝑄) + 𝑅) = 𝑁)
21 evenz 44087 . . . . . . . . . . . . 13 (𝑁 ∈ Even → 𝑁 ∈ ℤ)
2221zcnd 12076 . . . . . . . . . . . 12 (𝑁 ∈ Even → 𝑁 ∈ ℂ)
2322adantr 484 . . . . . . . . . . 11 ((𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ)) → 𝑁 ∈ ℂ)
24 prmz 16008 . . . . . . . . . . . . . 14 (𝑅 ∈ ℙ → 𝑅 ∈ ℤ)
2524zcnd 12076 . . . . . . . . . . . . 13 (𝑅 ∈ ℙ → 𝑅 ∈ ℂ)
26253ad2ant3 1132 . . . . . . . . . . . 12 ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) → 𝑅 ∈ ℂ)
2726adantl 485 . . . . . . . . . . 11 ((𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ)) → 𝑅 ∈ ℂ)
28 prmz 16008 . . . . . . . . . . . . . . 15 (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
29 prmz 16008 . . . . . . . . . . . . . . 15 (𝑄 ∈ ℙ → 𝑄 ∈ ℤ)
30 zaddcl 12010 . . . . . . . . . . . . . . 15 ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℤ) → (𝑃 + 𝑄) ∈ ℤ)
3128, 29, 30syl2an 598 . . . . . . . . . . . . . 14 ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → (𝑃 + 𝑄) ∈ ℤ)
3231zcnd 12076 . . . . . . . . . . . . 13 ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → (𝑃 + 𝑄) ∈ ℂ)
33323adant3 1129 . . . . . . . . . . . 12 ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) → (𝑃 + 𝑄) ∈ ℂ)
3433adantl 485 . . . . . . . . . . 11 ((𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ)) → (𝑃 + 𝑄) ∈ ℂ)
3523, 27, 34subadd2d 11005 . . . . . . . . . 10 ((𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ)) → ((𝑁𝑅) = (𝑃 + 𝑄) ↔ ((𝑃 + 𝑄) + 𝑅) = 𝑁))
3635biimprd 251 . . . . . . . . 9 ((𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ)) → (((𝑃 + 𝑄) + 𝑅) = 𝑁 → (𝑁𝑅) = (𝑃 + 𝑄)))
3720, 36syl5bi 245 . . . . . . . 8 ((𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ)) → (𝑁 = ((𝑃 + 𝑄) + 𝑅) → (𝑁𝑅) = (𝑃 + 𝑄)))
38373impia 1114 . . . . . . 7 ((𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) ∧ 𝑁 = ((𝑃 + 𝑄) + 𝑅)) → (𝑁𝑅) = (𝑃 + 𝑄))
3938adantl 485 . . . . . 6 ((¬ 𝑅 = 2 ∧ (𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) ∧ 𝑁 = ((𝑃 + 𝑄) + 𝑅))) → (𝑁𝑅) = (𝑃 + 𝑄))
40 odd2prm2 44175 . . . . . 6 (((𝑁𝑅) ∈ Odd ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑁𝑅) = (𝑃 + 𝑄)) → (𝑃 = 2 ∨ 𝑄 = 2))
4116, 19, 39, 40syl3anc 1368 . . . . 5 ((¬ 𝑅 = 2 ∧ (𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) ∧ 𝑁 = ((𝑃 + 𝑄) + 𝑅))) → (𝑃 = 2 ∨ 𝑄 = 2))
4241orcd 870 . . . 4 ((¬ 𝑅 = 2 ∧ (𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) ∧ 𝑁 = ((𝑃 + 𝑄) + 𝑅))) → ((𝑃 = 2 ∨ 𝑄 = 2) ∨ 𝑅 = 2))
4342ex 416 . . 3 𝑅 = 2 → ((𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) ∧ 𝑁 = ((𝑃 + 𝑄) + 𝑅)) → ((𝑃 = 2 ∨ 𝑄 = 2) ∨ 𝑅 = 2)))
442, 43pm2.61i 185 . 2 ((𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) ∧ 𝑁 = ((𝑃 + 𝑄) + 𝑅)) → ((𝑃 = 2 ∨ 𝑄 = 2) ∨ 𝑅 = 2))
45 df-3or 1085 . 2 ((𝑃 = 2 ∨ 𝑄 = 2 ∨ 𝑅 = 2) ↔ ((𝑃 = 2 ∨ 𝑄 = 2) ∨ 𝑅 = 2))
4644, 45sylibr 237 1 ((𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) ∧ 𝑁 = ((𝑃 + 𝑄) + 𝑅)) → (𝑃 = 2 ∨ 𝑄 = 2 ∨ 𝑅 = 2))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wo 844  w3o 1083  w3a 1084   = wceq 1538  wcel 2114  wne 3011  cdif 3905  {csn 4539  (class class class)co 7140  cc 10524   + caddc 10529  cmin 10859  2c2 11680  cz 11969  cprime 16004   Even ceven 44081   Odd codd 44082
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  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 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-sup 8894  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 14449  df-re 14450  df-im 14451  df-sqrt 14585  df-abs 14586  df-dvds 15599  df-prm 16005  df-even 44083  df-odd 44084
This theorem is referenced by:  mogoldbblem  44177
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