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Theorem even3prm2 42153
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 857 . . . 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 2944 . . . . . . . . . . . 12 (𝑅 ≠ 2 ↔ ¬ 𝑅 = 2)
4 eldifsn 4454 . . . . . . . . . . . . . 14 (𝑅 ∈ (ℙ ∖ {2}) ↔ (𝑅 ∈ ℙ ∧ 𝑅 ≠ 2))
5 oddprmALTV 42123 . . . . . . . . . . . . . . 15 (𝑅 ∈ (ℙ ∖ {2}) → 𝑅 ∈ Odd )
6 emoo 42138 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ Even ∧ 𝑅 ∈ Odd ) → (𝑁𝑅) ∈ Odd )
76expcom 398 . . . . . . . . . . . . . . 15 (𝑅 ∈ Odd → (𝑁 ∈ Even → (𝑁𝑅) ∈ Odd ))
85, 7syl 17 . . . . . . . . . . . . . 14 (𝑅 ∈ (ℙ ∖ {2}) → (𝑁 ∈ Even → (𝑁𝑅) ∈ Odd ))
94, 8sylbir 225 . . . . . . . . . . . . 13 ((𝑅 ∈ ℙ ∧ 𝑅 ≠ 2) → (𝑁 ∈ Even → (𝑁𝑅) ∈ Odd ))
109ex 397 . . . . . . . . . . . 12 (𝑅 ∈ ℙ → (𝑅 ≠ 2 → (𝑁 ∈ Even → (𝑁𝑅) ∈ Odd )))
113, 10syl5bir 233 . . . . . . . . . . 11 (𝑅 ∈ ℙ → (¬ 𝑅 = 2 → (𝑁 ∈ Even → (𝑁𝑅) ∈ Odd )))
1211com23 86 . . . . . . . . . 10 (𝑅 ∈ ℙ → (𝑁 ∈ Even → (¬ 𝑅 = 2 → (𝑁𝑅) ∈ Odd )))
13123ad2ant3 1129 . . . . . . . . 9 ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) → (𝑁 ∈ Even → (¬ 𝑅 = 2 → (𝑁𝑅) ∈ Odd )))
1413impcom 394 . . . . . . . 8 ((𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ)) → (¬ 𝑅 = 2 → (𝑁𝑅) ∈ Odd ))
15143adant3 1126 . . . . . . 7 ((𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) ∧ 𝑁 = ((𝑃 + 𝑄) + 𝑅)) → (¬ 𝑅 = 2 → (𝑁𝑅) ∈ Odd ))
1615impcom 394 . . . . . 6 ((¬ 𝑅 = 2 ∧ (𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) ∧ 𝑁 = ((𝑃 + 𝑄) + 𝑅))) → (𝑁𝑅) ∈ Odd )
17 3simpa 1142 . . . . . . . 8 ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) → (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ))
18173ad2ant2 1128 . . . . . . 7 ((𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) ∧ 𝑁 = ((𝑃 + 𝑄) + 𝑅)) → (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ))
1918adantl 467 . . . . . 6 ((¬ 𝑅 = 2 ∧ (𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) ∧ 𝑁 = ((𝑃 + 𝑄) + 𝑅))) → (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ))
20 eqcom 2778 . . . . . . . . 9 (𝑁 = ((𝑃 + 𝑄) + 𝑅) ↔ ((𝑃 + 𝑄) + 𝑅) = 𝑁)
21 evenz 42068 . . . . . . . . . . . . 13 (𝑁 ∈ Even → 𝑁 ∈ ℤ)
2221zcnd 11690 . . . . . . . . . . . 12 (𝑁 ∈ Even → 𝑁 ∈ ℂ)
2322adantr 466 . . . . . . . . . . 11 ((𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ)) → 𝑁 ∈ ℂ)
24 prmz 15596 . . . . . . . . . . . . . 14 (𝑅 ∈ ℙ → 𝑅 ∈ ℤ)
2524zcnd 11690 . . . . . . . . . . . . 13 (𝑅 ∈ ℙ → 𝑅 ∈ ℂ)
26253ad2ant3 1129 . . . . . . . . . . . 12 ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) → 𝑅 ∈ ℂ)
2726adantl 467 . . . . . . . . . . 11 ((𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ)) → 𝑅 ∈ ℂ)
28 prmz 15596 . . . . . . . . . . . . . . 15 (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
29 prmz 15596 . . . . . . . . . . . . . . 15 (𝑄 ∈ ℙ → 𝑄 ∈ ℤ)
30 zaddcl 11624 . . . . . . . . . . . . . . 15 ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℤ) → (𝑃 + 𝑄) ∈ ℤ)
3128, 29, 30syl2an 583 . . . . . . . . . . . . . 14 ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → (𝑃 + 𝑄) ∈ ℤ)
3231zcnd 11690 . . . . . . . . . . . . 13 ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → (𝑃 + 𝑄) ∈ ℂ)
33323adant3 1126 . . . . . . . . . . . 12 ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) → (𝑃 + 𝑄) ∈ ℂ)
3433adantl 467 . . . . . . . . . . 11 ((𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ)) → (𝑃 + 𝑄) ∈ ℂ)
3523, 27, 34subadd2d 10617 . . . . . . . . . 10 ((𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ)) → ((𝑁𝑅) = (𝑃 + 𝑄) ↔ ((𝑃 + 𝑄) + 𝑅) = 𝑁))
3635biimprd 238 . . . . . . . . 9 ((𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ)) → (((𝑃 + 𝑄) + 𝑅) = 𝑁 → (𝑁𝑅) = (𝑃 + 𝑄)))
3720, 36syl5bi 232 . . . . . . . 8 ((𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ)) → (𝑁 = ((𝑃 + 𝑄) + 𝑅) → (𝑁𝑅) = (𝑃 + 𝑄)))
38373impia 1109 . . . . . . 7 ((𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) ∧ 𝑁 = ((𝑃 + 𝑄) + 𝑅)) → (𝑁𝑅) = (𝑃 + 𝑄))
3938adantl 467 . . . . . 6 ((¬ 𝑅 = 2 ∧ (𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) ∧ 𝑁 = ((𝑃 + 𝑄) + 𝑅))) → (𝑁𝑅) = (𝑃 + 𝑄))
40 odd2prm2 42152 . . . . . 6 (((𝑁𝑅) ∈ Odd ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ (𝑁𝑅) = (𝑃 + 𝑄)) → (𝑃 = 2 ∨ 𝑄 = 2))
4116, 19, 39, 40syl3anc 1476 . . . . 5 ((¬ 𝑅 = 2 ∧ (𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) ∧ 𝑁 = ((𝑃 + 𝑄) + 𝑅))) → (𝑃 = 2 ∨ 𝑄 = 2))
4241orcd 862 . . . 4 ((¬ 𝑅 = 2 ∧ (𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) ∧ 𝑁 = ((𝑃 + 𝑄) + 𝑅))) → ((𝑃 = 2 ∨ 𝑄 = 2) ∨ 𝑅 = 2))
4342ex 397 . . 3 𝑅 = 2 → ((𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) ∧ 𝑁 = ((𝑃 + 𝑄) + 𝑅)) → ((𝑃 = 2 ∨ 𝑄 = 2) ∨ 𝑅 = 2)))
442, 43pm2.61i 176 . 2 ((𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) ∧ 𝑁 = ((𝑃 + 𝑄) + 𝑅)) → ((𝑃 = 2 ∨ 𝑄 = 2) ∨ 𝑅 = 2))
45 df-3or 1072 . 2 ((𝑃 = 2 ∨ 𝑄 = 2 ∨ 𝑅 = 2) ↔ ((𝑃 = 2 ∨ 𝑄 = 2) ∨ 𝑅 = 2))
4644, 45sylibr 224 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 382  wo 836  w3o 1070  w3a 1071   = wceq 1631  wcel 2145  wne 2943  cdif 3720  {csn 4317  (class class class)co 6796  cc 10140   + caddc 10145  cmin 10472  2c2 11276  cz 11584  cprime 15592   Even ceven 42062   Odd codd 42063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219  ax-pre-sup 10220
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-1o 7717  df-2o 7718  df-er 7900  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-sup 8508  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-div 10891  df-nn 11227  df-2 11285  df-3 11286  df-n0 11500  df-z 11585  df-uz 11894  df-rp 12036  df-seq 13009  df-exp 13068  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-dvds 15190  df-prm 15593  df-even 42064  df-odd 42065
This theorem is referenced by:  mogoldbblem  42154
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