Theorem odd2prm2 48194

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 2824	. . . . . 6 ⊢ (𝑁 = (𝑃 + 𝑄) → (𝑁 ∈ Odd ↔ (𝑃 + 𝑄) ∈ Odd ))
2		evennodd 48119	. . . . . . . . 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 2933	. . . . . . . . . . . 12 ⊢ (𝑃 ≠ 2 ↔ ¬ 𝑃 = 2)
5		eldifsn 4731	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ (ℙ ∖ {2}) ↔ (𝑃 ∈ ℙ ∧ 𝑃 ≠ 2))
6		oddprmALTV 48163	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ Odd )
7	5, 6	sylbir 235	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ ℙ ∧ 𝑃 ≠ 2) → 𝑃 ∈ Odd )
8	7	ex 412	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → (𝑃 ≠ 2 → 𝑃 ∈ Odd ))
9	4, 8	biimtrrid 243	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℙ → (¬ 𝑃 = 2 → 𝑃 ∈ Odd ))
10		df-ne 2933	. . . . . . . . . . . 12 ⊢ (𝑄 ≠ 2 ↔ ¬ 𝑄 = 2)
11		eldifsn 4731	. . . . . . . . . . . . . 14 ⊢ (𝑄 ∈ (ℙ ∖ {2}) ↔ (𝑄 ∈ ℙ ∧ 𝑄 ≠ 2))
12		oddprmALTV 48163	. . . . . . . . . . . . . 14 ⊢ (𝑄 ∈ (ℙ ∖ {2}) → 𝑄 ∈ Odd )
13	11, 12	sylbir 235	. . . . . . . . . . . . 13 ⊢ ((𝑄 ∈ ℙ ∧ 𝑄 ≠ 2) → 𝑄 ∈ Odd )
14	13	ex 412	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ ℙ → (𝑄 ≠ 2 → 𝑄 ∈ Odd ))
15	10, 14	biimtrrid 243	. . . . . . . . . . 11 ⊢ (𝑄 ∈ ℙ → (¬ 𝑄 = 2 → 𝑄 ∈ Odd ))
16	9, 15	im2anan9 621	. . . . . . . . . 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 48159	. . . . . . . . 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	biimtrdi 253	. . . . 5 ⊢ (𝑁 = (𝑃 + 𝑄) → (𝑁 ∈ Odd → ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → ((¬ 𝑃 = 2 ∧ ¬ 𝑄 = 2) → (𝑃 = 2 ∨ 𝑄 = 2)))))
23	22	3imp231 1113	. . . 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 868	. . 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 869	. . 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 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2932   ∖ cdif 3886  {csn 4567  (class class class)co 7367   + caddc 11041  2c2 12236  ℙcprime 16640   Even ceven 48100   Odd codd 48101

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-sup 9355  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-2 12244  df-3 12245  df-n0 12438  df-z 12525  df-uz 12789  df-rp 12943  df-seq 13964  df-exp 14024  df-cj 15061  df-re 15062  df-im 15063  df-sqrt 15197  df-abs 15198  df-dvds 16222  df-prm 16641  df-even 48102  df-odd 48103

This theorem is referenced by:  even3prm2  48195