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Theorem sgoldbaltlem1 40931
Description: Lemma 1 for sgoldbalt 40933: If an even number greater than 4 is the sum of two primes, one of the prime summands must be odd, i.e. not 2. (Contributed by AV, 22-Jul-2020.)
Assertion
Ref Expression
sgoldbaltlem1 ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → ((𝑁 ∈ Even ∧ 4 < 𝑁𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd ))

Proof of Theorem sgoldbaltlem1
StepHypRef Expression
1 prmnn 15307 . . . . . 6 (𝑄 ∈ ℙ → 𝑄 ∈ ℕ)
2 nneoALTV 40851 . . . . . . 7 (𝑄 ∈ ℕ → (𝑄 ∈ Even ↔ ¬ 𝑄 ∈ Odd ))
32bicomd 213 . . . . . 6 (𝑄 ∈ ℕ → (¬ 𝑄 ∈ Odd ↔ 𝑄 ∈ Even ))
41, 3syl 17 . . . . 5 (𝑄 ∈ ℙ → (¬ 𝑄 ∈ Odd ↔ 𝑄 ∈ Even ))
5 evenprm2 40891 . . . . 5 (𝑄 ∈ ℙ → (𝑄 ∈ Even ↔ 𝑄 = 2))
64, 5bitrd 268 . . . 4 (𝑄 ∈ ℙ → (¬ 𝑄 ∈ Odd ↔ 𝑄 = 2))
76adantl 482 . . 3 ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → (¬ 𝑄 ∈ Odd ↔ 𝑄 = 2))
8 oveq2 6613 . . . . . . . . 9 (𝑄 = 2 → (𝑃 + 𝑄) = (𝑃 + 2))
98eqeq2d 2636 . . . . . . . 8 (𝑄 = 2 → (𝑁 = (𝑃 + 𝑄) ↔ 𝑁 = (𝑃 + 2)))
109adantl 482 . . . . . . 7 ((𝑃 ∈ ℙ ∧ 𝑄 = 2) → (𝑁 = (𝑃 + 𝑄) ↔ 𝑁 = (𝑃 + 2)))
11103anbi3d 1402 . . . . . 6 ((𝑃 ∈ ℙ ∧ 𝑄 = 2) → ((𝑁 ∈ Even ∧ 4 < 𝑁𝑁 = (𝑃 + 𝑄)) ↔ (𝑁 ∈ Even ∧ 4 < 𝑁𝑁 = (𝑃 + 2))))
12 breq2 4622 . . . . . . . . . . . . 13 (𝑁 = (𝑃 + 2) → (4 < 𝑁 ↔ 4 < (𝑃 + 2)))
13 eleq1 2692 . . . . . . . . . . . . 13 (𝑁 = (𝑃 + 2) → (𝑁 ∈ Even ↔ (𝑃 + 2) ∈ Even ))
1412, 13anbi12d 746 . . . . . . . . . . . 12 (𝑁 = (𝑃 + 2) → ((4 < 𝑁𝑁 ∈ Even ) ↔ (4 < (𝑃 + 2) ∧ (𝑃 + 2) ∈ Even )))
15 prmz 15308 . . . . . . . . . . . . . . . 16 (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
16 2evenALTV 40871 . . . . . . . . . . . . . . . 16 2 ∈ Even
17 evensumeven 40884 . . . . . . . . . . . . . . . 16 ((𝑃 ∈ ℤ ∧ 2 ∈ Even ) → (𝑃 ∈ Even ↔ (𝑃 + 2) ∈ Even ))
1815, 16, 17sylancl 693 . . . . . . . . . . . . . . 15 (𝑃 ∈ ℙ → (𝑃 ∈ Even ↔ (𝑃 + 2) ∈ Even ))
19 evenprm2 40891 . . . . . . . . . . . . . . . 16 (𝑃 ∈ ℙ → (𝑃 ∈ Even ↔ 𝑃 = 2))
20 oveq1 6612 . . . . . . . . . . . . . . . . . . 19 (𝑃 = 2 → (𝑃 + 2) = (2 + 2))
21 2p2e4 11089 . . . . . . . . . . . . . . . . . . 19 (2 + 2) = 4
2220, 21syl6eq 2676 . . . . . . . . . . . . . . . . . 18 (𝑃 = 2 → (𝑃 + 2) = 4)
2322breq2d 4630 . . . . . . . . . . . . . . . . 17 (𝑃 = 2 → (4 < (𝑃 + 2) ↔ 4 < 4))
24 4re 11042 . . . . . . . . . . . . . . . . . . 19 4 ∈ ℝ
2524ltnri 10091 . . . . . . . . . . . . . . . . . 18 ¬ 4 < 4
2625pm2.21i 116 . . . . . . . . . . . . . . . . 17 (4 < 4 → 𝑄 ∈ Odd )
2723, 26syl6bi 243 . . . . . . . . . . . . . . . 16 (𝑃 = 2 → (4 < (𝑃 + 2) → 𝑄 ∈ Odd ))
2819, 27syl6bi 243 . . . . . . . . . . . . . . 15 (𝑃 ∈ ℙ → (𝑃 ∈ Even → (4 < (𝑃 + 2) → 𝑄 ∈ Odd )))
2918, 28sylbird 250 . . . . . . . . . . . . . 14 (𝑃 ∈ ℙ → ((𝑃 + 2) ∈ Even → (4 < (𝑃 + 2) → 𝑄 ∈ Odd )))
3029com13 88 . . . . . . . . . . . . 13 (4 < (𝑃 + 2) → ((𝑃 + 2) ∈ Even → (𝑃 ∈ ℙ → 𝑄 ∈ Odd )))
3130imp 445 . . . . . . . . . . . 12 ((4 < (𝑃 + 2) ∧ (𝑃 + 2) ∈ Even ) → (𝑃 ∈ ℙ → 𝑄 ∈ Odd ))
3214, 31syl6bi 243 . . . . . . . . . . 11 (𝑁 = (𝑃 + 2) → ((4 < 𝑁𝑁 ∈ Even ) → (𝑃 ∈ ℙ → 𝑄 ∈ Odd )))
3332expd 452 . . . . . . . . . 10 (𝑁 = (𝑃 + 2) → (4 < 𝑁 → (𝑁 ∈ Even → (𝑃 ∈ ℙ → 𝑄 ∈ Odd ))))
3433com13 88 . . . . . . . . 9 (𝑁 ∈ Even → (4 < 𝑁 → (𝑁 = (𝑃 + 2) → (𝑃 ∈ ℙ → 𝑄 ∈ Odd ))))
35343imp 1254 . . . . . . . 8 ((𝑁 ∈ Even ∧ 4 < 𝑁𝑁 = (𝑃 + 2)) → (𝑃 ∈ ℙ → 𝑄 ∈ Odd ))
3635com12 32 . . . . . . 7 (𝑃 ∈ ℙ → ((𝑁 ∈ Even ∧ 4 < 𝑁𝑁 = (𝑃 + 2)) → 𝑄 ∈ Odd ))
3736adantr 481 . . . . . 6 ((𝑃 ∈ ℙ ∧ 𝑄 = 2) → ((𝑁 ∈ Even ∧ 4 < 𝑁𝑁 = (𝑃 + 2)) → 𝑄 ∈ Odd ))
3811, 37sylbid 230 . . . . 5 ((𝑃 ∈ ℙ ∧ 𝑄 = 2) → ((𝑁 ∈ Even ∧ 4 < 𝑁𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd ))
3938ex 450 . . . 4 (𝑃 ∈ ℙ → (𝑄 = 2 → ((𝑁 ∈ Even ∧ 4 < 𝑁𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd )))
4039adantr 481 . . 3 ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → (𝑄 = 2 → ((𝑁 ∈ Even ∧ 4 < 𝑁𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd )))
417, 40sylbid 230 . 2 ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → (¬ 𝑄 ∈ Odd → ((𝑁 ∈ Even ∧ 4 < 𝑁𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd )))
42 ax-1 6 . 2 (𝑄 ∈ Odd → ((𝑁 ∈ Even ∧ 4 < 𝑁𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd ))
4341, 42pm2.61d2 172 1 ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → ((𝑁 ∈ Even ∧ 4 < 𝑁𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1992   class class class wbr 4618  (class class class)co 6605   + caddc 9884   < clt 10019  cn 10965  2c2 11015  4c4 11017  cz 11322  cprime 15304   Even ceven 40805   Odd codd 40806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958  ax-pre-sup 9959
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-2o 7507  df-oadd 7510  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-sup 8293  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-div 10630  df-nn 10966  df-2 11024  df-3 11025  df-4 11026  df-n0 11238  df-z 11323  df-uz 11632  df-rp 11777  df-seq 12739  df-exp 12798  df-cj 13768  df-re 13769  df-im 13770  df-sqrt 13904  df-abs 13905  df-dvds 14903  df-prm 15305  df-even 40807  df-odd 40808
This theorem is referenced by:  sgoldbaltlem2  40932
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