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Theorem sbgoldbaltlem1 43821
Description: Lemma 1 for sbgoldbalt 43823: 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
sbgoldbaltlem1 ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → ((𝑁 ∈ Even ∧ 4 < 𝑁𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd ))

Proof of Theorem sbgoldbaltlem1
StepHypRef Expression
1 prmnn 16006 . . . . . 6 (𝑄 ∈ ℙ → 𝑄 ∈ ℕ)
2 nneoALTV 43714 . . . . . . 7 (𝑄 ∈ ℕ → (𝑄 ∈ Even ↔ ¬ 𝑄 ∈ Odd ))
32bicomd 224 . . . . . 6 (𝑄 ∈ ℕ → (¬ 𝑄 ∈ Odd ↔ 𝑄 ∈ Even ))
41, 3syl 17 . . . . 5 (𝑄 ∈ ℙ → (¬ 𝑄 ∈ Odd ↔ 𝑄 ∈ Even ))
5 evenprm2 43756 . . . . 5 (𝑄 ∈ ℙ → (𝑄 ∈ Even ↔ 𝑄 = 2))
64, 5bitrd 280 . . . 4 (𝑄 ∈ ℙ → (¬ 𝑄 ∈ Odd ↔ 𝑄 = 2))
76adantl 482 . . 3 ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → (¬ 𝑄 ∈ Odd ↔ 𝑄 = 2))
8 oveq2 7153 . . . . . . . . 9 (𝑄 = 2 → (𝑃 + 𝑄) = (𝑃 + 2))
98eqeq2d 2829 . . . . . . . 8 (𝑄 = 2 → (𝑁 = (𝑃 + 𝑄) ↔ 𝑁 = (𝑃 + 2)))
109adantl 482 . . . . . . 7 ((𝑃 ∈ ℙ ∧ 𝑄 = 2) → (𝑁 = (𝑃 + 𝑄) ↔ 𝑁 = (𝑃 + 2)))
11103anbi3d 1433 . . . . . 6 ((𝑃 ∈ ℙ ∧ 𝑄 = 2) → ((𝑁 ∈ Even ∧ 4 < 𝑁𝑁 = (𝑃 + 𝑄)) ↔ (𝑁 ∈ Even ∧ 4 < 𝑁𝑁 = (𝑃 + 2))))
12 breq2 5061 . . . . . . . . . . . . 13 (𝑁 = (𝑃 + 2) → (4 < 𝑁 ↔ 4 < (𝑃 + 2)))
13 eleq1 2897 . . . . . . . . . . . . 13 (𝑁 = (𝑃 + 2) → (𝑁 ∈ Even ↔ (𝑃 + 2) ∈ Even ))
1412, 13anbi12d 630 . . . . . . . . . . . 12 (𝑁 = (𝑃 + 2) → ((4 < 𝑁𝑁 ∈ Even ) ↔ (4 < (𝑃 + 2) ∧ (𝑃 + 2) ∈ Even )))
15 prmz 16007 . . . . . . . . . . . . . . . 16 (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
16 2evenALTV 43734 . . . . . . . . . . . . . . . 16 2 ∈ Even
17 evensumeven 43749 . . . . . . . . . . . . . . . 16 ((𝑃 ∈ ℤ ∧ 2 ∈ Even ) → (𝑃 ∈ Even ↔ (𝑃 + 2) ∈ Even ))
1815, 16, 17sylancl 586 . . . . . . . . . . . . . . 15 (𝑃 ∈ ℙ → (𝑃 ∈ Even ↔ (𝑃 + 2) ∈ Even ))
19 evenprm2 43756 . . . . . . . . . . . . . . . 16 (𝑃 ∈ ℙ → (𝑃 ∈ Even ↔ 𝑃 = 2))
20 oveq1 7152 . . . . . . . . . . . . . . . . . . 19 (𝑃 = 2 → (𝑃 + 2) = (2 + 2))
21 2p2e4 11760 . . . . . . . . . . . . . . . . . . 19 (2 + 2) = 4
2220, 21syl6eq 2869 . . . . . . . . . . . . . . . . . 18 (𝑃 = 2 → (𝑃 + 2) = 4)
2322breq2d 5069 . . . . . . . . . . . . . . . . 17 (𝑃 = 2 → (4 < (𝑃 + 2) ↔ 4 < 4))
24 4re 11709 . . . . . . . . . . . . . . . . . . 19 4 ∈ ℝ
2524ltnri 10737 . . . . . . . . . . . . . . . . . 18 ¬ 4 < 4
2625pm2.21i 119 . . . . . . . . . . . . . . . . 17 (4 < 4 → 𝑄 ∈ Odd )
2723, 26syl6bi 254 . . . . . . . . . . . . . . . 16 (𝑃 = 2 → (4 < (𝑃 + 2) → 𝑄 ∈ Odd ))
2819, 27syl6bi 254 . . . . . . . . . . . . . . 15 (𝑃 ∈ ℙ → (𝑃 ∈ Even → (4 < (𝑃 + 2) → 𝑄 ∈ Odd )))
2918, 28sylbird 261 . . . . . . . . . . . . . 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 407 . . . . . . . . . . . 12 ((4 < (𝑃 + 2) ∧ (𝑃 + 2) ∈ Even ) → (𝑃 ∈ ℙ → 𝑄 ∈ Odd ))
3214, 31syl6bi 254 . . . . . . . . . . 11 (𝑁 = (𝑃 + 2) → ((4 < 𝑁𝑁 ∈ Even ) → (𝑃 ∈ ℙ → 𝑄 ∈ Odd )))
3332expd 416 . . . . . . . . . 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 1103 . . . . . . . 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 241 . . . . 5 ((𝑃 ∈ ℙ ∧ 𝑄 = 2) → ((𝑁 ∈ Even ∧ 4 < 𝑁𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd ))
3938ex 413 . . . 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 241 . 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 182 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 207  wa 396  w3a 1079   = wceq 1528  wcel 2105   class class class wbr 5057  (class class class)co 7145   + caddc 10528   < clt 10663  cn 11626  2c2 11680  4c4 11682  cz 11969  cprime 16003   Even ceven 43666   Odd codd 43667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-seq 13358  df-exp 13418  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-dvds 15596  df-prm 16004  df-even 43668  df-odd 43669
This theorem is referenced by:  sbgoldbaltlem2  43822
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