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Theorem sbgoldbaltlem1 42195
Description: Lemma 1 for sbgoldbalt 42197: 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 15595 . . . . . 6 (𝑄 ∈ ℙ → 𝑄 ∈ ℕ)
2 nneoALTV 42111 . . . . . . 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 42151 . . . . 5 (𝑄 ∈ ℙ → (𝑄 ∈ Even ↔ 𝑄 = 2))
64, 5bitrd 268 . . . 4 (𝑄 ∈ ℙ → (¬ 𝑄 ∈ Odd ↔ 𝑄 = 2))
76adantl 467 . . 3 ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → (¬ 𝑄 ∈ Odd ↔ 𝑄 = 2))
8 oveq2 6801 . . . . . . . . 9 (𝑄 = 2 → (𝑃 + 𝑄) = (𝑃 + 2))
98eqeq2d 2781 . . . . . . . 8 (𝑄 = 2 → (𝑁 = (𝑃 + 𝑄) ↔ 𝑁 = (𝑃 + 2)))
109adantl 467 . . . . . . 7 ((𝑃 ∈ ℙ ∧ 𝑄 = 2) → (𝑁 = (𝑃 + 𝑄) ↔ 𝑁 = (𝑃 + 2)))
11103anbi3d 1553 . . . . . 6 ((𝑃 ∈ ℙ ∧ 𝑄 = 2) → ((𝑁 ∈ Even ∧ 4 < 𝑁𝑁 = (𝑃 + 𝑄)) ↔ (𝑁 ∈ Even ∧ 4 < 𝑁𝑁 = (𝑃 + 2))))
12 breq2 4790 . . . . . . . . . . . . 13 (𝑁 = (𝑃 + 2) → (4 < 𝑁 ↔ 4 < (𝑃 + 2)))
13 eleq1 2838 . . . . . . . . . . . . 13 (𝑁 = (𝑃 + 2) → (𝑁 ∈ Even ↔ (𝑃 + 2) ∈ Even ))
1412, 13anbi12d 616 . . . . . . . . . . . 12 (𝑁 = (𝑃 + 2) → ((4 < 𝑁𝑁 ∈ Even ) ↔ (4 < (𝑃 + 2) ∧ (𝑃 + 2) ∈ Even )))
15 prmz 15596 . . . . . . . . . . . . . . . 16 (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
16 2evenALTV 42131 . . . . . . . . . . . . . . . 16 2 ∈ Even
17 evensumeven 42144 . . . . . . . . . . . . . . . 16 ((𝑃 ∈ ℤ ∧ 2 ∈ Even ) → (𝑃 ∈ Even ↔ (𝑃 + 2) ∈ Even ))
1815, 16, 17sylancl 574 . . . . . . . . . . . . . . 15 (𝑃 ∈ ℙ → (𝑃 ∈ Even ↔ (𝑃 + 2) ∈ Even ))
19 evenprm2 42151 . . . . . . . . . . . . . . . 16 (𝑃 ∈ ℙ → (𝑃 ∈ Even ↔ 𝑃 = 2))
20 oveq1 6800 . . . . . . . . . . . . . . . . . . 19 (𝑃 = 2 → (𝑃 + 2) = (2 + 2))
21 2p2e4 11346 . . . . . . . . . . . . . . . . . . 19 (2 + 2) = 4
2220, 21syl6eq 2821 . . . . . . . . . . . . . . . . . 18 (𝑃 = 2 → (𝑃 + 2) = 4)
2322breq2d 4798 . . . . . . . . . . . . . . . . 17 (𝑃 = 2 → (4 < (𝑃 + 2) ↔ 4 < 4))
24 4re 11299 . . . . . . . . . . . . . . . . . . 19 4 ∈ ℝ
2524ltnri 10348 . . . . . . . . . . . . . . . . . 18 ¬ 4 < 4
2625pm2.21i 117 . . . . . . . . . . . . . . . . 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 393 . . . . . . . . . . . 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 400 . . . . . . . . . 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 1101 . . . . . . . 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 466 . . . . . 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 397 . . . 4 (𝑃 ∈ ℙ → (𝑄 = 2 → ((𝑁 ∈ Even ∧ 4 < 𝑁𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd )))
4039adantr 466 . . 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 173 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 382  w3a 1071   = wceq 1631  wcel 2145   class class class wbr 4786  (class class class)co 6793   + caddc 10141   < clt 10276  cn 11222  2c2 11272  4c4 11274  cz 11579  cprime 15592   Even ceven 42065   Odd codd 42066
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 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  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 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-sup 8504  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-nn 11223  df-2 11281  df-3 11282  df-4 11283  df-n0 11495  df-z 11580  df-uz 11889  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 42067  df-odd 42068
This theorem is referenced by:  sbgoldbaltlem2  42196
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