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Theorem sbgoldbaltlem1 46061
Description: Lemma 1 for sbgoldbalt 46063: 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 16558 . . . . . 6 (𝑄 ∈ ℙ → 𝑄 ∈ ℕ)
2 nneoALTV 45954 . . . . . . 7 (𝑄 ∈ ℕ → (𝑄 ∈ Even ↔ ¬ 𝑄 ∈ Odd ))
32bicomd 222 . . . . . 6 (𝑄 ∈ ℕ → (¬ 𝑄 ∈ Odd ↔ 𝑄 ∈ Even ))
41, 3syl 17 . . . . 5 (𝑄 ∈ ℙ → (¬ 𝑄 ∈ Odd ↔ 𝑄 ∈ Even ))
5 evenprm2 45996 . . . . 5 (𝑄 ∈ ℙ → (𝑄 ∈ Even ↔ 𝑄 = 2))
64, 5bitrd 279 . . . 4 (𝑄 ∈ ℙ → (¬ 𝑄 ∈ Odd ↔ 𝑄 = 2))
76adantl 483 . . 3 ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → (¬ 𝑄 ∈ Odd ↔ 𝑄 = 2))
8 oveq2 7369 . . . . . . . . 9 (𝑄 = 2 → (𝑃 + 𝑄) = (𝑃 + 2))
98eqeq2d 2744 . . . . . . . 8 (𝑄 = 2 → (𝑁 = (𝑃 + 𝑄) ↔ 𝑁 = (𝑃 + 2)))
109adantl 483 . . . . . . 7 ((𝑃 ∈ ℙ ∧ 𝑄 = 2) → (𝑁 = (𝑃 + 𝑄) ↔ 𝑁 = (𝑃 + 2)))
11103anbi3d 1443 . . . . . 6 ((𝑃 ∈ ℙ ∧ 𝑄 = 2) → ((𝑁 ∈ Even ∧ 4 < 𝑁𝑁 = (𝑃 + 𝑄)) ↔ (𝑁 ∈ Even ∧ 4 < 𝑁𝑁 = (𝑃 + 2))))
12 breq2 5113 . . . . . . . . . . . . 13 (𝑁 = (𝑃 + 2) → (4 < 𝑁 ↔ 4 < (𝑃 + 2)))
13 eleq1 2822 . . . . . . . . . . . . 13 (𝑁 = (𝑃 + 2) → (𝑁 ∈ Even ↔ (𝑃 + 2) ∈ Even ))
1412, 13anbi12d 632 . . . . . . . . . . . 12 (𝑁 = (𝑃 + 2) → ((4 < 𝑁𝑁 ∈ Even ) ↔ (4 < (𝑃 + 2) ∧ (𝑃 + 2) ∈ Even )))
15 prmz 16559 . . . . . . . . . . . . . . . 16 (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
16 2evenALTV 45974 . . . . . . . . . . . . . . . 16 2 ∈ Even
17 evensumeven 45989 . . . . . . . . . . . . . . . 16 ((𝑃 ∈ ℤ ∧ 2 ∈ Even ) → (𝑃 ∈ Even ↔ (𝑃 + 2) ∈ Even ))
1815, 16, 17sylancl 587 . . . . . . . . . . . . . . 15 (𝑃 ∈ ℙ → (𝑃 ∈ Even ↔ (𝑃 + 2) ∈ Even ))
19 evenprm2 45996 . . . . . . . . . . . . . . . 16 (𝑃 ∈ ℙ → (𝑃 ∈ Even ↔ 𝑃 = 2))
20 oveq1 7368 . . . . . . . . . . . . . . . . . . 19 (𝑃 = 2 → (𝑃 + 2) = (2 + 2))
21 2p2e4 12296 . . . . . . . . . . . . . . . . . . 19 (2 + 2) = 4
2220, 21eqtrdi 2789 . . . . . . . . . . . . . . . . . 18 (𝑃 = 2 → (𝑃 + 2) = 4)
2322breq2d 5121 . . . . . . . . . . . . . . . . 17 (𝑃 = 2 → (4 < (𝑃 + 2) ↔ 4 < 4))
24 4re 12245 . . . . . . . . . . . . . . . . . . 19 4 ∈ ℝ
2524ltnri 11272 . . . . . . . . . . . . . . . . . 18 ¬ 4 < 4
2625pm2.21i 119 . . . . . . . . . . . . . . . . 17 (4 < 4 → 𝑄 ∈ Odd )
2723, 26syl6bi 253 . . . . . . . . . . . . . . . 16 (𝑃 = 2 → (4 < (𝑃 + 2) → 𝑄 ∈ Odd ))
2819, 27syl6bi 253 . . . . . . . . . . . . . . 15 (𝑃 ∈ ℙ → (𝑃 ∈ Even → (4 < (𝑃 + 2) → 𝑄 ∈ Odd )))
2918, 28sylbird 260 . . . . . . . . . . . . . 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 408 . . . . . . . . . . . 12 ((4 < (𝑃 + 2) ∧ (𝑃 + 2) ∈ Even ) → (𝑃 ∈ ℙ → 𝑄 ∈ Odd ))
3214, 31syl6bi 253 . . . . . . . . . . 11 (𝑁 = (𝑃 + 2) → ((4 < 𝑁𝑁 ∈ Even ) → (𝑃 ∈ ℙ → 𝑄 ∈ Odd )))
3332expd 417 . . . . . . . . . 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 1112 . . . . . . . 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 482 . . . . . 6 ((𝑃 ∈ ℙ ∧ 𝑄 = 2) → ((𝑁 ∈ Even ∧ 4 < 𝑁𝑁 = (𝑃 + 2)) → 𝑄 ∈ Odd ))
3811, 37sylbid 239 . . . . 5 ((𝑃 ∈ ℙ ∧ 𝑄 = 2) → ((𝑁 ∈ Even ∧ 4 < 𝑁𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd ))
3938ex 414 . . . 4 (𝑃 ∈ ℙ → (𝑄 = 2 → ((𝑁 ∈ Even ∧ 4 < 𝑁𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd )))
4039adantr 482 . . 3 ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → (𝑄 = 2 → ((𝑁 ∈ Even ∧ 4 < 𝑁𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd )))
417, 40sylbid 239 . 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 181 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 205  wa 397  w3a 1088   = wceq 1542  wcel 2107   class class class wbr 5109  (class class class)co 7361   + caddc 11062   < clt 11197  cn 12161  2c2 12216  4c4 12218  cz 12507  cprime 16555   Even ceven 45906   Odd codd 45907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-pre-sup 11137
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-2o 8417  df-er 8654  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-sup 9386  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-div 11821  df-nn 12162  df-2 12224  df-3 12225  df-4 12226  df-n0 12422  df-z 12508  df-uz 12772  df-rp 12924  df-seq 13916  df-exp 13977  df-cj 14993  df-re 14994  df-im 14995  df-sqrt 15129  df-abs 15130  df-dvds 16145  df-prm 16556  df-even 45908  df-odd 45909
This theorem is referenced by:  sbgoldbaltlem2  46062
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