Theorem sbgoldbaltlem1 49082

Description: Lemma 1 for sbgoldbalt 49084: 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 < 𝑁 ∧ 𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd ))

Proof of Theorem sbgoldbaltlem1

Step	Hyp	Ref	Expression
1		prmnn 16878	. . . . . 6 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℕ)
2		nneoALTV 48975	. . . . . . 7 ⊢ (𝑄 ∈ ℕ → (𝑄 ∈ Even ↔ ¬ 𝑄 ∈ Odd ))
3	2	bicomd 226	. . . . . 6 ⊢ (𝑄 ∈ ℕ → (¬ 𝑄 ∈ Odd ↔ 𝑄 ∈ Even ))
4	1, 3	syl 18	. . . . 5 ⊢ (𝑄 ∈ ℙ → (¬ 𝑄 ∈ Odd ↔ 𝑄 ∈ Even ))
5		evenprm2 49017	. . . . 5 ⊢ (𝑄 ∈ ℙ → (𝑄 ∈ Even ↔ 𝑄 = 2))
6	4, 5	bitrd 282	. . . 4 ⊢ (𝑄 ∈ ℙ → (¬ 𝑄 ∈ Odd ↔ 𝑄 = 2))
7	6	adantl 487	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → (¬ 𝑄 ∈ Odd ↔ 𝑄 = 2))
8		oveq2 7451	. . . . . . . . 9 ⊢ (𝑄 = 2 → (𝑃 + 𝑄) = (𝑃 + 2))
9	8	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑄 = 2 → (𝑁 = (𝑃 + 𝑄) ↔ 𝑁 = (𝑃 + 2)))
10	9	adantl 487	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 = 2) → (𝑁 = (𝑃 + 𝑄) ↔ 𝑁 = (𝑃 + 2)))
11	10	3anbi3d 1471	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 = 2) → ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 𝑄)) ↔ (𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 2))))
12		breq2 5110	. . . . . . . . . . . . 13 ⊢ (𝑁 = (𝑃 + 2) → (4 < 𝑁 ↔ 4 < (𝑃 + 2)))
13		eleq1 2852	. . . . . . . . . . . . 13 ⊢ (𝑁 = (𝑃 + 2) → (𝑁 ∈ Even ↔ (𝑃 + 2) ∈ Even ))
14	12, 13	anbi12d 644	. . . . . . . . . . . 12 ⊢ (𝑁 = (𝑃 + 2) → ((4 < 𝑁 ∧ 𝑁 ∈ Even ) ↔ (4 < (𝑃 + 2) ∧ (𝑃 + 2) ∈ Even )))
15		prmz 16879	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
16		2evenALTV 48995	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ Even
17		evensumeven 49010	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ ℤ ∧ 2 ∈ Even ) → (𝑃 ∈ Even ↔ (𝑃 + 2) ∈ Even ))
18	15, 16, 17	sylancl 598	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ Even ↔ (𝑃 + 2) ∈ Even ))
19		evenprm2 49017	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ Even ↔ 𝑃 = 2))
20		oveq1 7450	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 = 2 → (𝑃 + 2) = (2 + 2))
21		2p2e4 12504	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 + 2) = 4
22	20, 21	eqtrdi 2814	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 = 2 → (𝑃 + 2) = 4)
23	22	breq2d 5118	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 = 2 → (4 < (𝑃 + 2) ↔ 4 < 4))
24		4re 12454	. . . . . . . . . . . . . . . . . . 19 ⊢ 4 ∈ ℝ
25	24	ltnri 11446	. . . . . . . . . . . . . . . . . 18 ⊢ ¬ 4 < 4
26	25	pm2.21i 120	. . . . . . . . . . . . . . . . 17 ⊢ (4 < 4 → 𝑄 ∈ Odd )
27	23, 26	biimtrdi 256	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 = 2 → (4 < (𝑃 + 2) → 𝑄 ∈ Odd ))
28	19, 27	biimtrdi 256	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ Even → (4 < (𝑃 + 2) → 𝑄 ∈ Odd )))
29	18, 28	sylbird 263	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ ℙ → ((𝑃 + 2) ∈ Even → (4 < (𝑃 + 2) → 𝑄 ∈ Odd )))
30	29	com13 89	. . . . . . . . . . . . 13 ⊢ (4 < (𝑃 + 2) → ((𝑃 + 2) ∈ Even → (𝑃 ∈ ℙ → 𝑄 ∈ Odd )))
31	30	imp 412	. . . . . . . . . . . 12 ⊢ ((4 < (𝑃 + 2) ∧ (𝑃 + 2) ∈ Even ) → (𝑃 ∈ ℙ → 𝑄 ∈ Odd ))
32	14, 31	biimtrdi 256	. . . . . . . . . . 11 ⊢ (𝑁 = (𝑃 + 2) → ((4 < 𝑁 ∧ 𝑁 ∈ Even ) → (𝑃 ∈ ℙ → 𝑄 ∈ Odd )))
33	32	expd 421	. . . . . . . . . 10 ⊢ (𝑁 = (𝑃 + 2) → (4 < 𝑁 → (𝑁 ∈ Even → (𝑃 ∈ ℙ → 𝑄 ∈ Odd ))))
34	33	com13 89	. . . . . . . . 9 ⊢ (𝑁 ∈ Even → (4 < 𝑁 → (𝑁 = (𝑃 + 2) → (𝑃 ∈ ℙ → 𝑄 ∈ Odd ))))
35	34	3imp 1128	. . . . . . . 8 ⊢ ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 2)) → (𝑃 ∈ ℙ → 𝑄 ∈ Odd ))
36	35	com12 33	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 2)) → 𝑄 ∈ Odd ))
37	36	adantr 486	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 = 2) → ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 2)) → 𝑄 ∈ Odd ))
38	11, 37	sylbid 243	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 = 2) → ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd ))
39	38	ex 418	. . . 4 ⊢ (𝑃 ∈ ℙ → (𝑄 = 2 → ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd )))
40	39	adantr 486	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → (𝑄 = 2 → ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd )))
41	7, 40	sylbid 243	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → (¬ 𝑄 ∈ Odd → ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd )))
42		ax-1 6	. 2 ⊢ (𝑄 ∈ Odd → ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd ))
43	41, 42	pm2.61d2 183	1 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd ))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   class class class wbr 5106  (class class class)co 7443   + caddc 11230   < clt 11370  ℕcn 12362  2c2 12424  4c4 12426  ℤcz 12720  ℙcprime 16875   Even ceven 48927   Odd codd 48928

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304  ax-pre-sup 11305

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-2o 8505  df-er 8745  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-sup 9461  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-div 12001  df-nn 12363  df-2 12432  df-3 12433  df-4 12434  df-n0 12634  df-z 12721  df-uz 12993  df-rp 13148  df-seq 14172  df-exp 14232  df-cj 15295  df-re 15296  df-im 15297  df-sqrt 15431  df-abs 15432  df-dvds 16452  df-prm 16876  df-even 48929  df-odd 48930

This theorem is used by:  sbgoldbaltlem2  49083