Theorem sbgoldbaltlem1 48392

Description: Lemma 1 for sbgoldbalt 48394: 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 16718	. . . . . 6 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℕ)
2		nneoALTV 48285	. . . . . . 7 ⊢ (𝑄 ∈ ℕ → (𝑄 ∈ Even ↔ ¬ 𝑄 ∈ Odd ))
3	2	bicomd 225	. . . . . 6 ⊢ (𝑄 ∈ ℕ → (¬ 𝑄 ∈ Odd ↔ 𝑄 ∈ Even ))
4	1, 3	syl 17	. . . . 5 ⊢ (𝑄 ∈ ℙ → (¬ 𝑄 ∈ Odd ↔ 𝑄 ∈ Even ))
5		evenprm2 48327	. . . . 5 ⊢ (𝑄 ∈ ℙ → (𝑄 ∈ Even ↔ 𝑄 = 2))
6	4, 5	bitrd 281	. . . 4 ⊢ (𝑄 ∈ ℙ → (¬ 𝑄 ∈ Odd ↔ 𝑄 = 2))
7	6	adantl 485	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → (¬ 𝑄 ∈ Odd ↔ 𝑄 = 2))
8		oveq2 7404	. . . . . . . . 9 ⊢ (𝑄 = 2 → (𝑃 + 𝑄) = (𝑃 + 2))
9	8	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑄 = 2 → (𝑁 = (𝑃 + 𝑄) ↔ 𝑁 = (𝑃 + 2)))
10	9	adantl 485	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 = 2) → (𝑁 = (𝑃 + 𝑄) ↔ 𝑁 = (𝑃 + 2)))
11	10	3anbi3d 1464	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 = 2) → ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 𝑄)) ↔ (𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 2))))
12		breq2 5105	. . . . . . . . . . . . 13 ⊢ (𝑁 = (𝑃 + 2) → (4 < 𝑁 ↔ 4 < (𝑃 + 2)))
13		eleq1 2851	. . . . . . . . . . . . 13 ⊢ (𝑁 = (𝑃 + 2) → (𝑁 ∈ Even ↔ (𝑃 + 2) ∈ Even ))
14	12, 13	anbi12d 641	. . . . . . . . . . . 12 ⊢ (𝑁 = (𝑃 + 2) → ((4 < 𝑁 ∧ 𝑁 ∈ Even ) ↔ (4 < (𝑃 + 2) ∧ (𝑃 + 2) ∈ Even )))
15		prmz 16719	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
16		2evenALTV 48305	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ Even
17		evensumeven 48320	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ ℤ ∧ 2 ∈ Even ) → (𝑃 ∈ Even ↔ (𝑃 + 2) ∈ Even ))
18	15, 16, 17	sylancl 595	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ Even ↔ (𝑃 + 2) ∈ Even ))
19		evenprm2 48327	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ Even ↔ 𝑃 = 2))
20		oveq1 7403	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 = 2 → (𝑃 + 2) = (2 + 2))
21		2p2e4 12362	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 + 2) = 4
22	20, 21	eqtrdi 2814	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 = 2 → (𝑃 + 2) = 4)
23	22	breq2d 5113	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 = 2 → (4 < (𝑃 + 2) ↔ 4 < 4))
24		4re 12312	. . . . . . . . . . . . . . . . . . 19 ⊢ 4 ∈ ℝ
25	24	ltnri 11303	. . . . . . . . . . . . . . . . . 18 ⊢ ¬ 4 < 4
26	25	pm2.21i 119	. . . . . . . . . . . . . . . . 17 ⊢ (4 < 4 → 𝑄 ∈ Odd )
27	23, 26	biimtrdi 255	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 = 2 → (4 < (𝑃 + 2) → 𝑄 ∈ Odd ))
28	19, 27	biimtrdi 255	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ Even → (4 < (𝑃 + 2) → 𝑄 ∈ Odd )))
29	18, 28	sylbird 262	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ ℙ → ((𝑃 + 2) ∈ Even → (4 < (𝑃 + 2) → 𝑄 ∈ Odd )))
30	29	com13 88	. . . . . . . . . . . . 13 ⊢ (4 < (𝑃 + 2) → ((𝑃 + 2) ∈ Even → (𝑃 ∈ ℙ → 𝑄 ∈ Odd )))
31	30	imp 410	. . . . . . . . . . . 12 ⊢ ((4 < (𝑃 + 2) ∧ (𝑃 + 2) ∈ Even ) → (𝑃 ∈ ℙ → 𝑄 ∈ Odd ))
32	14, 31	biimtrdi 255	. . . . . . . . . . 11 ⊢ (𝑁 = (𝑃 + 2) → ((4 < 𝑁 ∧ 𝑁 ∈ Even ) → (𝑃 ∈ ℙ → 𝑄 ∈ Odd )))
33	32	expd 419	. . . . . . . . . 10 ⊢ (𝑁 = (𝑃 + 2) → (4 < 𝑁 → (𝑁 ∈ Even → (𝑃 ∈ ℙ → 𝑄 ∈ Odd ))))
34	33	com13 88	. . . . . . . . 9 ⊢ (𝑁 ∈ Even → (4 < 𝑁 → (𝑁 = (𝑃 + 2) → (𝑃 ∈ ℙ → 𝑄 ∈ Odd ))))
35	34	3imp 1124	. . . . . . . 8 ⊢ ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 2)) → (𝑃 ∈ ℙ → 𝑄 ∈ Odd ))
36	35	com12 32	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 2)) → 𝑄 ∈ Odd ))
37	36	adantr 484	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 = 2) → ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 2)) → 𝑄 ∈ Odd ))
38	11, 37	sylbid 242	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 = 2) → ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd ))
39	38	ex 416	. . . 4 ⊢ (𝑃 ∈ ℙ → (𝑄 = 2 → ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd )))
40	39	adantr 484	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → (𝑄 = 2 → ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd )))
41	7, 40	sylbid 242	. 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 182	1 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1561   ∈ wcel 2143   class class class wbr 5101  (class class class)co 7396   + caddc 11087   < clt 11227  ℕcn 12220  2c2 12282  4c4 12284  ℤcz 12578  ℙcprime 16715   Even ceven 48237   Odd codd 48238

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7718  ax-cnex 11140  ax-resscn 11141  ax-1cn 11142  ax-icn 11143  ax-addcl 11144  ax-addrcl 11145  ax-mulcl 11146  ax-mulrcl 11147  ax-mulcom 11148  ax-addass 11149  ax-mulass 11150  ax-distr 11151  ax-i2m1 11152  ax-1ne0 11153  ax-1rid 11154  ax-rnegex 11155  ax-rrecex 11156  ax-cnre 11157  ax-pre-lttri 11158  ax-pre-lttrn 11159  ax-pre-ltadd 11160  ax-pre-mulgt0 11161  ax-pre-sup 11162

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8678  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-sup 9386  df-pnf 11229  df-mnf 11230  df-xr 11231  df-ltxr 11232  df-le 11233  df-sub 11427  df-neg 11428  df-div 11856  df-nn 12221  df-2 12290  df-3 12291  df-4 12292  df-n0 12492  df-z 12579  df-uz 12850  df-rp 13004  df-seq 14025  df-exp 14085  df-cj 15136  df-re 15137  df-im 15138  df-sqrt 15272  df-abs 15273  df-dvds 16297  df-prm 16716  df-even 48239  df-odd 48240

This theorem is referenced by:  sbgoldbaltlem2  48393