Theorem sbgoldbaltlem1 46091

Description: Lemma 1 for sbgoldbalt 46093: 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 16561	. . . . . 6 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℕ)
2		nneoALTV 45984	. . . . . . 7 ⊢ (𝑄 ∈ ℕ → (𝑄 ∈ Even ↔ ¬ 𝑄 ∈ Odd ))
3	2	bicomd 222	. . . . . 6 ⊢ (𝑄 ∈ ℕ → (¬ 𝑄 ∈ Odd ↔ 𝑄 ∈ Even ))
4	1, 3	syl 17	. . . . 5 ⊢ (𝑄 ∈ ℙ → (¬ 𝑄 ∈ Odd ↔ 𝑄 ∈ Even ))
5		evenprm2 46026	. . . . 5 ⊢ (𝑄 ∈ ℙ → (𝑄 ∈ Even ↔ 𝑄 = 2))
6	4, 5	bitrd 278	. . . 4 ⊢ (𝑄 ∈ ℙ → (¬ 𝑄 ∈ Odd ↔ 𝑄 = 2))
7	6	adantl 482	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → (¬ 𝑄 ∈ Odd ↔ 𝑄 = 2))
8		oveq2 7370	. . . . . . . . 9 ⊢ (𝑄 = 2 → (𝑃 + 𝑄) = (𝑃 + 2))
9	8	eqeq2d 2742	. . . . . . . 8 ⊢ (𝑄 = 2 → (𝑁 = (𝑃 + 𝑄) ↔ 𝑁 = (𝑃 + 2)))
10	9	adantl 482	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 = 2) → (𝑁 = (𝑃 + 𝑄) ↔ 𝑁 = (𝑃 + 2)))
11	10	3anbi3d 1442	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 = 2) → ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 𝑄)) ↔ (𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 2))))
12		breq2 5114	. . . . . . . . . . . . 13 ⊢ (𝑁 = (𝑃 + 2) → (4 < 𝑁 ↔ 4 < (𝑃 + 2)))
13		eleq1 2820	. . . . . . . . . . . . 13 ⊢ (𝑁 = (𝑃 + 2) → (𝑁 ∈ Even ↔ (𝑃 + 2) ∈ Even ))
14	12, 13	anbi12d 631	. . . . . . . . . . . 12 ⊢ (𝑁 = (𝑃 + 2) → ((4 < 𝑁 ∧ 𝑁 ∈ Even ) ↔ (4 < (𝑃 + 2) ∧ (𝑃 + 2) ∈ Even )))
15		prmz 16562	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
16		2evenALTV 46004	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ Even
17		evensumeven 46019	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ ℤ ∧ 2 ∈ Even ) → (𝑃 ∈ Even ↔ (𝑃 + 2) ∈ Even ))
18	15, 16, 17	sylancl 586	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ Even ↔ (𝑃 + 2) ∈ Even ))
19		evenprm2 46026	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ Even ↔ 𝑃 = 2))
20		oveq1 7369	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 = 2 → (𝑃 + 2) = (2 + 2))
21		2p2e4 12297	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 + 2) = 4
22	20, 21	eqtrdi 2787	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 = 2 → (𝑃 + 2) = 4)
23	22	breq2d 5122	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 = 2 → (4 < (𝑃 + 2) ↔ 4 < 4))
24		4re 12246	. . . . . . . . . . . . . . . . . . 19 ⊢ 4 ∈ ℝ
25	24	ltnri 11273	. . . . . . . . . . . . . . . . . 18 ⊢ ¬ 4 < 4
26	25	pm2.21i 119	. . . . . . . . . . . . . . . . 17 ⊢ (4 < 4 → 𝑄 ∈ Odd )
27	23, 26	syl6bi 252	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 = 2 → (4 < (𝑃 + 2) → 𝑄 ∈ Odd ))
28	19, 27	syl6bi 252	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ Even → (4 < (𝑃 + 2) → 𝑄 ∈ Odd )))
29	18, 28	sylbird 259	. . . . . . . . . . . . . 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 407	. . . . . . . . . . . 12 ⊢ ((4 < (𝑃 + 2) ∧ (𝑃 + 2) ∈ Even ) → (𝑃 ∈ ℙ → 𝑄 ∈ Odd ))
32	14, 31	syl6bi 252	. . . . . . . . . . 11 ⊢ (𝑁 = (𝑃 + 2) → ((4 < 𝑁 ∧ 𝑁 ∈ Even ) → (𝑃 ∈ ℙ → 𝑄 ∈ Odd )))
33	32	expd 416	. . . . . . . . . 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 1111	. . . . . . . 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 481	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 = 2) → ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 2)) → 𝑄 ∈ Odd ))
38	11, 37	sylbid 239	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 = 2) → ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd ))
39	38	ex 413	. . . 4 ⊢ (𝑃 ∈ ℙ → (𝑄 = 2 → ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd )))
40	39	adantr 481	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → (𝑄 = 2 → ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd )))
41	7, 40	sylbid 239	. 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 181	1 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   class class class wbr 5110  (class class class)co 7362   + caddc 11063   < clt 11198  ℕcn 12162  2c2 12217  4c4 12219  ℤcz 12508  ℙcprime 16558   Even ceven 45936   Odd codd 45937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11116  ax-resscn 11117  ax-1cn 11118  ax-icn 11119  ax-addcl 11120  ax-addrcl 11121  ax-mulcl 11122  ax-mulrcl 11123  ax-mulcom 11124  ax-addass 11125  ax-mulass 11126  ax-distr 11127  ax-i2m1 11128  ax-1ne0 11129  ax-1rid 11130  ax-rnegex 11131  ax-rrecex 11132  ax-cnre 11133  ax-pre-lttri 11134  ax-pre-lttrn 11135  ax-pre-ltadd 11136  ax-pre-mulgt0 11137  ax-pre-sup 11138

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-sup 9387  df-pnf 11200  df-mnf 11201  df-xr 11202  df-ltxr 11203  df-le 11204  df-sub 11396  df-neg 11397  df-div 11822  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-n0 12423  df-z 12509  df-uz 12773  df-rp 12925  df-seq 13917  df-exp 13978  df-cj 14996  df-re 14997  df-im 14998  df-sqrt 15132  df-abs 15133  df-dvds 16148  df-prm 16559  df-even 45938  df-odd 45939

This theorem is referenced by:  sbgoldbaltlem2  46092