Theorem sbgoldbaltlem1 47784

Description: Lemma 1 for sbgoldbalt 47786: 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 16651	. . . . . 6 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℕ)
2		nneoALTV 47677	. . . . . . 7 ⊢ (𝑄 ∈ ℕ → (𝑄 ∈ Even ↔ ¬ 𝑄 ∈ Odd ))
3	2	bicomd 223	. . . . . 6 ⊢ (𝑄 ∈ ℕ → (¬ 𝑄 ∈ Odd ↔ 𝑄 ∈ Even ))
4	1, 3	syl 17	. . . . 5 ⊢ (𝑄 ∈ ℙ → (¬ 𝑄 ∈ Odd ↔ 𝑄 ∈ Even ))
5		evenprm2 47719	. . . . 5 ⊢ (𝑄 ∈ ℙ → (𝑄 ∈ Even ↔ 𝑄 = 2))
6	4, 5	bitrd 279	. . . 4 ⊢ (𝑄 ∈ ℙ → (¬ 𝑄 ∈ Odd ↔ 𝑄 = 2))
7	6	adantl 481	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → (¬ 𝑄 ∈ Odd ↔ 𝑄 = 2))
8		oveq2 7398	. . . . . . . . 9 ⊢ (𝑄 = 2 → (𝑃 + 𝑄) = (𝑃 + 2))
9	8	eqeq2d 2741	. . . . . . . 8 ⊢ (𝑄 = 2 → (𝑁 = (𝑃 + 𝑄) ↔ 𝑁 = (𝑃 + 2)))
10	9	adantl 481	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 = 2) → (𝑁 = (𝑃 + 𝑄) ↔ 𝑁 = (𝑃 + 2)))
11	10	3anbi3d 1444	. . . . . 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 2817	. . . . . . . . . . . . 13 ⊢ (𝑁 = (𝑃 + 2) → (𝑁 ∈ Even ↔ (𝑃 + 2) ∈ Even ))
14	12, 13	anbi12d 632	. . . . . . . . . . . 12 ⊢ (𝑁 = (𝑃 + 2) → ((4 < 𝑁 ∧ 𝑁 ∈ Even ) ↔ (4 < (𝑃 + 2) ∧ (𝑃 + 2) ∈ Even )))
15		prmz 16652	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
16		2evenALTV 47697	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ Even
17		evensumeven 47712	. . . . . . . . . . . . . . . 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 47719	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ Even ↔ 𝑃 = 2))
20		oveq1 7397	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 = 2 → (𝑃 + 2) = (2 + 2))
21		2p2e4 12323	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 + 2) = 4
22	20, 21	eqtrdi 2781	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 = 2 → (𝑃 + 2) = 4)
23	22	breq2d 5122	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 = 2 → (4 < (𝑃 + 2) ↔ 4 < 4))
24		4re 12277	. . . . . . . . . . . . . . . . . . 19 ⊢ 4 ∈ ℝ
25	24	ltnri 11290	. . . . . . . . . . . . . . . . . 18 ⊢ ¬ 4 < 4
26	25	pm2.21i 119	. . . . . . . . . . . . . . . . 17 ⊢ (4 < 4 → 𝑄 ∈ Odd )
27	23, 26	biimtrdi 253	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 = 2 → (4 < (𝑃 + 2) → 𝑄 ∈ Odd ))
28	19, 27	biimtrdi 253	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ Even → (4 < (𝑃 + 2) → 𝑄 ∈ Odd )))
29	18, 28	sylbird 260	. . . . . . . . . . . . . 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 406	. . . . . . . . . . . 12 ⊢ ((4 < (𝑃 + 2) ∧ (𝑃 + 2) ∈ Even ) → (𝑃 ∈ ℙ → 𝑄 ∈ Odd ))
32	14, 31	biimtrdi 253	. . . . . . . . . . 11 ⊢ (𝑁 = (𝑃 + 2) → ((4 < 𝑁 ∧ 𝑁 ∈ Even ) → (𝑃 ∈ ℙ → 𝑄 ∈ Odd )))
33	32	expd 415	. . . . . . . . . 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 1110	. . . . . . . 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 480	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 = 2) → ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 2)) → 𝑄 ∈ Odd ))
38	11, 37	sylbid 240	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 = 2) → ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd ))
39	38	ex 412	. . . 4 ⊢ (𝑃 ∈ ℙ → (𝑄 = 2 → ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd )))
40	39	adantr 480	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → (𝑄 = 2 → ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd )))
41	7, 40	sylbid 240	. 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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   class class class wbr 5110  (class class class)co 7390   + caddc 11078   < clt 11215  ℕcn 12193  2c2 12248  4c4 12250  ℤcz 12536  ℙcprime 16648   Even ceven 47629   Odd codd 47630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9400  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-n0 12450  df-z 12537  df-uz 12801  df-rp 12959  df-seq 13974  df-exp 14034  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-dvds 16230  df-prm 16649  df-even 47631  df-odd 47632

This theorem is referenced by:  sbgoldbaltlem2  47785