Theorem sbgoldbaltlem1 47703

Description: Lemma 1 for sbgoldbalt 47705: 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 16707	. . . . . 6 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℕ)
2		nneoALTV 47596	. . . . . . 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 47638	. . . . 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 7438	. . . . . . . . 9 ⊢ (𝑄 = 2 → (𝑃 + 𝑄) = (𝑃 + 2))
9	8	eqeq2d 2745	. . . . . . . 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 1441	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑄 = 2) → ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 𝑄)) ↔ (𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 2))))
12		breq2 5151	. . . . . . . . . . . . 13 ⊢ (𝑁 = (𝑃 + 2) → (4 < 𝑁 ↔ 4 < (𝑃 + 2)))
13		eleq1 2826	. . . . . . . . . . . . 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 16708	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
16		2evenALTV 47616	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ Even
17		evensumeven 47631	. . . . . . . . . . . . . . . 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 47638	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ Even ↔ 𝑃 = 2))
20		oveq1 7437	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 = 2 → (𝑃 + 2) = (2 + 2))
21		2p2e4 12398	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 + 2) = 4
22	20, 21	eqtrdi 2790	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 = 2 → (𝑃 + 2) = 4)
23	22	breq2d 5159	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 = 2 → (4 < (𝑃 + 2) ↔ 4 < 4))
24		4re 12347	. . . . . . . . . . . . . . . . . . 19 ⊢ 4 ∈ ℝ
25	24	ltnri 11367	. . . . . . . . . . . . . . . . . 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 1536   ∈ wcel 2105   class class class wbr 5147  (class class class)co 7430   + caddc 11155   < clt 11292  ℕcn 12263  2c2 12318  4c4 12320  ℤcz 12610  ℙcprime 16704   Even ceven 47548   Odd codd 47549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-n0 12524  df-z 12611  df-uz 12876  df-rp 13032  df-seq 14039  df-exp 14099  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-dvds 16287  df-prm 16705  df-even 47550  df-odd 47551

This theorem is referenced by:  sbgoldbaltlem2  47704