Theorem stgoldbwt 45228

Description: If the strong ternary Goldbach conjecture is valid, then the weak ternary Goldbach conjecture holds, too. (Contributed by AV, 27-Jul-2020.)

Assertion

Ref	Expression
stgoldbwt	⊢ (∀𝑛 ∈ Odd (7 < 𝑛 → 𝑛 ∈ GoldbachOdd ) → ∀𝑛 ∈ Odd (5 < 𝑛 → 𝑛 ∈ GoldbachOddW ))

Proof of Theorem stgoldbwt

Step	Hyp	Ref	Expression
1		pm3.35 800	. . . . . 6 ⊢ ((7 < 𝑛 ∧ (7 < 𝑛 → 𝑛 ∈ GoldbachOdd )) → 𝑛 ∈ GoldbachOdd )
2		gbogbow 45208	. . . . . . 7 ⊢ (𝑛 ∈ GoldbachOdd → 𝑛 ∈ GoldbachOddW )
3	2	a1d 25	. . . . . 6 ⊢ (𝑛 ∈ GoldbachOdd → (5 < 𝑛 → 𝑛 ∈ GoldbachOddW ))
4	1, 3	syl 17	. . . . 5 ⊢ ((7 < 𝑛 ∧ (7 < 𝑛 → 𝑛 ∈ GoldbachOdd )) → (5 < 𝑛 → 𝑛 ∈ GoldbachOddW ))
5	4	ex 413	. . . 4 ⊢ (7 < 𝑛 → ((7 < 𝑛 → 𝑛 ∈ GoldbachOdd ) → (5 < 𝑛 → 𝑛 ∈ GoldbachOddW )))
6	5	a1d 25	. . 3 ⊢ (7 < 𝑛 → (𝑛 ∈ Odd → ((7 < 𝑛 → 𝑛 ∈ GoldbachOdd ) → (5 < 𝑛 → 𝑛 ∈ GoldbachOddW ))))
7		oddz 45083	. . . . . . . 8 ⊢ (𝑛 ∈ Odd → 𝑛 ∈ ℤ)
8	7	zred 12426	. . . . . . 7 ⊢ (𝑛 ∈ Odd → 𝑛 ∈ ℝ)
9		7re 12066	. . . . . . . 8 ⊢ 7 ∈ ℝ
10	9	a1i 11	. . . . . . 7 ⊢ (𝑛 ∈ Odd → 7 ∈ ℝ)
11	8, 10	lenltd 11121	. . . . . 6 ⊢ (𝑛 ∈ Odd → (𝑛 ≤ 7 ↔ ¬ 7 < 𝑛))
12	8, 10	leloed 11118	. . . . . . . 8 ⊢ (𝑛 ∈ Odd → (𝑛 ≤ 7 ↔ (𝑛 < 7 ∨ 𝑛 = 7)))
13	7	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ Odd ∧ 5 < 𝑛) → 𝑛 ∈ ℤ)
14		6nn 12062	. . . . . . . . . . . . . . . . 17 ⊢ 6 ∈ ℕ
15	14	nnzi 12344	. . . . . . . . . . . . . . . 16 ⊢ 6 ∈ ℤ
16	13, 15	jctir 521	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (𝑛 ∈ ℤ ∧ 6 ∈ ℤ))
17	16	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑛 < 7 ∧ (𝑛 ∈ Odd ∧ 5 < 𝑛)) → (𝑛 ∈ ℤ ∧ 6 ∈ ℤ))
18		df-7 12041	. . . . . . . . . . . . . . . . 17 ⊢ 7 = (6 + 1)
19	18	breq2i 5082	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 < 7 ↔ 𝑛 < (6 + 1))
20	19	biimpi 215	. . . . . . . . . . . . . . 15 ⊢ (𝑛 < 7 → 𝑛 < (6 + 1))
21		df-6 12040	. . . . . . . . . . . . . . . 16 ⊢ 6 = (5 + 1)
22		5nn 12059	. . . . . . . . . . . . . . . . . . 19 ⊢ 5 ∈ ℕ
23	22	nnzi 12344	. . . . . . . . . . . . . . . . . 18 ⊢ 5 ∈ ℤ
24		zltp1le 12370	. . . . . . . . . . . . . . . . . 18 ⊢ ((5 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (5 < 𝑛 ↔ (5 + 1) ≤ 𝑛))
25	23, 7, 24	sylancr 587	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ Odd → (5 < 𝑛 ↔ (5 + 1) ≤ 𝑛))
26	25	biimpa 477	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (5 + 1) ≤ 𝑛)
27	21, 26	eqbrtrid 5109	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ Odd ∧ 5 < 𝑛) → 6 ≤ 𝑛)
28	20, 27	anim12ci 614	. . . . . . . . . . . . . 14 ⊢ ((𝑛 < 7 ∧ (𝑛 ∈ Odd ∧ 5 < 𝑛)) → (6 ≤ 𝑛 ∧ 𝑛 < (6 + 1)))
29		zgeltp1eq 44801	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℤ ∧ 6 ∈ ℤ) → ((6 ≤ 𝑛 ∧ 𝑛 < (6 + 1)) → 𝑛 = 6))
30	17, 28, 29	sylc 65	. . . . . . . . . . . . 13 ⊢ ((𝑛 < 7 ∧ (𝑛 ∈ Odd ∧ 5 < 𝑛)) → 𝑛 = 6)
31	30	orcd 870	. . . . . . . . . . . 12 ⊢ ((𝑛 < 7 ∧ (𝑛 ∈ Odd ∧ 5 < 𝑛)) → (𝑛 = 6 ∨ 𝑛 = 7))
32	31	ex 413	. . . . . . . . . . 11 ⊢ (𝑛 < 7 → ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (𝑛 = 6 ∨ 𝑛 = 7)))
33		olc 865	. . . . . . . . . . . 12 ⊢ (𝑛 = 7 → (𝑛 = 6 ∨ 𝑛 = 7))
34	33	a1d 25	. . . . . . . . . . 11 ⊢ (𝑛 = 7 → ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (𝑛 = 6 ∨ 𝑛 = 7)))
35	32, 34	jaoi 854	. . . . . . . . . 10 ⊢ ((𝑛 < 7 ∨ 𝑛 = 7) → ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (𝑛 = 6 ∨ 𝑛 = 7)))
36	35	expd 416	. . . . . . . . 9 ⊢ ((𝑛 < 7 ∨ 𝑛 = 7) → (𝑛 ∈ Odd → (5 < 𝑛 → (𝑛 = 6 ∨ 𝑛 = 7))))
37	36	com12 32	. . . . . . . 8 ⊢ (𝑛 ∈ Odd → ((𝑛 < 7 ∨ 𝑛 = 7) → (5 < 𝑛 → (𝑛 = 6 ∨ 𝑛 = 7))))
38	12, 37	sylbid 239	. . . . . . 7 ⊢ (𝑛 ∈ Odd → (𝑛 ≤ 7 → (5 < 𝑛 → (𝑛 = 6 ∨ 𝑛 = 7))))
39		eleq1 2826	. . . . . . . . . 10 ⊢ (𝑛 = 6 → (𝑛 ∈ Odd ↔ 6 ∈ Odd ))
40		6even 45163	. . . . . . . . . . 11 ⊢ 6 ∈ Even
41		evennodd 45095	. . . . . . . . . . . 12 ⊢ (6 ∈ Even → ¬ 6 ∈ Odd )
42	41	pm2.21d 121	. . . . . . . . . . 11 ⊢ (6 ∈ Even → (6 ∈ Odd → 𝑛 ∈ GoldbachOddW ))
43	40, 42	mp1i 13	. . . . . . . . . 10 ⊢ (𝑛 = 6 → (6 ∈ Odd → 𝑛 ∈ GoldbachOddW ))
44	39, 43	sylbid 239	. . . . . . . . 9 ⊢ (𝑛 = 6 → (𝑛 ∈ Odd → 𝑛 ∈ GoldbachOddW ))
45		7gbow 45224	. . . . . . . . . . 11 ⊢ 7 ∈ GoldbachOddW
46		eleq1 2826	. . . . . . . . . . 11 ⊢ (𝑛 = 7 → (𝑛 ∈ GoldbachOddW ↔ 7 ∈ GoldbachOddW ))
47	45, 46	mpbiri 257	. . . . . . . . . 10 ⊢ (𝑛 = 7 → 𝑛 ∈ GoldbachOddW )
48	47	a1d 25	. . . . . . . . 9 ⊢ (𝑛 = 7 → (𝑛 ∈ Odd → 𝑛 ∈ GoldbachOddW ))
49	44, 48	jaoi 854	. . . . . . . 8 ⊢ ((𝑛 = 6 ∨ 𝑛 = 7) → (𝑛 ∈ Odd → 𝑛 ∈ GoldbachOddW ))
50	49	com12 32	. . . . . . 7 ⊢ (𝑛 ∈ Odd → ((𝑛 = 6 ∨ 𝑛 = 7) → 𝑛 ∈ GoldbachOddW ))
51	38, 50	syl6d 75	. . . . . 6 ⊢ (𝑛 ∈ Odd → (𝑛 ≤ 7 → (5 < 𝑛 → 𝑛 ∈ GoldbachOddW )))
52	11, 51	sylbird 259	. . . . 5 ⊢ (𝑛 ∈ Odd → (¬ 7 < 𝑛 → (5 < 𝑛 → 𝑛 ∈ GoldbachOddW )))
53	52	com12 32	. . . 4 ⊢ (¬ 7 < 𝑛 → (𝑛 ∈ Odd → (5 < 𝑛 → 𝑛 ∈ GoldbachOddW )))
54	53	a1dd 50	. . 3 ⊢ (¬ 7 < 𝑛 → (𝑛 ∈ Odd → ((7 < 𝑛 → 𝑛 ∈ GoldbachOdd ) → (5 < 𝑛 → 𝑛 ∈ GoldbachOddW ))))
55	6, 54	pm2.61i 182	. 2 ⊢ (𝑛 ∈ Odd → ((7 < 𝑛 → 𝑛 ∈ GoldbachOdd ) → (5 < 𝑛 → 𝑛 ∈ GoldbachOddW )))
56	55	ralimia 3085	1 ⊢ (∀𝑛 ∈ Odd (7 < 𝑛 → 𝑛 ∈ GoldbachOdd ) → ∀𝑛 ∈ Odd (5 < 𝑛 → 𝑛 ∈ GoldbachOddW ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106  ∀wral 3064   class class class wbr 5074  (class class class)co 7275  ℝcr 10870  1c1 10872   + caddc 10874   < clt 11009   ≤ cle 11010  5c5 12031  6c6 12032  7c7 12033  ℤcz 12319   Even ceven 45076   Odd codd 45077   GoldbachOddW cgbow 45198   GoldbachOdd cgbo 45199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-sup 9201  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-n0 12234  df-z 12320  df-uz 12583  df-rp 12731  df-fz 13240  df-seq 13722  df-exp 13783  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-dvds 15964  df-prm 16377  df-even 45078  df-odd 45079  df-gbow 45201  df-gbo 45202

This theorem is referenced by:  stgoldbnnsum4prm  45255