Theorem stgoldbwt 46430

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		pm3.35 801	. . . . . 6 ⊢ ((7 < 𝑛 ∧ (7 < 𝑛 → 𝑛 ∈ GoldbachOdd )) → 𝑛 ∈ GoldbachOdd )
2		gbogbow 46410	. . . . . . 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 46285	. . . . . . . 8 ⊢ (𝑛 ∈ Odd → 𝑛 ∈ ℤ)
8	7	zred 12662	. . . . . . 7 ⊢ (𝑛 ∈ Odd → 𝑛 ∈ ℝ)
9		7re 12301	. . . . . . . 8 ⊢ 7 ∈ ℝ
10	9	a1i 11	. . . . . . 7 ⊢ (𝑛 ∈ Odd → 7 ∈ ℝ)
11	8, 10	lenltd 11356	. . . . . 6 ⊢ (𝑛 ∈ Odd → (𝑛 ≤ 7 ↔ ¬ 7 < 𝑛))
12	8, 10	leloed 11353	. . . . . . . 8 ⊢ (𝑛 ∈ Odd → (𝑛 ≤ 7 ↔ (𝑛 < 7 ∨ 𝑛 = 7)))
13	7	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ Odd ∧ 5 < 𝑛) → 𝑛 ∈ ℤ)
14		6nn 12297	. . . . . . . . . . . . . . . . 17 ⊢ 6 ∈ ℕ
15	14	nnzi 12582	. . . . . . . . . . . . . . . 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 12276	. . . . . . . . . . . . . . . . 17 ⊢ 7 = (6 + 1)
19	18	breq2i 5155	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 < 7 ↔ 𝑛 < (6 + 1))
20	19	biimpi 215	. . . . . . . . . . . . . . 15 ⊢ (𝑛 < 7 → 𝑛 < (6 + 1))
21		df-6 12275	. . . . . . . . . . . . . . . 16 ⊢ 6 = (5 + 1)
22		5nn 12294	. . . . . . . . . . . . . . . . . . 19 ⊢ 5 ∈ ℕ
23	22	nnzi 12582	. . . . . . . . . . . . . . . . . 18 ⊢ 5 ∈ ℤ
24		zltp1le 12608	. . . . . . . . . . . . . . . . . 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 5182	. . . . . . . . . . . . . . 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 46003	. . . . . . . . . . . . . 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 871	. . . . . . . . . . . 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 866	. . . . . . . . . . . 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 855	. . . . . . . . . 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 2821	. . . . . . . . . 10 ⊢ (𝑛 = 6 → (𝑛 ∈ Odd ↔ 6 ∈ Odd ))
40		6even 46365	. . . . . . . . . . 11 ⊢ 6 ∈ Even
41		evennodd 46297	. . . . . . . . . . . 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 46426	. . . . . . . . . . 11 ⊢ 7 ∈ GoldbachOddW
46		eleq1 2821	. . . . . . . . . . 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 855	. . . . . . . 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 3080	1 ⊢ (∀𝑛 ∈ Odd (7 < 𝑛 → 𝑛 ∈ GoldbachOdd ) → ∀𝑛 ∈ Odd (5 < 𝑛 → 𝑛 ∈ GoldbachOddW ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106  ∀wral 3061   class class class wbr 5147  (class class class)co 7405  ℝcr 11105  1c1 11107   + caddc 11109   < clt 11244   ≤ cle 11245  5c5 12266  6c6 12267  7c7 12268  ℤcz 12554   Even ceven 46278   Odd codd 46279   GoldbachOddW cgbow 46400   GoldbachOdd cgbo 46401

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 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184

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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-sup 9433  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-n0 12469  df-z 12555  df-uz 12819  df-rp 12971  df-fz 13481  df-seq 13963  df-exp 14024  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-dvds 16194  df-prm 16605  df-even 46280  df-odd 46281  df-gbow 46403  df-gbo 46404

This theorem is referenced by:  stgoldbnnsum4prm  46457