Theorem stgoldbwt 47650

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		pm3.35 802	. . . . . 6 ⊢ ((7 < 𝑛 ∧ (7 < 𝑛 → 𝑛 ∈ GoldbachOdd )) → 𝑛 ∈ GoldbachOdd )
2		gbogbow 47630	. . . . . . 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 412	. . . 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 47505	. . . . . . . 8 ⊢ (𝑛 ∈ Odd → 𝑛 ∈ ℤ)
8	7	zred 12747	. . . . . . 7 ⊢ (𝑛 ∈ Odd → 𝑛 ∈ ℝ)
9		7re 12386	. . . . . . . 8 ⊢ 7 ∈ ℝ
10	9	a1i 11	. . . . . . 7 ⊢ (𝑛 ∈ Odd → 7 ∈ ℝ)
11	8, 10	lenltd 11436	. . . . . 6 ⊢ (𝑛 ∈ Odd → (𝑛 ≤ 7 ↔ ¬ 7 < 𝑛))
12	8, 10	leloed 11433	. . . . . . . 8 ⊢ (𝑛 ∈ Odd → (𝑛 ≤ 7 ↔ (𝑛 < 7 ∨ 𝑛 = 7)))
13	7	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ Odd ∧ 5 < 𝑛) → 𝑛 ∈ ℤ)
14		6nn 12382	. . . . . . . . . . . . . . . . 17 ⊢ 6 ∈ ℕ
15	14	nnzi 12667	. . . . . . . . . . . . . . . 16 ⊢ 6 ∈ ℤ
16	13, 15	jctir 520	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (𝑛 ∈ ℤ ∧ 6 ∈ ℤ))
17	16	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑛 < 7 ∧ (𝑛 ∈ Odd ∧ 5 < 𝑛)) → (𝑛 ∈ ℤ ∧ 6 ∈ ℤ))
18		df-7 12361	. . . . . . . . . . . . . . . . 17 ⊢ 7 = (6 + 1)
19	18	breq2i 5174	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 < 7 ↔ 𝑛 < (6 + 1))
20	19	biimpi 216	. . . . . . . . . . . . . . 15 ⊢ (𝑛 < 7 → 𝑛 < (6 + 1))
21		df-6 12360	. . . . . . . . . . . . . . . 16 ⊢ 6 = (5 + 1)
22		5nn 12379	. . . . . . . . . . . . . . . . . . 19 ⊢ 5 ∈ ℕ
23	22	nnzi 12667	. . . . . . . . . . . . . . . . . 18 ⊢ 5 ∈ ℤ
24		zltp1le 12693	. . . . . . . . . . . . . . . . . 18 ⊢ ((5 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (5 < 𝑛 ↔ (5 + 1) ≤ 𝑛))
25	23, 7, 24	sylancr 586	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ Odd → (5 < 𝑛 ↔ (5 + 1) ≤ 𝑛))
26	25	biimpa 476	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (5 + 1) ≤ 𝑛)
27	21, 26	eqbrtrid 5201	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ Odd ∧ 5 < 𝑛) → 6 ≤ 𝑛)
28	20, 27	anim12ci 613	. . . . . . . . . . . . . 14 ⊢ ((𝑛 < 7 ∧ (𝑛 ∈ Odd ∧ 5 < 𝑛)) → (6 ≤ 𝑛 ∧ 𝑛 < (6 + 1)))
29		zgeltp1eq 47224	. . . . . . . . . . . . . 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 872	. . . . . . . . . . . 12 ⊢ ((𝑛 < 7 ∧ (𝑛 ∈ Odd ∧ 5 < 𝑛)) → (𝑛 = 6 ∨ 𝑛 = 7))
32	31	ex 412	. . . . . . . . . . 11 ⊢ (𝑛 < 7 → ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (𝑛 = 6 ∨ 𝑛 = 7)))
33		olc 867	. . . . . . . . . . . 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 856	. . . . . . . . . 10 ⊢ ((𝑛 < 7 ∨ 𝑛 = 7) → ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (𝑛 = 6 ∨ 𝑛 = 7)))
36	35	expd 415	. . . . . . . . 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 240	. . . . . . 7 ⊢ (𝑛 ∈ Odd → (𝑛 ≤ 7 → (5 < 𝑛 → (𝑛 = 6 ∨ 𝑛 = 7))))
39		eleq1 2832	. . . . . . . . . 10 ⊢ (𝑛 = 6 → (𝑛 ∈ Odd ↔ 6 ∈ Odd ))
40		6even 47585	. . . . . . . . . . 11 ⊢ 6 ∈ Even
41		evennodd 47517	. . . . . . . . . . . 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 240	. . . . . . . . 9 ⊢ (𝑛 = 6 → (𝑛 ∈ Odd → 𝑛 ∈ GoldbachOddW ))
45		7gbow 47646	. . . . . . . . . . 11 ⊢ 7 ∈ GoldbachOddW
46		eleq1 2832	. . . . . . . . . . 11 ⊢ (𝑛 = 7 → (𝑛 ∈ GoldbachOddW ↔ 7 ∈ GoldbachOddW ))
47	45, 46	mpbiri 258	. . . . . . . . . 10 ⊢ (𝑛 = 7 → 𝑛 ∈ GoldbachOddW )
48	47	a1d 25	. . . . . . . . 9 ⊢ (𝑛 = 7 → (𝑛 ∈ Odd → 𝑛 ∈ GoldbachOddW ))
49	44, 48	jaoi 856	. . . . . . . 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 260	. . . . 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 3086	1 ⊢ (∀𝑛 ∈ Odd (7 < 𝑛 → 𝑛 ∈ GoldbachOdd ) → ∀𝑛 ∈ Odd (5 < 𝑛 → 𝑛 ∈ GoldbachOddW ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 846   = wceq 1537   ∈ wcel 2108  ∀wral 3067   class class class wbr 5166  (class class class)co 7448  ℝcr 11183  1c1 11185   + caddc 11187   < clt 11324   ≤ cle 11325  5c5 12351  6c6 12352  7c7 12353  ℤcz 12639   Even ceven 47498   Odd codd 47499   GoldbachOddW cgbow 47620   GoldbachOdd cgbo 47621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-n0 12554  df-z 12640  df-uz 12904  df-rp 13058  df-fz 13568  df-seq 14053  df-exp 14113  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-dvds 16303  df-prm 16719  df-even 47500  df-odd 47501  df-gbow 47623  df-gbo 47624

This theorem is referenced by:  stgoldbnnsum4prm  47677