Theorem stgoldbwt 48601

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		pm3.35 815	. . . . . 6 ⊢ ((7 < 𝑛 ∧ (7 < 𝑛 → 𝑛 ∈ GoldbachOdd )) → 𝑛 ∈ GoldbachOdd )
2		gbogbow 48581	. . . . . . 7 ⊢ (𝑛 ∈ GoldbachOdd → 𝑛 ∈ GoldbachOddW )
3	2	a1d 26	. . . . . 6 ⊢ (𝑛 ∈ GoldbachOdd → (5 < 𝑛 → 𝑛 ∈ GoldbachOddW ))
4	1, 3	syl 18	. . . . 5 ⊢ ((7 < 𝑛 ∧ (7 < 𝑛 → 𝑛 ∈ GoldbachOdd )) → (5 < 𝑛 → 𝑛 ∈ GoldbachOddW ))
5	4	ex 418	. . . 4 ⊢ (7 < 𝑛 → ((7 < 𝑛 → 𝑛 ∈ GoldbachOdd ) → (5 < 𝑛 → 𝑛 ∈ GoldbachOddW )))
6	5	a1d 26	. . 3 ⊢ (7 < 𝑛 → (𝑛 ∈ Odd → ((7 < 𝑛 → 𝑛 ∈ GoldbachOdd ) → (5 < 𝑛 → 𝑛 ∈ GoldbachOddW ))))
7		oddz 48456	. . . . . . . 8 ⊢ (𝑛 ∈ Odd → 𝑛 ∈ ℤ)
8	7	zred 12718	. . . . . . 7 ⊢ (𝑛 ∈ Odd → 𝑛 ∈ ℝ)
9		7re 12351	. . . . . . . 8 ⊢ 7 ∈ ℝ
10	9	a1i 11	. . . . . . 7 ⊢ (𝑛 ∈ Odd → 7 ∈ ℝ)
11	8, 10	lenltd 11373	. . . . . 6 ⊢ (𝑛 ∈ Odd → (𝑛 ≤ 7 ↔ ¬ 7 < 𝑛))
12	8, 10	leloed 11370	. . . . . . . 8 ⊢ (𝑛 ∈ Odd → (𝑛 ≤ 7 ↔ (𝑛 < 7 ∨ 𝑛 = 7)))
13	7	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ Odd ∧ 5 < 𝑛) → 𝑛 ∈ ℤ)
14		6nn 12347	. . . . . . . . . . . . . . . . 17 ⊢ 6 ∈ ℕ
15	14	nnzi 12635	. . . . . . . . . . . . . . . 16 ⊢ 6 ∈ ℤ
16	13, 15	jctir 530	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (𝑛 ∈ ℤ ∧ 6 ∈ ℤ))
17	16	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝑛 < 7 ∧ (𝑛 ∈ Odd ∧ 5 < 𝑛)) → (𝑛 ∈ ℤ ∧ 6 ∈ ℤ))
18		df-7 12325	. . . . . . . . . . . . . . . . 17 ⊢ 7 = (6 + 1)
19	18	breq2i 5119	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 < 7 ↔ 𝑛 < (6 + 1))
20	19	biimpi 219	. . . . . . . . . . . . . . 15 ⊢ (𝑛 < 7 → 𝑛 < (6 + 1))
21		df-6 12324	. . . . . . . . . . . . . . . 16 ⊢ 6 = (5 + 1)
22		5nn 12344	. . . . . . . . . . . . . . . . . . 19 ⊢ 5 ∈ ℕ
23	22	nnzi 12635	. . . . . . . . . . . . . . . . . 18 ⊢ 5 ∈ ℤ
24		zltp1le 12661	. . . . . . . . . . . . . . . . . 18 ⊢ ((5 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (5 < 𝑛 ↔ (5 + 1) ≤ 𝑛))
25	23, 7, 24	sylancr 599	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ Odd → (5 < 𝑛 ↔ (5 + 1) ≤ 𝑛))
26	25	biimpa 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (5 + 1) ≤ 𝑛)
27	21, 26	eqbrtrid 5148	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ Odd ∧ 5 < 𝑛) → 6 ≤ 𝑛)
28	20, 27	anim12ci 626	. . . . . . . . . . . . . 14 ⊢ ((𝑛 < 7 ∧ (𝑛 ∈ Odd ∧ 5 < 𝑛)) → (6 ≤ 𝑛 ∧ 𝑛 < (6 + 1)))
29		zgeltp1eq 48106	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℤ ∧ 6 ∈ ℤ) → ((6 ≤ 𝑛 ∧ 𝑛 < (6 + 1)) → 𝑛 = 6))
30	17, 28, 29	sylc 66	. . . . . . . . . . . . 13 ⊢ ((𝑛 < 7 ∧ (𝑛 ∈ Odd ∧ 5 < 𝑛)) → 𝑛 = 6)
31	30	orcd 887	. . . . . . . . . . . 12 ⊢ ((𝑛 < 7 ∧ (𝑛 ∈ Odd ∧ 5 < 𝑛)) → (𝑛 = 6 ∨ 𝑛 = 7))
32	31	ex 418	. . . . . . . . . . 11 ⊢ (𝑛 < 7 → ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (𝑛 = 6 ∨ 𝑛 = 7)))
33		olc 882	. . . . . . . . . . . 12 ⊢ (𝑛 = 7 → (𝑛 = 6 ∨ 𝑛 = 7))
34	33	a1d 26	. . . . . . . . . . 11 ⊢ (𝑛 = 7 → ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (𝑛 = 6 ∨ 𝑛 = 7)))
35	32, 34	jaoi 871	. . . . . . . . . 10 ⊢ ((𝑛 < 7 ∨ 𝑛 = 7) → ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (𝑛 = 6 ∨ 𝑛 = 7)))
36	35	expd 421	. . . . . . . . 9 ⊢ ((𝑛 < 7 ∨ 𝑛 = 7) → (𝑛 ∈ Odd → (5 < 𝑛 → (𝑛 = 6 ∨ 𝑛 = 7))))
37	36	com12 33	. . . . . . . 8 ⊢ (𝑛 ∈ Odd → ((𝑛 < 7 ∨ 𝑛 = 7) → (5 < 𝑛 → (𝑛 = 6 ∨ 𝑛 = 7))))
38	12, 37	sylbid 243	. . . . . . 7 ⊢ (𝑛 ∈ Odd → (𝑛 ≤ 7 → (5 < 𝑛 → (𝑛 = 6 ∨ 𝑛 = 7))))
39		eleq1 2853	. . . . . . . . . 10 ⊢ (𝑛 = 6 → (𝑛 ∈ Odd ↔ 6 ∈ Odd ))
40		6even 48536	. . . . . . . . . . 11 ⊢ 6 ∈ Even
41		evennodd 48468	. . . . . . . . . . . 12 ⊢ (6 ∈ Even → ¬ 6 ∈ Odd )
42	41	pm2.21d 122	. . . . . . . . . . 11 ⊢ (6 ∈ Even → (6 ∈ Odd → 𝑛 ∈ GoldbachOddW ))
43	40, 42	mp1i 14	. . . . . . . . . 10 ⊢ (𝑛 = 6 → (6 ∈ Odd → 𝑛 ∈ GoldbachOddW ))
44	39, 43	sylbid 243	. . . . . . . . 9 ⊢ (𝑛 = 6 → (𝑛 ∈ Odd → 𝑛 ∈ GoldbachOddW ))
45		7gbow 48597	. . . . . . . . . . 11 ⊢ 7 ∈ GoldbachOddW
46		eleq1 2853	. . . . . . . . . . 11 ⊢ (𝑛 = 7 → (𝑛 ∈ GoldbachOddW ↔ 7 ∈ GoldbachOddW ))
47	45, 46	mpbiri 261	. . . . . . . . . 10 ⊢ (𝑛 = 7 → 𝑛 ∈ GoldbachOddW )
48	47	a1d 26	. . . . . . . . 9 ⊢ (𝑛 = 7 → (𝑛 ∈ Odd → 𝑛 ∈ GoldbachOddW ))
49	44, 48	jaoi 871	. . . . . . . 8 ⊢ ((𝑛 = 6 ∨ 𝑛 = 7) → (𝑛 ∈ Odd → 𝑛 ∈ GoldbachOddW ))
50	49	com12 33	. . . . . . 7 ⊢ (𝑛 ∈ Odd → ((𝑛 = 6 ∨ 𝑛 = 7) → 𝑛 ∈ GoldbachOddW ))
51	38, 50	syl6d 76	. . . . . 6 ⊢ (𝑛 ∈ Odd → (𝑛 ≤ 7 → (5 < 𝑛 → 𝑛 ∈ GoldbachOddW )))
52	11, 51	sylbird 263	. . . . 5 ⊢ (𝑛 ∈ Odd → (¬ 7 < 𝑛 → (5 < 𝑛 → 𝑛 ∈ GoldbachOddW )))
53	52	com12 33	. . . 4 ⊢ (¬ 7 < 𝑛 → (𝑛 ∈ Odd → (5 < 𝑛 → 𝑛 ∈ GoldbachOddW )))
54	53	a1dd 51	. . 3 ⊢ (¬ 7 < 𝑛 → (𝑛 ∈ Odd → ((7 < 𝑛 → 𝑛 ∈ GoldbachOdd ) → (5 < 𝑛 → 𝑛 ∈ GoldbachOddW ))))
55	6, 54	pm2.61i 184	. 2 ⊢ (𝑛 ∈ Odd → ((7 < 𝑛 → 𝑛 ∈ GoldbachOdd ) → (5 < 𝑛 → 𝑛 ∈ GoldbachOddW )))
56	55	ralimia 3101	1 ⊢ (∀𝑛 ∈ Odd (7 < 𝑛 → 𝑛 ∈ GoldbachOdd ) → ∀𝑛 ∈ Odd (5 < 𝑛 → 𝑛 ∈ GoldbachOddW ))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2146  ∀wral 3081   class class class wbr 5111  (class class class)co 7419  ℝcr 11116  1c1 11118   + caddc 11120   < clt 11260   ≤ cle 11261  5c5 12315  6c6 12316  7c7 12317  ℤcz 12608   Even ceven 48449   Odd codd 48450   GoldbachOddW cgbow 48571   GoldbachOdd cgbo 48572

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11173  ax-resscn 11174  ax-1cn 11175  ax-icn 11176  ax-addcl 11177  ax-addrcl 11178  ax-mulcl 11179  ax-mulrcl 11180  ax-mulcom 11181  ax-addass 11182  ax-mulass 11183  ax-distr 11184  ax-i2m1 11185  ax-1ne0 11186  ax-1rid 11187  ax-rnegex 11188  ax-rrecex 11189  ax-cnre 11190  ax-pre-lttri 11191  ax-pre-lttrn 11192  ax-pre-ltadd 11193  ax-pre-mulgt0 11194  ax-pre-sup 11195

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-er 8700  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-sup 9409  df-pnf 11262  df-mnf 11263  df-xr 11264  df-ltxr 11265  df-le 11266  df-sub 11460  df-neg 11461  df-div 11889  df-nn 12251  df-2 12320  df-3 12321  df-4 12322  df-5 12323  df-6 12324  df-7 12325  df-n0 12522  df-z 12609  df-uz 12881  df-rp 13035  df-fz 13554  df-seq 14058  df-exp 14118  df-cj 15176  df-re 15177  df-im 15178  df-sqrt 15312  df-abs 15313  df-dvds 16335  df-prm 16754  df-even 48451  df-odd 48452  df-gbow 48574  df-gbo 48575

This theorem is used by:  stgoldbnnsum4prm  48628