Theorem stgoldbwt 48399

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		pm3.35 812	. . . . . 6 ⊢ ((7 < 𝑛 ∧ (7 < 𝑛 → 𝑛 ∈ GoldbachOdd )) → 𝑛 ∈ GoldbachOdd )
2		gbogbow 48379	. . . . . . 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 416	. . . 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 48254	. . . . . . . 8 ⊢ (𝑛 ∈ Odd → 𝑛 ∈ ℤ)
8	7	zred 12678	. . . . . . 7 ⊢ (𝑛 ∈ Odd → 𝑛 ∈ ℝ)
9		7re 12312	. . . . . . . 8 ⊢ 7 ∈ ℝ
10	9	a1i 11	. . . . . . 7 ⊢ (𝑛 ∈ Odd → 7 ∈ ℝ)
11	8, 10	lenltd 11330	. . . . . 6 ⊢ (𝑛 ∈ Odd → (𝑛 ≤ 7 ↔ ¬ 7 < 𝑛))
12	8, 10	leloed 11327	. . . . . . . 8 ⊢ (𝑛 ∈ Odd → (𝑛 ≤ 7 ↔ (𝑛 < 7 ∨ 𝑛 = 7)))
13	7	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ Odd ∧ 5 < 𝑛) → 𝑛 ∈ ℤ)
14		6nn 12308	. . . . . . . . . . . . . . . . 17 ⊢ 6 ∈ ℕ
15	14	nnzi 12596	. . . . . . . . . . . . . . . 16 ⊢ 6 ∈ ℤ
16	13, 15	jctir 528	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (𝑛 ∈ ℤ ∧ 6 ∈ ℤ))
17	16	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((𝑛 < 7 ∧ (𝑛 ∈ Odd ∧ 5 < 𝑛)) → (𝑛 ∈ ℤ ∧ 6 ∈ ℤ))
18		df-7 12286	. . . . . . . . . . . . . . . . 17 ⊢ 7 = (6 + 1)
19	18	breq2i 5109	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 < 7 ↔ 𝑛 < (6 + 1))
20	19	biimpi 218	. . . . . . . . . . . . . . 15 ⊢ (𝑛 < 7 → 𝑛 < (6 + 1))
21		df-6 12285	. . . . . . . . . . . . . . . 16 ⊢ 6 = (5 + 1)
22		5nn 12305	. . . . . . . . . . . . . . . . . . 19 ⊢ 5 ∈ ℕ
23	22	nnzi 12596	. . . . . . . . . . . . . . . . . 18 ⊢ 5 ∈ ℤ
24		zltp1le 12622	. . . . . . . . . . . . . . . . . 18 ⊢ ((5 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (5 < 𝑛 ↔ (5 + 1) ≤ 𝑛))
25	23, 7, 24	sylancr 596	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ Odd → (5 < 𝑛 ↔ (5 + 1) ≤ 𝑛))
26	25	biimpa 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (5 + 1) ≤ 𝑛)
27	21, 26	eqbrtrid 5136	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ Odd ∧ 5 < 𝑛) → 6 ≤ 𝑛)
28	20, 27	anim12ci 623	. . . . . . . . . . . . . 14 ⊢ ((𝑛 < 7 ∧ (𝑛 ∈ Odd ∧ 5 < 𝑛)) → (6 ≤ 𝑛 ∧ 𝑛 < (6 + 1)))
29		zgeltp1eq 47904	. . . . . . . . . . . . . 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 884	. . . . . . . . . . . 12 ⊢ ((𝑛 < 7 ∧ (𝑛 ∈ Odd ∧ 5 < 𝑛)) → (𝑛 = 6 ∨ 𝑛 = 7))
32	31	ex 416	. . . . . . . . . . 11 ⊢ (𝑛 < 7 → ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (𝑛 = 6 ∨ 𝑛 = 7)))
33		olc 879	. . . . . . . . . . . 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 868	. . . . . . . . . 10 ⊢ ((𝑛 < 7 ∨ 𝑛 = 7) → ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (𝑛 = 6 ∨ 𝑛 = 7)))
36	35	expd 419	. . . . . . . . 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 242	. . . . . . 7 ⊢ (𝑛 ∈ Odd → (𝑛 ≤ 7 → (5 < 𝑛 → (𝑛 = 6 ∨ 𝑛 = 7))))
39		eleq1 2851	. . . . . . . . . 10 ⊢ (𝑛 = 6 → (𝑛 ∈ Odd ↔ 6 ∈ Odd ))
40		6even 48334	. . . . . . . . . . 11 ⊢ 6 ∈ Even
41		evennodd 48266	. . . . . . . . . . . 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 242	. . . . . . . . 9 ⊢ (𝑛 = 6 → (𝑛 ∈ Odd → 𝑛 ∈ GoldbachOddW ))
45		7gbow 48395	. . . . . . . . . . 11 ⊢ 7 ∈ GoldbachOddW
46		eleq1 2851	. . . . . . . . . . 11 ⊢ (𝑛 = 7 → (𝑛 ∈ GoldbachOddW ↔ 7 ∈ GoldbachOddW ))
47	45, 46	mpbiri 260	. . . . . . . . . 10 ⊢ (𝑛 = 7 → 𝑛 ∈ GoldbachOddW )
48	47	a1d 25	. . . . . . . . 9 ⊢ (𝑛 = 7 → (𝑛 ∈ Odd → 𝑛 ∈ GoldbachOddW ))
49	44, 48	jaoi 868	. . . . . . . 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 262	. . . . 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 183	. 2 ⊢ (𝑛 ∈ Odd → ((7 < 𝑛 → 𝑛 ∈ GoldbachOdd ) → (5 < 𝑛 → 𝑛 ∈ GoldbachOddW )))
56	55	ralimia 3097	1 ⊢ (∀𝑛 ∈ Odd (7 < 𝑛 → 𝑛 ∈ GoldbachOdd ) → ∀𝑛 ∈ Odd (5 < 𝑛 → 𝑛 ∈ GoldbachOddW ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   = wceq 1561   ∈ wcel 2143  ∀wral 3077   class class class wbr 5101  (class class class)co 7397  ℝcr 11073  1c1 11075   + caddc 11077   < clt 11217   ≤ cle 11218  5c5 12276  6c6 12277  7c7 12278  ℤcz 12569   Even ceven 48247   Odd codd 48248   GoldbachOddW cgbow 48369   GoldbachOdd cgbo 48370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719  ax-cnex 11130  ax-resscn 11131  ax-1cn 11132  ax-icn 11133  ax-addcl 11134  ax-addrcl 11135  ax-mulcl 11136  ax-mulrcl 11137  ax-mulcom 11138  ax-addass 11139  ax-mulass 11140  ax-distr 11141  ax-i2m1 11142  ax-1ne0 11143  ax-1rid 11144  ax-rnegex 11145  ax-rrecex 11146  ax-cnre 11147  ax-pre-lttri 11148  ax-pre-lttrn 11149  ax-pre-ltadd 11150  ax-pre-mulgt0 11151  ax-pre-sup 11152

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6289  df-ord 6350  df-on 6351  df-lim 6352  df-suc 6353  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-riota 7354  df-ov 7400  df-oprab 7401  df-mpo 7402  df-om 7848  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8382  df-1o 8438  df-2o 8439  df-er 8679  df-en 8929  df-dom 8930  df-sdom 8931  df-fin 8932  df-sup 9389  df-pnf 11219  df-mnf 11220  df-xr 11221  df-ltxr 11222  df-le 11223  df-sub 11417  df-neg 11418  df-div 11846  df-nn 12212  df-2 12281  df-3 12282  df-4 12283  df-5 12284  df-6 12285  df-7 12286  df-n0 12483  df-z 12570  df-uz 12841  df-rp 12995  df-fz 13514  df-seq 14016  df-exp 14076  df-cj 15127  df-re 15128  df-im 15129  df-sqrt 15263  df-abs 15264  df-dvds 16288  df-prm 16707  df-even 48249  df-odd 48250  df-gbow 48372  df-gbo 48373

This theorem is referenced by:  stgoldbnnsum4prm  48426