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Theorem stgoldbwt 44309
 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 < 𝑛𝑛 ∈ GoldbachOdd ) → ∀𝑛 ∈ Odd (5 < 𝑛𝑛 ∈ GoldbachOddW ))

Proof of Theorem stgoldbwt
StepHypRef Expression
1 pm3.35 802 . . . . . 6 ((7 < 𝑛 ∧ (7 < 𝑛𝑛 ∈ GoldbachOdd )) → 𝑛 ∈ GoldbachOdd )
2 gbogbow 44289 . . . . . . 7 (𝑛 ∈ GoldbachOdd → 𝑛 ∈ GoldbachOddW )
32a1d 25 . . . . . 6 (𝑛 ∈ GoldbachOdd → (5 < 𝑛𝑛 ∈ GoldbachOddW ))
41, 3syl 17 . . . . 5 ((7 < 𝑛 ∧ (7 < 𝑛𝑛 ∈ GoldbachOdd )) → (5 < 𝑛𝑛 ∈ GoldbachOddW ))
54ex 416 . . . 4 (7 < 𝑛 → ((7 < 𝑛𝑛 ∈ GoldbachOdd ) → (5 < 𝑛𝑛 ∈ GoldbachOddW )))
65a1d 25 . . 3 (7 < 𝑛 → (𝑛 ∈ Odd → ((7 < 𝑛𝑛 ∈ GoldbachOdd ) → (5 < 𝑛𝑛 ∈ GoldbachOddW ))))
7 oddz 44164 . . . . . . . 8 (𝑛 ∈ Odd → 𝑛 ∈ ℤ)
87zred 12077 . . . . . . 7 (𝑛 ∈ Odd → 𝑛 ∈ ℝ)
9 7re 11720 . . . . . . . 8 7 ∈ ℝ
109a1i 11 . . . . . . 7 (𝑛 ∈ Odd → 7 ∈ ℝ)
118, 10lenltd 10777 . . . . . 6 (𝑛 ∈ Odd → (𝑛 ≤ 7 ↔ ¬ 7 < 𝑛))
128, 10leloed 10774 . . . . . . . 8 (𝑛 ∈ Odd → (𝑛 ≤ 7 ↔ (𝑛 < 7 ∨ 𝑛 = 7)))
137adantr 484 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ Odd ∧ 5 < 𝑛) → 𝑛 ∈ ℤ)
14 6nn 11716 . . . . . . . . . . . . . . . . 17 6 ∈ ℕ
1514nnzi 11996 . . . . . . . . . . . . . . . 16 6 ∈ ℤ
1613, 15jctir 524 . . . . . . . . . . . . . . 15 ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (𝑛 ∈ ℤ ∧ 6 ∈ ℤ))
1716adantl 485 . . . . . . . . . . . . . 14 ((𝑛 < 7 ∧ (𝑛 ∈ Odd ∧ 5 < 𝑛)) → (𝑛 ∈ ℤ ∧ 6 ∈ ℤ))
18 df-7 11695 . . . . . . . . . . . . . . . . 17 7 = (6 + 1)
1918breq2i 5038 . . . . . . . . . . . . . . . 16 (𝑛 < 7 ↔ 𝑛 < (6 + 1))
2019biimpi 219 . . . . . . . . . . . . . . 15 (𝑛 < 7 → 𝑛 < (6 + 1))
21 df-6 11694 . . . . . . . . . . . . . . . 16 6 = (5 + 1)
22 5nn 11713 . . . . . . . . . . . . . . . . . . 19 5 ∈ ℕ
2322nnzi 11996 . . . . . . . . . . . . . . . . . 18 5 ∈ ℤ
24 zltp1le 12022 . . . . . . . . . . . . . . . . . 18 ((5 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (5 < 𝑛 ↔ (5 + 1) ≤ 𝑛))
2523, 7, 24sylancr 590 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ Odd → (5 < 𝑛 ↔ (5 + 1) ≤ 𝑛))
2625biimpa 480 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (5 + 1) ≤ 𝑛)
2721, 26eqbrtrid 5065 . . . . . . . . . . . . . . 15 ((𝑛 ∈ Odd ∧ 5 < 𝑛) → 6 ≤ 𝑛)
2820, 27anim12ci 616 . . . . . . . . . . . . . 14 ((𝑛 < 7 ∧ (𝑛 ∈ Odd ∧ 5 < 𝑛)) → (6 ≤ 𝑛𝑛 < (6 + 1)))
29 zgeltp1eq 43881 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℤ ∧ 6 ∈ ℤ) → ((6 ≤ 𝑛𝑛 < (6 + 1)) → 𝑛 = 6))
3017, 28, 29sylc 65 . . . . . . . . . . . . 13 ((𝑛 < 7 ∧ (𝑛 ∈ Odd ∧ 5 < 𝑛)) → 𝑛 = 6)
3130orcd 870 . . . . . . . . . . . 12 ((𝑛 < 7 ∧ (𝑛 ∈ Odd ∧ 5 < 𝑛)) → (𝑛 = 6 ∨ 𝑛 = 7))
3231ex 416 . . . . . . . . . . 11 (𝑛 < 7 → ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (𝑛 = 6 ∨ 𝑛 = 7)))
33 olc 865 . . . . . . . . . . . 12 (𝑛 = 7 → (𝑛 = 6 ∨ 𝑛 = 7))
3433a1d 25 . . . . . . . . . . 11 (𝑛 = 7 → ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (𝑛 = 6 ∨ 𝑛 = 7)))
3532, 34jaoi 854 . . . . . . . . . 10 ((𝑛 < 7 ∨ 𝑛 = 7) → ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (𝑛 = 6 ∨ 𝑛 = 7)))
3635expd 419 . . . . . . . . 9 ((𝑛 < 7 ∨ 𝑛 = 7) → (𝑛 ∈ Odd → (5 < 𝑛 → (𝑛 = 6 ∨ 𝑛 = 7))))
3736com12 32 . . . . . . . 8 (𝑛 ∈ Odd → ((𝑛 < 7 ∨ 𝑛 = 7) → (5 < 𝑛 → (𝑛 = 6 ∨ 𝑛 = 7))))
3812, 37sylbid 243 . . . . . . 7 (𝑛 ∈ Odd → (𝑛 ≤ 7 → (5 < 𝑛 → (𝑛 = 6 ∨ 𝑛 = 7))))
39 eleq1 2877 . . . . . . . . . 10 (𝑛 = 6 → (𝑛 ∈ Odd ↔ 6 ∈ Odd ))
40 6even 44244 . . . . . . . . . . 11 6 ∈ Even
41 evennodd 44176 . . . . . . . . . . . 12 (6 ∈ Even → ¬ 6 ∈ Odd )
4241pm2.21d 121 . . . . . . . . . . 11 (6 ∈ Even → (6 ∈ Odd → 𝑛 ∈ GoldbachOddW ))
4340, 42mp1i 13 . . . . . . . . . 10 (𝑛 = 6 → (6 ∈ Odd → 𝑛 ∈ GoldbachOddW ))
4439, 43sylbid 243 . . . . . . . . 9 (𝑛 = 6 → (𝑛 ∈ Odd → 𝑛 ∈ GoldbachOddW ))
45 7gbow 44305 . . . . . . . . . . 11 7 ∈ GoldbachOddW
46 eleq1 2877 . . . . . . . . . . 11 (𝑛 = 7 → (𝑛 ∈ GoldbachOddW ↔ 7 ∈ GoldbachOddW ))
4745, 46mpbiri 261 . . . . . . . . . 10 (𝑛 = 7 → 𝑛 ∈ GoldbachOddW )
4847a1d 25 . . . . . . . . 9 (𝑛 = 7 → (𝑛 ∈ Odd → 𝑛 ∈ GoldbachOddW ))
4944, 48jaoi 854 . . . . . . . 8 ((𝑛 = 6 ∨ 𝑛 = 7) → (𝑛 ∈ Odd → 𝑛 ∈ GoldbachOddW ))
5049com12 32 . . . . . . 7 (𝑛 ∈ Odd → ((𝑛 = 6 ∨ 𝑛 = 7) → 𝑛 ∈ GoldbachOddW ))
5138, 50syl6d 75 . . . . . 6 (𝑛 ∈ Odd → (𝑛 ≤ 7 → (5 < 𝑛𝑛 ∈ GoldbachOddW )))
5211, 51sylbird 263 . . . . 5 (𝑛 ∈ Odd → (¬ 7 < 𝑛 → (5 < 𝑛𝑛 ∈ GoldbachOddW )))
5352com12 32 . . . 4 (¬ 7 < 𝑛 → (𝑛 ∈ Odd → (5 < 𝑛𝑛 ∈ GoldbachOddW )))
5453a1dd 50 . . 3 (¬ 7 < 𝑛 → (𝑛 ∈ Odd → ((7 < 𝑛𝑛 ∈ GoldbachOdd ) → (5 < 𝑛𝑛 ∈ GoldbachOddW ))))
556, 54pm2.61i 185 . 2 (𝑛 ∈ Odd → ((7 < 𝑛𝑛 ∈ GoldbachOdd ) → (5 < 𝑛𝑛 ∈ GoldbachOddW )))
5655ralimia 3126 1 (∀𝑛 ∈ Odd (7 < 𝑛𝑛 ∈ GoldbachOdd ) → ∀𝑛 ∈ Odd (5 < 𝑛𝑛 ∈ GoldbachOddW ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  ∀wral 3106   class class class wbr 5030  (class class class)co 7135  ℝcr 10527  1c1 10529   + caddc 10531   < clt 10666   ≤ cle 10667  5c5 11685  6c6 11686  7c7 11687  ℤcz 11971   Even ceven 44157   Odd codd 44158   GoldbachOddW cgbow 44279   GoldbachOdd cgbo 44280 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443  ax-cnex 10584  ax-resscn 10585  ax-1cn 10586  ax-icn 10587  ax-addcl 10588  ax-addrcl 10589  ax-mulcl 10590  ax-mulrcl 10591  ax-mulcom 10592  ax-addass 10593  ax-mulass 10594  ax-distr 10595  ax-i2m1 10596  ax-1ne0 10597  ax-1rid 10598  ax-rnegex 10599  ax-rrecex 10600  ax-cnre 10601  ax-pre-lttri 10602  ax-pre-lttrn 10603  ax-pre-ltadd 10604  ax-pre-mulgt0 10605  ax-pre-sup 10606 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7563  df-1st 7673  df-2nd 7674  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-1o 8087  df-2o 8088  df-er 8274  df-en 8495  df-dom 8496  df-sdom 8497  df-fin 8498  df-sup 8892  df-pnf 10668  df-mnf 10669  df-xr 10670  df-ltxr 10671  df-le 10672  df-sub 10863  df-neg 10864  df-div 11289  df-nn 11628  df-2 11690  df-3 11691  df-4 11692  df-5 11693  df-6 11694  df-7 11695  df-n0 11888  df-z 11972  df-uz 12234  df-rp 12380  df-fz 12888  df-seq 13367  df-exp 13428  df-cj 14452  df-re 14453  df-im 14454  df-sqrt 14588  df-abs 14589  df-dvds 15602  df-prm 16008  df-even 44159  df-odd 44160  df-gbow 44282  df-gbo 44283 This theorem is referenced by:  stgoldbnnsum4prm  44336
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