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Theorem stgoldbwt 41989
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 pm3.35 610 . . . . . 6 ((7 < 𝑛 ∧ (7 < 𝑛𝑛 ∈ GoldbachOdd )) → 𝑛 ∈ GoldbachOdd )
2 gbogbow 41969 . . . . . . 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 449 . . . 4 (7 < 𝑛 → ((7 < 𝑛𝑛 ∈ GoldbachOdd ) → (5 < 𝑛𝑛 ∈ GoldbachOddW )))
65a1d 25 . . 3 (7 < 𝑛 → (𝑛 ∈ Odd → ((7 < 𝑛𝑛 ∈ GoldbachOdd ) → (5 < 𝑛𝑛 ∈ GoldbachOddW ))))
7 oddz 41869 . . . . . . . 8 (𝑛 ∈ Odd → 𝑛 ∈ ℤ)
87zred 11520 . . . . . . 7 (𝑛 ∈ Odd → 𝑛 ∈ ℝ)
9 7re 11141 . . . . . . . 8 7 ∈ ℝ
109a1i 11 . . . . . . 7 (𝑛 ∈ Odd → 7 ∈ ℝ)
118, 10lenltd 10221 . . . . . 6 (𝑛 ∈ Odd → (𝑛 ≤ 7 ↔ ¬ 7 < 𝑛))
128, 10leloed 10218 . . . . . . . 8 (𝑛 ∈ Odd → (𝑛 ≤ 7 ↔ (𝑛 < 7 ∨ 𝑛 = 7)))
137adantr 480 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ Odd ∧ 5 < 𝑛) → 𝑛 ∈ ℤ)
14 6nn 11227 . . . . . . . . . . . . . . . . 17 6 ∈ ℕ
1514nnzi 11439 . . . . . . . . . . . . . . . 16 6 ∈ ℤ
1613, 15jctir 560 . . . . . . . . . . . . . . 15 ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (𝑛 ∈ ℤ ∧ 6 ∈ ℤ))
1716adantl 481 . . . . . . . . . . . . . 14 ((𝑛 < 7 ∧ (𝑛 ∈ Odd ∧ 5 < 𝑛)) → (𝑛 ∈ ℤ ∧ 6 ∈ ℤ))
18 df-7 11122 . . . . . . . . . . . . . . . . 17 7 = (6 + 1)
1918breq2i 4693 . . . . . . . . . . . . . . . 16 (𝑛 < 7 ↔ 𝑛 < (6 + 1))
2019biimpi 206 . . . . . . . . . . . . . . 15 (𝑛 < 7 → 𝑛 < (6 + 1))
21 df-6 11121 . . . . . . . . . . . . . . . 16 6 = (5 + 1)
22 5nn 11226 . . . . . . . . . . . . . . . . . . 19 5 ∈ ℕ
2322nnzi 11439 . . . . . . . . . . . . . . . . . 18 5 ∈ ℤ
24 zltp1le 11465 . . . . . . . . . . . . . . . . . 18 ((5 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (5 < 𝑛 ↔ (5 + 1) ≤ 𝑛))
2523, 7, 24sylancr 696 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ Odd → (5 < 𝑛 ↔ (5 + 1) ≤ 𝑛))
2625biimpa 500 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (5 + 1) ≤ 𝑛)
2721, 26syl5eqbr 4720 . . . . . . . . . . . . . . 15 ((𝑛 ∈ Odd ∧ 5 < 𝑛) → 6 ≤ 𝑛)
2820, 27anim12ci 590 . . . . . . . . . . . . . 14 ((𝑛 < 7 ∧ (𝑛 ∈ Odd ∧ 5 < 𝑛)) → (6 ≤ 𝑛𝑛 < (6 + 1)))
29 zgeltp1eq 41643 . . . . . . . . . . . . . 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 406 . . . . . . . . . . . 12 ((𝑛 < 7 ∧ (𝑛 ∈ Odd ∧ 5 < 𝑛)) → (𝑛 = 6 ∨ 𝑛 = 7))
3231ex 449 . . . . . . . . . . 11 (𝑛 < 7 → ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (𝑛 = 6 ∨ 𝑛 = 7)))
33 olc 398 . . . . . . . . . . . 12 (𝑛 = 7 → (𝑛 = 6 ∨ 𝑛 = 7))
3433a1d 25 . . . . . . . . . . 11 (𝑛 = 7 → ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (𝑛 = 6 ∨ 𝑛 = 7)))
3532, 34jaoi 393 . . . . . . . . . 10 ((𝑛 < 7 ∨ 𝑛 = 7) → ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (𝑛 = 6 ∨ 𝑛 = 7)))
3635expd 451 . . . . . . . . 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 230 . . . . . . 7 (𝑛 ∈ Odd → (𝑛 ≤ 7 → (5 < 𝑛 → (𝑛 = 6 ∨ 𝑛 = 7))))
39 eleq1 2718 . . . . . . . . . 10 (𝑛 = 6 → (𝑛 ∈ Odd ↔ 6 ∈ Odd ))
40 6even 41945 . . . . . . . . . . 11 6 ∈ Even
41 evennodd 41881 . . . . . . . . . . . 12 (6 ∈ Even → ¬ 6 ∈ Odd )
4241pm2.21d 118 . . . . . . . . . . 11 (6 ∈ Even → (6 ∈ Odd → 𝑛 ∈ GoldbachOddW ))
4340, 42mp1i 13 . . . . . . . . . 10 (𝑛 = 6 → (6 ∈ Odd → 𝑛 ∈ GoldbachOddW ))
4439, 43sylbid 230 . . . . . . . . 9 (𝑛 = 6 → (𝑛 ∈ Odd → 𝑛 ∈ GoldbachOddW ))
45 7gbow 41985 . . . . . . . . . . 11 7 ∈ GoldbachOddW
46 eleq1 2718 . . . . . . . . . . 11 (𝑛 = 7 → (𝑛 ∈ GoldbachOddW ↔ 7 ∈ GoldbachOddW ))
4745, 46mpbiri 248 . . . . . . . . . 10 (𝑛 = 7 → 𝑛 ∈ GoldbachOddW )
4847a1d 25 . . . . . . . . 9 (𝑛 = 7 → (𝑛 ∈ Odd → 𝑛 ∈ GoldbachOddW ))
4944, 48jaoi 393 . . . . . . . 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 250 . . . . 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 176 . 2 (𝑛 ∈ Odd → ((7 < 𝑛𝑛 ∈ GoldbachOdd ) → (5 < 𝑛𝑛 ∈ GoldbachOddW )))
5655ralimia 2979 1 (∀𝑛 ∈ Odd (7 < 𝑛𝑛 ∈ GoldbachOdd ) → ∀𝑛 ∈ Odd (5 < 𝑛𝑛 ∈ GoldbachOddW ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wcel 2030  wral 2941   class class class wbr 4685  (class class class)co 6690  cr 9973  1c1 9975   + caddc 9977   < clt 10112  cle 10113  5c5 11111  6c6 11112  7c7 11113  cz 11415   Even ceven 41862   Odd codd 41863   GoldbachOddW cgbow 41959   GoldbachOdd cgbo 41960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-fz 12365  df-seq 12842  df-exp 12901  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-dvds 15028  df-prm 15433  df-even 41864  df-odd 41865  df-gbow 41962  df-gbo 41963
This theorem is referenced by:  stgoldbnnsum4prm  42016
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