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Theorem stgoldbwt 43818
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 pm3.35 799 . . . . . 6 ((7 < 𝑛 ∧ (7 < 𝑛𝑛 ∈ GoldbachOdd )) → 𝑛 ∈ GoldbachOdd )
2 gbogbow 43798 . . . . . . 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 413 . . . 4 (7 < 𝑛 → ((7 < 𝑛𝑛 ∈ GoldbachOdd ) → (5 < 𝑛𝑛 ∈ GoldbachOddW )))
65a1d 25 . . 3 (7 < 𝑛 → (𝑛 ∈ Odd → ((7 < 𝑛𝑛 ∈ GoldbachOdd ) → (5 < 𝑛𝑛 ∈ GoldbachOddW ))))
7 oddz 43673 . . . . . . . 8 (𝑛 ∈ Odd → 𝑛 ∈ ℤ)
87zred 12075 . . . . . . 7 (𝑛 ∈ Odd → 𝑛 ∈ ℝ)
9 7re 11718 . . . . . . . 8 7 ∈ ℝ
109a1i 11 . . . . . . 7 (𝑛 ∈ Odd → 7 ∈ ℝ)
118, 10lenltd 10774 . . . . . 6 (𝑛 ∈ Odd → (𝑛 ≤ 7 ↔ ¬ 7 < 𝑛))
128, 10leloed 10771 . . . . . . . 8 (𝑛 ∈ Odd → (𝑛 ≤ 7 ↔ (𝑛 < 7 ∨ 𝑛 = 7)))
137adantr 481 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ Odd ∧ 5 < 𝑛) → 𝑛 ∈ ℤ)
14 6nn 11714 . . . . . . . . . . . . . . . . 17 6 ∈ ℕ
1514nnzi 11994 . . . . . . . . . . . . . . . 16 6 ∈ ℤ
1613, 15jctir 521 . . . . . . . . . . . . . . 15 ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (𝑛 ∈ ℤ ∧ 6 ∈ ℤ))
1716adantl 482 . . . . . . . . . . . . . 14 ((𝑛 < 7 ∧ (𝑛 ∈ Odd ∧ 5 < 𝑛)) → (𝑛 ∈ ℤ ∧ 6 ∈ ℤ))
18 df-7 11693 . . . . . . . . . . . . . . . . 17 7 = (6 + 1)
1918breq2i 5065 . . . . . . . . . . . . . . . 16 (𝑛 < 7 ↔ 𝑛 < (6 + 1))
2019biimpi 217 . . . . . . . . . . . . . . 15 (𝑛 < 7 → 𝑛 < (6 + 1))
21 df-6 11692 . . . . . . . . . . . . . . . 16 6 = (5 + 1)
22 5nn 11711 . . . . . . . . . . . . . . . . . . 19 5 ∈ ℕ
2322nnzi 11994 . . . . . . . . . . . . . . . . . 18 5 ∈ ℤ
24 zltp1le 12020 . . . . . . . . . . . . . . . . . 18 ((5 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (5 < 𝑛 ↔ (5 + 1) ≤ 𝑛))
2523, 7, 24sylancr 587 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ Odd → (5 < 𝑛 ↔ (5 + 1) ≤ 𝑛))
2625biimpa 477 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (5 + 1) ≤ 𝑛)
2721, 26eqbrtrid 5092 . . . . . . . . . . . . . . 15 ((𝑛 ∈ Odd ∧ 5 < 𝑛) → 6 ≤ 𝑛)
2820, 27anim12ci 613 . . . . . . . . . . . . . 14 ((𝑛 < 7 ∧ (𝑛 ∈ Odd ∧ 5 < 𝑛)) → (6 ≤ 𝑛𝑛 < (6 + 1)))
29 zgeltp1eq 43386 . . . . . . . . . . . . . 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 869 . . . . . . . . . . . 12 ((𝑛 < 7 ∧ (𝑛 ∈ Odd ∧ 5 < 𝑛)) → (𝑛 = 6 ∨ 𝑛 = 7))
3231ex 413 . . . . . . . . . . 11 (𝑛 < 7 → ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (𝑛 = 6 ∨ 𝑛 = 7)))
33 olc 862 . . . . . . . . . . . 12 (𝑛 = 7 → (𝑛 = 6 ∨ 𝑛 = 7))
3433a1d 25 . . . . . . . . . . 11 (𝑛 = 7 → ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (𝑛 = 6 ∨ 𝑛 = 7)))
3532, 34jaoi 851 . . . . . . . . . 10 ((𝑛 < 7 ∨ 𝑛 = 7) → ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (𝑛 = 6 ∨ 𝑛 = 7)))
3635expd 416 . . . . . . . . 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 241 . . . . . . 7 (𝑛 ∈ Odd → (𝑛 ≤ 7 → (5 < 𝑛 → (𝑛 = 6 ∨ 𝑛 = 7))))
39 eleq1 2897 . . . . . . . . . 10 (𝑛 = 6 → (𝑛 ∈ Odd ↔ 6 ∈ Odd ))
40 6even 43753 . . . . . . . . . . 11 6 ∈ Even
41 evennodd 43685 . . . . . . . . . . . 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 241 . . . . . . . . 9 (𝑛 = 6 → (𝑛 ∈ Odd → 𝑛 ∈ GoldbachOddW ))
45 7gbow 43814 . . . . . . . . . . 11 7 ∈ GoldbachOddW
46 eleq1 2897 . . . . . . . . . . 11 (𝑛 = 7 → (𝑛 ∈ GoldbachOddW ↔ 7 ∈ GoldbachOddW ))
4745, 46mpbiri 259 . . . . . . . . . 10 (𝑛 = 7 → 𝑛 ∈ GoldbachOddW )
4847a1d 25 . . . . . . . . 9 (𝑛 = 7 → (𝑛 ∈ Odd → 𝑛 ∈ GoldbachOddW ))
4944, 48jaoi 851 . . . . . . . 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 261 . . . . 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 183 . 2 (𝑛 ∈ Odd → ((7 < 𝑛𝑛 ∈ GoldbachOdd ) → (5 < 𝑛𝑛 ∈ GoldbachOddW )))
5655ralimia 3155 1 (∀𝑛 ∈ Odd (7 < 𝑛𝑛 ∈ GoldbachOdd ) → ∀𝑛 ∈ Odd (5 < 𝑛𝑛 ∈ GoldbachOddW ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841   = wceq 1528  wcel 2105  wral 3135   class class class wbr 5057  (class class class)co 7145  cr 10524  1c1 10526   + caddc 10528   < clt 10663  cle 10664  5c5 11683  6c6 11684  7c7 11685  cz 11969   Even ceven 43666   Odd codd 43667   GoldbachOddW cgbow 43788   GoldbachOdd cgbo 43789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12881  df-seq 13358  df-exp 13418  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-dvds 15596  df-prm 16004  df-even 43668  df-odd 43669  df-gbow 43791  df-gbo 43792
This theorem is referenced by:  stgoldbnnsum4prm  43845
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