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Theorem stgoldbwt 46042
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 46022 . . . . . . 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 414 . . . 4 (7 < 𝑛 → ((7 < 𝑛𝑛 ∈ GoldbachOdd ) → (5 < 𝑛𝑛 ∈ GoldbachOddW )))
65a1d 25 . . 3 (7 < 𝑛 → (𝑛 ∈ Odd → ((7 < 𝑛𝑛 ∈ GoldbachOdd ) → (5 < 𝑛𝑛 ∈ GoldbachOddW ))))
7 oddz 45897 . . . . . . . 8 (𝑛 ∈ Odd → 𝑛 ∈ ℤ)
87zred 12614 . . . . . . 7 (𝑛 ∈ Odd → 𝑛 ∈ ℝ)
9 7re 12253 . . . . . . . 8 7 ∈ ℝ
109a1i 11 . . . . . . 7 (𝑛 ∈ Odd → 7 ∈ ℝ)
118, 10lenltd 11308 . . . . . 6 (𝑛 ∈ Odd → (𝑛 ≤ 7 ↔ ¬ 7 < 𝑛))
128, 10leloed 11305 . . . . . . . 8 (𝑛 ∈ Odd → (𝑛 ≤ 7 ↔ (𝑛 < 7 ∨ 𝑛 = 7)))
137adantr 482 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ Odd ∧ 5 < 𝑛) → 𝑛 ∈ ℤ)
14 6nn 12249 . . . . . . . . . . . . . . . . 17 6 ∈ ℕ
1514nnzi 12534 . . . . . . . . . . . . . . . 16 6 ∈ ℤ
1613, 15jctir 522 . . . . . . . . . . . . . . 15 ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (𝑛 ∈ ℤ ∧ 6 ∈ ℤ))
1716adantl 483 . . . . . . . . . . . . . 14 ((𝑛 < 7 ∧ (𝑛 ∈ Odd ∧ 5 < 𝑛)) → (𝑛 ∈ ℤ ∧ 6 ∈ ℤ))
18 df-7 12228 . . . . . . . . . . . . . . . . 17 7 = (6 + 1)
1918breq2i 5118 . . . . . . . . . . . . . . . 16 (𝑛 < 7 ↔ 𝑛 < (6 + 1))
2019biimpi 215 . . . . . . . . . . . . . . 15 (𝑛 < 7 → 𝑛 < (6 + 1))
21 df-6 12227 . . . . . . . . . . . . . . . 16 6 = (5 + 1)
22 5nn 12246 . . . . . . . . . . . . . . . . . . 19 5 ∈ ℕ
2322nnzi 12534 . . . . . . . . . . . . . . . . . 18 5 ∈ ℤ
24 zltp1le 12560 . . . . . . . . . . . . . . . . . 18 ((5 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (5 < 𝑛 ↔ (5 + 1) ≤ 𝑛))
2523, 7, 24sylancr 588 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ Odd → (5 < 𝑛 ↔ (5 + 1) ≤ 𝑛))
2625biimpa 478 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (5 + 1) ≤ 𝑛)
2721, 26eqbrtrid 5145 . . . . . . . . . . . . . . 15 ((𝑛 ∈ Odd ∧ 5 < 𝑛) → 6 ≤ 𝑛)
2820, 27anim12ci 615 . . . . . . . . . . . . . 14 ((𝑛 < 7 ∧ (𝑛 ∈ Odd ∧ 5 < 𝑛)) → (6 ≤ 𝑛𝑛 < (6 + 1)))
29 zgeltp1eq 45615 . . . . . . . . . . . . . 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 872 . . . . . . . . . . . 12 ((𝑛 < 7 ∧ (𝑛 ∈ Odd ∧ 5 < 𝑛)) → (𝑛 = 6 ∨ 𝑛 = 7))
3231ex 414 . . . . . . . . . . 11 (𝑛 < 7 → ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (𝑛 = 6 ∨ 𝑛 = 7)))
33 olc 867 . . . . . . . . . . . 12 (𝑛 = 7 → (𝑛 = 6 ∨ 𝑛 = 7))
3433a1d 25 . . . . . . . . . . 11 (𝑛 = 7 → ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (𝑛 = 6 ∨ 𝑛 = 7)))
3532, 34jaoi 856 . . . . . . . . . 10 ((𝑛 < 7 ∨ 𝑛 = 7) → ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (𝑛 = 6 ∨ 𝑛 = 7)))
3635expd 417 . . . . . . . . 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 239 . . . . . . 7 (𝑛 ∈ Odd → (𝑛 ≤ 7 → (5 < 𝑛 → (𝑛 = 6 ∨ 𝑛 = 7))))
39 eleq1 2826 . . . . . . . . . 10 (𝑛 = 6 → (𝑛 ∈ Odd ↔ 6 ∈ Odd ))
40 6even 45977 . . . . . . . . . . 11 6 ∈ Even
41 evennodd 45909 . . . . . . . . . . . 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 239 . . . . . . . . 9 (𝑛 = 6 → (𝑛 ∈ Odd → 𝑛 ∈ GoldbachOddW ))
45 7gbow 46038 . . . . . . . . . . 11 7 ∈ GoldbachOddW
46 eleq1 2826 . . . . . . . . . . 11 (𝑛 = 7 → (𝑛 ∈ GoldbachOddW ↔ 7 ∈ GoldbachOddW ))
4745, 46mpbiri 258 . . . . . . . . . 10 (𝑛 = 7 → 𝑛 ∈ GoldbachOddW )
4847a1d 25 . . . . . . . . 9 (𝑛 = 7 → (𝑛 ∈ Odd → 𝑛 ∈ GoldbachOddW ))
4944, 48jaoi 856 . . . . . . . 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 260 . . . . 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 182 . 2 (𝑛 ∈ Odd → ((7 < 𝑛𝑛 ∈ GoldbachOdd ) → (5 < 𝑛𝑛 ∈ GoldbachOddW )))
5655ralimia 3084 1 (∀𝑛 ∈ Odd (7 < 𝑛𝑛 ∈ GoldbachOdd ) → ∀𝑛 ∈ Odd (5 < 𝑛𝑛 ∈ GoldbachOddW ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846   = wceq 1542  wcel 2107  wral 3065   class class class wbr 5110  (class class class)co 7362  cr 11057  1c1 11059   + caddc 11061   < clt 11196  cle 11197  5c5 12218  6c6 12219  7c7 12220  cz 12506   Even ceven 45890   Odd codd 45891   GoldbachOddW cgbow 46012   GoldbachOdd cgbo 46013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135  ax-pre-sup 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-sup 9385  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-div 11820  df-nn 12161  df-2 12223  df-3 12224  df-4 12225  df-5 12226  df-6 12227  df-7 12228  df-n0 12421  df-z 12507  df-uz 12771  df-rp 12923  df-fz 13432  df-seq 13914  df-exp 13975  df-cj 14991  df-re 14992  df-im 14993  df-sqrt 15127  df-abs 15128  df-dvds 16144  df-prm 16555  df-even 45892  df-odd 45893  df-gbow 46015  df-gbo 46016
This theorem is referenced by:  stgoldbnnsum4prm  46069
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