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Theorem stgoldbwt 40198
 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 ∈ GoldbachOddALTV ) → ∀𝑛 ∈ Odd (5 < 𝑛𝑛 ∈ GoldbachOdd ))

Proof of Theorem stgoldbwt
StepHypRef Expression
1 pm3.35 609 . . . . . 6 ((7 < 𝑛 ∧ (7 < 𝑛𝑛 ∈ GoldbachOddALTV )) → 𝑛 ∈ GoldbachOddALTV )
2 gboagbo 40178 . . . . . . 7 (𝑛 ∈ GoldbachOddALTV → 𝑛 ∈ GoldbachOdd )
32a1d 25 . . . . . 6 (𝑛 ∈ GoldbachOddALTV → (5 < 𝑛𝑛 ∈ GoldbachOdd ))
41, 3syl 17 . . . . 5 ((7 < 𝑛 ∧ (7 < 𝑛𝑛 ∈ GoldbachOddALTV )) → (5 < 𝑛𝑛 ∈ GoldbachOdd ))
54ex 449 . . . 4 (7 < 𝑛 → ((7 < 𝑛𝑛 ∈ GoldbachOddALTV ) → (5 < 𝑛𝑛 ∈ GoldbachOdd )))
65a1d 25 . . 3 (7 < 𝑛 → (𝑛 ∈ Odd → ((7 < 𝑛𝑛 ∈ GoldbachOddALTV ) → (5 < 𝑛𝑛 ∈ GoldbachOdd ))))
7 oddz 40082 . . . . . . . 8 (𝑛 ∈ Odd → 𝑛 ∈ ℤ)
87zred 11358 . . . . . . 7 (𝑛 ∈ Odd → 𝑛 ∈ ℝ)
9 7re 10980 . . . . . . . 8 7 ∈ ℝ
109a1i 11 . . . . . . 7 (𝑛 ∈ Odd → 7 ∈ ℝ)
118, 10lenltd 10062 . . . . . 6 (𝑛 ∈ Odd → (𝑛 ≤ 7 ↔ ¬ 7 < 𝑛))
128, 10leloed 10059 . . . . . . . 8 (𝑛 ∈ Odd → (𝑛 ≤ 7 ↔ (𝑛 < 7 ∨ 𝑛 = 7)))
137adantr 480 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ Odd ∧ 5 < 𝑛) → 𝑛 ∈ ℤ)
14 6nn 11066 . . . . . . . . . . . . . . . . 17 6 ∈ ℕ
1514nnzi 11278 . . . . . . . . . . . . . . . 16 6 ∈ ℤ
1613, 15jctir 559 . . . . . . . . . . . . . . 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 10961 . . . . . . . . . . . . . . . . 17 7 = (6 + 1)
1918breq2i 4591 . . . . . . . . . . . . . . . 16 (𝑛 < 7 ↔ 𝑛 < (6 + 1))
2019biimpi 205 . . . . . . . . . . . . . . 15 (𝑛 < 7 → 𝑛 < (6 + 1))
21 df-6 10960 . . . . . . . . . . . . . . . 16 6 = (5 + 1)
22 5nn 11065 . . . . . . . . . . . . . . . . . . 19 5 ∈ ℕ
2322nnzi 11278 . . . . . . . . . . . . . . . . . 18 5 ∈ ℤ
24 zltp1le 11304 . . . . . . . . . . . . . . . . . 18 ((5 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (5 < 𝑛 ↔ (5 + 1) ≤ 𝑛))
2523, 7, 24sylancr 694 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ Odd → (5 < 𝑛 ↔ (5 + 1) ≤ 𝑛))
2625biimpa 500 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ Odd ∧ 5 < 𝑛) → (5 + 1) ≤ 𝑛)
2721, 26syl5eqbr 4618 . . . . . . . . . . . . . . 15 ((𝑛 ∈ Odd ∧ 5 < 𝑛) → 6 ≤ 𝑛)
2820, 27anim12ci 589 . . . . . . . . . . . . . 14 ((𝑛 < 7 ∧ (𝑛 ∈ Odd ∧ 5 < 𝑛)) → (6 ≤ 𝑛𝑛 < (6 + 1)))
29 zgeltp1eq 39943 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℤ ∧ 6 ∈ ℤ) → ((6 ≤ 𝑛𝑛 < (6 + 1)) → 𝑛 = 6))
3017, 28, 29sylc 63 . . . . . . . . . . . . 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 229 . . . . . . 7 (𝑛 ∈ Odd → (𝑛 ≤ 7 → (5 < 𝑛 → (𝑛 = 6 ∨ 𝑛 = 7))))
39 eleq1 2676 . . . . . . . . . 10 (𝑛 = 6 → (𝑛 ∈ Odd ↔ 6 ∈ Odd ))
40 6even 40158 . . . . . . . . . . 11 6 ∈ Even
41 evennodd 40094 . . . . . . . . . . . 12 (6 ∈ Even → ¬ 6 ∈ Odd )
4241pm2.21d 117 . . . . . . . . . . 11 (6 ∈ Even → (6 ∈ Odd → 𝑛 ∈ GoldbachOdd ))
4340, 42mp1i 13 . . . . . . . . . 10 (𝑛 = 6 → (6 ∈ Odd → 𝑛 ∈ GoldbachOdd ))
4439, 43sylbid 229 . . . . . . . . 9 (𝑛 = 6 → (𝑛 ∈ Odd → 𝑛 ∈ GoldbachOdd ))
45 7gbo 40194 . . . . . . . . . . 11 7 ∈ GoldbachOdd
46 eleq1 2676 . . . . . . . . . . 11 (𝑛 = 7 → (𝑛 ∈ GoldbachOdd ↔ 7 ∈ GoldbachOdd ))
4745, 46mpbiri 247 . . . . . . . . . 10 (𝑛 = 7 → 𝑛 ∈ GoldbachOdd )
4847a1d 25 . . . . . . . . 9 (𝑛 = 7 → (𝑛 ∈ Odd → 𝑛 ∈ GoldbachOdd ))
4944, 48jaoi 393 . . . . . . . 8 ((𝑛 = 6 ∨ 𝑛 = 7) → (𝑛 ∈ Odd → 𝑛 ∈ GoldbachOdd ))
5049com12 32 . . . . . . 7 (𝑛 ∈ Odd → ((𝑛 = 6 ∨ 𝑛 = 7) → 𝑛 ∈ GoldbachOdd ))
5138, 50syl6d 73 . . . . . 6 (𝑛 ∈ Odd → (𝑛 ≤ 7 → (5 < 𝑛𝑛 ∈ GoldbachOdd )))
5211, 51sylbird 249 . . . . 5 (𝑛 ∈ Odd → (¬ 7 < 𝑛 → (5 < 𝑛𝑛 ∈ GoldbachOdd )))
5352com12 32 . . . 4 (¬ 7 < 𝑛 → (𝑛 ∈ Odd → (5 < 𝑛𝑛 ∈ GoldbachOdd )))
5453a1dd 48 . . 3 (¬ 7 < 𝑛 → (𝑛 ∈ Odd → ((7 < 𝑛𝑛 ∈ GoldbachOddALTV ) → (5 < 𝑛𝑛 ∈ GoldbachOdd ))))
556, 54pm2.61i 175 . 2 (𝑛 ∈ Odd → ((7 < 𝑛𝑛 ∈ GoldbachOddALTV ) → (5 < 𝑛𝑛 ∈ GoldbachOdd )))
5655ralimia 2934 1 (∀𝑛 ∈ Odd (7 < 𝑛𝑛 ∈ GoldbachOddALTV ) → ∀𝑛 ∈ Odd (5 < 𝑛𝑛 ∈ GoldbachOdd ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   class class class wbr 4583  (class class class)co 6549  ℝcr 9814  1c1 9816   + caddc 9818   < clt 9953   ≤ cle 9954  5c5 10950  6c6 10951  7c7 10952  ℤcz 11254   Even ceven 40075   Odd codd 40076   GoldbachOdd cgbo 40168   GoldbachOddALTV cgboa 40169 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-dvds 14822  df-prm 15224  df-even 40077  df-odd 40078  df-gbo 40171  df-gboa 40172 This theorem is referenced by:  stgoldbnnsum4prm  40219
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