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Theorem bgoldbachlt 45937
Description: The binary Goldbach conjecture is valid for small even numbers (i.e. for all even numbers less than or equal to a fixed big 𝑚). This is verified for m = 4 x 10^18 by Oliveira e Silva, see ax-bgbltosilva 45934. (Contributed by AV, 3-Aug-2020.) (Revised by AV, 9-Sep-2021.)
Assertion
Ref Expression
bgoldbachlt 𝑚 ∈ ℕ ((4 · (10↑18)) ≤ 𝑚 ∧ ∀𝑛 ∈ Even ((4 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven ))
Distinct variable group:   𝑚,𝑛

Proof of Theorem bgoldbachlt
StepHypRef Expression
1 4nn 12232 . . 3 4 ∈ ℕ
2 10nn 12630 . . . 4 10 ∈ ℕ
3 1nn0 12425 . . . . 5 1 ∈ ℕ0
4 8nn0 12432 . . . . 5 8 ∈ ℕ0
53, 4deccl 12629 . . . 4 18 ∈ ℕ0
6 nnexpcl 13972 . . . 4 ((10 ∈ ℕ ∧ 18 ∈ ℕ0) → (10↑18) ∈ ℕ)
72, 5, 6mp2an 690 . . 3 (10↑18) ∈ ℕ
81, 7nnmulcli 12174 . 2 (4 · (10↑18)) ∈ ℕ
9 id 22 . . 3 ((4 · (10↑18)) ∈ ℕ → (4 · (10↑18)) ∈ ℕ)
10 breq2 5107 . . . . 5 (𝑚 = (4 · (10↑18)) → ((4 · (10↑18)) ≤ 𝑚 ↔ (4 · (10↑18)) ≤ (4 · (10↑18))))
11 breq2 5107 . . . . . . . 8 (𝑚 = (4 · (10↑18)) → (𝑛 < 𝑚𝑛 < (4 · (10↑18))))
1211anbi2d 629 . . . . . . 7 (𝑚 = (4 · (10↑18)) → ((4 < 𝑛𝑛 < 𝑚) ↔ (4 < 𝑛𝑛 < (4 · (10↑18)))))
1312imbi1d 341 . . . . . 6 (𝑚 = (4 · (10↑18)) → (((4 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven ) ↔ ((4 < 𝑛𝑛 < (4 · (10↑18))) → 𝑛 ∈ GoldbachEven )))
1413ralbidv 3172 . . . . 5 (𝑚 = (4 · (10↑18)) → (∀𝑛 ∈ Even ((4 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven ) ↔ ∀𝑛 ∈ Even ((4 < 𝑛𝑛 < (4 · (10↑18))) → 𝑛 ∈ GoldbachEven )))
1510, 14anbi12d 631 . . . 4 (𝑚 = (4 · (10↑18)) → (((4 · (10↑18)) ≤ 𝑚 ∧ ∀𝑛 ∈ Even ((4 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven )) ↔ ((4 · (10↑18)) ≤ (4 · (10↑18)) ∧ ∀𝑛 ∈ Even ((4 < 𝑛𝑛 < (4 · (10↑18))) → 𝑛 ∈ GoldbachEven ))))
1615adantl 482 . . 3 (((4 · (10↑18)) ∈ ℕ ∧ 𝑚 = (4 · (10↑18))) → (((4 · (10↑18)) ≤ 𝑚 ∧ ∀𝑛 ∈ Even ((4 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven )) ↔ ((4 · (10↑18)) ≤ (4 · (10↑18)) ∧ ∀𝑛 ∈ Even ((4 < 𝑛𝑛 < (4 · (10↑18))) → 𝑛 ∈ GoldbachEven ))))
17 nnre 12156 . . . . 5 ((4 · (10↑18)) ∈ ℕ → (4 · (10↑18)) ∈ ℝ)
1817leidd 11717 . . . 4 ((4 · (10↑18)) ∈ ℕ → (4 · (10↑18)) ≤ (4 · (10↑18)))
19 simplr 767 . . . . . . 7 ((((4 · (10↑18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛𝑛 < (4 · (10↑18)))) → 𝑛 ∈ Even )
20 simprl 769 . . . . . . 7 ((((4 · (10↑18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛𝑛 < (4 · (10↑18)))) → 4 < 𝑛)
21 evenz 45754 . . . . . . . . . . 11 (𝑛 ∈ Even → 𝑛 ∈ ℤ)
2221zred 12603 . . . . . . . . . 10 (𝑛 ∈ Even → 𝑛 ∈ ℝ)
23 ltle 11239 . . . . . . . . . 10 ((𝑛 ∈ ℝ ∧ (4 · (10↑18)) ∈ ℝ) → (𝑛 < (4 · (10↑18)) → 𝑛 ≤ (4 · (10↑18))))
2422, 17, 23syl2anr 597 . . . . . . . . 9 (((4 · (10↑18)) ∈ ℕ ∧ 𝑛 ∈ Even ) → (𝑛 < (4 · (10↑18)) → 𝑛 ≤ (4 · (10↑18))))
2524a1d 25 . . . . . . . 8 (((4 · (10↑18)) ∈ ℕ ∧ 𝑛 ∈ Even ) → (4 < 𝑛 → (𝑛 < (4 · (10↑18)) → 𝑛 ≤ (4 · (10↑18)))))
2625imp32 419 . . . . . . 7 ((((4 · (10↑18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛𝑛 < (4 · (10↑18)))) → 𝑛 ≤ (4 · (10↑18)))
27 ax-bgbltosilva 45934 . . . . . . 7 ((𝑛 ∈ Even ∧ 4 < 𝑛𝑛 ≤ (4 · (10↑18))) → 𝑛 ∈ GoldbachEven )
2819, 20, 26, 27syl3anc 1371 . . . . . 6 ((((4 · (10↑18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛𝑛 < (4 · (10↑18)))) → 𝑛 ∈ GoldbachEven )
2928ex 413 . . . . 5 (((4 · (10↑18)) ∈ ℕ ∧ 𝑛 ∈ Even ) → ((4 < 𝑛𝑛 < (4 · (10↑18))) → 𝑛 ∈ GoldbachEven ))
3029ralrimiva 3141 . . . 4 ((4 · (10↑18)) ∈ ℕ → ∀𝑛 ∈ Even ((4 < 𝑛𝑛 < (4 · (10↑18))) → 𝑛 ∈ GoldbachEven ))
3118, 30jca 512 . . 3 ((4 · (10↑18)) ∈ ℕ → ((4 · (10↑18)) ≤ (4 · (10↑18)) ∧ ∀𝑛 ∈ Even ((4 < 𝑛𝑛 < (4 · (10↑18))) → 𝑛 ∈ GoldbachEven )))
329, 16, 31rspcedvd 3581 . 2 ((4 · (10↑18)) ∈ ℕ → ∃𝑚 ∈ ℕ ((4 · (10↑18)) ≤ 𝑚 ∧ ∀𝑛 ∈ Even ((4 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven )))
338, 32ax-mp 5 1 𝑚 ∈ ℕ ((4 · (10↑18)) ≤ 𝑚 ∧ ∀𝑛 ∈ Even ((4 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3062  wrex 3071   class class class wbr 5103  (class class class)co 7353  cr 11046  0cc0 11047  1c1 11048   · cmul 11052   < clt 11185  cle 11186  cn 12149  4c4 12206  8c8 12210  0cn0 12409  cdc 12614  cexp 13959   Even ceven 45748   GoldbachEven cgbe 45869
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668  ax-cnex 11103  ax-resscn 11104  ax-1cn 11105  ax-icn 11106  ax-addcl 11107  ax-addrcl 11108  ax-mulcl 11109  ax-mulrcl 11110  ax-mulcom 11111  ax-addass 11112  ax-mulass 11113  ax-distr 11114  ax-i2m1 11115  ax-1ne0 11116  ax-1rid 11117  ax-rnegex 11118  ax-rrecex 11119  ax-cnre 11120  ax-pre-lttri 11121  ax-pre-lttrn 11122  ax-pre-ltadd 11123  ax-pre-mulgt0 11124  ax-bgbltosilva 45934
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7309  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7799  df-2nd 7918  df-frecs 8208  df-wrecs 8239  df-recs 8313  df-rdg 8352  df-er 8644  df-en 8880  df-dom 8881  df-sdom 8882  df-pnf 11187  df-mnf 11188  df-xr 11189  df-ltxr 11190  df-le 11191  df-sub 11383  df-neg 11384  df-nn 12150  df-2 12212  df-3 12213  df-4 12214  df-5 12215  df-6 12216  df-7 12217  df-8 12218  df-9 12219  df-n0 12410  df-z 12496  df-dec 12615  df-uz 12760  df-seq 13899  df-exp 13960  df-even 45750
This theorem is referenced by: (None)
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