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Theorem bgoldbachlt 42230
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 42227. (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 11400 . . 3 4 ∈ ℕ
2 10nn 11727 . . . 4 10 ∈ ℕ
3 1nn0 11521 . . . . 5 1 ∈ ℕ0
4 8nn0 11528 . . . . 5 8 ∈ ℕ0
53, 4deccl 11725 . . . 4 18 ∈ ℕ0
6 nnexpcl 13088 . . . 4 ((10 ∈ ℕ ∧ 18 ∈ ℕ0) → (10↑18) ∈ ℕ)
72, 5, 6mp2an 710 . . 3 (10↑18) ∈ ℕ
81, 7nnmulcli 11257 . 2 (4 · (10↑18)) ∈ ℕ
9 id 22 . . 3 ((4 · (10↑18)) ∈ ℕ → (4 · (10↑18)) ∈ ℕ)
10 breq2 4809 . . . . 5 (𝑚 = (4 · (10↑18)) → ((4 · (10↑18)) ≤ 𝑚 ↔ (4 · (10↑18)) ≤ (4 · (10↑18))))
11 breq2 4809 . . . . . . . 8 (𝑚 = (4 · (10↑18)) → (𝑛 < 𝑚𝑛 < (4 · (10↑18))))
1211anbi2d 742 . . . . . . 7 (𝑚 = (4 · (10↑18)) → ((4 < 𝑛𝑛 < 𝑚) ↔ (4 < 𝑛𝑛 < (4 · (10↑18)))))
1312imbi1d 330 . . . . . 6 (𝑚 = (4 · (10↑18)) → (((4 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven ) ↔ ((4 < 𝑛𝑛 < (4 · (10↑18))) → 𝑛 ∈ GoldbachEven )))
1413ralbidv 3125 . . . . 5 (𝑚 = (4 · (10↑18)) → (∀𝑛 ∈ Even ((4 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven ) ↔ ∀𝑛 ∈ Even ((4 < 𝑛𝑛 < (4 · (10↑18))) → 𝑛 ∈ GoldbachEven )))
1510, 14anbi12d 749 . . . 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 473 . . 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 11240 . . . . 5 ((4 · (10↑18)) ∈ ℕ → (4 · (10↑18)) ∈ ℝ)
1817leidd 10807 . . . 4 ((4 · (10↑18)) ∈ ℕ → (4 · (10↑18)) ≤ (4 · (10↑18)))
19 simplr 809 . . . . . . 7 ((((4 · (10↑18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛𝑛 < (4 · (10↑18)))) → 𝑛 ∈ Even )
20 simprl 811 . . . . . . 7 ((((4 · (10↑18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛𝑛 < (4 · (10↑18)))) → 4 < 𝑛)
21 evenz 42072 . . . . . . . . . . 11 (𝑛 ∈ Even → 𝑛 ∈ ℤ)
2221zred 11695 . . . . . . . . . 10 (𝑛 ∈ Even → 𝑛 ∈ ℝ)
23 ltle 10339 . . . . . . . . . 10 ((𝑛 ∈ ℝ ∧ (4 · (10↑18)) ∈ ℝ) → (𝑛 < (4 · (10↑18)) → 𝑛 ≤ (4 · (10↑18))))
2422, 17, 23syl2anr 496 . . . . . . . . 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 448 . . . . . . 7 ((((4 · (10↑18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛𝑛 < (4 · (10↑18)))) → 𝑛 ≤ (4 · (10↑18)))
27 ax-bgbltosilva 42227 . . . . . . 7 ((𝑛 ∈ Even ∧ 4 < 𝑛𝑛 ≤ (4 · (10↑18))) → 𝑛 ∈ GoldbachEven )
2819, 20, 26, 27syl3anc 1477 . . . . . 6 ((((4 · (10↑18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛𝑛 < (4 · (10↑18)))) → 𝑛 ∈ GoldbachEven )
2928ex 449 . . . . 5 (((4 · (10↑18)) ∈ ℕ ∧ 𝑛 ∈ Even ) → ((4 < 𝑛𝑛 < (4 · (10↑18))) → 𝑛 ∈ GoldbachEven ))
3029ralrimiva 3105 . . . 4 ((4 · (10↑18)) ∈ ℕ → ∀𝑛 ∈ Even ((4 < 𝑛𝑛 < (4 · (10↑18))) → 𝑛 ∈ GoldbachEven ))
3118, 30jca 555 . . 3 ((4 · (10↑18)) ∈ ℕ → ((4 · (10↑18)) ≤ (4 · (10↑18)) ∧ ∀𝑛 ∈ Even ((4 < 𝑛𝑛 < (4 · (10↑18))) → 𝑛 ∈ GoldbachEven )))
329, 16, 31rspcedvd 3457 . 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 196  wa 383   = wceq 1632  wcel 2140  wral 3051  wrex 3052   class class class wbr 4805  (class class class)co 6815  cr 10148  0cc0 10149  1c1 10150   · cmul 10154   < clt 10287  cle 10288  cn 11233  4c4 11285  8c8 11289  0cn0 11505  cdc 11706  cexp 13075   Even ceven 42066   GoldbachEven cgbe 42162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116  ax-cnex 10205  ax-resscn 10206  ax-1cn 10207  ax-icn 10208  ax-addcl 10209  ax-addrcl 10210  ax-mulcl 10211  ax-mulrcl 10212  ax-mulcom 10213  ax-addass 10214  ax-mulass 10215  ax-distr 10216  ax-i2m1 10217  ax-1ne0 10218  ax-1rid 10219  ax-rnegex 10220  ax-rrecex 10221  ax-cnre 10222  ax-pre-lttri 10223  ax-pre-lttrn 10224  ax-pre-ltadd 10225  ax-pre-mulgt0 10226  ax-bgbltosilva 42227
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-tr 4906  df-id 5175  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-we 5228  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-pred 5842  df-ord 5888  df-on 5889  df-lim 5890  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-riota 6776  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-om 7233  df-2nd 7336  df-wrecs 7578  df-recs 7639  df-rdg 7677  df-er 7914  df-en 8125  df-dom 8126  df-sdom 8127  df-pnf 10289  df-mnf 10290  df-xr 10291  df-ltxr 10292  df-le 10293  df-sub 10481  df-neg 10482  df-nn 11234  df-2 11292  df-3 11293  df-4 11294  df-5 11295  df-6 11296  df-7 11297  df-8 11298  df-9 11299  df-n0 11506  df-z 11591  df-dec 11707  df-uz 11901  df-seq 13017  df-exp 13076  df-even 42068
This theorem is referenced by: (None)
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