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Theorem bgoldbachlt 42211
 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 42208. (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 11379 . . 3 4 ∈ ℕ
2 10nn 11706 . . . 4 10 ∈ ℕ
3 1nn0 11500 . . . . 5 1 ∈ ℕ0
4 8nn0 11507 . . . . 5 8 ∈ ℕ0
53, 4deccl 11704 . . . 4 18 ∈ ℕ0
6 nnexpcl 13067 . . . 4 ((10 ∈ ℕ ∧ 18 ∈ ℕ0) → (10↑18) ∈ ℕ)
72, 5, 6mp2an 710 . . 3 (10↑18) ∈ ℕ
81, 7nnmulcli 11236 . 2 (4 · (10↑18)) ∈ ℕ
9 id 22 . . 3 ((4 · (10↑18)) ∈ ℕ → (4 · (10↑18)) ∈ ℕ)
10 breq2 4808 . . . . 5 (𝑚 = (4 · (10↑18)) → ((4 · (10↑18)) ≤ 𝑚 ↔ (4 · (10↑18)) ≤ (4 · (10↑18))))
11 breq2 4808 . . . . . . . 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 3124 . . . . 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 11219 . . . . 5 ((4 · (10↑18)) ∈ ℕ → (4 · (10↑18)) ∈ ℝ)
1817leidd 10786 . . . 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 42053 . . . . . . . . . . 11 (𝑛 ∈ Even → 𝑛 ∈ ℤ)
2221zred 11674 . . . . . . . . . 10 (𝑛 ∈ Even → 𝑛 ∈ ℝ)
23 ltle 10318 . . . . . . . . . 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 42208 . . . . . . 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 3104 . . . 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 3456 . 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 2139  ∀wral 3050  ∃wrex 3051   class class class wbr 4804  (class class class)co 6813  ℝcr 10127  0cc0 10128  1c1 10129   · cmul 10133   < clt 10266   ≤ cle 10267  ℕcn 11212  4c4 11264  8c8 11268  ℕ0cn0 11484  ;cdc 11685  ↑cexp 13054   Even ceven 42047   GoldbachEven cgbe 42143 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 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-bgbltosilva 42208 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 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-7 11276  df-8 11277  df-9 11278  df-n0 11485  df-z 11570  df-dec 11686  df-uz 11880  df-seq 12996  df-exp 13055  df-even 42049 This theorem is referenced by: (None)
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