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Theorem bgoldbachltOLD 42235
Description: Obsolete version of bgoldbachlt 42229 as of 9-Sep-2021. (Contributed by AV, 3-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bgoldbachltOLD 𝑚 ∈ ℕ ((4 · (10↑18)) ≤ 𝑚 ∧ ∀𝑛 ∈ Even ((4 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven ))
Distinct variable group:   𝑚,𝑛

Proof of Theorem bgoldbachltOLD
StepHypRef Expression
1 4nn 11399 . . 3 4 ∈ ℕ
2 10nnOLD 11405 . . . 4 10 ∈ ℕ
3 1nn0 11520 . . . . . 6 1 ∈ ℕ0
4 8nn 11403 . . . . . 6 8 ∈ ℕ
53, 4decnncl 11730 . . . . 5 18 ∈ ℕ
65nnnn0i 11512 . . . 4 18 ∈ ℕ0
7 nnexpcl 13087 . . . 4 ((10 ∈ ℕ ∧ 18 ∈ ℕ0) → (10↑18) ∈ ℕ)
82, 6, 7mp2an 710 . . 3 (10↑18) ∈ ℕ
91, 8nnmulcli 11256 . 2 (4 · (10↑18)) ∈ ℕ
10 id 22 . . 3 ((4 · (10↑18)) ∈ ℕ → (4 · (10↑18)) ∈ ℕ)
11 breq2 4808 . . . . 5 (𝑚 = (4 · (10↑18)) → ((4 · (10↑18)) ≤ 𝑚 ↔ (4 · (10↑18)) ≤ (4 · (10↑18))))
12 breq2 4808 . . . . . . . 8 (𝑚 = (4 · (10↑18)) → (𝑛 < 𝑚𝑛 < (4 · (10↑18))))
1312anbi2d 742 . . . . . . 7 (𝑚 = (4 · (10↑18)) → ((4 < 𝑛𝑛 < 𝑚) ↔ (4 < 𝑛𝑛 < (4 · (10↑18)))))
1413imbi1d 330 . . . . . 6 (𝑚 = (4 · (10↑18)) → (((4 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven ) ↔ ((4 < 𝑛𝑛 < (4 · (10↑18))) → 𝑛 ∈ GoldbachEven )))
1514ralbidv 3124 . . . . 5 (𝑚 = (4 · (10↑18)) → (∀𝑛 ∈ Even ((4 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven ) ↔ ∀𝑛 ∈ Even ((4 < 𝑛𝑛 < (4 · (10↑18))) → 𝑛 ∈ GoldbachEven )))
1611, 15anbi12d 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 ))))
1716adantl 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 ))))
18 nnre 11239 . . . . 5 ((4 · (10↑18)) ∈ ℕ → (4 · (10↑18)) ∈ ℝ)
1918leidd 10806 . . . 4 ((4 · (10↑18)) ∈ ℕ → (4 · (10↑18)) ≤ (4 · (10↑18)))
20 simplr 809 . . . . . . 7 ((((4 · (10↑18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛𝑛 < (4 · (10↑18)))) → 𝑛 ∈ Even )
21 simprl 811 . . . . . . 7 ((((4 · (10↑18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛𝑛 < (4 · (10↑18)))) → 4 < 𝑛)
22 evenz 42071 . . . . . . . . . . 11 (𝑛 ∈ Even → 𝑛 ∈ ℤ)
2322zred 11694 . . . . . . . . . 10 (𝑛 ∈ Even → 𝑛 ∈ ℝ)
24 ltle 10338 . . . . . . . . . 10 ((𝑛 ∈ ℝ ∧ (4 · (10↑18)) ∈ ℝ) → (𝑛 < (4 · (10↑18)) → 𝑛 ≤ (4 · (10↑18))))
2523, 18, 24syl2anr 496 . . . . . . . . 9 (((4 · (10↑18)) ∈ ℕ ∧ 𝑛 ∈ Even ) → (𝑛 < (4 · (10↑18)) → 𝑛 ≤ (4 · (10↑18))))
2625a1d 25 . . . . . . . 8 (((4 · (10↑18)) ∈ ℕ ∧ 𝑛 ∈ Even ) → (4 < 𝑛 → (𝑛 < (4 · (10↑18)) → 𝑛 ≤ (4 · (10↑18)))))
2726imp32 448 . . . . . . 7 ((((4 · (10↑18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛𝑛 < (4 · (10↑18)))) → 𝑛 ≤ (4 · (10↑18)))
28 ax-bgbltosilvaOLD 42234 . . . . . . 7 ((𝑛 ∈ Even ∧ 4 < 𝑛𝑛 ≤ (4 · (10↑18))) → 𝑛 ∈ GoldbachEven )
2920, 21, 27, 28syl3anc 1477 . . . . . 6 ((((4 · (10↑18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛𝑛 < (4 · (10↑18)))) → 𝑛 ∈ GoldbachEven )
3029ex 449 . . . . 5 (((4 · (10↑18)) ∈ ℕ ∧ 𝑛 ∈ Even ) → ((4 < 𝑛𝑛 < (4 · (10↑18))) → 𝑛 ∈ GoldbachEven ))
3130ralrimiva 3104 . . . 4 ((4 · (10↑18)) ∈ ℕ → ∀𝑛 ∈ Even ((4 < 𝑛𝑛 < (4 · (10↑18))) → 𝑛 ∈ GoldbachEven ))
3219, 31jca 555 . . 3 ((4 · (10↑18)) ∈ ℕ → ((4 · (10↑18)) ≤ (4 · (10↑18)) ∧ ∀𝑛 ∈ Even ((4 < 𝑛𝑛 < (4 · (10↑18))) → 𝑛 ∈ GoldbachEven )))
3310, 17, 32rspcedvd 3456 . 2 ((4 · (10↑18)) ∈ ℕ → ∃𝑚 ∈ ℕ ((4 · (10↑18)) ≤ 𝑚 ∧ ∀𝑛 ∈ Even ((4 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven )))
349, 33ax-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 6814  cr 10147  1c1 10149   · cmul 10153   < clt 10286  cle 10287  cn 11232  4c4 11284  8c8 11288  10c10 11290  0cn0 11504  cdc 11705  cexp 13074   Even ceven 42065   GoldbachEven cgbe 42161
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 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225  ax-bgbltosilvaOLD 42234
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 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-2 11291  df-3 11292  df-4 11293  df-5 11294  df-6 11295  df-7 11296  df-8 11297  df-9 11298  df-10OLD 11299  df-n0 11505  df-z 11590  df-dec 11706  df-uz 11900  df-seq 13016  df-exp 13075  df-even 42067
This theorem is referenced by: (None)
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