Theorem bgoldbachlt 49114

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 49111. (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 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven ))

Distinct variable group:   𝑚,𝑛

Proof of Theorem bgoldbachlt

Step	Hyp	Ref	Expression
1		4nn 12450	. . 3 ⊢ 4 ∈ ℕ
2		10nn 12858	. . . 4 ⊢ ;10 ∈ ℕ
3		1nn0 12646	. . . . 5 ⊢ 1 ∈ ℕ0
4		8nn0 12653	. . . . 5 ⊢ 8 ∈ ℕ0
5	3, 4	deccl 12853	. . . 4 ⊢ ;18 ∈ ℕ0
6		nnexpcl 14241	. . . 4 ⊢ ((;10 ∈ ℕ ∧ ;18 ∈ ℕ0) → (;10↑;18) ∈ ℕ)
7	2, 5, 6	mp2an 705	. . 3 ⊢ (;10↑;18) ∈ ℕ
8	1, 7	nnmulcli 12384	. 2 ⊢ (4 · (;10↑;18)) ∈ ℕ
9		id 23	. . 3 ⊢ ((4 · (;10↑;18)) ∈ ℕ → (4 · (;10↑;18)) ∈ ℕ)
10		breq2 5110	. . . . 5 ⊢ (𝑚 = (4 · (;10↑;18)) → ((4 · (;10↑;18)) ≤ 𝑚 ↔ (4 · (;10↑;18)) ≤ (4 · (;10↑;18))))
11		breq2 5110	. . . . . . . 8 ⊢ (𝑚 = (4 · (;10↑;18)) → (𝑛 < 𝑚 ↔ 𝑛 < (4 · (;10↑;18))))
12	11	anbi2d 642	. . . . . . 7 ⊢ (𝑚 = (4 · (;10↑;18)) → ((4 < 𝑛 ∧ 𝑛 < 𝑚) ↔ (4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18)))))
13	12	imbi1d 344	. . . . . 6 ⊢ (𝑚 = (4 · (;10↑;18)) → (((4 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven ) ↔ ((4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18))) → 𝑛 ∈ GoldbachEven )))
14	13	ralbidv 3189	. . . . 5 ⊢ (𝑚 = (4 · (;10↑;18)) → (∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven ) ↔ ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18))) → 𝑛 ∈ GoldbachEven )))
15	10, 14	anbi12d 644	. . . 4 ⊢ (𝑚 = (4 · (;10↑;18)) → (((4 · (;10↑;18)) ≤ 𝑚 ∧ ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven )) ↔ ((4 · (;10↑;18)) ≤ (4 · (;10↑;18)) ∧ ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18))) → 𝑛 ∈ GoldbachEven ))))
16	15	adantl 487	. . 3 ⊢ (((4 · (;10↑;18)) ∈ ℕ ∧ 𝑚 = (4 · (;10↑;18))) → (((4 · (;10↑;18)) ≤ 𝑚 ∧ ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven )) ↔ ((4 · (;10↑;18)) ≤ (4 · (;10↑;18)) ∧ ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18))) → 𝑛 ∈ GoldbachEven ))))
17		nnre 12366	. . . . 5 ⊢ ((4 · (;10↑;18)) ∈ ℕ → (4 · (;10↑;18)) ∈ ℝ)
18	17	leidd 11906	. . . 4 ⊢ ((4 · (;10↑;18)) ∈ ℕ → (4 · (;10↑;18)) ≤ (4 · (;10↑;18)))
19		simplr 781	. . . . . . 7 ⊢ ((((4 · (;10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18)))) → 𝑛 ∈ Even )
20		simprl 783	. . . . . . 7 ⊢ ((((4 · (;10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18)))) → 4 < 𝑛)
21		evenz 48931	. . . . . . . . . . 11 ⊢ (𝑛 ∈ Even → 𝑛 ∈ ℤ)
22	21	zred 12827	. . . . . . . . . 10 ⊢ (𝑛 ∈ Even → 𝑛 ∈ ℝ)
23		ltle 11422	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℝ ∧ (4 · (;10↑;18)) ∈ ℝ) → (𝑛 < (4 · (;10↑;18)) → 𝑛 ≤ (4 · (;10↑;18))))
24	22, 17, 23	syl2anr 609	. . . . . . . . 9 ⊢ (((4 · (;10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) → (𝑛 < (4 · (;10↑;18)) → 𝑛 ≤ (4 · (;10↑;18))))
25	24	a1d 26	. . . . . . . 8 ⊢ (((4 · (;10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) → (4 < 𝑛 → (𝑛 < (4 · (;10↑;18)) → 𝑛 ≤ (4 · (;10↑;18)))))
26	25	imp32 424	. . . . . . 7 ⊢ ((((4 · (;10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18)))) → 𝑛 ≤ (4 · (;10↑;18)))
27		ax-bgbltosilva 49111	. . . . . . 7 ⊢ ((𝑛 ∈ Even ∧ 4 < 𝑛 ∧ 𝑛 ≤ (4 · (;10↑;18))) → 𝑛 ∈ GoldbachEven )
28	19, 20, 26, 27	syl3anc 1399	. . . . . 6 ⊢ ((((4 · (;10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18)))) → 𝑛 ∈ GoldbachEven )
29	28	ex 418	. . . . 5 ⊢ (((4 · (;10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) → ((4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18))) → 𝑛 ∈ GoldbachEven ))
30	29	ralrimiva 3158	. . . 4 ⊢ ((4 · (;10↑;18)) ∈ ℕ → ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18))) → 𝑛 ∈ GoldbachEven ))
31	18, 30	jca 521	. . 3 ⊢ ((4 · (;10↑;18)) ∈ ℕ → ((4 · (;10↑;18)) ≤ (4 · (;10↑;18)) ∧ ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18))) → 𝑛 ∈ GoldbachEven )))
32	9, 16, 31	rspcedvd 3582	. 2 ⊢ ((4 · (;10↑;18)) ∈ ℕ → ∃𝑚 ∈ ℕ ((4 · (;10↑;18)) ≤ 𝑚 ∧ ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven )))
33	8, 32	ax-mp 5	1 ⊢ ∃𝑚 ∈ ℕ ((4 · (;10↑;18)) ≤ 𝑚 ∧ ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   class class class wbr 5106  (class class class)co 7440  ℝcr 11223  0cc0 11224  1c1 11225   · cmul 11229   < clt 11367   ≤ cle 11368  ℕcn 12359  4c4 12423  8c8 12427  ℕ₀cn0 12630  ;cdc 12838  ↑cexp 14228   Even ceven 48925   GoldbachEven cgbe 49046

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301  ax-bgbltosilva 49111

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-er 8742  df-en 8999  df-dom 9000  df-sdom 9001  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-nn 12360  df-2 12429  df-3 12430  df-4 12431  df-5 12432  df-6 12433  df-7 12434  df-8 12435  df-9 12436  df-n0 12631  df-z 12718  df-dec 12839  df-uz 12990  df-seq 14169  df-exp 14229  df-even 48927

This theorem is used by: (None)