Theorem bgoldbachlt 47738

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 47735. (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 12347	. . 3 ⊢ 4 ∈ ℕ
2		10nn 12747	. . . 4 ⊢ ;10 ∈ ℕ
3		1nn0 12540	. . . . 5 ⊢ 1 ∈ ℕ0
4		8nn0 12547	. . . . 5 ⊢ 8 ∈ ℕ0
5	3, 4	deccl 12746	. . . 4 ⊢ ;18 ∈ ℕ0
6		nnexpcl 14112	. . . 4 ⊢ ((;10 ∈ ℕ ∧ ;18 ∈ ℕ0) → (;10↑;18) ∈ ℕ)
7	2, 5, 6	mp2an 692	. . 3 ⊢ (;10↑;18) ∈ ℕ
8	1, 7	nnmulcli 12289	. 2 ⊢ (4 · (;10↑;18)) ∈ ℕ
9		id 22	. . 3 ⊢ ((4 · (;10↑;18)) ∈ ℕ → (4 · (;10↑;18)) ∈ ℕ)
10		breq2 5152	. . . . 5 ⊢ (𝑚 = (4 · (;10↑;18)) → ((4 · (;10↑;18)) ≤ 𝑚 ↔ (4 · (;10↑;18)) ≤ (4 · (;10↑;18))))
11		breq2 5152	. . . . . . . 8 ⊢ (𝑚 = (4 · (;10↑;18)) → (𝑛 < 𝑚 ↔ 𝑛 < (4 · (;10↑;18))))
12	11	anbi2d 630	. . . . . . 7 ⊢ (𝑚 = (4 · (;10↑;18)) → ((4 < 𝑛 ∧ 𝑛 < 𝑚) ↔ (4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18)))))
13	12	imbi1d 341	. . . . . 6 ⊢ (𝑚 = (4 · (;10↑;18)) → (((4 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven ) ↔ ((4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18))) → 𝑛 ∈ GoldbachEven )))
14	13	ralbidv 3176	. . . . 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 632	. . . 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 481	. . 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 12271	. . . . 5 ⊢ ((4 · (;10↑;18)) ∈ ℕ → (4 · (;10↑;18)) ∈ ℝ)
18	17	leidd 11827	. . . 4 ⊢ ((4 · (;10↑;18)) ∈ ℕ → (4 · (;10↑;18)) ≤ (4 · (;10↑;18)))
19		simplr 769	. . . . . . 7 ⊢ ((((4 · (;10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18)))) → 𝑛 ∈ Even )
20		simprl 771	. . . . . . 7 ⊢ ((((4 · (;10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18)))) → 4 < 𝑛)
21		evenz 47555	. . . . . . . . . . 11 ⊢ (𝑛 ∈ Even → 𝑛 ∈ ℤ)
22	21	zred 12720	. . . . . . . . . 10 ⊢ (𝑛 ∈ Even → 𝑛 ∈ ℝ)
23		ltle 11347	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℝ ∧ (4 · (;10↑;18)) ∈ ℝ) → (𝑛 < (4 · (;10↑;18)) → 𝑛 ≤ (4 · (;10↑;18))))
24	22, 17, 23	syl2anr 597	. . . . . . . . 9 ⊢ (((4 · (;10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) → (𝑛 < (4 · (;10↑;18)) → 𝑛 ≤ (4 · (;10↑;18))))
25	24	a1d 25	. . . . . . . 8 ⊢ (((4 · (;10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) → (4 < 𝑛 → (𝑛 < (4 · (;10↑;18)) → 𝑛 ≤ (4 · (;10↑;18)))))
26	25	imp32 418	. . . . . . 7 ⊢ ((((4 · (;10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18)))) → 𝑛 ≤ (4 · (;10↑;18)))
27		ax-bgbltosilva 47735	. . . . . . 7 ⊢ ((𝑛 ∈ Even ∧ 4 < 𝑛 ∧ 𝑛 ≤ (4 · (;10↑;18))) → 𝑛 ∈ GoldbachEven )
28	19, 20, 26, 27	syl3anc 1370	. . . . . 6 ⊢ ((((4 · (;10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18)))) → 𝑛 ∈ GoldbachEven )
29	28	ex 412	. . . . 5 ⊢ (((4 · (;10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) → ((4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18))) → 𝑛 ∈ GoldbachEven ))
30	29	ralrimiva 3144	. . . 4 ⊢ ((4 · (;10↑;18)) ∈ ℕ → ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18))) → 𝑛 ∈ GoldbachEven ))
31	18, 30	jca 511	. . 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 3624	. 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 ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068   class class class wbr 5148  (class class class)co 7431  ℝcr 11152  0cc0 11153  1c1 11154   · cmul 11158   < clt 11293   ≤ cle 11294  ℕcn 12264  4c4 12321  8c8 12325  ℕ₀cn0 12524  ;cdc 12731  ↑cexp 14099   Even ceven 47549   GoldbachEven cgbe 47670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-bgbltosilva 47735

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-seq 14040  df-exp 14100  df-even 47551

This theorem is referenced by: (None)