Theorem bgoldbachlt 46481

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 46478. (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 12295	. . 3 ⊢ 4 ∈ ℕ
2		10nn 12693	. . . 4 ⊢ ;10 ∈ ℕ
3		1nn0 12488	. . . . 5 ⊢ 1 ∈ ℕ0
4		8nn0 12495	. . . . 5 ⊢ 8 ∈ ℕ0
5	3, 4	deccl 12692	. . . 4 ⊢ ;18 ∈ ℕ0
6		nnexpcl 14040	. . . 4 ⊢ ((;10 ∈ ℕ ∧ ;18 ∈ ℕ0) → (;10↑;18) ∈ ℕ)
7	2, 5, 6	mp2an 691	. . 3 ⊢ (;10↑;18) ∈ ℕ
8	1, 7	nnmulcli 12237	. 2 ⊢ (4 · (;10↑;18)) ∈ ℕ
9		id 22	. . 3 ⊢ ((4 · (;10↑;18)) ∈ ℕ → (4 · (;10↑;18)) ∈ ℕ)
10		breq2 5153	. . . . 5 ⊢ (𝑚 = (4 · (;10↑;18)) → ((4 · (;10↑;18)) ≤ 𝑚 ↔ (4 · (;10↑;18)) ≤ (4 · (;10↑;18))))
11		breq2 5153	. . . . . . . 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 342	. . . . . 6 ⊢ (𝑚 = (4 · (;10↑;18)) → (((4 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven ) ↔ ((4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18))) → 𝑛 ∈ GoldbachEven )))
14	13	ralbidv 3178	. . . . 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 483	. . 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 12219	. . . . 5 ⊢ ((4 · (;10↑;18)) ∈ ℕ → (4 · (;10↑;18)) ∈ ℝ)
18	17	leidd 11780	. . . 4 ⊢ ((4 · (;10↑;18)) ∈ ℕ → (4 · (;10↑;18)) ≤ (4 · (;10↑;18)))
19		simplr 768	. . . . . . 7 ⊢ ((((4 · (;10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18)))) → 𝑛 ∈ Even )
20		simprl 770	. . . . . . 7 ⊢ ((((4 · (;10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18)))) → 4 < 𝑛)
21		evenz 46298	. . . . . . . . . . 11 ⊢ (𝑛 ∈ Even → 𝑛 ∈ ℤ)
22	21	zred 12666	. . . . . . . . . 10 ⊢ (𝑛 ∈ Even → 𝑛 ∈ ℝ)
23		ltle 11302	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℝ ∧ (4 · (;10↑;18)) ∈ ℝ) → (𝑛 < (4 · (;10↑;18)) → 𝑛 ≤ (4 · (;10↑;18))))
24	22, 17, 23	syl2anr 598	. . . . . . . . 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 420	. . . . . . 7 ⊢ ((((4 · (;10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18)))) → 𝑛 ≤ (4 · (;10↑;18)))
27		ax-bgbltosilva 46478	. . . . . . 7 ⊢ ((𝑛 ∈ Even ∧ 4 < 𝑛 ∧ 𝑛 ≤ (4 · (;10↑;18))) → 𝑛 ∈ GoldbachEven )
28	19, 20, 26, 27	syl3anc 1372	. . . . . 6 ⊢ ((((4 · (;10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18)))) → 𝑛 ∈ GoldbachEven )
29	28	ex 414	. . . . 5 ⊢ (((4 · (;10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) → ((4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18))) → 𝑛 ∈ GoldbachEven ))
30	29	ralrimiva 3147	. . . 4 ⊢ ((4 · (;10↑;18)) ∈ ℕ → ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18))) → 𝑛 ∈ GoldbachEven ))
31	18, 30	jca 513	. . 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 3615	. 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 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071   class class class wbr 5149  (class class class)co 7409  ℝcr 11109  0cc0 11110  1c1 11111   · cmul 11115   < clt 11248   ≤ cle 11249  ℕcn 12212  4c4 12269  8c8 12273  ℕ₀cn0 12472  ;cdc 12677  ↑cexp 14027   Even ceven 46292   GoldbachEven cgbe 46413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-bgbltosilva 46478

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-dec 12678  df-uz 12823  df-seq 13967  df-exp 14028  df-even 46294

This theorem is referenced by: (None)