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 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven ))

Distinct variable group:   𝑚,𝑛

Proof of Theorem bgoldbachltOLD

Step	Hyp	Ref	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
5	3, 4	decnncl 11730	. . . . 5 ⊢ ;18 ∈ ℕ
6	5	nnnn0i 11512	. . . 4 ⊢ ;18 ∈ ℕ0
7		nnexpcl 13087	. . . 4 ⊢ ((10 ∈ ℕ ∧ ;18 ∈ ℕ0) → (10↑;18) ∈ ℕ)
8	2, 6, 7	mp2an 710	. . 3 ⊢ (10↑;18) ∈ ℕ
9	1, 8	nnmulcli 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 < 𝑚 ↔ 𝑛 < (4 · (10↑;18))))
13	12	anbi2d 742	. . . . . . 7 ⊢ (𝑚 = (4 · (10↑;18)) → ((4 < 𝑛 ∧ 𝑛 < 𝑚) ↔ (4 < 𝑛 ∧ 𝑛 < (4 · (10↑;18)))))
14	13	imbi1d 330	. . . . . 6 ⊢ (𝑚 = (4 · (10↑;18)) → (((4 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven ) ↔ ((4 < 𝑛 ∧ 𝑛 < (4 · (10↑;18))) → 𝑛 ∈ GoldbachEven )))
15	14	ralbidv 3124	. . . . 5 ⊢ (𝑚 = (4 · (10↑;18)) → (∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven ) ↔ ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < (4 · (10↑;18))) → 𝑛 ∈ GoldbachEven )))
16	11, 15	anbi12d 749	. . . 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 ))))
17	16	adantl 473	. . 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 ))))
18		nnre 11239	. . . . 5 ⊢ ((4 · (10↑;18)) ∈ ℕ → (4 · (10↑;18)) ∈ ℝ)
19	18	leidd 10806	. . . 4 ⊢ ((4 · (10↑;18)) ∈ ℕ → (4 · (10↑;18)) ≤ (4 · (10↑;18)))
20		simplr 809	. . . . . . 7 ⊢ ((((4 · (10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛 ∧ 𝑛 < (4 · (10↑;18)))) → 𝑛 ∈ Even )
21		simprl 811	. . . . . . 7 ⊢ ((((4 · (10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛 ∧ 𝑛 < (4 · (10↑;18)))) → 4 < 𝑛)
22		evenz 42071	. . . . . . . . . . 11 ⊢ (𝑛 ∈ Even → 𝑛 ∈ ℤ)
23	22	zred 11694	. . . . . . . . . 10 ⊢ (𝑛 ∈ Even → 𝑛 ∈ ℝ)
24		ltle 10338	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℝ ∧ (4 · (10↑;18)) ∈ ℝ) → (𝑛 < (4 · (10↑;18)) → 𝑛 ≤ (4 · (10↑;18))))
25	23, 18, 24	syl2anr 496	. . . . . . . . 9 ⊢ (((4 · (10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) → (𝑛 < (4 · (10↑;18)) → 𝑛 ≤ (4 · (10↑;18))))
26	25	a1d 25	. . . . . . . 8 ⊢ (((4 · (10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) → (4 < 𝑛 → (𝑛 < (4 · (10↑;18)) → 𝑛 ≤ (4 · (10↑;18)))))
27	26	imp32 448	. . . . . . 7 ⊢ ((((4 · (10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛 ∧ 𝑛 < (4 · (10↑;18)))) → 𝑛 ≤ (4 · (10↑;18)))
28		ax-bgbltosilvaOLD 42234	. . . . . . 7 ⊢ ((𝑛 ∈ Even ∧ 4 < 𝑛 ∧ 𝑛 ≤ (4 · (10↑;18))) → 𝑛 ∈ GoldbachEven )
29	20, 21, 27, 28	syl3anc 1477	. . . . . 6 ⊢ ((((4 · (10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛 ∧ 𝑛 < (4 · (10↑;18)))) → 𝑛 ∈ GoldbachEven )
30	29	ex 449	. . . . 5 ⊢ (((4 · (10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) → ((4 < 𝑛 ∧ 𝑛 < (4 · (10↑;18))) → 𝑛 ∈ GoldbachEven ))
31	30	ralrimiva 3104	. . . 4 ⊢ ((4 · (10↑;18)) ∈ ℕ → ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < (4 · (10↑;18))) → 𝑛 ∈ GoldbachEven ))
32	19, 31	jca 555	. . 3 ⊢ ((4 · (10↑;18)) ∈ ℕ → ((4 · (10↑;18)) ≤ (4 · (10↑;18)) ∧ ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < (4 · (10↑;18))) → 𝑛 ∈ GoldbachEven )))
33	10, 17, 32	rspcedvd 3456	. 2 ⊢ ((4 · (10↑;18)) ∈ ℕ → ∃𝑚 ∈ ℕ ((4 · (10↑;18)) ≤ 𝑚 ∧ ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven )))
34	9, 33	ax-mp 5	1 ⊢ ∃𝑚 ∈ ℕ ((4 · (10↑;18)) ≤ 𝑚 ∧ ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven ))

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  ℝcr 10147  1c1 10149   · cmul 10153   < clt 10286   ≤ cle 10287  ℕcn 11232  4c4 11284  8c8 11288  10c10 11290  ℕ₀cn0 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)