Theorem bgoldbachltOLD 41707

Description: Obsolete version of bgoldbachlt 41701 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 11187	. . 3 ⊢ 4 ∈ ℕ
2		10nnOLD 11193	. . . 4 ⊢ 10 ∈ ℕ
3		1nn0 11308	. . . . . 6 ⊢ 1 ∈ ℕ0
4		8nn 11191	. . . . . 6 ⊢ 8 ∈ ℕ
5	3, 4	decnncl 11518	. . . . 5 ⊢ ;18 ∈ ℕ
6	5	nnnn0i 11300	. . . 4 ⊢ ;18 ∈ ℕ0
7		nnexpcl 12873	. . . 4 ⊢ ((10 ∈ ℕ ∧ ;18 ∈ ℕ0) → (10↑;18) ∈ ℕ)
8	2, 6, 7	mp2an 708	. . 3 ⊢ (10↑;18) ∈ ℕ
9	1, 8	nnmulcli 11044	. 2 ⊢ (4 · (10↑;18)) ∈ ℕ
10		id 22	. . 3 ⊢ ((4 · (10↑;18)) ∈ ℕ → (4 · (10↑;18)) ∈ ℕ)
11		breq2 4657	. . . . 5 ⊢ (𝑚 = (4 · (10↑;18)) → ((4 · (10↑;18)) ≤ 𝑚 ↔ (4 · (10↑;18)) ≤ (4 · (10↑;18))))
12		breq2 4657	. . . . . . . 8 ⊢ (𝑚 = (4 · (10↑;18)) → (𝑛 < 𝑚 ↔ 𝑛 < (4 · (10↑;18))))
13	12	anbi2d 740	. . . . . . 7 ⊢ (𝑚 = (4 · (10↑;18)) → ((4 < 𝑛 ∧ 𝑛 < 𝑚) ↔ (4 < 𝑛 ∧ 𝑛 < (4 · (10↑;18)))))
14	13	imbi1d 331	. . . . . 6 ⊢ (𝑚 = (4 · (10↑;18)) → (((4 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven ) ↔ ((4 < 𝑛 ∧ 𝑛 < (4 · (10↑;18))) → 𝑛 ∈ GoldbachEven )))
15	14	ralbidv 2986	. . . . 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 747	. . . 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 482	. . 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 11027	. . . . 5 ⊢ ((4 · (10↑;18)) ∈ ℕ → (4 · (10↑;18)) ∈ ℝ)
19	18	leidd 10594	. . . 4 ⊢ ((4 · (10↑;18)) ∈ ℕ → (4 · (10↑;18)) ≤ (4 · (10↑;18)))
20		simplr 792	. . . . . . 7 ⊢ ((((4 · (10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛 ∧ 𝑛 < (4 · (10↑;18)))) → 𝑛 ∈ Even )
21		simprl 794	. . . . . . 7 ⊢ ((((4 · (10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛 ∧ 𝑛 < (4 · (10↑;18)))) → 4 < 𝑛)
22		evenz 41543	. . . . . . . . . . 11 ⊢ (𝑛 ∈ Even → 𝑛 ∈ ℤ)
23	22	zred 11482	. . . . . . . . . 10 ⊢ (𝑛 ∈ Even → 𝑛 ∈ ℝ)
24		ltle 10126	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℝ ∧ (4 · (10↑;18)) ∈ ℝ) → (𝑛 < (4 · (10↑;18)) → 𝑛 ≤ (4 · (10↑;18))))
25	23, 18, 24	syl2anr 495	. . . . . . . . 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 449	. . . . . . 7 ⊢ ((((4 · (10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛 ∧ 𝑛 < (4 · (10↑;18)))) → 𝑛 ≤ (4 · (10↑;18)))
28		ax-bgbltosilvaOLD 41706	. . . . . . 7 ⊢ ((𝑛 ∈ Even ∧ 4 < 𝑛 ∧ 𝑛 ≤ (4 · (10↑;18))) → 𝑛 ∈ GoldbachEven )
29	20, 21, 27, 28	syl3anc 1326	. . . . . 6 ⊢ ((((4 · (10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛 ∧ 𝑛 < (4 · (10↑;18)))) → 𝑛 ∈ GoldbachEven )
30	29	ex 450	. . . . 5 ⊢ (((4 · (10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) → ((4 < 𝑛 ∧ 𝑛 < (4 · (10↑;18))) → 𝑛 ∈ GoldbachEven ))
31	30	ralrimiva 2966	. . . 4 ⊢ ((4 · (10↑;18)) ∈ ℕ → ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < (4 · (10↑;18))) → 𝑛 ∈ GoldbachEven ))
32	19, 31	jca 554	. . 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 3317	. 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 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   class class class wbr 4653  (class class class)co 6650  ℝcr 9935  1c1 9937   · cmul 9941   < clt 10074   ≤ cle 10075  ℕcn 11020  4c4 11072  8c8 11076  10c10 11078  ℕ₀cn0 11292  ;cdc 11493  ↑cexp 12860   Even ceven 41537   GoldbachEven cgbe 41633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-bgbltosilvaOLD 41706

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-10OLD 11087  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-seq 12802  df-exp 12861  df-even 41539

This theorem is referenced by: (None)