2mulprm - Metamath Proof Explorer

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Theorem 2mulprm 16696

Description: A multiple of two is prime iff the multiplier is one. (Contributed by AV, 8-Jun-2023.)

**Assertion**
Ref	Expression
2mulprm	⊢ (𝐴 ∈ ℤ → ((2 · 𝐴) ∈ ℙ ↔ 𝐴 = 1))

**Proof of Theorem 2mulprm**
Step	Hyp	Ref	Expression
1		zre 12616	. . . . . 6 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
2		0red 11269	. . . . . 6 ⊢ (𝐴 ∈ ℤ → 0 ∈ ℝ)
3	1, 2	leloed 11409	. . . . 5 ⊢ (𝐴 ∈ ℤ → (𝐴 ≤ 0 ↔ (𝐴 < 0 ∨ 𝐴 = 0)))
4		prmnn 16677	. . . . . . . . . 10 ⊢ ((2 · 𝐴) ∈ ℙ → (2 · 𝐴) ∈ ℕ)
5		nnnn0 12533	. . . . . . . . . 10 ⊢ ((2 · 𝐴) ∈ ℕ → (2 · 𝐴) ∈ ℕ₀)
6		nn0ge0 12551	. . . . . . . . . 10 ⊢ ((2 · 𝐴) ∈ ℕ₀ → 0 ≤ (2 · 𝐴))
7		2pos 12369	. . . . . . . . . . . . . . . . . 18 ⊢ 0 < 2
8	7	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℤ → 0 < 2)
9	8	anim1i 613	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → (0 < 2 ∧ 𝐴 < 0))
10	9	olcd 872	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → ((2 < 0 ∧ 0 < 𝐴) ∨ (0 < 2 ∧ 𝐴 < 0)))
11		2re 12340	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ
12	11	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → 2 ∈ ℝ)
13	1	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → 𝐴 ∈ ℝ)
14	12, 13	mul2lt0bi 13136	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → ((2 · 𝐴) < 0 ↔ ((2 < 0 ∧ 0 < 𝐴) ∨ (0 < 2 ∧ 𝐴 < 0))))
15	10, 14	mpbird 256	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → (2 · 𝐴) < 0)
16	12, 13	remulcld 11296	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → (2 · 𝐴) ∈ ℝ)
17		0red 11269	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → 0 ∈ ℝ)
18	16, 17	ltnled 11413	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → ((2 · 𝐴) < 0 ↔ ¬ 0 ≤ (2 · 𝐴)))
19	15, 18	mpbid 231	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → ¬ 0 ≤ (2 · 𝐴))
20	19	ex 411	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℤ → (𝐴 < 0 → ¬ 0 ≤ (2 · 𝐴)))
21	20	con2d 134	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ → (0 ≤ (2 · 𝐴) → ¬ 𝐴 < 0))
22	21	com12 32	. . . . . . . . . 10 ⊢ (0 ≤ (2 · 𝐴) → (𝐴 ∈ ℤ → ¬ 𝐴 < 0))
23	4, 5, 6, 22	4syl 19	. . . . . . . . 9 ⊢ ((2 · 𝐴) ∈ ℙ → (𝐴 ∈ ℤ → ¬ 𝐴 < 0))
24	23	com12 32	. . . . . . . 8 ⊢ (𝐴 ∈ ℤ → ((2 · 𝐴) ∈ ℙ → ¬ 𝐴 < 0))
25	24	con2d 134	. . . . . . 7 ⊢ (𝐴 ∈ ℤ → (𝐴 < 0 → ¬ (2 · 𝐴) ∈ ℙ))
26	25	a1dd 50	. . . . . 6 ⊢ (𝐴 ∈ ℤ → (𝐴 < 0 → (¬ 𝐴 = 1 → ¬ (2 · 𝐴) ∈ ℙ)))
27		oveq2 7434	. . . . . . . . 9 ⊢ (𝐴 = 0 → (2 · 𝐴) = (2 · 0))
28		2t0e0 12435	. . . . . . . . 9 ⊢ (2 · 0) = 0
29	27, 28	eqtrdi 2782	. . . . . . . 8 ⊢ (𝐴 = 0 → (2 · 𝐴) = 0)
30		0nprm 16681	. . . . . . . . 9 ⊢ ¬ 0 ∈ ℙ
31	30	a1i 11	. . . . . . . 8 ⊢ (𝐴 = 0 → ¬ 0 ∈ ℙ)
32	29, 31	eqneltrd 2846	. . . . . . 7 ⊢ (𝐴 = 0 → ¬ (2 · 𝐴) ∈ ℙ)
33	32	a1i13 27	. . . . . 6 ⊢ (𝐴 ∈ ℤ → (𝐴 = 0 → (¬ 𝐴 = 1 → ¬ (2 · 𝐴) ∈ ℙ)))
34	26, 33	jaod 857	. . . . 5 ⊢ (𝐴 ∈ ℤ → ((𝐴 < 0 ∨ 𝐴 = 0) → (¬ 𝐴 = 1 → ¬ (2 · 𝐴) ∈ ℙ)))
35	3, 34	sylbid 239	. . . 4 ⊢ (𝐴 ∈ ℤ → (𝐴 ≤ 0 → (¬ 𝐴 = 1 → ¬ (2 · 𝐴) ∈ ℙ)))
36		2z 12648	. . . . . . 7 ⊢ 2 ∈ ℤ
37		uzid 12891	. . . . . . 7 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ_≥‘2))
38	36, 37	ax-mp 5	. . . . . 6 ⊢ 2 ∈ (ℤ_≥‘2)
39	36	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 1 ≤ 𝐴 ∧ ¬ 𝐴 = 1) → 2 ∈ ℤ)
40		simp1 1133	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 1 ≤ 𝐴 ∧ ¬ 𝐴 = 1) → 𝐴 ∈ ℤ)
41		df-ne 2931	. . . . . . . . 9 ⊢ (𝐴 ≠ 1 ↔ ¬ 𝐴 = 1)
42		1red 11267	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℤ → 1 ∈ ℝ)
43	42, 1	ltlend 11411	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ → (1 < 𝐴 ↔ (1 ≤ 𝐴 ∧ 𝐴 ≠ 1)))
44		1zzd 12647	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℤ → 1 ∈ ℤ)
45		zltp1le 12666	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (1 < 𝐴 ↔ (1 + 1) ≤ 𝐴))
46	44, 45	mpancom 686	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℤ → (1 < 𝐴 ↔ (1 + 1) ≤ 𝐴))
47	46	biimpd 228	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℤ → (1 < 𝐴 → (1 + 1) ≤ 𝐴))
48		df-2 12329	. . . . . . . . . . . . 13 ⊢ 2 = (1 + 1)
49	48	breq1i 5162	. . . . . . . . . . . 12 ⊢ (2 ≤ 𝐴 ↔ (1 + 1) ≤ 𝐴)
50	47, 49	imbitrrdi 251	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ → (1 < 𝐴 → 2 ≤ 𝐴))
51	43, 50	sylbird 259	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℤ → ((1 ≤ 𝐴 ∧ 𝐴 ≠ 1) → 2 ≤ 𝐴))
52	51	expdimp 451	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 1 ≤ 𝐴) → (𝐴 ≠ 1 → 2 ≤ 𝐴))
53	41, 52	biimtrrid 242	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 1 ≤ 𝐴) → (¬ 𝐴 = 1 → 2 ≤ 𝐴))
54	53	3impia 1114	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 1 ≤ 𝐴 ∧ ¬ 𝐴 = 1) → 2 ≤ 𝐴)
55		eluz2 12882	. . . . . . 7 ⊢ (𝐴 ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 2 ≤ 𝐴))
56	39, 40, 54, 55	syl3anbrc 1340	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 1 ≤ 𝐴 ∧ ¬ 𝐴 = 1) → 𝐴 ∈ (ℤ_≥‘2))
57		nprm 16691	. . . . . 6 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 𝐴 ∈ (ℤ_≥‘2)) → ¬ (2 · 𝐴) ∈ ℙ)
58	38, 56, 57	sylancr 585	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 1 ≤ 𝐴 ∧ ¬ 𝐴 = 1) → ¬ (2 · 𝐴) ∈ ℙ)
59	58	3exp 1116	. . . 4 ⊢ (𝐴 ∈ ℤ → (1 ≤ 𝐴 → (¬ 𝐴 = 1 → ¬ (2 · 𝐴) ∈ ℙ)))
60		zle0orge1 12629	. . . 4 ⊢ (𝐴 ∈ ℤ → (𝐴 ≤ 0 ∨ 1 ≤ 𝐴))
61	35, 59, 60	mpjaod 858	. . 3 ⊢ (𝐴 ∈ ℤ → (¬ 𝐴 = 1 → ¬ (2 · 𝐴) ∈ ℙ))
62	61	con4d 115	. 2 ⊢ (𝐴 ∈ ℤ → ((2 · 𝐴) ∈ ℙ → 𝐴 = 1))
63		oveq2 7434	. . . 4 ⊢ (𝐴 = 1 → (2 · 𝐴) = (2 · 1))
64		2t1e2 12429	. . . 4 ⊢ (2 · 1) = 2
65	63, 64	eqtrdi 2782	. . 3 ⊢ (𝐴 = 1 → (2 · 𝐴) = 2)
66		2prm 16695	. . 3 ⊢ 2 ∈ ℙ
67	65, 66	eqeltrdi 2834	. 2 ⊢ (𝐴 = 1 → (2 · 𝐴) ∈ ℙ)
68	62, 67	impbid1 224	1 ⊢ (𝐴 ∈ ℤ → ((2 · 𝐴) ∈ ℙ ↔ 𝐴 = 1))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 class class class wbr 5155 ‘cfv 6556 (class class class)co 7426 ℝcr 11159 0cc0 11160 1c1 11161 + caddc 11163 · cmul 11165 < clt 11300 ≤ cle 11301 ℕcn 12266 2c2 12321 ℕ₀cn0 12526 ℤcz 12612 ℤ_≥cuz 12876 ℙcprime 16674

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-cnex 11216 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237 ax-pre-sup 11238

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8005 df-2nd 8006 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-1o 8498 df-2o 8499 df-er 8736 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-sup 9487 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 df-div 11924 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12613 df-uz 12877 df-rp 13031 df-fz 13541 df-seq 14024 df-exp 14084 df-cj 15106 df-re 15107 df-im 15108 df-sqrt 15242 df-abs 15243 df-dvds 16259 df-prm 16675

This theorem is referenced by: 2sqreultlem 27479 2sqreunnltlem 27482

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