2mulprm - Metamath Proof Explorer

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Theorem 2mulprm 16620

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 12492	. . . . . 6 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
2		0red 11135	. . . . . 6 ⊢ (𝐴 ∈ ℤ → 0 ∈ ℝ)
3	1, 2	leloed 11276	. . . . 5 ⊢ (𝐴 ∈ ℤ → (𝐴 ≤ 0 ↔ (𝐴 < 0 ∨ 𝐴 = 0)))
4		prmnn 16601	. . . . . . . . . 10 ⊢ ((2 · 𝐴) ∈ ℙ → (2 · 𝐴) ∈ ℕ)
5		nnnn0 12408	. . . . . . . . . 10 ⊢ ((2 · 𝐴) ∈ ℕ → (2 · 𝐴) ∈ ℕ₀)
6		nn0ge0 12426	. . . . . . . . . 10 ⊢ ((2 · 𝐴) ∈ ℕ₀ → 0 ≤ (2 · 𝐴))
7		2pos 12248	. . . . . . . . . . . . . . . . . 18 ⊢ 0 < 2
8	7	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℤ → 0 < 2)
9	8	anim1i 615	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → (0 < 2 ∧ 𝐴 < 0))
10	9	olcd 874	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → ((2 < 0 ∧ 0 < 𝐴) ∨ (0 < 2 ∧ 𝐴 < 0)))
11		2re 12219	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ
12	11	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → 2 ∈ ℝ)
13	1	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → 𝐴 ∈ ℝ)
14	12, 13	mul2lt0bi 13013	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → ((2 · 𝐴) < 0 ↔ ((2 < 0 ∧ 0 < 𝐴) ∨ (0 < 2 ∧ 𝐴 < 0))))
15	10, 14	mpbird 257	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → (2 · 𝐴) < 0)
16	12, 13	remulcld 11162	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → (2 · 𝐴) ∈ ℝ)
17		0red 11135	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → 0 ∈ ℝ)
18	16, 17	ltnled 11280	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → ((2 · 𝐴) < 0 ↔ ¬ 0 ≤ (2 · 𝐴)))
19	15, 18	mpbid 232	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → ¬ 0 ≤ (2 · 𝐴))
20	19	ex 412	. . . . . . . . . . . 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 7366	. . . . . . . . 9 ⊢ (𝐴 = 0 → (2 · 𝐴) = (2 · 0))
28		2t0e0 12309	. . . . . . . . 9 ⊢ (2 · 0) = 0
29	27, 28	eqtrdi 2787	. . . . . . . 8 ⊢ (𝐴 = 0 → (2 · 𝐴) = 0)
30		0nprm 16605	. . . . . . . . 9 ⊢ ¬ 0 ∈ ℙ
31	30	a1i 11	. . . . . . . 8 ⊢ (𝐴 = 0 → ¬ 0 ∈ ℙ)
32	29, 31	eqneltrd 2856	. . . . . . 7 ⊢ (𝐴 = 0 → ¬ (2 · 𝐴) ∈ ℙ)
33	32	a1i13 27	. . . . . 6 ⊢ (𝐴 ∈ ℤ → (𝐴 = 0 → (¬ 𝐴 = 1 → ¬ (2 · 𝐴) ∈ ℙ)))
34	26, 33	jaod 859	. . . . 5 ⊢ (𝐴 ∈ ℤ → ((𝐴 < 0 ∨ 𝐴 = 0) → (¬ 𝐴 = 1 → ¬ (2 · 𝐴) ∈ ℙ)))
35	3, 34	sylbid 240	. . . 4 ⊢ (𝐴 ∈ ℤ → (𝐴 ≤ 0 → (¬ 𝐴 = 1 → ¬ (2 · 𝐴) ∈ ℙ)))
36		2z 12523	. . . . . . 7 ⊢ 2 ∈ ℤ
37		uzid 12766	. . . . . . 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 1136	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 1 ≤ 𝐴 ∧ ¬ 𝐴 = 1) → 𝐴 ∈ ℤ)
41		df-ne 2933	. . . . . . . . 9 ⊢ (𝐴 ≠ 1 ↔ ¬ 𝐴 = 1)
42		1red 11133	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℤ → 1 ∈ ℝ)
43	42, 1	ltlend 11278	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ → (1 < 𝐴 ↔ (1 ≤ 𝐴 ∧ 𝐴 ≠ 1)))
44		1zzd 12522	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℤ → 1 ∈ ℤ)
45		zltp1le 12541	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (1 < 𝐴 ↔ (1 + 1) ≤ 𝐴))
46	44, 45	mpancom 688	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℤ → (1 < 𝐴 ↔ (1 + 1) ≤ 𝐴))
47	46	biimpd 229	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℤ → (1 < 𝐴 → (1 + 1) ≤ 𝐴))
48		df-2 12208	. . . . . . . . . . . . 13 ⊢ 2 = (1 + 1)
49	48	breq1i 5105	. . . . . . . . . . . 12 ⊢ (2 ≤ 𝐴 ↔ (1 + 1) ≤ 𝐴)
50	47, 49	imbitrrdi 252	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ → (1 < 𝐴 → 2 ≤ 𝐴))
51	43, 50	sylbird 260	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℤ → ((1 ≤ 𝐴 ∧ 𝐴 ≠ 1) → 2 ≤ 𝐴))
52	51	expdimp 452	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 1 ≤ 𝐴) → (𝐴 ≠ 1 → 2 ≤ 𝐴))
53	41, 52	biimtrrid 243	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 1 ≤ 𝐴) → (¬ 𝐴 = 1 → 2 ≤ 𝐴))
54	53	3impia 1117	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 1 ≤ 𝐴 ∧ ¬ 𝐴 = 1) → 2 ≤ 𝐴)
55		eluz2 12757	. . . . . . 7 ⊢ (𝐴 ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 2 ≤ 𝐴))
56	39, 40, 54, 55	syl3anbrc 1344	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 1 ≤ 𝐴 ∧ ¬ 𝐴 = 1) → 𝐴 ∈ (ℤ_≥‘2))
57		nprm 16615	. . . . . 6 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 𝐴 ∈ (ℤ_≥‘2)) → ¬ (2 · 𝐴) ∈ ℙ)
58	38, 56, 57	sylancr 587	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 1 ≤ 𝐴 ∧ ¬ 𝐴 = 1) → ¬ (2 · 𝐴) ∈ ℙ)
59	58	3exp 1119	. . . 4 ⊢ (𝐴 ∈ ℤ → (1 ≤ 𝐴 → (¬ 𝐴 = 1 → ¬ (2 · 𝐴) ∈ ℙ)))
60		zle0orge1 12505	. . . 4 ⊢ (𝐴 ∈ ℤ → (𝐴 ≤ 0 ∨ 1 ≤ 𝐴))
61	35, 59, 60	mpjaod 860	. . 3 ⊢ (𝐴 ∈ ℤ → (¬ 𝐴 = 1 → ¬ (2 · 𝐴) ∈ ℙ))
62	61	con4d 115	. 2 ⊢ (𝐴 ∈ ℤ → ((2 · 𝐴) ∈ ℙ → 𝐴 = 1))
63		oveq2 7366	. . . 4 ⊢ (𝐴 = 1 → (2 · 𝐴) = (2 · 1))
64		2t1e2 12303	. . . 4 ⊢ (2 · 1) = 2
65	63, 64	eqtrdi 2787	. . 3 ⊢ (𝐴 = 1 → (2 · 𝐴) = 2)
66		2prm 16619	. . 3 ⊢ 2 ∈ ℙ
67	65, 66	eqeltrdi 2844	. 2 ⊢ (𝐴 = 1 → (2 · 𝐴) ∈ ℙ)
68	62, 67	impbid1 225	1 ⊢ (𝐴 ∈ ℤ → ((2 · 𝐴) ∈ ℙ ↔ 𝐴 = 1))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 class class class wbr 5098 ‘cfv 6492 (class class class)co 7358 ℝcr 11025 0cc0 11026 1c1 11027 + caddc 11029 · cmul 11031 < clt 11166 ≤ cle 11167 ℕcn 12145 2c2 12200 ℕ₀cn0 12401 ℤcz 12488 ℤ_≥cuz 12751 ℙcprime 16598

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9345 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-n0 12402 df-z 12489 df-uz 12752 df-rp 12906 df-fz 13424 df-seq 13925 df-exp 13985 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-dvds 16180 df-prm 16599

This theorem is referenced by: 2sqreultlem 27414 2sqreunnltlem 27417

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