2mulprm - Metamath Proof Explorer

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Theorem 2mulprm 16726

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 12614	. . . . . 6 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
2		0red 11261	. . . . . 6 ⊢ (𝐴 ∈ ℤ → 0 ∈ ℝ)
3	1, 2	leloed 11401	. . . . 5 ⊢ (𝐴 ∈ ℤ → (𝐴 ≤ 0 ↔ (𝐴 < 0 ∨ 𝐴 = 0)))
4		prmnn 16707	. . . . . . . . . 10 ⊢ ((2 · 𝐴) ∈ ℙ → (2 · 𝐴) ∈ ℕ)
5		nnnn0 12530	. . . . . . . . . 10 ⊢ ((2 · 𝐴) ∈ ℕ → (2 · 𝐴) ∈ ℕ₀)
6		nn0ge0 12548	. . . . . . . . . 10 ⊢ ((2 · 𝐴) ∈ ℕ₀ → 0 ≤ (2 · 𝐴))
7		2pos 12366	. . . . . . . . . . . . . . . . . 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 12337	. . . . . . . . . . . . . . . . 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 13138	. . . . . . . . . . . . . . 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 11288	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → (2 · 𝐴) ∈ ℝ)
17		0red 11261	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → 0 ∈ ℝ)
18	16, 17	ltnled 11405	. . . . . . . . . . . . . 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 7438	. . . . . . . . 9 ⊢ (𝐴 = 0 → (2 · 𝐴) = (2 · 0))
28		2t0e0 12432	. . . . . . . . 9 ⊢ (2 · 0) = 0
29	27, 28	eqtrdi 2790	. . . . . . . 8 ⊢ (𝐴 = 0 → (2 · 𝐴) = 0)
30		0nprm 16711	. . . . . . . . 9 ⊢ ¬ 0 ∈ ℙ
31	30	a1i 11	. . . . . . . 8 ⊢ (𝐴 = 0 → ¬ 0 ∈ ℙ)
32	29, 31	eqneltrd 2858	. . . . . . 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 12646	. . . . . . 7 ⊢ 2 ∈ ℤ
37		uzid 12890	. . . . . . 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 1135	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 1 ≤ 𝐴 ∧ ¬ 𝐴 = 1) → 𝐴 ∈ ℤ)
41		df-ne 2938	. . . . . . . . 9 ⊢ (𝐴 ≠ 1 ↔ ¬ 𝐴 = 1)
42		1red 11259	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℤ → 1 ∈ ℝ)
43	42, 1	ltlend 11403	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ → (1 < 𝐴 ↔ (1 ≤ 𝐴 ∧ 𝐴 ≠ 1)))
44		1zzd 12645	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℤ → 1 ∈ ℤ)
45		zltp1le 12664	. . . . . . . . . . . . . 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 12326	. . . . . . . . . . . . 13 ⊢ 2 = (1 + 1)
49	48	breq1i 5154	. . . . . . . . . . . 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 1116	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 1 ≤ 𝐴 ∧ ¬ 𝐴 = 1) → 2 ≤ 𝐴)
55		eluz2 12881	. . . . . . 7 ⊢ (𝐴 ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 2 ≤ 𝐴))
56	39, 40, 54, 55	syl3anbrc 1342	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 1 ≤ 𝐴 ∧ ¬ 𝐴 = 1) → 𝐴 ∈ (ℤ_≥‘2))
57		nprm 16721	. . . . . 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 1118	. . . 4 ⊢ (𝐴 ∈ ℤ → (1 ≤ 𝐴 → (¬ 𝐴 = 1 → ¬ (2 · 𝐴) ∈ ℙ)))
60		zle0orge1 12627	. . . 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 7438	. . . 4 ⊢ (𝐴 = 1 → (2 · 𝐴) = (2 · 1))
64		2t1e2 12426	. . . 4 ⊢ (2 · 1) = 2
65	63, 64	eqtrdi 2790	. . 3 ⊢ (𝐴 = 1 → (2 · 𝐴) = 2)
66		2prm 16725	. . 3 ⊢ 2 ∈ ℙ
67	65, 66	eqeltrdi 2846	. 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 1536 ∈ wcel 2105 ≠ wne 2937 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 ℝcr 11151 0cc0 11152 1c1 11153 + caddc 11155 · cmul 11157 < clt 11292 ≤ cle 11293 ℕcn 12263 2c2 12318 ℕ₀cn0 12523 ℤcz 12610 ℤ_≥cuz 12875 ℙcprime 16704

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-fz 13544 df-seq 14039 df-exp 14099 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-dvds 16287 df-prm 16705

This theorem is referenced by: 2sqreultlem 27505 2sqreunnltlem 27508

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