2mulprm - Metamath Proof Explorer

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Theorem 2mulprm 16569

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 12503	. . . . . 6 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
2		0red 11158	. . . . . 6 ⊢ (𝐴 ∈ ℤ → 0 ∈ ℝ)
3	1, 2	leloed 11298	. . . . 5 ⊢ (𝐴 ∈ ℤ → (𝐴 ≤ 0 ↔ (𝐴 < 0 ∨ 𝐴 = 0)))
4		prmnn 16550	. . . . . . . . . 10 ⊢ ((2 · 𝐴) ∈ ℙ → (2 · 𝐴) ∈ ℕ)
5		nnnn0 12420	. . . . . . . . . 10 ⊢ ((2 · 𝐴) ∈ ℕ → (2 · 𝐴) ∈ ℕ₀)
6		nn0ge0 12438	. . . . . . . . . . 11 ⊢ ((2 · 𝐴) ∈ ℕ₀ → 0 ≤ (2 · 𝐴))
7		2pos 12256	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 < 2
8	7	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ℤ → 0 < 2)
9	8	anim1i 615	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → (0 < 2 ∧ 𝐴 < 0))
10	9	olcd 872	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → ((2 < 0 ∧ 0 < 𝐴) ∨ (0 < 2 ∧ 𝐴 < 0)))
11		2re 12227	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℝ
12	11	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → 2 ∈ ℝ)
13	1	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → 𝐴 ∈ ℝ)
14	12, 13	mul2lt0bi 13021	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → ((2 · 𝐴) < 0 ↔ ((2 < 0 ∧ 0 < 𝐴) ∨ (0 < 2 ∧ 𝐴 < 0))))
15	10, 14	mpbird 256	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → (2 · 𝐴) < 0)
16	12, 13	remulcld 11185	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → (2 · 𝐴) ∈ ℝ)
17		0red 11158	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → 0 ∈ ℝ)
18	16, 17	ltnled 11302	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → ((2 · 𝐴) < 0 ↔ ¬ 0 ≤ (2 · 𝐴)))
19	15, 18	mpbid 231	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → ¬ 0 ≤ (2 · 𝐴))
20	19	ex 413	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℤ → (𝐴 < 0 → ¬ 0 ≤ (2 · 𝐴)))
21	20	con2d 134	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℤ → (0 ≤ (2 · 𝐴) → ¬ 𝐴 < 0))
22	21	com12 32	. . . . . . . . . . 11 ⊢ (0 ≤ (2 · 𝐴) → (𝐴 ∈ ℤ → ¬ 𝐴 < 0))
23	6, 22	syl 17	. . . . . . . . . 10 ⊢ ((2 · 𝐴) ∈ ℕ₀ → (𝐴 ∈ ℤ → ¬ 𝐴 < 0))
24	4, 5, 23	3syl 18	. . . . . . . . 9 ⊢ ((2 · 𝐴) ∈ ℙ → (𝐴 ∈ ℤ → ¬ 𝐴 < 0))
25	24	com12 32	. . . . . . . 8 ⊢ (𝐴 ∈ ℤ → ((2 · 𝐴) ∈ ℙ → ¬ 𝐴 < 0))
26	25	con2d 134	. . . . . . 7 ⊢ (𝐴 ∈ ℤ → (𝐴 < 0 → ¬ (2 · 𝐴) ∈ ℙ))
27	26	a1dd 50	. . . . . 6 ⊢ (𝐴 ∈ ℤ → (𝐴 < 0 → (¬ 𝐴 = 1 → ¬ (2 · 𝐴) ∈ ℙ)))
28		oveq2 7365	. . . . . . . . 9 ⊢ (𝐴 = 0 → (2 · 𝐴) = (2 · 0))
29		2t0e0 12322	. . . . . . . . 9 ⊢ (2 · 0) = 0
30	28, 29	eqtrdi 2792	. . . . . . . 8 ⊢ (𝐴 = 0 → (2 · 𝐴) = 0)
31		0nprm 16554	. . . . . . . . 9 ⊢ ¬ 0 ∈ ℙ
32	31	a1i 11	. . . . . . . 8 ⊢ (𝐴 = 0 → ¬ 0 ∈ ℙ)
33	30, 32	eqneltrd 2857	. . . . . . 7 ⊢ (𝐴 = 0 → ¬ (2 · 𝐴) ∈ ℙ)
34	33	a1i13 27	. . . . . 6 ⊢ (𝐴 ∈ ℤ → (𝐴 = 0 → (¬ 𝐴 = 1 → ¬ (2 · 𝐴) ∈ ℙ)))
35	27, 34	jaod 857	. . . . 5 ⊢ (𝐴 ∈ ℤ → ((𝐴 < 0 ∨ 𝐴 = 0) → (¬ 𝐴 = 1 → ¬ (2 · 𝐴) ∈ ℙ)))
36	3, 35	sylbid 239	. . . 4 ⊢ (𝐴 ∈ ℤ → (𝐴 ≤ 0 → (¬ 𝐴 = 1 → ¬ (2 · 𝐴) ∈ ℙ)))
37		2z 12535	. . . . . . 7 ⊢ 2 ∈ ℤ
38		uzid 12778	. . . . . . 7 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ_≥‘2))
39	37, 38	ax-mp 5	. . . . . 6 ⊢ 2 ∈ (ℤ_≥‘2)
40	37	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 1 ≤ 𝐴 ∧ ¬ 𝐴 = 1) → 2 ∈ ℤ)
41		simp1 1136	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 1 ≤ 𝐴 ∧ ¬ 𝐴 = 1) → 𝐴 ∈ ℤ)
42		df-ne 2944	. . . . . . . . 9 ⊢ (𝐴 ≠ 1 ↔ ¬ 𝐴 = 1)
43		1red 11156	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℤ → 1 ∈ ℝ)
44	43, 1	ltlend 11300	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ → (1 < 𝐴 ↔ (1 ≤ 𝐴 ∧ 𝐴 ≠ 1)))
45		1zzd 12534	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℤ → 1 ∈ ℤ)
46		zltp1le 12553	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (1 < 𝐴 ↔ (1 + 1) ≤ 𝐴))
47	45, 46	mpancom 686	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℤ → (1 < 𝐴 ↔ (1 + 1) ≤ 𝐴))
48	47	biimpd 228	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℤ → (1 < 𝐴 → (1 + 1) ≤ 𝐴))
49		df-2 12216	. . . . . . . . . . . . 13 ⊢ 2 = (1 + 1)
50	49	breq1i 5112	. . . . . . . . . . . 12 ⊢ (2 ≤ 𝐴 ↔ (1 + 1) ≤ 𝐴)
51	48, 50	syl6ibr 251	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ → (1 < 𝐴 → 2 ≤ 𝐴))
52	44, 51	sylbird 259	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℤ → ((1 ≤ 𝐴 ∧ 𝐴 ≠ 1) → 2 ≤ 𝐴))
53	52	expdimp 453	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 1 ≤ 𝐴) → (𝐴 ≠ 1 → 2 ≤ 𝐴))
54	42, 53	biimtrrid 242	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 1 ≤ 𝐴) → (¬ 𝐴 = 1 → 2 ≤ 𝐴))
55	54	3impia 1117	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 1 ≤ 𝐴 ∧ ¬ 𝐴 = 1) → 2 ≤ 𝐴)
56		eluz2 12769	. . . . . . 7 ⊢ (𝐴 ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 2 ≤ 𝐴))
57	40, 41, 55, 56	syl3anbrc 1343	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 1 ≤ 𝐴 ∧ ¬ 𝐴 = 1) → 𝐴 ∈ (ℤ_≥‘2))
58		nprm 16564	. . . . . 6 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 𝐴 ∈ (ℤ_≥‘2)) → ¬ (2 · 𝐴) ∈ ℙ)
59	39, 57, 58	sylancr 587	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 1 ≤ 𝐴 ∧ ¬ 𝐴 = 1) → ¬ (2 · 𝐴) ∈ ℙ)
60	59	3exp 1119	. . . 4 ⊢ (𝐴 ∈ ℤ → (1 ≤ 𝐴 → (¬ 𝐴 = 1 → ¬ (2 · 𝐴) ∈ ℙ)))
61		zle0orge1 12516	. . . 4 ⊢ (𝐴 ∈ ℤ → (𝐴 ≤ 0 ∨ 1 ≤ 𝐴))
62	36, 60, 61	mpjaod 858	. . 3 ⊢ (𝐴 ∈ ℤ → (¬ 𝐴 = 1 → ¬ (2 · 𝐴) ∈ ℙ))
63	62	con4d 115	. 2 ⊢ (𝐴 ∈ ℤ → ((2 · 𝐴) ∈ ℙ → 𝐴 = 1))
64		oveq2 7365	. . . 4 ⊢ (𝐴 = 1 → (2 · 𝐴) = (2 · 1))
65		2t1e2 12316	. . . 4 ⊢ (2 · 1) = 2
66	64, 65	eqtrdi 2792	. . 3 ⊢ (𝐴 = 1 → (2 · 𝐴) = 2)
67		2prm 16568	. . 3 ⊢ 2 ∈ ℙ
68	66, 67	eqeltrdi 2846	. 2 ⊢ (𝐴 = 1 → (2 · 𝐴) ∈ ℙ)
69	63, 68	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 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 class class class wbr 5105 ‘cfv 6496 (class class class)co 7357 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 · cmul 11056 < clt 11189 ≤ cle 11190 ℕcn 12153 2c2 12208 ℕ₀cn0 12413 ℤcz 12499 ℤ_≥cuz 12763 ℙcprime 16547

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-z 12500 df-uz 12764 df-rp 12916 df-fz 13425 df-seq 13907 df-exp 13968 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-dvds 16137 df-prm 16548

This theorem is referenced by: 2sqreultlem 26795 2sqreunnltlem 26798

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