2mulprm - Metamath Proof Explorer

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Theorem 2mulprm 16710

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 12569	. . . . . 6 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
2		0red 11181	. . . . . 6 ⊢ (𝐴 ∈ ℤ → 0 ∈ ℝ)
3	1, 2	leloed 11323	. . . . 5 ⊢ (𝐴 ∈ ℤ → (𝐴 ≤ 0 ↔ (𝐴 < 0 ∨ 𝐴 = 0)))
4		prmnn 16691	. . . . . . . . . 10 ⊢ ((2 · 𝐴) ∈ ℙ → (2 · 𝐴) ∈ ℕ)
5		nnnn0 12485	. . . . . . . . . 10 ⊢ ((2 · 𝐴) ∈ ℕ → (2 · 𝐴) ∈ ℕ₀)
6		nn0ge0 12503	. . . . . . . . . 10 ⊢ ((2 · 𝐴) ∈ ℕ₀ → 0 ≤ (2 · 𝐴))
7		2pos 12319	. . . . . . . . . . . . . . . . . 18 ⊢ 0 < 2
8	7	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℤ → 0 < 2)
9	8	anim1i 624	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → (0 < 2 ∧ 𝐴 < 0))
10	9	olcd 885	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → ((2 < 0 ∧ 0 < 𝐴) ∨ (0 < 2 ∧ 𝐴 < 0)))
11		2re 12289	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ
12	11	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → 2 ∈ ℝ)
13	1	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → 𝐴 ∈ ℝ)
14	12, 13	mul2lt0bi 13098	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → ((2 · 𝐴) < 0 ↔ ((2 < 0 ∧ 0 < 𝐴) ∨ (0 < 2 ∧ 𝐴 < 0))))
15	10, 14	mpbird 259	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → (2 · 𝐴) < 0)
16	12, 13	remulcld 11209	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → (2 · 𝐴) ∈ ℝ)
17		0red 11181	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → 0 ∈ ℝ)
18	16, 17	ltnled 11327	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → ((2 · 𝐴) < 0 ↔ ¬ 0 ≤ (2 · 𝐴)))
19	15, 18	mpbid 234	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 < 0) → ¬ 0 ≤ (2 · 𝐴))
20	19	ex 416	. . . . . . . . . . . 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 7400	. . . . . . . . 9 ⊢ (𝐴 = 0 → (2 · 𝐴) = (2 · 0))
28		2t0e0 12385	. . . . . . . . 9 ⊢ (2 · 0) = 0
29	27, 28	eqtrdi 2812	. . . . . . . 8 ⊢ (𝐴 = 0 → (2 · 𝐴) = 0)
30		0nprm 16695	. . . . . . . . 9 ⊢ ¬ 0 ∈ ℙ
31	30	a1i 11	. . . . . . . 8 ⊢ (𝐴 = 0 → ¬ 0 ∈ ℙ)
32	29, 31	eqneltrd 2881	. . . . . . 7 ⊢ (𝐴 = 0 → ¬ (2 · 𝐴) ∈ ℙ)
33	32	a1i13 27	. . . . . 6 ⊢ (𝐴 ∈ ℤ → (𝐴 = 0 → (¬ 𝐴 = 1 → ¬ (2 · 𝐴) ∈ ℙ)))
34	26, 33	jaod 870	. . . . 5 ⊢ (𝐴 ∈ ℤ → ((𝐴 < 0 ∨ 𝐴 = 0) → (¬ 𝐴 = 1 → ¬ (2 · 𝐴) ∈ ℙ)))
35	3, 34	sylbid 242	. . . 4 ⊢ (𝐴 ∈ ℤ → (𝐴 ≤ 0 → (¬ 𝐴 = 1 → ¬ (2 · 𝐴) ∈ ℙ)))
36		2z 12600	. . . . . . 7 ⊢ 2 ∈ ℤ
37		uzid 12851	. . . . . . 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 1148	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 1 ≤ 𝐴 ∧ ¬ 𝐴 = 1) → 𝐴 ∈ ℤ)
41		df-ne 2957	. . . . . . . . 9 ⊢ (𝐴 ≠ 1 ↔ ¬ 𝐴 = 1)
42		1red 11179	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℤ → 1 ∈ ℝ)
43	42, 1	ltlend 11325	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ → (1 < 𝐴 ↔ (1 ≤ 𝐴 ∧ 𝐴 ≠ 1)))
44		1zzd 12599	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℤ → 1 ∈ ℤ)
45		zltp1le 12618	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (1 < 𝐴 ↔ (1 + 1) ≤ 𝐴))
46	44, 45	mpancom 698	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℤ → (1 < 𝐴 ↔ (1 + 1) ≤ 𝐴))
47	46	biimpd 231	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℤ → (1 < 𝐴 → (1 + 1) ≤ 𝐴))
48		df-2 12277	. . . . . . . . . . . . 13 ⊢ 2 = (1 + 1)
49	48	breq1i 5106	. . . . . . . . . . . 12 ⊢ (2 ≤ 𝐴 ↔ (1 + 1) ≤ 𝐴)
50	47, 49	imbitrrdi 254	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ → (1 < 𝐴 → 2 ≤ 𝐴))
51	43, 50	sylbird 262	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℤ → ((1 ≤ 𝐴 ∧ 𝐴 ≠ 1) → 2 ≤ 𝐴))
52	51	expdimp 456	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 1 ≤ 𝐴) → (𝐴 ≠ 1 → 2 ≤ 𝐴))
53	41, 52	biimtrrid 245	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 1 ≤ 𝐴) → (¬ 𝐴 = 1 → 2 ≤ 𝐴))
54	53	3impia 1129	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 1 ≤ 𝐴 ∧ ¬ 𝐴 = 1) → 2 ≤ 𝐴)
55		eluz2 12842	. . . . . . 7 ⊢ (𝐴 ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 2 ≤ 𝐴))
56	39, 40, 54, 55	syl3anbrc 1356	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 1 ≤ 𝐴 ∧ ¬ 𝐴 = 1) → 𝐴 ∈ (ℤ_≥‘2))
57		nprm 16705	. . . . . 6 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 𝐴 ∈ (ℤ_≥‘2)) → ¬ (2 · 𝐴) ∈ ℙ)
58	38, 56, 57	sylancr 596	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 1 ≤ 𝐴 ∧ ¬ 𝐴 = 1) → ¬ (2 · 𝐴) ∈ ℙ)
59	58	3exp 1131	. . . 4 ⊢ (𝐴 ∈ ℤ → (1 ≤ 𝐴 → (¬ 𝐴 = 1 → ¬ (2 · 𝐴) ∈ ℙ)))
60		zle0orge1 12582	. . . 4 ⊢ (𝐴 ∈ ℤ → (𝐴 ≤ 0 ∨ 1 ≤ 𝐴))
61	35, 59, 60	mpjaod 871	. . 3 ⊢ (𝐴 ∈ ℤ → (¬ 𝐴 = 1 → ¬ (2 · 𝐴) ∈ ℙ))
62	61	con4d 115	. 2 ⊢ (𝐴 ∈ ℤ → ((2 · 𝐴) ∈ ℙ → 𝐴 = 1))
63		oveq2 7400	. . . 4 ⊢ (𝐴 = 1 → (2 · 𝐴) = (2 · 1))
64		2t1e2 12377	. . . 4 ⊢ (2 · 1) = 2
65	63, 64	eqtrdi 2812	. . 3 ⊢ (𝐴 = 1 → (2 · 𝐴) = 2)
66		2prm 16709	. . 3 ⊢ 2 ∈ ℙ
67	65, 66	eqeltrdi 2869	. 2 ⊢ (𝐴 = 1 → (2 · 𝐴) ∈ ℙ)
68	62, 67	impbid1 227	1 ⊢ (𝐴 ∈ ℤ → ((2 · 𝐴) ∈ ℙ ↔ 𝐴 = 1))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 ∨ wo 858 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 class class class wbr 5099 ‘cfv 6517 (class class class)co 7392 ℝcr 11069 0cc0 11070 1c1 11071 + caddc 11073 · cmul 11075 < clt 11213 ≤ cle 11214 ℕcn 12207 2c2 12269 ℕ₀cn0 12478 ℤcz 12565 ℤ_≥cuz 12836 ℙcprime 16688

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147 ax-pre-sup 11148

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-iun 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-om 7843 df-1st 7966 df-2nd 7967 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-2o 8433 df-er 8673 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-sup 9385 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12208 df-2 12277 df-3 12278 df-n0 12479 df-z 12566 df-uz 12837 df-rp 12991 df-fz 13510 df-seq 14012 df-exp 14072 df-cj 15109 df-re 15110 df-im 15111 df-sqrt 15245 df-abs 15246 df-dvds 16270 df-prm 16689

This theorem is referenced by: 2sqreultlem 27488 2sqreunnltlem 27491

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