Theorem nprmmul2 48815

Description: Special factorization of a non-prime integer greater than 3. (Contributed by AV, 6-Apr-2026.)

Assertion

Ref	Expression
nprmmul2	⊢ (𝑁 ∈ (ℤ≥‘4) → (𝑁 ∉ ℙ ↔ ∃𝑎 ∈ (2..^𝑁)∃𝑏 ∈ (2..^𝑁)(𝑎 ≤ 𝑏 ∧ 𝑁 = (𝑎 · 𝑏))))

Distinct variable group:   𝑁,𝑎,𝑏

Proof of Theorem nprmmul2

Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nprmmul1 48814	. 2 ⊢ (𝑁 ∈ (ℤ≥‘4) → (𝑁 ∉ ℙ ↔ ∃𝑚 ∈ (2..^𝑁)∃𝑛 ∈ (2..^𝑁)𝑁 = (𝑚 · 𝑛)))
2		breq1 5109	. . . . . . . 8 ⊢ (𝑎 = 𝑛 → (𝑎 ≤ 𝑏 ↔ 𝑛 ≤ 𝑏))
3		oveq1 7450	. . . . . . . . 9 ⊢ (𝑎 = 𝑛 → (𝑎 · 𝑏) = (𝑛 · 𝑏))
4	3	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑎 = 𝑛 → (𝑁 = (𝑎 · 𝑏) ↔ 𝑁 = (𝑛 · 𝑏)))
5	2, 4	anbi12d 644	. . . . . . 7 ⊢ (𝑎 = 𝑛 → ((𝑎 ≤ 𝑏 ∧ 𝑁 = (𝑎 · 𝑏)) ↔ (𝑛 ≤ 𝑏 ∧ 𝑁 = (𝑛 · 𝑏))))
6		breq2 5110	. . . . . . . 8 ⊢ (𝑏 = 𝑚 → (𝑛 ≤ 𝑏 ↔ 𝑛 ≤ 𝑚))
7		oveq2 7451	. . . . . . . . 9 ⊢ (𝑏 = 𝑚 → (𝑛 · 𝑏) = (𝑛 · 𝑚))
8	7	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑏 = 𝑚 → (𝑁 = (𝑛 · 𝑏) ↔ 𝑁 = (𝑛 · 𝑚)))
9	6, 8	anbi12d 644	. . . . . . 7 ⊢ (𝑏 = 𝑚 → ((𝑛 ≤ 𝑏 ∧ 𝑁 = (𝑛 · 𝑏)) ↔ (𝑛 ≤ 𝑚 ∧ 𝑁 = (𝑛 · 𝑚))))
10		simp1rr 1258	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) ∧ 𝑛 ≤ 𝑚 ∧ 𝑁 = (𝑚 · 𝑛)) → 𝑛 ∈ (2..^𝑁))
11		simp1rl 1257	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) ∧ 𝑛 ≤ 𝑚 ∧ 𝑁 = (𝑚 · 𝑛)) → 𝑚 ∈ (2..^𝑁))
12		simp2 1155	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) ∧ 𝑛 ≤ 𝑚 ∧ 𝑁 = (𝑚 · 𝑛)) → 𝑛 ≤ 𝑚)
13		elfzo2nn 48604	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (2..^𝑁) → 𝑚 ∈ ℕ)
14		elfzo2nn 48604	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (2..^𝑁) → 𝑛 ∈ ℕ)
15		nnmulcom 12423	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝑚 · 𝑛) = (𝑛 · 𝑚))
16	13, 14, 15	syl2an 608	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁)) → (𝑚 · 𝑛) = (𝑛 · 𝑚))
17	16	eqeq2d 2774	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁)) → (𝑁 = (𝑚 · 𝑛) ↔ 𝑁 = (𝑛 · 𝑚)))
18	17	biimpd 232	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁)) → (𝑁 = (𝑚 · 𝑛) → 𝑁 = (𝑛 · 𝑚)))
19	18	adantl 487	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) → (𝑁 = (𝑚 · 𝑛) → 𝑁 = (𝑛 · 𝑚)))
20	19	imp 412	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) ∧ 𝑁 = (𝑚 · 𝑛)) → 𝑁 = (𝑛 · 𝑚))
21	20	3adant2 1149	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) ∧ 𝑛 ≤ 𝑚 ∧ 𝑁 = (𝑚 · 𝑛)) → 𝑁 = (𝑛 · 𝑚))
22	12, 21	jca 521	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) ∧ 𝑛 ≤ 𝑚 ∧ 𝑁 = (𝑚 · 𝑛)) → (𝑛 ≤ 𝑚 ∧ 𝑁 = (𝑛 · 𝑚)))
23	5, 9, 10, 11, 22	2rspcedvdw 3593	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) ∧ 𝑛 ≤ 𝑚 ∧ 𝑁 = (𝑚 · 𝑛)) → ∃𝑎 ∈ (2..^𝑁)∃𝑏 ∈ (2..^𝑁)(𝑎 ≤ 𝑏 ∧ 𝑁 = (𝑎 · 𝑏)))
24	23	3exp 1137	. . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) → (𝑛 ≤ 𝑚 → (𝑁 = (𝑚 · 𝑛) → ∃𝑎 ∈ (2..^𝑁)∃𝑏 ∈ (2..^𝑁)(𝑎 ≤ 𝑏 ∧ 𝑁 = (𝑎 · 𝑏)))))
25		breq1 5109	. . . . . . . 8 ⊢ (𝑎 = 𝑚 → (𝑎 ≤ 𝑏 ↔ 𝑚 ≤ 𝑏))
26		oveq1 7450	. . . . . . . . 9 ⊢ (𝑎 = 𝑚 → (𝑎 · 𝑏) = (𝑚 · 𝑏))
27	26	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑎 = 𝑚 → (𝑁 = (𝑎 · 𝑏) ↔ 𝑁 = (𝑚 · 𝑏)))
28	25, 27	anbi12d 644	. . . . . . 7 ⊢ (𝑎 = 𝑚 → ((𝑎 ≤ 𝑏 ∧ 𝑁 = (𝑎 · 𝑏)) ↔ (𝑚 ≤ 𝑏 ∧ 𝑁 = (𝑚 · 𝑏))))
29		breq2 5110	. . . . . . . 8 ⊢ (𝑏 = 𝑛 → (𝑚 ≤ 𝑏 ↔ 𝑚 ≤ 𝑛))
30		oveq2 7451	. . . . . . . . 9 ⊢ (𝑏 = 𝑛 → (𝑚 · 𝑏) = (𝑚 · 𝑛))
31	30	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑏 = 𝑛 → (𝑁 = (𝑚 · 𝑏) ↔ 𝑁 = (𝑚 · 𝑛)))
32	29, 31	anbi12d 644	. . . . . . 7 ⊢ (𝑏 = 𝑛 → ((𝑚 ≤ 𝑏 ∧ 𝑁 = (𝑚 · 𝑏)) ↔ (𝑚 ≤ 𝑛 ∧ 𝑁 = (𝑚 · 𝑛))))
33		simp1rl 1257	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) ∧ 𝑚 ≤ 𝑛 ∧ 𝑁 = (𝑚 · 𝑛)) → 𝑚 ∈ (2..^𝑁))
34		simp1rr 1258	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) ∧ 𝑚 ≤ 𝑛 ∧ 𝑁 = (𝑚 · 𝑛)) → 𝑛 ∈ (2..^𝑁))
35		3simpc 1168	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) ∧ 𝑚 ≤ 𝑛 ∧ 𝑁 = (𝑚 · 𝑛)) → (𝑚 ≤ 𝑛 ∧ 𝑁 = (𝑚 · 𝑛)))
36	28, 32, 33, 34, 35	2rspcedvdw 3593	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) ∧ 𝑚 ≤ 𝑛 ∧ 𝑁 = (𝑚 · 𝑛)) → ∃𝑎 ∈ (2..^𝑁)∃𝑏 ∈ (2..^𝑁)(𝑎 ≤ 𝑏 ∧ 𝑁 = (𝑎 · 𝑏)))
37	36	3exp 1137	. . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) → (𝑚 ≤ 𝑛 → (𝑁 = (𝑚 · 𝑛) → ∃𝑎 ∈ (2..^𝑁)∃𝑏 ∈ (2..^𝑁)(𝑎 ≤ 𝑏 ∧ 𝑁 = (𝑎 · 𝑏)))))
38		elfzoelz 13820	. . . . . . . . 9 ⊢ (𝑚 ∈ (2..^𝑁) → 𝑚 ∈ ℤ)
39	38	zred 12830	. . . . . . . 8 ⊢ (𝑚 ∈ (2..^𝑁) → 𝑚 ∈ ℝ)
40		elfzoelz 13820	. . . . . . . . 9 ⊢ (𝑛 ∈ (2..^𝑁) → 𝑛 ∈ ℤ)
41	40	zred 12830	. . . . . . . 8 ⊢ (𝑛 ∈ (2..^𝑁) → 𝑛 ∈ ℝ)
42	39, 41	anim12ci 626	. . . . . . 7 ⊢ ((𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁)) → (𝑛 ∈ ℝ ∧ 𝑚 ∈ ℝ))
43	42	adantl 487	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) → (𝑛 ∈ ℝ ∧ 𝑚 ∈ ℝ))
44		letric 11437	. . . . . 6 ⊢ ((𝑛 ∈ ℝ ∧ 𝑚 ∈ ℝ) → (𝑛 ≤ 𝑚 ∨ 𝑚 ≤ 𝑛))
45	43, 44	syl 18	. . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) → (𝑛 ≤ 𝑚 ∨ 𝑚 ≤ 𝑛))
46	24, 37, 45	mpjaod 874	. . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) → (𝑁 = (𝑚 · 𝑛) → ∃𝑎 ∈ (2..^𝑁)∃𝑏 ∈ (2..^𝑁)(𝑎 ≤ 𝑏 ∧ 𝑁 = (𝑎 · 𝑏))))
47	46	rexlimdvva 3223	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘4) → (∃𝑚 ∈ (2..^𝑁)∃𝑛 ∈ (2..^𝑁)𝑁 = (𝑚 · 𝑛) → ∃𝑎 ∈ (2..^𝑁)∃𝑏 ∈ (2..^𝑁)(𝑎 ≤ 𝑏 ∧ 𝑁 = (𝑎 · 𝑏))))
48		oveq1 7450	. . . . . . . 8 ⊢ (𝑚 = 𝑎 → (𝑚 · 𝑛) = (𝑎 · 𝑛))
49	48	eqeq2d 2774	. . . . . . 7 ⊢ (𝑚 = 𝑎 → (𝑁 = (𝑚 · 𝑛) ↔ 𝑁 = (𝑎 · 𝑛)))
50		oveq2 7451	. . . . . . . 8 ⊢ (𝑛 = 𝑏 → (𝑎 · 𝑛) = (𝑎 · 𝑏))
51	50	eqeq2d 2774	. . . . . . 7 ⊢ (𝑛 = 𝑏 → (𝑁 = (𝑎 · 𝑛) ↔ 𝑁 = (𝑎 · 𝑏)))
52		simplrl 789	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑎 ∈ (2..^𝑁) ∧ 𝑏 ∈ (2..^𝑁))) ∧ 𝑁 = (𝑎 · 𝑏)) → 𝑎 ∈ (2..^𝑁))
53		simplrr 790	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑎 ∈ (2..^𝑁) ∧ 𝑏 ∈ (2..^𝑁))) ∧ 𝑁 = (𝑎 · 𝑏)) → 𝑏 ∈ (2..^𝑁))
54		simpr 490	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑎 ∈ (2..^𝑁) ∧ 𝑏 ∈ (2..^𝑁))) ∧ 𝑁 = (𝑎 · 𝑏)) → 𝑁 = (𝑎 · 𝑏))
55	49, 51, 52, 53, 54	2rspcedvdw 3593	. . . . . 6 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑎 ∈ (2..^𝑁) ∧ 𝑏 ∈ (2..^𝑁))) ∧ 𝑁 = (𝑎 · 𝑏)) → ∃𝑚 ∈ (2..^𝑁)∃𝑛 ∈ (2..^𝑁)𝑁 = (𝑚 · 𝑛))
56	55	ex 418	. . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘4) ∧ (𝑎 ∈ (2..^𝑁) ∧ 𝑏 ∈ (2..^𝑁))) → (𝑁 = (𝑎 · 𝑏) → ∃𝑚 ∈ (2..^𝑁)∃𝑛 ∈ (2..^𝑁)𝑁 = (𝑚 · 𝑛)))
57	56	adantld 496	. . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘4) ∧ (𝑎 ∈ (2..^𝑁) ∧ 𝑏 ∈ (2..^𝑁))) → ((𝑎 ≤ 𝑏 ∧ 𝑁 = (𝑎 · 𝑏)) → ∃𝑚 ∈ (2..^𝑁)∃𝑛 ∈ (2..^𝑁)𝑁 = (𝑚 · 𝑛)))
58	57	rexlimdvva 3223	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘4) → (∃𝑎 ∈ (2..^𝑁)∃𝑏 ∈ (2..^𝑁)(𝑎 ≤ 𝑏 ∧ 𝑁 = (𝑎 · 𝑏)) → ∃𝑚 ∈ (2..^𝑁)∃𝑛 ∈ (2..^𝑁)𝑁 = (𝑚 · 𝑛)))
59	47, 58	impbid 215	. 2 ⊢ (𝑁 ∈ (ℤ≥‘4) → (∃𝑚 ∈ (2..^𝑁)∃𝑛 ∈ (2..^𝑁)𝑁 = (𝑚 · 𝑛) ↔ ∃𝑎 ∈ (2..^𝑁)∃𝑏 ∈ (2..^𝑁)(𝑎 ≤ 𝑏 ∧ 𝑁 = (𝑎 · 𝑏))))
60	1, 59	bitrd 282	1 ⊢ (𝑁 ∈ (ℤ≥‘4) → (𝑁 ∉ ℙ ↔ ∃𝑎 ∈ (2..^𝑁)∃𝑏 ∈ (2..^𝑁)(𝑎 ≤ 𝑏 ∧ 𝑁 = (𝑎 · 𝑏))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ∉ wnel 3065  ∃wrex 3090   class class class wbr 5106  ‘cfv 6558  (class class class)co 7443  ℝcr 11226   · cmul 11232   ≤ cle 11371  ℕcn 12362  2c2 12424  4c4 12426  ℤ_≥cuz 12992  ..^cfzo 13815  ℙcprime 16875

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304  ax-pre-sup 11305

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-2o 8505  df-er 8745  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-sup 9461  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-div 12001  df-nn 12363  df-2 12432  df-3 12433  df-4 12434  df-n0 12634  df-z 12721  df-uz 12993  df-rp 13148  df-fz 13667  df-fzo 13816  df-seq 14172  df-exp 14232  df-cj 15295  df-re 15296  df-im 15297  df-sqrt 15431  df-abs 15432  df-dvds 16452  df-prm 16876

This theorem is used by:  nprmmul3  48816