Theorem nprmmul2 48305

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 48304	. 2 ⊢ (𝑁 ∈ (ℤ≥‘4) → (𝑁 ∉ ℙ ↔ ∃𝑚 ∈ (2..^𝑁)∃𝑛 ∈ (2..^𝑁)𝑁 = (𝑚 · 𝑛)))
2		breq1 5111	. . . . . . . 8 ⊢ (𝑎 = 𝑛 → (𝑎 ≤ 𝑏 ↔ 𝑛 ≤ 𝑏))
3		oveq1 7419	. . . . . . . . 9 ⊢ (𝑎 = 𝑛 → (𝑎 · 𝑏) = (𝑛 · 𝑏))
4	3	eqeq2d 2773	. . . . . . . 8 ⊢ (𝑎 = 𝑛 → (𝑁 = (𝑎 · 𝑏) ↔ 𝑁 = (𝑛 · 𝑏)))
5	2, 4	anbi12d 643	. . . . . . 7 ⊢ (𝑎 = 𝑛 → ((𝑎 ≤ 𝑏 ∧ 𝑁 = (𝑎 · 𝑏)) ↔ (𝑛 ≤ 𝑏 ∧ 𝑁 = (𝑛 · 𝑏))))
6		breq2 5112	. . . . . . . 8 ⊢ (𝑏 = 𝑚 → (𝑛 ≤ 𝑏 ↔ 𝑛 ≤ 𝑚))
7		oveq2 7420	. . . . . . . . 9 ⊢ (𝑏 = 𝑚 → (𝑛 · 𝑏) = (𝑛 · 𝑚))
8	7	eqeq2d 2773	. . . . . . . 8 ⊢ (𝑏 = 𝑚 → (𝑁 = (𝑛 · 𝑏) ↔ 𝑁 = (𝑛 · 𝑚)))
9	6, 8	anbi12d 643	. . . . . . 7 ⊢ (𝑏 = 𝑚 → ((𝑛 ≤ 𝑏 ∧ 𝑁 = (𝑛 · 𝑏)) ↔ (𝑛 ≤ 𝑚 ∧ 𝑁 = (𝑛 · 𝑚))))
10		simp1rr 1257	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) ∧ 𝑛 ≤ 𝑚 ∧ 𝑁 = (𝑚 · 𝑛)) → 𝑛 ∈ (2..^𝑁))
11		simp1rl 1256	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) ∧ 𝑛 ≤ 𝑚 ∧ 𝑁 = (𝑚 · 𝑛)) → 𝑚 ∈ (2..^𝑁))
12		simp2 1154	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) ∧ 𝑛 ≤ 𝑚 ∧ 𝑁 = (𝑚 · 𝑛)) → 𝑛 ≤ 𝑚)
13		elfzo2nn 48094	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (2..^𝑁) → 𝑚 ∈ ℕ)
14		elfzo2nn 48094	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (2..^𝑁) → 𝑛 ∈ ℕ)
15		nnmulcom 12300	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝑚 · 𝑛) = (𝑛 · 𝑚))
16	13, 14, 15	syl2an 607	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁)) → (𝑚 · 𝑛) = (𝑛 · 𝑚))
17	16	eqeq2d 2773	. . . . . . . . . . . 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 486	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) → (𝑁 = (𝑚 · 𝑛) → 𝑁 = (𝑛 · 𝑚)))
20	19	imp 411	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) ∧ 𝑁 = (𝑚 · 𝑛)) → 𝑁 = (𝑛 · 𝑚))
21	20	3adant2 1148	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) ∧ 𝑛 ≤ 𝑚 ∧ 𝑁 = (𝑚 · 𝑛)) → 𝑁 = (𝑛 · 𝑚))
22	12, 21	jca 520	. . . . . . 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 3594	. . . . . 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 1136	. . . . 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 5111	. . . . . . . 8 ⊢ (𝑎 = 𝑚 → (𝑎 ≤ 𝑏 ↔ 𝑚 ≤ 𝑏))
26		oveq1 7419	. . . . . . . . 9 ⊢ (𝑎 = 𝑚 → (𝑎 · 𝑏) = (𝑚 · 𝑏))
27	26	eqeq2d 2773	. . . . . . . 8 ⊢ (𝑎 = 𝑚 → (𝑁 = (𝑎 · 𝑏) ↔ 𝑁 = (𝑚 · 𝑏)))
28	25, 27	anbi12d 643	. . . . . . 7 ⊢ (𝑎 = 𝑚 → ((𝑎 ≤ 𝑏 ∧ 𝑁 = (𝑎 · 𝑏)) ↔ (𝑚 ≤ 𝑏 ∧ 𝑁 = (𝑚 · 𝑏))))
29		breq2 5112	. . . . . . . 8 ⊢ (𝑏 = 𝑛 → (𝑚 ≤ 𝑏 ↔ 𝑚 ≤ 𝑛))
30		oveq2 7420	. . . . . . . . 9 ⊢ (𝑏 = 𝑛 → (𝑚 · 𝑏) = (𝑚 · 𝑛))
31	30	eqeq2d 2773	. . . . . . . 8 ⊢ (𝑏 = 𝑛 → (𝑁 = (𝑚 · 𝑏) ↔ 𝑁 = (𝑚 · 𝑛)))
32	29, 31	anbi12d 643	. . . . . . 7 ⊢ (𝑏 = 𝑛 → ((𝑚 ≤ 𝑏 ∧ 𝑁 = (𝑚 · 𝑏)) ↔ (𝑚 ≤ 𝑛 ∧ 𝑁 = (𝑚 · 𝑛))))
33		simp1rl 1256	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) ∧ 𝑚 ≤ 𝑛 ∧ 𝑁 = (𝑚 · 𝑛)) → 𝑚 ∈ (2..^𝑁))
34		simp1rr 1257	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) ∧ 𝑚 ≤ 𝑛 ∧ 𝑁 = (𝑚 · 𝑛)) → 𝑛 ∈ (2..^𝑁))
35		3simpc 1167	. . . . . . 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 3594	. . . . . 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 1136	. . . . 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 13694	. . . . . . . . 9 ⊢ (𝑚 ∈ (2..^𝑁) → 𝑚 ∈ ℤ)
39	38	zred 12706	. . . . . . . 8 ⊢ (𝑚 ∈ (2..^𝑁) → 𝑚 ∈ ℝ)
40		elfzoelz 13694	. . . . . . . . 9 ⊢ (𝑛 ∈ (2..^𝑁) → 𝑛 ∈ ℤ)
41	40	zred 12706	. . . . . . . 8 ⊢ (𝑛 ∈ (2..^𝑁) → 𝑛 ∈ ℝ)
42	39, 41	anim12ci 625	. . . . . . 7 ⊢ ((𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁)) → (𝑛 ∈ ℝ ∧ 𝑚 ∈ ℝ))
43	42	adantl 486	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) → (𝑛 ∈ ℝ ∧ 𝑚 ∈ ℝ))
44		letric 11316	. . . . . 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 873	. . . 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 3221	. . 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 7419	. . . . . . . 8 ⊢ (𝑚 = 𝑎 → (𝑚 · 𝑛) = (𝑎 · 𝑛))
49	48	eqeq2d 2773	. . . . . . 7 ⊢ (𝑚 = 𝑎 → (𝑁 = (𝑚 · 𝑛) ↔ 𝑁 = (𝑎 · 𝑛)))
50		oveq2 7420	. . . . . . . 8 ⊢ (𝑛 = 𝑏 → (𝑎 · 𝑛) = (𝑎 · 𝑏))
51	50	eqeq2d 2773	. . . . . . 7 ⊢ (𝑛 = 𝑏 → (𝑁 = (𝑎 · 𝑛) ↔ 𝑁 = (𝑎 · 𝑏)))
52		simplrl 788	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑎 ∈ (2..^𝑁) ∧ 𝑏 ∈ (2..^𝑁))) ∧ 𝑁 = (𝑎 · 𝑏)) → 𝑎 ∈ (2..^𝑁))
53		simplrr 789	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑎 ∈ (2..^𝑁) ∧ 𝑏 ∈ (2..^𝑁))) ∧ 𝑁 = (𝑎 · 𝑏)) → 𝑏 ∈ (2..^𝑁))
54		simpr 489	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑎 ∈ (2..^𝑁) ∧ 𝑏 ∈ (2..^𝑁))) ∧ 𝑁 = (𝑎 · 𝑏)) → 𝑁 = (𝑎 · 𝑏))
55	49, 51, 52, 53, 54	2rspcedvdw 3594	. . . . . 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 417	. . . . 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 495	. . . 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 3221	. . 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 400   ∨ wo 860   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142   ∉ wnel 3063  ∃wrex 3088   class class class wbr 5108  ‘cfv 6536  (class class class)co 7412  ℝcr 11105   · cmul 11111   ≤ cle 11250  ℕcn 12239  2c2 12301  4c4 12303  ℤ_≥cuz 12868  ..^cfzo 13689  ℙcprime 16735

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451  df-2o 8452  df-er 8692  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9400  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11449  df-neg 11450  df-div 11878  df-nn 12240  df-2 12309  df-3 12310  df-4 12311  df-n0 12511  df-z 12598  df-uz 12869  df-rp 13023  df-fz 13542  df-fzo 13690  df-seq 14045  df-exp 14105  df-cj 15157  df-re 15158  df-im 15159  df-sqrt 15293  df-abs 15294  df-dvds 16317  df-prm 16736

This theorem is used by:  nprmmul3  48306