Theorem nprmmul2 48004

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 48003	. 2 ⊢ (𝑁 ∈ (ℤ≥‘4) → (𝑁 ∉ ℙ ↔ ∃𝑚 ∈ (2..^𝑁)∃𝑛 ∈ (2..^𝑁)𝑁 = (𝑚 · 𝑛)))
2		breq1 5082	. . . . . . . 8 ⊢ (𝑎 = 𝑛 → (𝑎 ≤ 𝑏 ↔ 𝑛 ≤ 𝑏))
3		oveq1 7370	. . . . . . . . 9 ⊢ (𝑎 = 𝑛 → (𝑎 · 𝑏) = (𝑛 · 𝑏))
4	3	eqeq2d 2751	. . . . . . . 8 ⊢ (𝑎 = 𝑛 → (𝑁 = (𝑎 · 𝑏) ↔ 𝑁 = (𝑛 · 𝑏)))
5	2, 4	anbi12d 638	. . . . . . 7 ⊢ (𝑎 = 𝑛 → ((𝑎 ≤ 𝑏 ∧ 𝑁 = (𝑎 · 𝑏)) ↔ (𝑛 ≤ 𝑏 ∧ 𝑁 = (𝑛 · 𝑏))))
6		breq2 5083	. . . . . . . 8 ⊢ (𝑏 = 𝑚 → (𝑛 ≤ 𝑏 ↔ 𝑛 ≤ 𝑚))
7		oveq2 7371	. . . . . . . . 9 ⊢ (𝑏 = 𝑚 → (𝑛 · 𝑏) = (𝑛 · 𝑚))
8	7	eqeq2d 2751	. . . . . . . 8 ⊢ (𝑏 = 𝑚 → (𝑁 = (𝑛 · 𝑏) ↔ 𝑁 = (𝑛 · 𝑚)))
9	6, 8	anbi12d 638	. . . . . . 7 ⊢ (𝑏 = 𝑚 → ((𝑛 ≤ 𝑏 ∧ 𝑁 = (𝑛 · 𝑏)) ↔ (𝑛 ≤ 𝑚 ∧ 𝑁 = (𝑛 · 𝑚))))
10		simp1rr 1246	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) ∧ 𝑛 ≤ 𝑚 ∧ 𝑁 = (𝑚 · 𝑛)) → 𝑛 ∈ (2..^𝑁))
11		simp1rl 1245	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) ∧ 𝑛 ≤ 𝑚 ∧ 𝑁 = (𝑚 · 𝑛)) → 𝑚 ∈ (2..^𝑁))
12		simp2 1143	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) ∧ 𝑛 ≤ 𝑚 ∧ 𝑁 = (𝑚 · 𝑛)) → 𝑛 ≤ 𝑚)
13		elfzo2nn 47793	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (2..^𝑁) → 𝑚 ∈ ℕ)
14		elfzo2nn 47793	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (2..^𝑁) → 𝑛 ∈ ℕ)
15		nnmulcom 12233	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝑚 · 𝑛) = (𝑛 · 𝑚))
16	13, 14, 15	syl2an 602	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁)) → (𝑚 · 𝑛) = (𝑛 · 𝑚))
17	16	eqeq2d 2751	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁)) → (𝑁 = (𝑚 · 𝑛) ↔ 𝑁 = (𝑛 · 𝑚)))
18	17	biimpd 230	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁)) → (𝑁 = (𝑚 · 𝑛) → 𝑁 = (𝑛 · 𝑚)))
19	18	adantl 482	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) → (𝑁 = (𝑚 · 𝑛) → 𝑁 = (𝑛 · 𝑚)))
20	19	imp 407	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) ∧ 𝑁 = (𝑚 · 𝑛)) → 𝑁 = (𝑛 · 𝑚))
21	20	3adant2 1137	. . . . . . . 8 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) ∧ 𝑛 ≤ 𝑚 ∧ 𝑁 = (𝑚 · 𝑛)) → 𝑁 = (𝑛 · 𝑚))
22	12, 21	jca 516	. . . . . . 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 3581	. . . . . 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 1125	. . . . 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 5082	. . . . . . . 8 ⊢ (𝑎 = 𝑚 → (𝑎 ≤ 𝑏 ↔ 𝑚 ≤ 𝑏))
26		oveq1 7370	. . . . . . . . 9 ⊢ (𝑎 = 𝑚 → (𝑎 · 𝑏) = (𝑚 · 𝑏))
27	26	eqeq2d 2751	. . . . . . . 8 ⊢ (𝑎 = 𝑚 → (𝑁 = (𝑎 · 𝑏) ↔ 𝑁 = (𝑚 · 𝑏)))
28	25, 27	anbi12d 638	. . . . . . 7 ⊢ (𝑎 = 𝑚 → ((𝑎 ≤ 𝑏 ∧ 𝑁 = (𝑎 · 𝑏)) ↔ (𝑚 ≤ 𝑏 ∧ 𝑁 = (𝑚 · 𝑏))))
29		breq2 5083	. . . . . . . 8 ⊢ (𝑏 = 𝑛 → (𝑚 ≤ 𝑏 ↔ 𝑚 ≤ 𝑛))
30		oveq2 7371	. . . . . . . . 9 ⊢ (𝑏 = 𝑛 → (𝑚 · 𝑏) = (𝑚 · 𝑛))
31	30	eqeq2d 2751	. . . . . . . 8 ⊢ (𝑏 = 𝑛 → (𝑁 = (𝑚 · 𝑏) ↔ 𝑁 = (𝑚 · 𝑛)))
32	29, 31	anbi12d 638	. . . . . . 7 ⊢ (𝑏 = 𝑛 → ((𝑚 ≤ 𝑏 ∧ 𝑁 = (𝑚 · 𝑏)) ↔ (𝑚 ≤ 𝑛 ∧ 𝑁 = (𝑚 · 𝑛))))
33		simp1rl 1245	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) ∧ 𝑚 ≤ 𝑛 ∧ 𝑁 = (𝑚 · 𝑛)) → 𝑚 ∈ (2..^𝑁))
34		simp1rr 1246	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) ∧ 𝑚 ≤ 𝑛 ∧ 𝑁 = (𝑚 · 𝑛)) → 𝑛 ∈ (2..^𝑁))
35		3simpc 1156	. . . . . . 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 3581	. . . . . 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 1125	. . . . 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 13611	. . . . . . . . 9 ⊢ (𝑚 ∈ (2..^𝑁) → 𝑚 ∈ ℤ)
39	38	zred 12631	. . . . . . . 8 ⊢ (𝑚 ∈ (2..^𝑁) → 𝑚 ∈ ℝ)
40		elfzoelz 13611	. . . . . . . . 9 ⊢ (𝑛 ∈ (2..^𝑁) → 𝑛 ∈ ℤ)
41	40	zred 12631	. . . . . . . 8 ⊢ (𝑛 ∈ (2..^𝑁) → 𝑛 ∈ ℝ)
42	39, 41	anim12ci 620	. . . . . . 7 ⊢ ((𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁)) → (𝑛 ∈ ℝ ∧ 𝑚 ∈ ℝ))
43	42	adantl 482	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) → (𝑛 ∈ ℝ ∧ 𝑚 ∈ ℝ))
44		letric 11244	. . . . . 6 ⊢ ((𝑛 ∈ ℝ ∧ 𝑚 ∈ ℝ) → (𝑛 ≤ 𝑚 ∨ 𝑚 ≤ 𝑛))
45	43, 44	syl 17	. . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘4) ∧ (𝑚 ∈ (2..^𝑁) ∧ 𝑛 ∈ (2..^𝑁))) → (𝑛 ≤ 𝑚 ∨ 𝑚 ≤ 𝑛))
46	24, 37, 45	mpjaod 866	. . . 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 3197	. . 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 7370	. . . . . . . 8 ⊢ (𝑚 = 𝑎 → (𝑚 · 𝑛) = (𝑎 · 𝑛))
49	48	eqeq2d 2751	. . . . . . 7 ⊢ (𝑚 = 𝑎 → (𝑁 = (𝑚 · 𝑛) ↔ 𝑁 = (𝑎 · 𝑛)))
50		oveq2 7371	. . . . . . . 8 ⊢ (𝑛 = 𝑏 → (𝑎 · 𝑛) = (𝑎 · 𝑏))
51	50	eqeq2d 2751	. . . . . . 7 ⊢ (𝑛 = 𝑏 → (𝑁 = (𝑎 · 𝑛) ↔ 𝑁 = (𝑎 · 𝑏)))
52		simplrl 782	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑎 ∈ (2..^𝑁) ∧ 𝑏 ∈ (2..^𝑁))) ∧ 𝑁 = (𝑎 · 𝑏)) → 𝑎 ∈ (2..^𝑁))
53		simplrr 783	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑎 ∈ (2..^𝑁) ∧ 𝑏 ∈ (2..^𝑁))) ∧ 𝑁 = (𝑎 · 𝑏)) → 𝑏 ∈ (2..^𝑁))
54		simpr 485	. . . . . . 7 ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ (𝑎 ∈ (2..^𝑁) ∧ 𝑏 ∈ (2..^𝑁))) ∧ 𝑁 = (𝑎 · 𝑏)) → 𝑁 = (𝑎 · 𝑏))
55	49, 51, 52, 53, 54	2rspcedvdw 3581	. . . . . 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 413	. . . . 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 491	. . . 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 3197	. . 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 213	. 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 280	1 ⊢ (𝑁 ∈ (ℤ≥‘4) → (𝑁 ∉ ℙ ↔ ∃𝑎 ∈ (2..^𝑁)∃𝑏 ∈ (2..^𝑁)(𝑎 ≤ 𝑏 ∧ 𝑁 = (𝑎 · 𝑏))))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ∉ wnel 3039  ∃wrex 3064   class class class wbr 5079  ‘cfv 6492  (class class class)co 7363  ℝcr 11035   · cmul 11041   ≤ cle 11178  ℕcn 12172  2c2 12234  4c4 12236  ℤ_≥cuz 12786  ..^cfzo 13606  ℙcprime 16638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113  ax-pre-sup 11114

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-2o 8403  df-er 8640  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-sup 9352  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-div 11806  df-nn 12173  df-2 12242  df-3 12243  df-4 12244  df-n0 12436  df-z 12523  df-uz 12787  df-rp 12941  df-fz 13460  df-fzo 13607  df-seq 13962  df-exp 14022  df-cj 15059  df-re 15060  df-im 15061  df-sqrt 15195  df-abs 15196  df-dvds 16220  df-prm 16639

This theorem is referenced by:  nprmmul3  48005