Theorem fzm1ndvds

Description: No number between 1 and M - 1 divides M . (Contributed by Mario Carneiro, 24-Jan-2015)

	Ref	Expression
Assertion	fzm1ndvds	⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ( 1 ... ( 𝑀 − 1 ) ) ) → ¬ 𝑀 ∥ 𝑁 )

Proof

Step	Hyp	Ref	Expression
1		elfzle2	⊢ ( 𝑁 ∈ ( 1 ... ( 𝑀 − 1 ) ) → 𝑁 ≤ ( 𝑀 − 1 ) )
2	1	adantl	⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ( 1 ... ( 𝑀 − 1 ) ) ) → 𝑁 ≤ ( 𝑀 − 1 ) )
3		elfzelz	⊢ ( 𝑁 ∈ ( 1 ... ( 𝑀 − 1 ) ) → 𝑁 ∈ ℤ )
4	3	adantl	⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ( 1 ... ( 𝑀 − 1 ) ) ) → 𝑁 ∈ ℤ )
5		nnz	⊢ ( 𝑀 ∈ ℕ → 𝑀 ∈ ℤ )
6	5	adantr	⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ( 1 ... ( 𝑀 − 1 ) ) ) → 𝑀 ∈ ℤ )
7		zltlem1	⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ) → ( 𝑁 < 𝑀 ↔ 𝑁 ≤ ( 𝑀 − 1 ) ) )
8	4 6 7	syl2anc	⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ( 1 ... ( 𝑀 − 1 ) ) ) → ( 𝑁 < 𝑀 ↔ 𝑁 ≤ ( 𝑀 − 1 ) ) )
9	2 8	mpbird	⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ( 1 ... ( 𝑀 − 1 ) ) ) → 𝑁 < 𝑀 )
10		elfznn	⊢ ( 𝑁 ∈ ( 1 ... ( 𝑀 − 1 ) ) → 𝑁 ∈ ℕ )
11	10	adantl	⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ( 1 ... ( 𝑀 − 1 ) ) ) → 𝑁 ∈ ℕ )
12	11	nnred	⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ( 1 ... ( 𝑀 − 1 ) ) ) → 𝑁 ∈ ℝ )
13		nnre	⊢ ( 𝑀 ∈ ℕ → 𝑀 ∈ ℝ )
14	13	adantr	⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ( 1 ... ( 𝑀 − 1 ) ) ) → 𝑀 ∈ ℝ )
15	12 14	ltnled	⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ( 1 ... ( 𝑀 − 1 ) ) ) → ( 𝑁 < 𝑀 ↔ ¬ 𝑀 ≤ 𝑁 ) )
16	9 15	mpbid	⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ( 1 ... ( 𝑀 − 1 ) ) ) → ¬ 𝑀 ≤ 𝑁 )
17		dvdsle	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ ) → ( 𝑀 ∥ 𝑁 → 𝑀 ≤ 𝑁 ) )
18	6 11 17	syl2anc	⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ( 1 ... ( 𝑀 − 1 ) ) ) → ( 𝑀 ∥ 𝑁 → 𝑀 ≤ 𝑁 ) )
19	16 18	mtod	⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ( 1 ... ( 𝑀 − 1 ) ) ) → ¬ 𝑀 ∥ 𝑁 )