Theorem explt1d

Description: A nonnegative real number less than one raised to a positive integer is less than one. (Contributed by SN, 3-Jul-2025)

	Ref	Expression
Hypotheses	explt1d.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	explt1d.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
	explt1d.0	⊢ ( 𝜑 → 0 ≤ 𝐴 )
	explt1d.1	⊢ ( 𝜑 → 𝐴 < 1 )
Assertion	explt1d	⊢ ( 𝜑 → ( 𝐴 ↑ 𝑁 ) < 1 )

Proof

Step	Hyp	Ref	Expression
1		explt1d.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		explt1d.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
3		explt1d.0	⊢ ( 𝜑 → 0 ≤ 𝐴 )
4		explt1d.1	⊢ ( 𝜑 → 𝐴 < 1 )
5		oveq1	⊢ ( 𝐴 = 0 → ( 𝐴 ↑ 𝑁 ) = ( 0 ↑ 𝑁 ) )
6	5	breq1d	⊢ ( 𝐴 = 0 → ( ( 𝐴 ↑ 𝑁 ) < ( 1 ↑ 𝑁 ) ↔ ( 0 ↑ 𝑁 ) < ( 1 ↑ 𝑁 ) ) )
7	1	adantr	⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → 𝐴 ∈ ℝ )
8	3	adantr	⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → 0 ≤ 𝐴 )
9		simpr	⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → 𝐴 ≠ 0 )
10	7 8 9	ne0gt0d	⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → 0 < 𝐴 )
11	7 10	elrpd	⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → 𝐴 ∈ ℝ+ )
12		simpr	⊢ ( ( 𝜑 ∧ 𝐴 ∈ ℝ+ ) → 𝐴 ∈ ℝ+ )
13		1rp	⊢ 1 ∈ ℝ+
14	13	a1i	⊢ ( ( 𝜑 ∧ 𝐴 ∈ ℝ+ ) → 1 ∈ ℝ+ )
15	2	adantr	⊢ ( ( 𝜑 ∧ 𝐴 ∈ ℝ+ ) → 𝑁 ∈ ℕ )
16	4	adantr	⊢ ( ( 𝜑 ∧ 𝐴 ∈ ℝ+ ) → 𝐴 < 1 )
17	12 14 15 16	ltexp1dd	⊢ ( ( 𝜑 ∧ 𝐴 ∈ ℝ+ ) → ( 𝐴 ↑ 𝑁 ) < ( 1 ↑ 𝑁 ) )
18	11 17	syldan	⊢ ( ( 𝜑 ∧ 𝐴 ≠ 0 ) → ( 𝐴 ↑ 𝑁 ) < ( 1 ↑ 𝑁 ) )
19		0lt1	⊢ 0 < 1
20	19	a1i	⊢ ( 𝜑 → 0 < 1 )
21	2	0expd	⊢ ( 𝜑 → ( 0 ↑ 𝑁 ) = 0 )
22	2	nnzd	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
23		1exp	⊢ ( 𝑁 ∈ ℤ → ( 1 ↑ 𝑁 ) = 1 )
24	22 23	syl	⊢ ( 𝜑 → ( 1 ↑ 𝑁 ) = 1 )
25	20 21 24	3brtr4d	⊢ ( 𝜑 → ( 0 ↑ 𝑁 ) < ( 1 ↑ 𝑁 ) )
26	6 18 25	pm2.61ne	⊢ ( 𝜑 → ( 𝐴 ↑ 𝑁 ) < ( 1 ↑ 𝑁 ) )
27	26 24	breqtrd	⊢ ( 𝜑 → ( 𝐴 ↑ 𝑁 ) < 1 )