Metamath Proof Explorer


Theorem exp11nnd

Description: sq11d for positive real bases and positive integer exponents. The base cannot be generalized much further, since if N is even then we have A ^ N = -u A ^ N . (Contributed by SN, 14-Sep-2023)

Ref Expression
Hypotheses exp11nnd.1 ( 𝜑𝐴 ∈ ℝ+ )
exp11nnd.2 ( 𝜑𝐵 ∈ ℝ+ )
exp11nnd.3 ( 𝜑𝑁 ∈ ℕ )
exp11nnd.4 ( 𝜑 → ( 𝐴𝑁 ) = ( 𝐵𝑁 ) )
Assertion exp11nnd ( 𝜑𝐴 = 𝐵 )

Proof

Step Hyp Ref Expression
1 exp11nnd.1 ( 𝜑𝐴 ∈ ℝ+ )
2 exp11nnd.2 ( 𝜑𝐵 ∈ ℝ+ )
3 exp11nnd.3 ( 𝜑𝑁 ∈ ℕ )
4 exp11nnd.4 ( 𝜑 → ( 𝐴𝑁 ) = ( 𝐵𝑁 ) )
5 1 rpred ( 𝜑𝐴 ∈ ℝ )
6 3 nnnn0d ( 𝜑𝑁 ∈ ℕ0 )
7 5 6 reexpcld ( 𝜑 → ( 𝐴𝑁 ) ∈ ℝ )
8 2 rpred ( 𝜑𝐵 ∈ ℝ )
9 8 6 reexpcld ( 𝜑 → ( 𝐵𝑁 ) ∈ ℝ )
10 7 9 lttri3d ( 𝜑 → ( ( 𝐴𝑁 ) = ( 𝐵𝑁 ) ↔ ( ¬ ( 𝐴𝑁 ) < ( 𝐵𝑁 ) ∧ ¬ ( 𝐵𝑁 ) < ( 𝐴𝑁 ) ) ) )
11 4 10 mpbid ( 𝜑 → ( ¬ ( 𝐴𝑁 ) < ( 𝐵𝑁 ) ∧ ¬ ( 𝐵𝑁 ) < ( 𝐴𝑁 ) ) )
12 1 2 3 ltexp1d ( 𝜑 → ( 𝐴 < 𝐵 ↔ ( 𝐴𝑁 ) < ( 𝐵𝑁 ) ) )
13 12 notbid ( 𝜑 → ( ¬ 𝐴 < 𝐵 ↔ ¬ ( 𝐴𝑁 ) < ( 𝐵𝑁 ) ) )
14 2 1 3 ltexp1d ( 𝜑 → ( 𝐵 < 𝐴 ↔ ( 𝐵𝑁 ) < ( 𝐴𝑁 ) ) )
15 14 notbid ( 𝜑 → ( ¬ 𝐵 < 𝐴 ↔ ¬ ( 𝐵𝑁 ) < ( 𝐴𝑁 ) ) )
16 13 15 anbi12d ( 𝜑 → ( ( ¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴 ) ↔ ( ¬ ( 𝐴𝑁 ) < ( 𝐵𝑁 ) ∧ ¬ ( 𝐵𝑁 ) < ( 𝐴𝑁 ) ) ) )
17 11 16 mpbird ( 𝜑 → ( ¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴 ) )
18 5 8 lttri3d ( 𝜑 → ( 𝐴 = 𝐵 ↔ ( ¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴 ) ) )
19 17 18 mpbird ( 𝜑𝐴 = 𝐵 )