Theorem exps0

Description: Surreal exponentiation to zero. (Contributed by Scott Fenton, 24-Jul-2025)

	Ref	Expression
Assertion	exps0	⊢ ( 𝐴 ∈ No → ( 𝐴 ↑s 0s ) = 1s )

Proof

Step	Hyp	Ref	Expression
1		0zs	⊢ 0s ∈ ℤs
2		expsval	⊢ ( ( 𝐴 ∈ No ∧ 0s ∈ ℤs ) → ( 𝐴 ↑s 0s ) = if ( 0s = 0s , 1s , if ( 0s <s 0s , ( seqs 1s ( ·s , ( ℕs × { 𝐴 } ) ) ‘ 0s ) , ( 1s /su ( seqs 1s ( ·s , ( ℕs × { 𝐴 } ) ) ‘ ( -us ‘ 0s ) ) ) ) ) )
3	1 2	mpan2	⊢ ( 𝐴 ∈ No → ( 𝐴 ↑s 0s ) = if ( 0s = 0s , 1s , if ( 0s <s 0s , ( seqs 1s ( ·s , ( ℕs × { 𝐴 } ) ) ‘ 0s ) , ( 1s /su ( seqs 1s ( ·s , ( ℕs × { 𝐴 } ) ) ‘ ( -us ‘ 0s ) ) ) ) ) )
4		eqid	⊢ 0s = 0s
5	4	iftruei	⊢ if ( 0s = 0s , 1s , if ( 0s <s 0s , ( seqs 1s ( ·s , ( ℕs × { 𝐴 } ) ) ‘ 0s ) , ( 1s /su ( seqs 1s ( ·s , ( ℕs × { 𝐴 } ) ) ‘ ( -us ‘ 0s ) ) ) ) ) = 1s
6	3 5	eqtrdi	⊢ ( 𝐴 ∈ No → ( 𝐴 ↑s 0s ) = 1s )