Theorem naddass1

Description: Natural addition of ordinal numbers is associative when the third element is 1. (Contributed by RP, 1-Jan-2025)

	Ref	Expression
Assertion	naddass1	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝐴 +no 𝐵 ) +no 1o ) = ( 𝐴 +no ( 𝐵 +no 1o ) ) )

Proof

Step	Hyp	Ref	Expression
1		naddsuc2	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 +no suc 𝐵 ) = suc ( 𝐴 +no 𝐵 ) )
2		nadd1suc	⊢ ( 𝐵 ∈ On → ( 𝐵 +no 1o ) = suc 𝐵 )
3	2	adantl	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐵 +no 1o ) = suc 𝐵 )
4	3	oveq2d	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 +no ( 𝐵 +no 1o ) ) = ( 𝐴 +no suc 𝐵 ) )
5		naddcl	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 +no 𝐵 ) ∈ On )
6		nadd1suc	⊢ ( ( 𝐴 +no 𝐵 ) ∈ On → ( ( 𝐴 +no 𝐵 ) +no 1o ) = suc ( 𝐴 +no 𝐵 ) )
7	5 6	syl	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝐴 +no 𝐵 ) +no 1o ) = suc ( 𝐴 +no 𝐵 ) )
8	1 4 7	3eqtr4rd	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝐴 +no 𝐵 ) +no 1o ) = ( 𝐴 +no ( 𝐵 +no 1o ) ) )