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 e. On /\ B e. On ) -> ( ( A +no B ) +no 1o ) = ( A +no ( B +no 1o ) ) )

Proof

Step	Hyp	Ref	Expression
1		naddsuc2	|- ( ( A e. On /\ B e. On ) -> ( A +no suc B ) = suc ( A +no B ) )
2		nadd1suc	|- ( B e. On -> ( B +no 1o ) = suc B )
3	2	adantl	|- ( ( A e. On /\ B e. On ) -> ( B +no 1o ) = suc B )
4	3	oveq2d	|- ( ( A e. On /\ B e. On ) -> ( A +no ( B +no 1o ) ) = ( A +no suc B ) )
5		naddcl	|- ( ( A e. On /\ B e. On ) -> ( A +no B ) e. On )
6		nadd1suc	|- ( ( A +no B ) e. On -> ( ( A +no B ) +no 1o ) = suc ( A +no B ) )
7	5 6	syl	|- ( ( A e. On /\ B e. On ) -> ( ( A +no B ) +no 1o ) = suc ( A +no B ) )
8	1 4 7	3eqtr4rd	|- ( ( A e. On /\ B e. On ) -> ( ( A +no B ) +no 1o ) = ( A +no ( B +no 1o ) ) )