Theorem hashnnsuc

Description: The # function on _om turns successor into adding 1. (Contributed by Eric Schmidt, 7-Jul-2026)

	Ref	Expression
Assertion	hashnnsuc	|- ( A e. _om -> ( # ` suc A ) = ( ( # ` A ) + 1 ) )

Proof

Step	Hyp	Ref	Expression
1		nnon	|- ( A e. _om -> A e. On )
2		oa1suc	|- ( A e. On -> ( A +o 1o ) = suc A )
3	1 2	syl	|- ( A e. _om -> ( A +o 1o ) = suc A )
4	3	fveq2d	|- ( A e. _om -> ( # ` ( A +o 1o ) ) = ( # ` suc A ) )
5		1onn	|- 1o e. _om
6		hashnna	|- ( ( A e. _om /\ 1o e. _om ) -> ( # ` ( A +o 1o ) ) = ( ( # ` A ) + ( # ` 1o ) ) )
7	5 6	mpan2	|- ( A e. _om -> ( # ` ( A +o 1o ) ) = ( ( # ` A ) + ( # ` 1o ) ) )
8	4 7	eqtr3d	|- ( A e. _om -> ( # ` suc A ) = ( ( # ` A ) + ( # ` 1o ) ) )
9		hash1	|- ( # ` 1o ) = 1
10	9	oveq2i	|- ( ( # ` A ) + ( # ` 1o ) ) = ( ( # ` A ) + 1 )
11	8 10	eqtrdi	|- ( A e. _om -> ( # ` suc A ) = ( ( # ` A ) + 1 ) )