Theorem pw2divsidd

Description: Identity law for division over powers of two. (Contributed by Scott Fenton, 21-Feb-2026)

	Ref	Expression
Hypothesis	pw2divs0d.1	|- ( ph -> N e. NN0_s )
Assertion	pw2divsidd	|- ( ph -> ( ( 2s ^su N ) /su ( 2s ^su N ) ) = 1s )

Proof

Step	Hyp	Ref	Expression
1		pw2divs0d.1	|- ( ph -> N e. NN0_s )
2		2sno	|- 2s e. No
3		expscl	|- ( ( 2s e. No /\ N e. NN0_s ) -> ( 2s ^su N ) e. No )
4	2 1 3	sylancr	|- ( ph -> ( 2s ^su N ) e. No )
5	4	mulsridd	|- ( ph -> ( ( 2s ^su N ) x.s 1s ) = ( 2s ^su N ) )
6		1sno	|- 1s e. No
7	6	a1i	|- ( ph -> 1s e. No )
8	4 7 1	pw2divsmuld	|- ( ph -> ( ( ( 2s ^su N ) /su ( 2s ^su N ) ) = 1s <-> ( ( 2s ^su N ) x.s 1s ) = ( 2s ^su N ) ) )
9	5 8	mpbird	|- ( ph -> ( ( 2s ^su N ) /su ( 2s ^su N ) ) = 1s )