Theorem pw2divsidd

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

	Ref	Expression
Hypothesis	pw2divs0d.1	$$⊢\phi \to N\in {\mathbb{N}}_{\mathrm{0s}}$$
Assertion	pw2divsidd	Could not format assertion : No typesetting found for |- ( ph -> ( ( 2s ^su N ) /su ( 2s ^su N ) ) = 1s ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		pw2divs0d.1	$$⊢\phi \to N\in {\mathbb{N}}_{\mathrm{0s}}$$
2		2sno	Could not format 2s e. No : No typesetting found for |- 2s e. No with typecode |-
3		expscl	Could not format ( ( 2s e. No /\ N e. NN0_s ) -> ( 2s ^su N ) e. No ) : No typesetting found for |- ( ( 2s e. No /\ N e. NN0_s ) -> ( 2s ^su N ) e. No ) with typecode |-
4	2 1 3	sylancr	Could not format ( ph -> ( 2s ^su N ) e. No ) : No typesetting found for |- ( ph -> ( 2s ^su N ) e. No ) with typecode |-
5	4	mulsridd	Could not format ( ph -> ( ( 2s ^su N ) x.s 1s ) = ( 2s ^su N ) ) : No typesetting found for |- ( ph -> ( ( 2s ^su N ) x.s 1s ) = ( 2s ^su N ) ) with typecode |-
6		1sno	$$⊢1_s\in \mathrm{No}$$
7	6	a1i	$$⊢\phi \to 1_s\in \mathrm{No}$$
8	4 7 1	pw2divsmuld	Could not format ( ph -> ( ( ( 2s ^su N ) /su ( 2s ^su N ) ) = 1s <-> ( ( 2s ^su N ) x.s 1s ) = ( 2s ^su N ) ) ) : No typesetting found for |- ( ph -> ( ( ( 2s ^su N ) /su ( 2s ^su N ) ) = 1s <-> ( ( 2s ^su N ) x.s 1s ) = ( 2s ^su N ) ) ) with typecode |-
9	5 8	mpbird	Could not format ( ph -> ( ( 2s ^su N ) /su ( 2s ^su N ) ) = 1s ) : No typesetting found for |- ( ph -> ( ( 2s ^su N ) /su ( 2s ^su N ) ) = 1s ) with typecode |-