Theorem dvdsexpad

Description: Deduction associated with dvdsexpim . (Contributed by SN, 21-Aug-2024)

	Ref	Expression
Hypotheses	dvdsexpad.1	|- ( ph -> A e. ZZ )
	dvdsexpad.2	|- ( ph -> B e. ZZ )
	dvdsexpad.3	|- ( ph -> N e. NN0 )
	dvdsexpad.5	|- ( ph -> A || B )
Assertion	dvdsexpad	|- ( ph -> ( A ^ N ) || ( B ^ N ) )

Proof

Step	Hyp	Ref	Expression
1		dvdsexpad.1	|- ( ph -> A e. ZZ )
2		dvdsexpad.2	|- ( ph -> B e. ZZ )
3		dvdsexpad.3	|- ( ph -> N e. NN0 )
4		dvdsexpad.5	|- ( ph -> A || B )
5		dvdsexpim	|- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> ( A || B -> ( A ^ N ) || ( B ^ N ) ) )
6	1 2 3 5	syl3anc	|- ( ph -> ( A || B -> ( A ^ N ) || ( B ^ N ) ) )
7	4 6	mpd	|- ( ph -> ( A ^ N ) || ( B ^ N ) )