Theorem divgcdz

Description: An integer divided by the gcd of it and a nonzero integer is an integer. (Contributed by AV, 11-Jul-2021)

	Ref	Expression
Assertion	divgcdz	|- ( ( A e. ZZ /\ B e. ZZ /\ B =/= 0 ) -> ( A / ( A gcd B ) ) e. ZZ )

Proof

Step	Hyp	Ref	Expression
1		gcddvds	|- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) )
2	1	3adant3	|- ( ( A e. ZZ /\ B e. ZZ /\ B =/= 0 ) -> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) )
3	2	simpld	|- ( ( A e. ZZ /\ B e. ZZ /\ B =/= 0 ) -> ( A gcd B ) || A )
4		gcd2n0cl	|- ( ( A e. ZZ /\ B e. ZZ /\ B =/= 0 ) -> ( A gcd B ) e. NN )
5		nnz	|- ( ( A gcd B ) e. NN -> ( A gcd B ) e. ZZ )
6		nnne0	|- ( ( A gcd B ) e. NN -> ( A gcd B ) =/= 0 )
7	5 6	jca	|- ( ( A gcd B ) e. NN -> ( ( A gcd B ) e. ZZ /\ ( A gcd B ) =/= 0 ) )
8	4 7	syl	|- ( ( A e. ZZ /\ B e. ZZ /\ B =/= 0 ) -> ( ( A gcd B ) e. ZZ /\ ( A gcd B ) =/= 0 ) )
9		simp1	|- ( ( A e. ZZ /\ B e. ZZ /\ B =/= 0 ) -> A e. ZZ )
10		df-3an	|- ( ( ( A gcd B ) e. ZZ /\ ( A gcd B ) =/= 0 /\ A e. ZZ ) <-> ( ( ( A gcd B ) e. ZZ /\ ( A gcd B ) =/= 0 ) /\ A e. ZZ ) )
11	8 9 10	sylanbrc	|- ( ( A e. ZZ /\ B e. ZZ /\ B =/= 0 ) -> ( ( A gcd B ) e. ZZ /\ ( A gcd B ) =/= 0 /\ A e. ZZ ) )
12		dvdsval2	|- ( ( ( A gcd B ) e. ZZ /\ ( A gcd B ) =/= 0 /\ A e. ZZ ) -> ( ( A gcd B ) || A <-> ( A / ( A gcd B ) ) e. ZZ ) )
13	11 12	syl	|- ( ( A e. ZZ /\ B e. ZZ /\ B =/= 0 ) -> ( ( A gcd B ) || A <-> ( A / ( A gcd B ) ) e. ZZ ) )
14	3 13	mpbid	|- ( ( A e. ZZ /\ B e. ZZ /\ B =/= 0 ) -> ( A / ( A gcd B ) ) e. ZZ )