Theorem isdomn5

Description: The right conjunct in the right hand side of the equivalence of isdomn is logically equivalent to a less symmetric version where one of the variables is restricted to be nonzero. (Contributed by SN, 16-Sep-2024)

	Ref	Expression
Assertion	isdomn5	|- ( A. a e. B A. b e. B ( ( a .x. b ) = .0. -> ( a = .0. \/ b = .0. ) ) <-> A. a e. ( B \ { .0. } ) A. b e. B ( ( a .x. b ) = .0. -> b = .0. ) )

Proof

Step	Hyp	Ref	Expression
1		bi2.04	|- ( ( -. a = .0. -> ( ( a .x. b ) = .0. -> b = .0. ) ) <-> ( ( a .x. b ) = .0. -> ( -. a = .0. -> b = .0. ) ) )
2		df-ne	|- ( a =/= .0. <-> -. a = .0. )
3	2	imbi1i	|- ( ( a =/= .0. -> ( ( a .x. b ) = .0. -> b = .0. ) ) <-> ( -. a = .0. -> ( ( a .x. b ) = .0. -> b = .0. ) ) )
4		df-or	|- ( ( a = .0. \/ b = .0. ) <-> ( -. a = .0. -> b = .0. ) )
5	4	imbi2i	|- ( ( ( a .x. b ) = .0. -> ( a = .0. \/ b = .0. ) ) <-> ( ( a .x. b ) = .0. -> ( -. a = .0. -> b = .0. ) ) )
6	1 3 5	3bitr4ri	|- ( ( ( a .x. b ) = .0. -> ( a = .0. \/ b = .0. ) ) <-> ( a =/= .0. -> ( ( a .x. b ) = .0. -> b = .0. ) ) )
7	6	2ralbii	|- ( A. a e. B A. b e. B ( ( a .x. b ) = .0. -> ( a = .0. \/ b = .0. ) ) <-> A. a e. B A. b e. B ( a =/= .0. -> ( ( a .x. b ) = .0. -> b = .0. ) ) )
8		r19.21v	|- ( A. b e. B ( a =/= .0. -> ( ( a .x. b ) = .0. -> b = .0. ) ) <-> ( a =/= .0. -> A. b e. B ( ( a .x. b ) = .0. -> b = .0. ) ) )
9	8	ralbii	|- ( A. a e. B A. b e. B ( a =/= .0. -> ( ( a .x. b ) = .0. -> b = .0. ) ) <-> A. a e. B ( a =/= .0. -> A. b e. B ( ( a .x. b ) = .0. -> b = .0. ) ) )
10		raldifsnb	|- ( A. a e. B ( a =/= .0. -> A. b e. B ( ( a .x. b ) = .0. -> b = .0. ) ) <-> A. a e. ( B \ { .0. } ) A. b e. B ( ( a .x. b ) = .0. -> b = .0. ) )
11	7 9 10	3bitri	|- ( A. a e. B A. b e. B ( ( a .x. b ) = .0. -> ( a = .0. \/ b = .0. ) ) <-> A. a e. ( B \ { .0. } ) A. b e. B ( ( a .x. b ) = .0. -> b = .0. ) )