Theorem fzo0to3tp

Description: A half-open integer range from 0 to 3 is an unordered triple. (Contributed by Alexander van der Vekens, 9-Nov-2017)

	Ref	Expression
Assertion	fzo0to3tp	|- ( 0 ..^ 3 ) = { 0 , 1 , 2 }

Proof

Step	Hyp	Ref	Expression
1		3z	|- 3 e. ZZ
2		fzoval	|- ( 3 e. ZZ -> ( 0 ..^ 3 ) = ( 0 ... ( 3 - 1 ) ) )
3	1 2	ax-mp	|- ( 0 ..^ 3 ) = ( 0 ... ( 3 - 1 ) )
4		3m1e2	|- ( 3 - 1 ) = 2
5		2cn	|- 2 e. CC
6	5	addid2i	|- ( 0 + 2 ) = 2
7	4 6	eqtr4i	|- ( 3 - 1 ) = ( 0 + 2 )
8	7	oveq2i	|- ( 0 ... ( 3 - 1 ) ) = ( 0 ... ( 0 + 2 ) )
9		0z	|- 0 e. ZZ
10		fztp	|- ( 0 e. ZZ -> ( 0 ... ( 0 + 2 ) ) = { 0 , ( 0 + 1 ) , ( 0 + 2 ) } )
11		eqidd	|- ( 0 e. ZZ -> 0 = 0 )
12		0p1e1	|- ( 0 + 1 ) = 1
13	12	a1i	|- ( 0 e. ZZ -> ( 0 + 1 ) = 1 )
14	6	a1i	|- ( 0 e. ZZ -> ( 0 + 2 ) = 2 )
15	11 13 14	tpeq123d	|- ( 0 e. ZZ -> { 0 , ( 0 + 1 ) , ( 0 + 2 ) } = { 0 , 1 , 2 } )
16	10 15	eqtrd	|- ( 0 e. ZZ -> ( 0 ... ( 0 + 2 ) ) = { 0 , 1 , 2 } )
17	9 16	ax-mp	|- ( 0 ... ( 0 + 2 ) ) = { 0 , 1 , 2 }
18	3 8 17	3eqtri	|- ( 0 ..^ 3 ) = { 0 , 1 , 2 }