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 ∈ ℤ
2		fzoval	⊢ ( 3 ∈ ℤ → ( 0 ..^ 3 ) = ( 0 ... ( 3 − 1 ) ) )
3	1 2	ax-mp	⊢ ( 0 ..^ 3 ) = ( 0 ... ( 3 − 1 ) )
4		3m1e2	⊢ ( 3 − 1 ) = 2
5		2cn	⊢ 2 ∈ ℂ
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 ∈ ℤ
10		fztp	⊢ ( 0 ∈ ℤ → ( 0 ... ( 0 + 2 ) ) = { 0 , ( 0 + 1 ) , ( 0 + 2 ) } )
11		eqidd	⊢ ( 0 ∈ ℤ → 0 = 0 )
12		0p1e1	⊢ ( 0 + 1 ) = 1
13	12	a1i	⊢ ( 0 ∈ ℤ → ( 0 + 1 ) = 1 )
14	6	a1i	⊢ ( 0 ∈ ℤ → ( 0 + 2 ) = 2 )
15	11 13 14	tpeq123d	⊢ ( 0 ∈ ℤ → { 0 , ( 0 + 1 ) , ( 0 + 2 ) } = { 0 , 1 , 2 } )
16	10 15	eqtrd	⊢ ( 0 ∈ ℤ → ( 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 }