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Metamath Proof Explorer

Theorem fzo1to4tp

Description: A half-open integer range from 1 to 4 is an unordered triple. (Contributed by AV, 28-Jul-2021)

Ref Expression
Assertion fzo1to4tp ( 1 ..^ 4 ) = { 1 , 2 , 3 }

Proof

Step Hyp Ref Expression
1 4z ⊢ 4 ∈ ℤ
2 fzoval ⊢ ( 4 ∈ ℤ → ( 1 ..^ 4 ) = ( 1 ... ( 4 − 1 ) ) )
3 1 2 ax-mp ⊢ ( 1 ..^ 4 ) = ( 1 ... ( 4 − 1 ) )
4 4m1e3 ⊢ ( 4 − 1 ) = 3
5 df-3 ⊢ 3 = ( 2 + 1 )
6 2cn ⊢ 2 ∈ ℂ
7 ax-1cn ⊢ 1 ∈ ℂ
8 6 7 addcomi ⊢ ( 2 + 1 ) = ( 1 + 2 )
9 4 5 8 3eqtri ⊢ ( 4 − 1 ) = ( 1 + 2 )
10 9 oveq2i ⊢ ( 1 ... ( 4 − 1 ) ) = ( 1 ... ( 1 + 2 ) )
11 1z ⊢ 1 ∈ ℤ
12 fztp ⊢ ( 1 ∈ ℤ → ( 1 ... ( 1 + 2 ) ) = { 1 , ( 1 + 1 ) , ( 1 + 2 ) } )
13 eqidd ⊢ ( 1 ∈ ℤ → 1 = 1 )
14 1p1e2 ⊢ ( 1 + 1 ) = 2
15 14 a1i ⊢ ( 1 ∈ ℤ → ( 1 + 1 ) = 2 )
16 1p2e3 ⊢ ( 1 + 2 ) = 3
17 16 a1i ⊢ ( 1 ∈ ℤ → ( 1 + 2 ) = 3 )
18 13 15 17 tpeq123d ⊢ ( 1 ∈ ℤ → { 1 , ( 1 + 1 ) , ( 1 + 2 ) } = { 1 , 2 , 3 } )
19 12 18 eqtrd ⊢ ( 1 ∈ ℤ → ( 1 ... ( 1 + 2 ) ) = { 1 , 2 , 3 } )
20 11 19 ax-mp ⊢ ( 1 ... ( 1 + 2 ) ) = { 1 , 2 , 3 }
21 3 10 20 3eqtri ⊢ ( 1 ..^ 4 ) = { 1 , 2 , 3 }