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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