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Metamath Proof Explorer

Theorem fz0to4untppr

Description: An integer range from 0 to 4 is the union of a triple and a pair. (Contributed by Alexander van der Vekens, 13-Aug-2017) (Proof shortened by AV, 7-Aug-2025)

Ref Expression
Assertion fz0to4untppr ( 0 ... 4 ) = ( { 0 , 1 , 2 } ∪ { 3 , 4 } )

Proof

Step Hyp Ref Expression
1 2p1e3 ⊢ ( 2 + 1 ) = 3
2 0z ⊢ 0 ∈ ℤ
3 3z ⊢ 3 ∈ ℤ
4 0re ⊢ 0 ∈ ℝ
5 3re ⊢ 3 ∈ ℝ
6 3pos ⊢ 0 < 3
7 4 5 6 ltleii ⊢ 0 ≤ 3
8 eluz2 ⊢ ( 3 ∈ ( ℤ≥ ‘ 0 ) ↔ ( 0 ∈ ℤ ∧ 3 ∈ ℤ ∧ 0 ≤ 3 ) )
9 2 3 7 8 mpbir3an ⊢ 3 ∈ ( ℤ≥ ‘ 0 )
10 1 9 eqeltri ⊢ ( 2 + 1 ) ∈ ( ℤ≥ ‘ 0 )
11 2z ⊢ 2 ∈ ℤ
12 4z ⊢ 4 ∈ ℤ
13 2re ⊢ 2 ∈ ℝ
14 4re ⊢ 4 ∈ ℝ
15 2lt4 ⊢ 2 < 4
16 13 14 15 ltleii ⊢ 2 ≤ 4
17 eluz2 ⊢ ( 4 ∈ ( ℤ≥ ‘ 2 ) ↔ ( 2 ∈ ℤ ∧ 4 ∈ ℤ ∧ 2 ≤ 4 ) )
18 11 12 16 17 mpbir3an ⊢ 4 ∈ ( ℤ≥ ‘ 2 )
19 fzsplit2 ⊢ ( ( ( 2 + 1 ) ∈ ( ℤ≥ ‘ 0 ) ∧ 4 ∈ ( ℤ≥ ‘ 2 ) ) → ( 0 ... 4 ) = ( ( 0 ... 2 ) ∪ ( ( 2 + 1 ) ... 4 ) ) )
20 10 18 19 mp2an ⊢ ( 0 ... 4 ) = ( ( 0 ... 2 ) ∪ ( ( 2 + 1 ) ... 4 ) )
21 fz0tp ⊢ ( 0 ... 2 ) = { 0 , 1 , 2 }
22 1 oveq1i ⊢ ( ( 2 + 1 ) ... 4 ) = ( 3 ... 4 )
23 df-4 ⊢ 4 = ( 3 + 1 )
24 23 oveq2i ⊢ ( 3 ... 4 ) = ( 3 ... ( 3 + 1 ) )
25 fzpr ⊢ ( 3 ∈ ℤ → ( 3 ... ( 3 + 1 ) ) = { 3 , ( 3 + 1 ) } )
26 3 25 ax-mp ⊢ ( 3 ... ( 3 + 1 ) ) = { 3 , ( 3 + 1 ) }
27 3p1e4 ⊢ ( 3 + 1 ) = 4
28 27 preq2i ⊢ { 3 , ( 3 + 1 ) } = { 3 , 4 }
29 24 26 28 3eqtri ⊢ ( 3 ... 4 ) = { 3 , 4 }
30 22 29 eqtri ⊢ ( ( 2 + 1 ) ... 4 ) = { 3 , 4 }
31 21 30 uneq12i ⊢ ( ( 0 ... 2 ) ∪ ( ( 2 + 1 ) ... 4 ) ) = ( { 0 , 1 , 2 } ∪ { 3 , 4 } )
32 20 31 eqtri ⊢ ( 0 ... 4 ) = ( { 0 , 1 , 2 } ∪ { 3 , 4 } )