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Metamath Proof Explorer

Theorem fz0to5un2tp

Description: An integer range from 0 to 5 is the union of two triples. (Contributed by AV, 30-Jul-2025)

Ref Expression
Assertion fz0to5un2tp ( 0 ... 5 ) = ( { 0 , 1 , 2 } ∪ { 3 , 4 , 5 } )

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 5nn0 ⊢ 5 ∈ ℕ0
13 12 nn0zi ⊢ 5 ∈ ℤ
14 2re ⊢ 2 ∈ ℝ
15 5re ⊢ 5 ∈ ℝ
16 2lt5 ⊢ 2 < 5
17 14 15 16 ltleii ⊢ 2 ≤ 5
18 eluz2 ⊢ ( 5 ∈ ( ℤ≥ ‘ 2 ) ↔ ( 2 ∈ ℤ ∧ 5 ∈ ℤ ∧ 2 ≤ 5 ) )
19 11 13 17 18 mpbir3an ⊢ 5 ∈ ( ℤ≥ ‘ 2 )
20 fzsplit2 ⊢ ( ( ( 2 + 1 ) ∈ ( ℤ≥ ‘ 0 ) ∧ 5 ∈ ( ℤ≥ ‘ 2 ) ) → ( 0 ... 5 ) = ( ( 0 ... 2 ) ∪ ( ( 2 + 1 ) ... 5 ) ) )
21 10 19 20 mp2an ⊢ ( 0 ... 5 ) = ( ( 0 ... 2 ) ∪ ( ( 2 + 1 ) ... 5 ) )
22 fz0tp ⊢ ( 0 ... 2 ) = { 0 , 1 , 2 }
23 1 oveq1i ⊢ ( ( 2 + 1 ) ... 5 ) = ( 3 ... 5 )
24 3p2e5 ⊢ ( 3 + 2 ) = 5
25 24 eqcomi ⊢ 5 = ( 3 + 2 )
26 25 oveq2i ⊢ ( 3 ... 5 ) = ( 3 ... ( 3 + 2 ) )
27 fztp ⊢ ( 3 ∈ ℤ → ( 3 ... ( 3 + 2 ) ) = { 3 , ( 3 + 1 ) , ( 3 + 2 ) } )
28 3 27 ax-mp ⊢ ( 3 ... ( 3 + 2 ) ) = { 3 , ( 3 + 1 ) , ( 3 + 2 ) }
29 eqid ⊢ 3 = 3
30 id ⊢ ( 3 = 3 → 3 = 3 )
31 3p1e4 ⊢ ( 3 + 1 ) = 4
32 31 a1i ⊢ ( 3 = 3 → ( 3 + 1 ) = 4 )
33 24 a1i ⊢ ( 3 = 3 → ( 3 + 2 ) = 5 )
34 30 32 33 tpeq123d ⊢ ( 3 = 3 → { 3 , ( 3 + 1 ) , ( 3 + 2 ) } = { 3 , 4 , 5 } )
35 29 34 ax-mp ⊢ { 3 , ( 3 + 1 ) , ( 3 + 2 ) } = { 3 , 4 , 5 }
36 26 28 35 3eqtri ⊢ ( 3 ... 5 ) = { 3 , 4 , 5 }
37 23 36 eqtri ⊢ ( ( 2 + 1 ) ... 5 ) = { 3 , 4 , 5 }
38 22 37 uneq12i ⊢ ( ( 0 ... 2 ) ∪ ( ( 2 + 1 ) ... 5 ) ) = ( { 0 , 1 , 2 } ∪ { 3 , 4 , 5 } )
39 21 38 eqtri ⊢ ( 0 ... 5 ) = ( { 0 , 1 , 2 } ∪ { 3 , 4 , 5 } )