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Metamath Proof Explorer

Theorem 1fzopredsuc

Description: Join 0 and a successor to the beginning and the end of an open integer interval starting at 1. (Contributed by AV, 14-Jul-2020)

Ref Expression
Assertion 1fzopredsuc ( 𝑁 ∈ ℕ0 → ( 0 ... 𝑁 ) = ( ( { 0 } ∪ ( 1 ..^ 𝑁 ) ) ∪ { 𝑁 } ) )

Proof

Step Hyp Ref Expression
1 elnn0uz ⊢ ( 𝑁 ∈ ℕ0 ↔ 𝑁 ∈ ( ℤ≥ ‘ 0 ) )
2 fzopredsuc ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 0 ) → ( 0 ... 𝑁 ) = ( ( { 0 } ∪ ( ( 0 + 1 ) ..^ 𝑁 ) ) ∪ { 𝑁 } ) )
3 0p1e1 ⊢ ( 0 + 1 ) = 1
4 3 oveq1i ⊢ ( ( 0 + 1 ) ..^ 𝑁 ) = ( 1 ..^ 𝑁 )
5 4 uneq2i ⊢ ( { 0 } ∪ ( ( 0 + 1 ) ..^ 𝑁 ) ) = ( { 0 } ∪ ( 1 ..^ 𝑁 ) )
6 5 uneq1i ⊢ ( ( { 0 } ∪ ( ( 0 + 1 ) ..^ 𝑁 ) ) ∪ { 𝑁 } ) = ( ( { 0 } ∪ ( 1 ..^ 𝑁 ) ) ∪ { 𝑁 } )
7 2 6 eqtrdi ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 0 ) → ( 0 ... 𝑁 ) = ( ( { 0 } ∪ ( 1 ..^ 𝑁 ) ) ∪ { 𝑁 } ) )
8 1 7 sylbi ⊢ ( 𝑁 ∈ ℕ0 → ( 0 ... 𝑁 ) = ( ( { 0 } ∪ ( 1 ..^ 𝑁 ) ) ∪ { 𝑁 } ) )