Metamath Proof Explorer

Theorem iimulcn

Description: Multiplication is a continuous function on the unit interval. (Contributed by Mario Carneiro, 8-Jun-2014) Avoid ax-mulf . (Revised by GG, 16-Mar-2025)

Ref Expression
Assertion iimulcn ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 · 𝑦 ) ) ∈ ( ( II ×t II ) Cn II )

Proof

Step Hyp Ref Expression
1 eqid ⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld )
2 1 dfii3 ⊢ II = ( ( TopOpen ‘ ℂfld ) ↾t ( 0 [,] 1 ) )
3 1 cnfldtopon ⊢ ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ )
4 3 a1i ⊢ ( ⊤ → ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ ) )
5 unitsscn ⊢ ( 0 [,] 1 ) ⊆ ℂ
6 5 a1i ⊢ ( ⊤ → ( 0 [,] 1 ) ⊆ ℂ )
7 1 mpomulcn ⊢ ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 · 𝑦 ) ) ∈ ( ( ( TopOpen ‘ ℂfld ) ×t ( TopOpen ‘ ℂfld ) ) Cn ( TopOpen ‘ ℂfld ) )
8 7 a1i ⊢ ( ⊤ → ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 · 𝑦 ) ) ∈ ( ( ( TopOpen ‘ ℂfld ) ×t ( TopOpen ‘ ℂfld ) ) Cn ( TopOpen ‘ ℂfld ) ) )
9 2 4 6 2 4 6 8 cnmpt2res ⊢ ( ⊤ → ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 · 𝑦 ) ) ∈ ( ( II ×t II ) Cn ( TopOpen ‘ ℂfld ) ) )
10 9 mptru ⊢ ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 · 𝑦 ) ) ∈ ( ( II ×t II ) Cn ( TopOpen ‘ ℂfld ) )
11 iimulcl ⊢ ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] 1 ) ) → ( 𝑥 · 𝑦 ) ∈ ( 0 [,] 1 ) )
12 11 rgen2 ⊢ ∀ 𝑥 ∈ ( 0 [,] 1 ) ∀ 𝑦 ∈ ( 0 [,] 1 ) ( 𝑥 · 𝑦 ) ∈ ( 0 [,] 1 )
13 eqid ⊢ ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 · 𝑦 ) ) = ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 · 𝑦 ) )
14 13 fmpo ⊢ ( ∀ 𝑥 ∈ ( 0 [,] 1 ) ∀ 𝑦 ∈ ( 0 [,] 1 ) ( 𝑥 · 𝑦 ) ∈ ( 0 [,] 1 ) ↔ ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 · 𝑦 ) ) : ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ⟶ ( 0 [,] 1 ) )
15 frn ⊢ ( ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 · 𝑦 ) ) : ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ⟶ ( 0 [,] 1 ) → ran ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 · 𝑦 ) ) ⊆ ( 0 [,] 1 ) )
16 14 15 sylbi ⊢ ( ∀ 𝑥 ∈ ( 0 [,] 1 ) ∀ 𝑦 ∈ ( 0 [,] 1 ) ( 𝑥 · 𝑦 ) ∈ ( 0 [,] 1 ) → ran ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 · 𝑦 ) ) ⊆ ( 0 [,] 1 ) )
17 12 16 ax-mp ⊢ ran ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 · 𝑦 ) ) ⊆ ( 0 [,] 1 )
18 cnrest2 ⊢ ( ( ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ ) ∧ ran ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 · 𝑦 ) ) ⊆ ( 0 [,] 1 ) ∧ ( 0 [,] 1 ) ⊆ ℂ ) → ( ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 · 𝑦 ) ) ∈ ( ( II ×t II ) Cn ( TopOpen ‘ ℂfld ) ) ↔ ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 · 𝑦 ) ) ∈ ( ( II ×t II ) Cn ( ( TopOpen ‘ ℂfld ) ↾t ( 0 [,] 1 ) ) ) ) )
19 3 17 5 18 mp3an ⊢ ( ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 · 𝑦 ) ) ∈ ( ( II ×t II ) Cn ( TopOpen ‘ ℂfld ) ) ↔ ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 · 𝑦 ) ) ∈ ( ( II ×t II ) Cn ( ( TopOpen ‘ ℂfld ) ↾t ( 0 [,] 1 ) ) ) )
20 10 19 mpbi ⊢ ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 · 𝑦 ) ) ∈ ( ( II ×t II ) Cn ( ( TopOpen ‘ ℂfld ) ↾t ( 0 [,] 1 ) ) )
21 2 oveq2i ⊢ ( ( II ×t II ) Cn II ) = ( ( II ×t II ) Cn ( ( TopOpen ‘ ℂfld ) ↾t ( 0 [,] 1 ) ) )
22 20 21 eleqtrri ⊢ ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 · 𝑦 ) ) ∈ ( ( II ×t II ) Cn II )