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Metamath Proof Explorer

Theorem xrtgcntopre

Description: The standard topologies on the extended reals and on the complex numbers, coincide when restricted to the reals. (Contributed by Glauco Siliprandi, 5-Feb-2022)

Ref Expression
Assertion xrtgcntopre ⊢ ordTop ⁡ ≤ ↾ 𝑡 ℝ = TopOpen ⁡ ℂ fld ↾ 𝑡 ℝ

Proof

Step Hyp Ref Expression
1 eqid ⊢ ordTop ⁡ ≤ ↾ 𝑡 ℝ = ordTop ⁡ ≤ ↾ 𝑡 ℝ
2 1 xrtgioo ⊢ topGen ⁡ ran ⁡ . = ordTop ⁡ ≤ ↾ 𝑡 ℝ
3 tgioo4 ⊢ topGen ⁡ ran ⁡ . = TopOpen ⁡ ℂ fld ↾ 𝑡 ℝ
4 2 3 eqtr3i ⊢ ordTop ⁡ ≤ ↾ 𝑡 ℝ = TopOpen ⁡ ℂ fld ↾ 𝑡 ℝ