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Metamath Proof Explorer

Theorem rlimcn2

Description: Image of a limit under a continuous map, two-arg version. (Contributed by Mario Carneiro, 17-Sep-2014)

Ref Expression
Hypotheses rlimcn2.1a ⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → 𝐵 ∈ 𝑋 )
rlimcn2.1b ⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → 𝐶 ∈ 𝑌 )
rlimcn2.2a ⊢ ( 𝜑 → 𝑅 ∈ 𝑋 )
rlimcn2.2b ⊢ ( 𝜑 → 𝑆 ∈ 𝑌 )
rlimcn2.3a ⊢ ( 𝜑 → ( 𝑧 ∈ 𝐴 ↦ 𝐵 ) ⇝𝑟 𝑅 )
rlimcn2.3b ⊢ ( 𝜑 → ( 𝑧 ∈ 𝐴 ↦ 𝐶 ) ⇝𝑟 𝑆 )
rlimcn2.4 ⊢ ( 𝜑 → 𝐹 : ( 𝑋 × 𝑌 ) ⟶ ℂ )
rlimcn2.5 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ∃ 𝑟 ∈ ℝ+ ∃ 𝑠 ∈ ℝ+ ∀ 𝑢 ∈ 𝑋 ∀ 𝑣 ∈ 𝑌 ( ( ( abs ‘ ( 𝑢 − 𝑅 ) ) < 𝑟 ∧ ( abs ‘ ( 𝑣 − 𝑆 ) ) < 𝑠 ) → ( abs ‘ ( ( 𝑢 𝐹 𝑣 ) − ( 𝑅 𝐹 𝑆 ) ) ) < 𝑥 ) )
Assertion rlimcn2 ( 𝜑 → ( 𝑧 ∈ 𝐴 ↦ ( 𝐵 𝐹 𝐶 ) ) ⇝𝑟 ( 𝑅 𝐹 𝑆 ) )

Proof

Step Hyp Ref Expression
1 rlimcn2.1a ⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → 𝐵 ∈ 𝑋 )
2 rlimcn2.1b ⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → 𝐶 ∈ 𝑌 )
3 rlimcn2.2a ⊢ ( 𝜑 → 𝑅 ∈ 𝑋 )
4 rlimcn2.2b ⊢ ( 𝜑 → 𝑆 ∈ 𝑌 )
5 rlimcn2.3a ⊢ ( 𝜑 → ( 𝑧 ∈ 𝐴 ↦ 𝐵 ) ⇝𝑟 𝑅 )
6 rlimcn2.3b ⊢ ( 𝜑 → ( 𝑧 ∈ 𝐴 ↦ 𝐶 ) ⇝𝑟 𝑆 )
7 rlimcn2.4 ⊢ ( 𝜑 → 𝐹 : ( 𝑋 × 𝑌 ) ⟶ ℂ )
8 rlimcn2.5 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ∃ 𝑟 ∈ ℝ+ ∃ 𝑠 ∈ ℝ+ ∀ 𝑢 ∈ 𝑋 ∀ 𝑣 ∈ 𝑌 ( ( ( abs ‘ ( 𝑢 − 𝑅 ) ) < 𝑟 ∧ ( abs ‘ ( 𝑣 − 𝑆 ) ) < 𝑠 ) → ( abs ‘ ( ( 𝑢 𝐹 𝑣 ) − ( 𝑅 𝐹 𝑆 ) ) ) < 𝑥 ) )
9 7 adantr ⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → 𝐹 : ( 𝑋 × 𝑌 ) ⟶ ℂ )
10 9 1 2 fovcdmd ⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → ( 𝐵 𝐹 𝐶 ) ∈ ℂ )
11 7 3 4 fovcdmd ⊢ ( 𝜑 → ( 𝑅 𝐹 𝑆 ) ∈ ℂ )
12 1 2 10 11 5 6 8 rlimcn3 ⊢ ( 𝜑 → ( 𝑧 ∈ 𝐴 ↦ ( 𝐵 𝐹 𝐶 ) ) ⇝𝑟 ( 𝑅 𝐹 𝑆 ) )