Metamath Proof Explorer

Theorem ulmclm

Description: A uniform limit of functions converges pointwise. (Contributed by Mario Carneiro, 27-Feb-2015)

Ref Expression
Hypotheses ulmclm.z ⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
ulmclm.m ⊢ ( 𝜑 → 𝑀 ∈ ℤ )
ulmclm.f ⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ( ℂ ↑m 𝑆 ) )
ulmclm.a ⊢ ( 𝜑 → 𝐴 ∈ 𝑆 )
ulmclm.h ⊢ ( 𝜑 → 𝐻 ∈ 𝑊 )
ulmclm.e ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( ( 𝐹 ‘ 𝑘 ) ‘ 𝐴 ) = ( 𝐻 ‘ 𝑘 ) )
ulmclm.u ⊢ ( 𝜑 → 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 )
Assertion ulmclm ( 𝜑 → 𝐻 ⇝ ( 𝐺 ‘ 𝐴 ) )

Proof

Step Hyp Ref Expression
1 ulmclm.z ⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
2 ulmclm.m ⊢ ( 𝜑 → 𝑀 ∈ ℤ )
3 ulmclm.f ⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ( ℂ ↑m 𝑆 ) )
4 ulmclm.a ⊢ ( 𝜑 → 𝐴 ∈ 𝑆 )
5 ulmclm.h ⊢ ( 𝜑 → 𝐻 ∈ 𝑊 )
6 ulmclm.e ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( ( 𝐹 ‘ 𝑘 ) ‘ 𝐴 ) = ( 𝐻 ‘ 𝑘 ) )
7 ulmclm.u ⊢ ( 𝜑 → 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 )
8 fveq2 ⊢ ( 𝑧 = 𝐴 → ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) = ( ( 𝐹 ‘ 𝑘 ) ‘ 𝐴 ) )
9 fveq2 ⊢ ( 𝑧 = 𝐴 → ( 𝐺 ‘ 𝑧 ) = ( 𝐺 ‘ 𝐴 ) )
10 8 9 oveq12d ⊢ ( 𝑧 = 𝐴 → ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) = ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝐴 ) − ( 𝐺 ‘ 𝐴 ) ) )
11 10 fveq2d ⊢ ( 𝑧 = 𝐴 → ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) = ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝐴 ) − ( 𝐺 ‘ 𝐴 ) ) ) )
12 11 breq1d ⊢ ( 𝑧 = 𝐴 → ( ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 𝑥 ↔ ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝐴 ) − ( 𝐺 ‘ 𝐴 ) ) ) < 𝑥 ) )
13 12 rspcv ⊢ ( 𝐴 ∈ 𝑆 → ( ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 𝑥 → ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝐴 ) − ( 𝐺 ‘ 𝐴 ) ) ) < 𝑥 ) )
14 4 13 syl ⊢ ( 𝜑 → ( ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 𝑥 → ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝐴 ) − ( 𝐺 ‘ 𝐴 ) ) ) < 𝑥 ) )
15 14 ralimdv ⊢ ( 𝜑 → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 𝑥 → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝐴 ) − ( 𝐺 ‘ 𝐴 ) ) ) < 𝑥 ) )
16 15 reximdv ⊢ ( 𝜑 → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 𝑥 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝐴 ) − ( 𝐺 ‘ 𝐴 ) ) ) < 𝑥 ) )
17 16 ralimdv ⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 𝑥 → ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝐴 ) − ( 𝐺 ‘ 𝐴 ) ) ) < 𝑥 ) )
18 eqidd ⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆 ) ) → ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) = ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) )
19 eqidd ⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝑆 ) → ( 𝐺 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑧 ) )
20 ulmcl ⊢ ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 → 𝐺 : 𝑆 ⟶ ℂ )
21 7 20 syl ⊢ ( 𝜑 → 𝐺 : 𝑆 ⟶ ℂ )
22 ulmscl ⊢ ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 → 𝑆 ∈ V )
23 7 22 syl ⊢ ( 𝜑 → 𝑆 ∈ V )
24 1 2 3 18 19 21 23 ulm2 ⊢ ( 𝜑 → ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) − ( 𝐺 ‘ 𝑧 ) ) ) < 𝑥 ) )
25 6 eqcomd ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐻 ‘ 𝑘 ) = ( ( 𝐹 ‘ 𝑘 ) ‘ 𝐴 ) )
26 21 4 ffvelcdmd ⊢ ( 𝜑 → ( 𝐺 ‘ 𝐴 ) ∈ ℂ )
27 3 ffvelcdmda ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ( ℂ ↑m 𝑆 ) )
28 elmapi ⊢ ( ( 𝐹 ‘ 𝑘 ) ∈ ( ℂ ↑m 𝑆 ) → ( 𝐹 ‘ 𝑘 ) : 𝑆 ⟶ ℂ )
29 27 28 syl ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) : 𝑆 ⟶ ℂ )
30 4 adantr ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → 𝐴 ∈ 𝑆 )
31 29 30 ffvelcdmd ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( ( 𝐹 ‘ 𝑘 ) ‘ 𝐴 ) ∈ ℂ )
32 1 2 5 25 26 31 clim2c ⊢ ( 𝜑 → ( 𝐻 ⇝ ( 𝐺 ‘ 𝐴 ) ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( abs ‘ ( ( ( 𝐹 ‘ 𝑘 ) ‘ 𝐴 ) − ( 𝐺 ‘ 𝐴 ) ) ) < 𝑥 ) )
33 17 24 32 3imtr4d ⊢ ( 𝜑 → ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 → 𝐻 ⇝ ( 𝐺 ‘ 𝐴 ) ) )
34 7 33 mpd ⊢ ( 𝜑 → 𝐻 ⇝ ( 𝐺 ‘ 𝐴 ) )