Theorem ulmi

Description: The uniform limit property. (Contributed by Mario Carneiro, 27-Feb-2015)

	Ref	Expression
Hypotheses	ulm2.z	⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
	ulm2.m	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
	ulm2.f	⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ( ℂ ↑m 𝑆 ) )
	ulm2.b	⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆 ) ) → ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) = 𝐵 )
	ulm2.a	⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝑆 ) → ( 𝐺 ‘ 𝑧 ) = 𝐴 )
	ulmi.u	⊢ ( 𝜑 → 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 )
	ulmi.c	⊢ ( 𝜑 → 𝐶 ∈ ℝ+ )
Assertion	ulmi	⊢ ( 𝜑 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		ulm2.z	⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
2		ulm2.m	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
3		ulm2.f	⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ( ℂ ↑m 𝑆 ) )
4		ulm2.b	⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆 ) ) → ( ( 𝐹 ‘ 𝑘 ) ‘ 𝑧 ) = 𝐵 )
5		ulm2.a	⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝑆 ) → ( 𝐺 ‘ 𝑧 ) = 𝐴 )
6		ulmi.u	⊢ ( 𝜑 → 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 )
7		ulmi.c	⊢ ( 𝜑 → 𝐶 ∈ ℝ+ )
8		breq2	⊢ ( 𝑥 = 𝐶 → ( ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝑥 ↔ ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝐶 ) )
9	8	ralbidv	⊢ ( 𝑥 = 𝐶 → ( ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝑥 ↔ ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝐶 ) )
10	9	rexralbidv	⊢ ( 𝑥 = 𝐶 → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝑥 ↔ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝐶 ) )
11		ulmcl	⊢ ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 → 𝐺 : 𝑆 ⟶ ℂ )
12	6 11	syl	⊢ ( 𝜑 → 𝐺 : 𝑆 ⟶ ℂ )
13		ulmscl	⊢ ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 → 𝑆 ∈ V )
14	6 13	syl	⊢ ( 𝜑 → 𝑆 ∈ V )
15	1 2 3 4 5 12 14	ulm2	⊢ ( 𝜑 → ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝑥 ) )
16	6 15	mpbid	⊢ ( 𝜑 → ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝑥 )
17	10 16 7	rspcdva	⊢ ( 𝜑 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ∀ 𝑧 ∈ 𝑆 ( abs ‘ ( 𝐵 − 𝐴 ) ) < 𝐶 )