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Theorem ulmval 24179
Description: Express the predicate: The sequence of functions 𝐹 converges uniformly to 𝐺 on 𝑆. (Contributed by Mario Carneiro, 26-Feb-2015.)
Assertion
Ref Expression
ulmval (𝑆𝑉 → (𝐹(⇝𝑢𝑆)𝐺 ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥)))
Distinct variable groups:   𝑗,𝑘,𝑛,𝑥,𝑧,𝐹   𝑗,𝐺,𝑘,𝑛,𝑥,𝑧   𝑆,𝑗,𝑘,𝑛,𝑥,𝑧   𝑛,𝑉
Allowed substitution hints:   𝑉(𝑥,𝑧,𝑗,𝑘)

Proof of Theorem ulmval
Dummy variables 𝑓 𝑦 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ulmrel 24177 . . . 4 Rel (⇝𝑢𝑆)
2 brrelex12 5189 . . . 4 ((Rel (⇝𝑢𝑆) ∧ 𝐹(⇝𝑢𝑆)𝐺) → (𝐹 ∈ V ∧ 𝐺 ∈ V))
31, 2mpan 706 . . 3 (𝐹(⇝𝑢𝑆)𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
43a1i 11 . 2 (𝑆𝑉 → (𝐹(⇝𝑢𝑆)𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V)))
5 3simpa 1078 . . . 4 ((𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥) → (𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝐺:𝑆⟶ℂ))
6 fvex 6239 . . . . . . 7 (ℤ𝑛) ∈ V
7 fex 6530 . . . . . . 7 ((𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ (ℤ𝑛) ∈ V) → 𝐹 ∈ V)
86, 7mpan2 707 . . . . . 6 (𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) → 𝐹 ∈ V)
98a1i 11 . . . . 5 (𝑆𝑉 → (𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) → 𝐹 ∈ V))
10 fex 6530 . . . . . 6 ((𝐺:𝑆⟶ℂ ∧ 𝑆𝑉) → 𝐺 ∈ V)
1110expcom 450 . . . . 5 (𝑆𝑉 → (𝐺:𝑆⟶ℂ → 𝐺 ∈ V))
129, 11anim12d 585 . . . 4 (𝑆𝑉 → ((𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝐺:𝑆⟶ℂ) → (𝐹 ∈ V ∧ 𝐺 ∈ V)))
135, 12syl5 34 . . 3 (𝑆𝑉 → ((𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥) → (𝐹 ∈ V ∧ 𝐺 ∈ V)))
1413rexlimdvw 3063 . 2 (𝑆𝑉 → (∃𝑛 ∈ ℤ (𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥) → (𝐹 ∈ V ∧ 𝐺 ∈ V)))
15 elex 3243 . . . . . 6 (𝑆𝑉𝑆 ∈ V)
16 simpr1 1087 . . . . . . . . . . . . 13 ((𝑆𝑉 ∧ (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥)) → 𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆))
17 uzssz 11745 . . . . . . . . . . . . 13 (ℤ𝑛) ⊆ ℤ
18 ovex 6718 . . . . . . . . . . . . . 14 (ℂ ↑𝑚 𝑆) ∈ V
19 zex 11424 . . . . . . . . . . . . . 14 ℤ ∈ V
20 elpm2r 7917 . . . . . . . . . . . . . 14 ((((ℂ ↑𝑚 𝑆) ∈ V ∧ ℤ ∈ V) ∧ (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ (ℤ𝑛) ⊆ ℤ)) → 𝑓 ∈ ((ℂ ↑𝑚 𝑆) ↑pm ℤ))
2118, 19, 20mpanl12 718 . . . . . . . . . . . . 13 ((𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ (ℤ𝑛) ⊆ ℤ) → 𝑓 ∈ ((ℂ ↑𝑚 𝑆) ↑pm ℤ))
2216, 17, 21sylancl 695 . . . . . . . . . . . 12 ((𝑆𝑉 ∧ (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥)) → 𝑓 ∈ ((ℂ ↑𝑚 𝑆) ↑pm ℤ))
23 simpr2 1088 . . . . . . . . . . . . 13 ((𝑆𝑉 ∧ (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥)) → 𝑦:𝑆⟶ℂ)
24 cnex 10055 . . . . . . . . . . . . . 14 ℂ ∈ V
25 simpl 472 . . . . . . . . . . . . . 14 ((𝑆𝑉 ∧ (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥)) → 𝑆𝑉)
26 elmapg 7912 . . . . . . . . . . . . . 14 ((ℂ ∈ V ∧ 𝑆𝑉) → (𝑦 ∈ (ℂ ↑𝑚 𝑆) ↔ 𝑦:𝑆⟶ℂ))
2724, 25, 26sylancr 696 . . . . . . . . . . . . 13 ((𝑆𝑉 ∧ (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥)) → (𝑦 ∈ (ℂ ↑𝑚 𝑆) ↔ 𝑦:𝑆⟶ℂ))
2823, 27mpbird 247 . . . . . . . . . . . 12 ((𝑆𝑉 ∧ (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥)) → 𝑦 ∈ (ℂ ↑𝑚 𝑆))
2922, 28jca 553 . . . . . . . . . . 11 ((𝑆𝑉 ∧ (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥)) → (𝑓 ∈ ((ℂ ↑𝑚 𝑆) ↑pm ℤ) ∧ 𝑦 ∈ (ℂ ↑𝑚 𝑆)))
3029ex 449 . . . . . . . . . 10 (𝑆𝑉 → ((𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥) → (𝑓 ∈ ((ℂ ↑𝑚 𝑆) ↑pm ℤ) ∧ 𝑦 ∈ (ℂ ↑𝑚 𝑆))))
3130rexlimdvw 3063 . . . . . . . . 9 (𝑆𝑉 → (∃𝑛 ∈ ℤ (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥) → (𝑓 ∈ ((ℂ ↑𝑚 𝑆) ↑pm ℤ) ∧ 𝑦 ∈ (ℂ ↑𝑚 𝑆))))
3231ssopab2dv 5033 . . . . . . . 8 (𝑆𝑉 → {⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥)} ⊆ {⟨𝑓, 𝑦⟩ ∣ (𝑓 ∈ ((ℂ ↑𝑚 𝑆) ↑pm ℤ) ∧ 𝑦 ∈ (ℂ ↑𝑚 𝑆))})
33 df-xp 5149 . . . . . . . 8 (((ℂ ↑𝑚 𝑆) ↑pm ℤ) × (ℂ ↑𝑚 𝑆)) = {⟨𝑓, 𝑦⟩ ∣ (𝑓 ∈ ((ℂ ↑𝑚 𝑆) ↑pm ℤ) ∧ 𝑦 ∈ (ℂ ↑𝑚 𝑆))}
3432, 33syl6sseqr 3685 . . . . . . 7 (𝑆𝑉 → {⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥)} ⊆ (((ℂ ↑𝑚 𝑆) ↑pm ℤ) × (ℂ ↑𝑚 𝑆)))
35 ovex 6718 . . . . . . . . 9 ((ℂ ↑𝑚 𝑆) ↑pm ℤ) ∈ V
3635, 18xpex 7004 . . . . . . . 8 (((ℂ ↑𝑚 𝑆) ↑pm ℤ) × (ℂ ↑𝑚 𝑆)) ∈ V
3736ssex 4835 . . . . . . 7 ({⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥)} ⊆ (((ℂ ↑𝑚 𝑆) ↑pm ℤ) × (ℂ ↑𝑚 𝑆)) → {⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥)} ∈ V)
3834, 37syl 17 . . . . . 6 (𝑆𝑉 → {⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥)} ∈ V)
39 oveq2 6698 . . . . . . . . . . 11 (𝑠 = 𝑆 → (ℂ ↑𝑚 𝑠) = (ℂ ↑𝑚 𝑆))
4039feq3d 6070 . . . . . . . . . 10 (𝑠 = 𝑆 → (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑠) ↔ 𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆)))
41 feq2 6065 . . . . . . . . . 10 (𝑠 = 𝑆 → (𝑦:𝑠⟶ℂ ↔ 𝑦:𝑆⟶ℂ))
42 raleq 3168 . . . . . . . . . . . 12 (𝑠 = 𝑆 → (∀𝑧𝑠 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥 ↔ ∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥))
4342rexralbidv 3087 . . . . . . . . . . 11 (𝑠 = 𝑆 → (∃𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑠 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥 ↔ ∃𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥))
4443ralbidv 3015 . . . . . . . . . 10 (𝑠 = 𝑆 → (∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑠 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥 ↔ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥))
4540, 41, 443anbi123d 1439 . . . . . . . . 9 (𝑠 = 𝑆 → ((𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑠) ∧ 𝑦:𝑠⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑠 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥) ↔ (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥)))
4645rexbidv 3081 . . . . . . . 8 (𝑠 = 𝑆 → (∃𝑛 ∈ ℤ (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑠) ∧ 𝑦:𝑠⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑠 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥) ↔ ∃𝑛 ∈ ℤ (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥)))
4746opabbidv 4749 . . . . . . 7 (𝑠 = 𝑆 → {⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑠) ∧ 𝑦:𝑠⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑠 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥)} = {⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥)})
48 df-ulm 24176 . . . . . . 7 𝑢 = (𝑠 ∈ V ↦ {⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑠) ∧ 𝑦:𝑠⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑠 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥)})
4947, 48fvmptg 6319 . . . . . 6 ((𝑆 ∈ V ∧ {⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥)} ∈ V) → (⇝𝑢𝑆) = {⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥)})
5015, 38, 49syl2anc 694 . . . . 5 (𝑆𝑉 → (⇝𝑢𝑆) = {⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥)})
5150breqd 4696 . . . 4 (𝑆𝑉 → (𝐹(⇝𝑢𝑆)𝐺𝐹{⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥)}𝐺))
52 simpl 472 . . . . . . . 8 ((𝑓 = 𝐹𝑦 = 𝐺) → 𝑓 = 𝐹)
5352feq1d 6068 . . . . . . 7 ((𝑓 = 𝐹𝑦 = 𝐺) → (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ↔ 𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆)))
54 simpr 476 . . . . . . . 8 ((𝑓 = 𝐹𝑦 = 𝐺) → 𝑦 = 𝐺)
5554feq1d 6068 . . . . . . 7 ((𝑓 = 𝐹𝑦 = 𝐺) → (𝑦:𝑆⟶ℂ ↔ 𝐺:𝑆⟶ℂ))
5652fveq1d 6231 . . . . . . . . . . . . . 14 ((𝑓 = 𝐹𝑦 = 𝐺) → (𝑓𝑘) = (𝐹𝑘))
5756fveq1d 6231 . . . . . . . . . . . . 13 ((𝑓 = 𝐹𝑦 = 𝐺) → ((𝑓𝑘)‘𝑧) = ((𝐹𝑘)‘𝑧))
5854fveq1d 6231 . . . . . . . . . . . . 13 ((𝑓 = 𝐹𝑦 = 𝐺) → (𝑦𝑧) = (𝐺𝑧))
5957, 58oveq12d 6708 . . . . . . . . . . . 12 ((𝑓 = 𝐹𝑦 = 𝐺) → (((𝑓𝑘)‘𝑧) − (𝑦𝑧)) = (((𝐹𝑘)‘𝑧) − (𝐺𝑧)))
6059fveq2d 6233 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑦 = 𝐺) → (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) = (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))))
6160breq1d 4695 . . . . . . . . . 10 ((𝑓 = 𝐹𝑦 = 𝐺) → ((abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥 ↔ (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥))
6261ralbidv 3015 . . . . . . . . 9 ((𝑓 = 𝐹𝑦 = 𝐺) → (∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥 ↔ ∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥))
6362rexralbidv 3087 . . . . . . . 8 ((𝑓 = 𝐹𝑦 = 𝐺) → (∃𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥 ↔ ∃𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥))
6463ralbidv 3015 . . . . . . 7 ((𝑓 = 𝐹𝑦 = 𝐺) → (∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥 ↔ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥))
6553, 55, 643anbi123d 1439 . . . . . 6 ((𝑓 = 𝐹𝑦 = 𝐺) → ((𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥) ↔ (𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥)))
6665rexbidv 3081 . . . . 5 ((𝑓 = 𝐹𝑦 = 𝐺) → (∃𝑛 ∈ ℤ (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥) ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥)))
67 eqid 2651 . . . . 5 {⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥)} = {⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥)}
6866, 67brabga 5018 . . . 4 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹{⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥)}𝐺 ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥)))
6951, 68sylan9bb 736 . . 3 ((𝑆𝑉 ∧ (𝐹 ∈ V ∧ 𝐺 ∈ V)) → (𝐹(⇝𝑢𝑆)𝐺 ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥)))
7069ex 449 . 2 (𝑆𝑉 → ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹(⇝𝑢𝑆)𝐺 ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥))))
714, 14, 70pm5.21ndd 368 1 (𝑆𝑉 → (𝐹(⇝𝑢𝑆)𝐺 ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  wss 3607   class class class wbr 4685  {copab 4745   × cxp 5141  Rel wrel 5148  wf 5922  cfv 5926  (class class class)co 6690  𝑚 cmap 7899  pm cpm 7900  cc 9972   < clt 10112  cmin 10304  cz 11415  cuz 11725  +crp 11870  abscabs 14018  𝑢culm 24175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-pm 7902  df-neg 10307  df-z 11416  df-uz 11726  df-ulm 24176
This theorem is referenced by:  ulmcl  24180  ulmf  24181  ulm2  24184
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