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Theorem ulmval 23883
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 23881 . . . 4 Rel (⇝𝑢𝑆)
2 brrelex12 5069 . . . 4 ((Rel (⇝𝑢𝑆) ∧ 𝐹(⇝𝑢𝑆)𝐺) → (𝐹 ∈ V ∧ 𝐺 ∈ V))
31, 2mpan 701 . . 3 (𝐹(⇝𝑢𝑆)𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
43a1i 11 . 2 (𝑆𝑉 → (𝐹(⇝𝑢𝑆)𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V)))
5 3simpa 1050 . . . 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 6098 . . . . . . 7 (ℤ𝑛) ∈ V
7 fex 6372 . . . . . . 7 ((𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ (ℤ𝑛) ∈ V) → 𝐹 ∈ V)
86, 7mpan2 702 . . . . . 6 (𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) → 𝐹 ∈ V)
98a1i 11 . . . . 5 (𝑆𝑉 → (𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) → 𝐹 ∈ V))
10 fex 6372 . . . . . 6 ((𝐺:𝑆⟶ℂ ∧ 𝑆𝑉) → 𝐺 ∈ V)
1110expcom 449 . . . . 5 (𝑆𝑉 → (𝐺:𝑆⟶ℂ → 𝐺 ∈ V))
129, 11anim12d 583 . . . 4 (𝑆𝑉 → ((𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝐺:𝑆⟶ℂ) → (𝐹 ∈ V ∧ 𝐺 ∈ V)))
135, 12syl5 33 . . 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 3015 . 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 3184 . . . . . 6 (𝑆𝑉𝑆 ∈ V)
16 simpr1 1059 . . . . . . . . . . . . 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 11542 . . . . . . . . . . . . 13 (ℤ𝑛) ⊆ ℤ
18 ovex 6555 . . . . . . . . . . . . . 14 (ℂ ↑𝑚 𝑆) ∈ V
19 zex 11222 . . . . . . . . . . . . . 14 ℤ ∈ V
20 elpm2r 7739 . . . . . . . . . . . . . 14 ((((ℂ ↑𝑚 𝑆) ∈ V ∧ ℤ ∈ V) ∧ (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ (ℤ𝑛) ⊆ ℤ)) → 𝑓 ∈ ((ℂ ↑𝑚 𝑆) ↑pm ℤ))
2118, 19, 20mpanl12 713 . . . . . . . . . . . . 13 ((𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ (ℤ𝑛) ⊆ ℤ) → 𝑓 ∈ ((ℂ ↑𝑚 𝑆) ↑pm ℤ))
2216, 17, 21sylancl 692 . . . . . . . . . . . 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 1060 . . . . . . . . . . . . 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 9874 . . . . . . . . . . . . . 14 ℂ ∈ V
25 simpl 471 . . . . . . . . . . . . . 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 7735 . . . . . . . . . . . . . 14 ((ℂ ∈ V ∧ 𝑆𝑉) → (𝑦 ∈ (ℂ ↑𝑚 𝑆) ↔ 𝑦:𝑆⟶ℂ))
2724, 25, 26sylancr 693 . . . . . . . . . . . . 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 245 . . . . . . . . . . . 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 552 . . . . . . . . . . 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 448 . . . . . . . . . 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 3015 . . . . . . . . 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 4919 . . . . . . . 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 5034 . . . . . . . 8 (((ℂ ↑𝑚 𝑆) ↑pm ℤ) × (ℂ ↑𝑚 𝑆)) = {⟨𝑓, 𝑦⟩ ∣ (𝑓 ∈ ((ℂ ↑𝑚 𝑆) ↑pm ℤ) ∧ 𝑦 ∈ (ℂ ↑𝑚 𝑆))}
3432, 33syl6sseqr 3614 . . . . . . 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 6555 . . . . . . . . 9 ((ℂ ↑𝑚 𝑆) ↑pm ℤ) ∈ V
3635, 18xpex 6838 . . . . . . . 8 (((ℂ ↑𝑚 𝑆) ↑pm ℤ) × (ℂ ↑𝑚 𝑆)) ∈ V
3736ssex 4725 . . . . . . 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 6535 . . . . . . . . . . 11 (𝑠 = 𝑆 → (ℂ ↑𝑚 𝑠) = (ℂ ↑𝑚 𝑆))
4039feq3d 5931 . . . . . . . . . 10 (𝑠 = 𝑆 → (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑠) ↔ 𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆)))
41 feq2 5926 . . . . . . . . . 10 (𝑠 = 𝑆 → (𝑦:𝑠⟶ℂ ↔ 𝑦:𝑆⟶ℂ))
42 raleq 3114 . . . . . . . . . . . 12 (𝑠 = 𝑆 → (∀𝑧𝑠 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥 ↔ ∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥))
4342rexralbidv 3039 . . . . . . . . . . 11 (𝑠 = 𝑆 → (∃𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑠 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥 ↔ ∃𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥))
4443ralbidv 2968 . . . . . . . . . 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 1390 . . . . . . . . 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 3033 . . . . . . . 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 4642 . . . . . . 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 23880 . . . . . . 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 6174 . . . . . 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 690 . . . . 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 4588 . . . 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 471 . . . . . . . 8 ((𝑓 = 𝐹𝑦 = 𝐺) → 𝑓 = 𝐹)
5352feq1d 5929 . . . . . . 7 ((𝑓 = 𝐹𝑦 = 𝐺) → (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ↔ 𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆)))
54 simpr 475 . . . . . . . 8 ((𝑓 = 𝐹𝑦 = 𝐺) → 𝑦 = 𝐺)
5554feq1d 5929 . . . . . . 7 ((𝑓 = 𝐹𝑦 = 𝐺) → (𝑦:𝑆⟶ℂ ↔ 𝐺:𝑆⟶ℂ))
5652fveq1d 6090 . . . . . . . . . . . . . 14 ((𝑓 = 𝐹𝑦 = 𝐺) → (𝑓𝑘) = (𝐹𝑘))
5756fveq1d 6090 . . . . . . . . . . . . 13 ((𝑓 = 𝐹𝑦 = 𝐺) → ((𝑓𝑘)‘𝑧) = ((𝐹𝑘)‘𝑧))
5854fveq1d 6090 . . . . . . . . . . . . 13 ((𝑓 = 𝐹𝑦 = 𝐺) → (𝑦𝑧) = (𝐺𝑧))
5957, 58oveq12d 6545 . . . . . . . . . . . 12 ((𝑓 = 𝐹𝑦 = 𝐺) → (((𝑓𝑘)‘𝑧) − (𝑦𝑧)) = (((𝐹𝑘)‘𝑧) − (𝐺𝑧)))
6059fveq2d 6092 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑦 = 𝐺) → (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) = (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))))
6160breq1d 4587 . . . . . . . . . 10 ((𝑓 = 𝐹𝑦 = 𝐺) → ((abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥 ↔ (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥))
6261ralbidv 2968 . . . . . . . . 9 ((𝑓 = 𝐹𝑦 = 𝐺) → (∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥 ↔ ∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥))
6362rexralbidv 3039 . . . . . . . 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 2968 . . . . . . 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 1390 . . . . . 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 3033 . . . . 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 2609 . . . . 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 4904 . . . 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 731 . . 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 448 . 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 367 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 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wral 2895  wrex 2896  Vcvv 3172  wss 3539   class class class wbr 4577  {copab 4636   × cxp 5026  Rel wrel 5033  wf 5786  cfv 5790  (class class class)co 6527  𝑚 cmap 7722  pm cpm 7723  cc 9791   < clt 9931  cmin 10118  cz 11213  cuz 11522  +crp 11667  abscabs 13771  𝑢culm 23879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-cnex 9849  ax-resscn 9850
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-map 7724  df-pm 7725  df-neg 10121  df-z 11214  df-uz 11523  df-ulm 23880
This theorem is referenced by:  ulmcl  23884  ulmf  23885  ulm2  23888
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