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Theorem ulmval 26423
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 26421 . . . 4 Rel (⇝𝑢𝑆)
21brrelex12i 5740 . . 3 (𝐹(⇝𝑢𝑆)𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
32a1i 11 . 2 (𝑆𝑉 → (𝐹(⇝𝑢𝑆)𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V)))
4 3simpa 1149 . . . 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))
5 fvex 6919 . . . . . . 7 (ℤ𝑛) ∈ V
6 fex 7246 . . . . . . 7 ((𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆) ∧ (ℤ𝑛) ∈ V) → 𝐹 ∈ V)
75, 6mpan2 691 . . . . . 6 (𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆) → 𝐹 ∈ V)
87a1i 11 . . . . 5 (𝑆𝑉 → (𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆) → 𝐹 ∈ V))
9 fex 7246 . . . . . 6 ((𝐺:𝑆⟶ℂ ∧ 𝑆𝑉) → 𝐺 ∈ V)
109expcom 413 . . . . 5 (𝑆𝑉 → (𝐺:𝑆⟶ℂ → 𝐺 ∈ V))
118, 10anim12d 609 . . . 4 (𝑆𝑉 → ((𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ) → (𝐹 ∈ V ∧ 𝐺 ∈ V)))
124, 11syl5 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)))
1312rexlimdvw 3160 . 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)))
14 df-ulm 26420 . . . . . 6 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)})
15 oveq2 7439 . . . . . . . . . 10 (𝑠 = 𝑆 → (ℂ ↑m 𝑠) = (ℂ ↑m 𝑆))
1615feq3d 6723 . . . . . . . . 9 (𝑠 = 𝑆 → (𝑓:(ℤ𝑛)⟶(ℂ ↑m 𝑠) ↔ 𝑓:(ℤ𝑛)⟶(ℂ ↑m 𝑆)))
17 feq2 6717 . . . . . . . . 9 (𝑠 = 𝑆 → (𝑦:𝑠⟶ℂ ↔ 𝑦:𝑆⟶ℂ))
18 raleq 3323 . . . . . . . . . . 11 (𝑠 = 𝑆 → (∀𝑧𝑠 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥 ↔ ∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥))
1918rexralbidv 3223 . . . . . . . . . 10 (𝑠 = 𝑆 → (∃𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑠 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥 ↔ ∃𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥))
2019ralbidv 3178 . . . . . . . . 9 (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))
2116, 17, 203anbi123d 1438 . . . . . . . 8 (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)))
2221rexbidv 3179 . . . . . . 7 (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)))
2322opabbidv 5209 . . . . . 6 (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)})
24 elex 3501 . . . . . 6 (𝑆𝑉𝑆 ∈ V)
25 simpr1 1195 . . . . . . . . . . . . 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))
26 uzssz 12899 . . . . . . . . . . . . 13 (ℤ𝑛) ⊆ ℤ
27 ovex 7464 . . . . . . . . . . . . . 14 (ℂ ↑m 𝑆) ∈ V
28 zex 12622 . . . . . . . . . . . . . 14 ℤ ∈ V
29 elpm2r 8885 . . . . . . . . . . . . . 14 ((((ℂ ↑m 𝑆) ∈ V ∧ ℤ ∈ V) ∧ (𝑓:(ℤ𝑛)⟶(ℂ ↑m 𝑆) ∧ (ℤ𝑛) ⊆ ℤ)) → 𝑓 ∈ ((ℂ ↑m 𝑆) ↑pm ℤ))
3027, 28, 29mpanl12 702 . . . . . . . . . . . . 13 ((𝑓:(ℤ𝑛)⟶(ℂ ↑m 𝑆) ∧ (ℤ𝑛) ⊆ ℤ) → 𝑓 ∈ ((ℂ ↑m 𝑆) ↑pm ℤ))
3125, 26, 30sylancl 586 . . . . . . . . . . . 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))
32 simpr2 1196 . . . . . . . . . . . . 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)
33 cnex 11236 . . . . . . . . . . . . . 14 ℂ ∈ V
34 simpl 482 . . . . . . . . . . . . . 14 ((𝑆𝑉 ∧ (𝑓:(ℤ𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥)) → 𝑆𝑉)
35 elmapg 8879 . . . . . . . . . . . . . 14 ((ℂ ∈ V ∧ 𝑆𝑉) → (𝑦 ∈ (ℂ ↑m 𝑆) ↔ 𝑦:𝑆⟶ℂ))
3633, 34, 35sylancr 587 . . . . . . . . . . . . 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))
3732, 36mpbird 257 . . . . . . . . . . . 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))
3831, 37jca 511 . . . . . . . . . . 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)))
3938ex 412 . . . . . . . . . 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))))
4039rexlimdvw 3160 . . . . . . . . 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))))
4140ssopab2dv 5556 . . . . . . . 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))})
42 df-xp 5691 . . . . . . . 8 (((ℂ ↑m 𝑆) ↑pm ℤ) × (ℂ ↑m 𝑆)) = {⟨𝑓, 𝑦⟩ ∣ (𝑓 ∈ ((ℂ ↑m 𝑆) ↑pm ℤ) ∧ 𝑦 ∈ (ℂ ↑m 𝑆))}
4341, 42sseqtrrdi 4025 . . . . . . 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)))
44 ovex 7464 . . . . . . . . 9 ((ℂ ↑m 𝑆) ↑pm ℤ) ∈ V
4544, 27xpex 7773 . . . . . . . 8 (((ℂ ↑m 𝑆) ↑pm ℤ) × (ℂ ↑m 𝑆)) ∈ V
4645ssex 5321 . . . . . . 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)
4743, 46syl 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)
4814, 23, 24, 47fvmptd3 7039 . . . . 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)})
4948breqd 5154 . . . 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))
50 simpl 482 . . . . . . . 8 ((𝑓 = 𝐹𝑦 = 𝐺) → 𝑓 = 𝐹)
5150feq1d 6720 . . . . . . 7 ((𝑓 = 𝐹𝑦 = 𝐺) → (𝑓:(ℤ𝑛)⟶(ℂ ↑m 𝑆) ↔ 𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆)))
52 simpr 484 . . . . . . . 8 ((𝑓 = 𝐹𝑦 = 𝐺) → 𝑦 = 𝐺)
5352feq1d 6720 . . . . . . 7 ((𝑓 = 𝐹𝑦 = 𝐺) → (𝑦:𝑆⟶ℂ ↔ 𝐺:𝑆⟶ℂ))
5450fveq1d 6908 . . . . . . . . . . . . . 14 ((𝑓 = 𝐹𝑦 = 𝐺) → (𝑓𝑘) = (𝐹𝑘))
5554fveq1d 6908 . . . . . . . . . . . . 13 ((𝑓 = 𝐹𝑦 = 𝐺) → ((𝑓𝑘)‘𝑧) = ((𝐹𝑘)‘𝑧))
5652fveq1d 6908 . . . . . . . . . . . . 13 ((𝑓 = 𝐹𝑦 = 𝐺) → (𝑦𝑧) = (𝐺𝑧))
5755, 56oveq12d 7449 . . . . . . . . . . . 12 ((𝑓 = 𝐹𝑦 = 𝐺) → (((𝑓𝑘)‘𝑧) − (𝑦𝑧)) = (((𝐹𝑘)‘𝑧) − (𝐺𝑧)))
5857fveq2d 6910 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑦 = 𝐺) → (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) = (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))))
5958breq1d 5153 . . . . . . . . . 10 ((𝑓 = 𝐹𝑦 = 𝐺) → ((abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥 ↔ (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥))
6059ralbidv 3178 . . . . . . . . 9 ((𝑓 = 𝐹𝑦 = 𝐺) → (∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥 ↔ ∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥))
6160rexralbidv 3223 . . . . . . . 8 ((𝑓 = 𝐹𝑦 = 𝐺) → (∃𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥 ↔ ∃𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥))
6261ralbidv 3178 . . . . . . 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))
6351, 53, 623anbi123d 1438 . . . . . 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)))
6463rexbidv 3179 . . . . 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)))
65 eqid 2737 . . . . 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)}
6664, 65brabga 5539 . . . 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)))
6749, 66sylan9bb 509 . . 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)))
6867ex 412 . 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))))
693, 13, 68pm5.21ndd 379 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  wss 3951   class class class wbr 5143  {copab 5205   × cxp 5683  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866  pm cpm 8867  cc 11153   < clt 11295  cmin 11492  cz 12613  cuz 12878  +crp 13034  abscabs 15273  𝑢culm 26419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-pm 8869  df-neg 11495  df-z 12614  df-uz 12879  df-ulm 26420
This theorem is referenced by:  ulmcl  26424  ulmf  26425  ulm2  26428
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