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Theorem ulmval 24979
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 24977 . . . 4 Rel (⇝𝑢𝑆)
21brrelex12i 5575 . . 3 (𝐹(⇝𝑢𝑆)𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
32a1i 11 . 2 (𝑆𝑉 → (𝐹(⇝𝑢𝑆)𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V)))
4 3simpa 1145 . . . 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 6662 . . . . . . 7 (ℤ𝑛) ∈ V
6 fex 6970 . . . . . . 7 ((𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆) ∧ (ℤ𝑛) ∈ V) → 𝐹 ∈ V)
75, 6mpan2 690 . . . . . 6 (𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆) → 𝐹 ∈ V)
87a1i 11 . . . . 5 (𝑆𝑉 → (𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆) → 𝐹 ∈ V))
9 fex 6970 . . . . . 6 ((𝐺:𝑆⟶ℂ ∧ 𝑆𝑉) → 𝐺 ∈ V)
109expcom 417 . . . . 5 (𝑆𝑉 → (𝐺:𝑆⟶ℂ → 𝐺 ∈ V))
118, 10anim12d 611 . . . 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 3252 . 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 24976 . . . . . 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 7147 . . . . . . . . . 10 (𝑠 = 𝑆 → (ℂ ↑m 𝑠) = (ℂ ↑m 𝑆))
1615feq3d 6478 . . . . . . . . 9 (𝑠 = 𝑆 → (𝑓:(ℤ𝑛)⟶(ℂ ↑m 𝑠) ↔ 𝑓:(ℤ𝑛)⟶(ℂ ↑m 𝑆)))
17 feq2 6473 . . . . . . . . 9 (𝑠 = 𝑆 → (𝑦:𝑠⟶ℂ ↔ 𝑦:𝑆⟶ℂ))
18 raleq 3361 . . . . . . . . . . 11 (𝑠 = 𝑆 → (∀𝑧𝑠 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥 ↔ ∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥))
1918rexralbidv 3263 . . . . . . . . . 10 (𝑠 = 𝑆 → (∃𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑠 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥 ↔ ∃𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥))
2019ralbidv 3165 . . . . . . . . 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 1433 . . . . . . . 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 3259 . . . . . . 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 5099 . . . . . 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 3462 . . . . . 6 (𝑆𝑉𝑆 ∈ V)
25 simpr1 1191 . . . . . . . . . . . . 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 12256 . . . . . . . . . . . . 13 (ℤ𝑛) ⊆ ℤ
27 ovex 7172 . . . . . . . . . . . . . 14 (ℂ ↑m 𝑆) ∈ V
28 zex 11982 . . . . . . . . . . . . . 14 ℤ ∈ V
29 elpm2r 8411 . . . . . . . . . . . . . 14 ((((ℂ ↑m 𝑆) ∈ V ∧ ℤ ∈ V) ∧ (𝑓:(ℤ𝑛)⟶(ℂ ↑m 𝑆) ∧ (ℤ𝑛) ⊆ ℤ)) → 𝑓 ∈ ((ℂ ↑m 𝑆) ↑pm ℤ))
3027, 28, 29mpanl12 701 . . . . . . . . . . . . 13 ((𝑓:(ℤ𝑛)⟶(ℂ ↑m 𝑆) ∧ (ℤ𝑛) ⊆ ℤ) → 𝑓 ∈ ((ℂ ↑m 𝑆) ↑pm ℤ))
3125, 26, 30sylancl 589 . . . . . . . . . . . 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 1192 . . . . . . . . . . . . 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 10611 . . . . . . . . . . . . . 14 ℂ ∈ V
34 simpl 486 . . . . . . . . . . . . . 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 8406 . . . . . . . . . . . . . 14 ((ℂ ∈ V ∧ 𝑆𝑉) → (𝑦 ∈ (ℂ ↑m 𝑆) ↔ 𝑦:𝑆⟶ℂ))
3633, 34, 35sylancr 590 . . . . . . . . . . . . 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 260 . . . . . . . . . . . 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 515 . . . . . . . . . . 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 416 . . . . . . . . . 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 3252 . . . . . . . . 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 5406 . . . . . . . 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 5529 . . . . . . . 8 (((ℂ ↑m 𝑆) ↑pm ℤ) × (ℂ ↑m 𝑆)) = {⟨𝑓, 𝑦⟩ ∣ (𝑓 ∈ ((ℂ ↑m 𝑆) ↑pm ℤ) ∧ 𝑦 ∈ (ℂ ↑m 𝑆))}
4341, 42sseqtrrdi 3969 . . . . . . 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 7172 . . . . . . . . 9 ((ℂ ↑m 𝑆) ↑pm ℤ) ∈ V
4544, 27xpex 7460 . . . . . . . 8 (((ℂ ↑m 𝑆) ↑pm ℤ) × (ℂ ↑m 𝑆)) ∈ V
4645ssex 5192 . . . . . . 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 6772 . . . . 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 5044 . . . 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 486 . . . . . . . 8 ((𝑓 = 𝐹𝑦 = 𝐺) → 𝑓 = 𝐹)
5150feq1d 6476 . . . . . . 7 ((𝑓 = 𝐹𝑦 = 𝐺) → (𝑓:(ℤ𝑛)⟶(ℂ ↑m 𝑆) ↔ 𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆)))
52 simpr 488 . . . . . . . 8 ((𝑓 = 𝐹𝑦 = 𝐺) → 𝑦 = 𝐺)
5352feq1d 6476 . . . . . . 7 ((𝑓 = 𝐹𝑦 = 𝐺) → (𝑦:𝑆⟶ℂ ↔ 𝐺:𝑆⟶ℂ))
5450fveq1d 6651 . . . . . . . . . . . . . 14 ((𝑓 = 𝐹𝑦 = 𝐺) → (𝑓𝑘) = (𝐹𝑘))
5554fveq1d 6651 . . . . . . . . . . . . 13 ((𝑓 = 𝐹𝑦 = 𝐺) → ((𝑓𝑘)‘𝑧) = ((𝐹𝑘)‘𝑧))
5652fveq1d 6651 . . . . . . . . . . . . 13 ((𝑓 = 𝐹𝑦 = 𝐺) → (𝑦𝑧) = (𝐺𝑧))
5755, 56oveq12d 7157 . . . . . . . . . . . 12 ((𝑓 = 𝐹𝑦 = 𝐺) → (((𝑓𝑘)‘𝑧) − (𝑦𝑧)) = (((𝐹𝑘)‘𝑧) − (𝐺𝑧)))
5857fveq2d 6653 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑦 = 𝐺) → (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) = (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))))
5958breq1d 5043 . . . . . . . . . 10 ((𝑓 = 𝐹𝑦 = 𝐺) → ((abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥 ↔ (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥))
6059ralbidv 3165 . . . . . . . . 9 ((𝑓 = 𝐹𝑦 = 𝐺) → (∀𝑧𝑆 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥 ↔ ∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥))
6160rexralbidv 3263 . . . . . . . 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 3165 . . . . . . 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 1433 . . . . . 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 3259 . . . . 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 2801 . . . . 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 5389 . . . 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 513 . . 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 416 . 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 384 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wral 3109  wrex 3110  Vcvv 3444  wss 3884   class class class wbr 5033  {copab 5095   × cxp 5521  wf 6324  cfv 6328  (class class class)co 7139  m cmap 8393  pm cpm 8394  cc 10528   < clt 10668  cmin 10863  cz 11973  cuz 12235  +crp 12381  abscabs 14589  𝑢culm 24975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-map 8395  df-pm 8396  df-neg 10866  df-z 11974  df-uz 12236  df-ulm 24976
This theorem is referenced by:  ulmcl  24980  ulmf  24981  ulm2  24984
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