Theorem ulmval 26357

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	⊢ (𝑆 ∈ 𝑉 → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 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.

Step	Hyp	Ref	Expression
1		ulmrel 26355	. . . 4 ⊢ Rel (⇝𝑢‘𝑆)
2	1	brrelex12i 5687	. . 3 ⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
3	2	a1i 11	. 2 ⊢ (𝑆 ∈ 𝑉 → (𝐹(⇝𝑢‘𝑆)𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V)))
4		3simpa 1149	. . . 4 ⊢ ((𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) → (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ))
5		fvex 6855	. . . . . . 7 ⊢ (ℤ≥‘𝑛) ∈ V
6		fex 7182	. . . . . . 7 ⊢ ((𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ (ℤ≥‘𝑛) ∈ V) → 𝐹 ∈ V)
7	5, 6	mpan2 692	. . . . . 6 ⊢ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) → 𝐹 ∈ V)
8	7	a1i 11	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) → 𝐹 ∈ V))
9		fex 7182	. . . . . 6 ⊢ ((𝐺:𝑆⟶ℂ ∧ 𝑆 ∈ 𝑉) → 𝐺 ∈ V)
10	9	expcom 413	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → (𝐺:𝑆⟶ℂ → 𝐺 ∈ V))
11	8, 10	anim12d 610	. . . 4 ⊢ (𝑆 ∈ 𝑉 → ((𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ) → (𝐹 ∈ V ∧ 𝐺 ∈ V)))
12	4, 11	syl5 34	. . 3 ⊢ (𝑆 ∈ 𝑉 → ((𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) → (𝐹 ∈ V ∧ 𝐺 ∈ V)))
13	12	rexlimdvw 3144	. 2 ⊢ (𝑆 ∈ 𝑉 → (∃𝑛 ∈ ℤ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) → (𝐹 ∈ V ∧ 𝐺 ∈ V)))
14		df-ulm 26354	. . . . . 6 ⊢ ⇝𝑢 = (𝑠 ∈ V ↦ {⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑠) ∧ 𝑦:𝑠⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑠 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)})
15		oveq2 7376	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (ℂ ↑m 𝑠) = (ℂ ↑m 𝑆))
16	15	feq3d 6655	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑠) ↔ 𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆)))
17		feq2 6649	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (𝑦:𝑠⟶ℂ ↔ 𝑦:𝑆⟶ℂ))
18		raleq 3295	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (∀𝑧 ∈ 𝑠 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥 ↔ ∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥))
19	18	rexralbidv 3204	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑠 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥 ↔ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥))
20	19	ralbidv 3161	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑠 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥))
21	16, 17, 20	3anbi123d 1439	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → ((𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑠) ∧ 𝑦:𝑠⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑠 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥) ↔ (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)))
22	21	rexbidv 3162	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑠) ∧ 𝑦:𝑠⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑠 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥) ↔ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)))
23	22	opabbidv 5166	. . . . . 6 ⊢ (𝑠 = 𝑆 → {⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑠) ∧ 𝑦:𝑠⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑠 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)} = {⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)})
24		elex 3463	. . . . . 6 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
25		simpr1 1196	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)) → 𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆))
26		uzssz 12784	. . . . . . . . . . . . 13 ⊢ (ℤ≥‘𝑛) ⊆ ℤ
27		ovex 7401	. . . . . . . . . . . . . 14 ⊢ (ℂ ↑m 𝑆) ∈ V
28		zex 12509	. . . . . . . . . . . . . 14 ⊢ ℤ ∈ V
29		elpm2r 8794	. . . . . . . . . . . . . 14 ⊢ ((((ℂ ↑m 𝑆) ∈ V ∧ ℤ ∈ V) ∧ (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ (ℤ≥‘𝑛) ⊆ ℤ)) → 𝑓 ∈ ((ℂ ↑m 𝑆) ↑pm ℤ))
30	27, 28, 29	mpanl12 703	. . . . . . . . . . . . 13 ⊢ ((𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ (ℤ≥‘𝑛) ⊆ ℤ) → 𝑓 ∈ ((ℂ ↑m 𝑆) ↑pm ℤ))
31	25, 26, 30	sylancl 587	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)) → 𝑓 ∈ ((ℂ ↑m 𝑆) ↑pm ℤ))
32		simpr2 1197	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)) → 𝑦:𝑆⟶ℂ)
33		cnex 11119	. . . . . . . . . . . . . 14 ⊢ ℂ ∈ V
34		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)) → 𝑆 ∈ 𝑉)
35		elmapg 8788	. . . . . . . . . . . . . 14 ⊢ ((ℂ ∈ V ∧ 𝑆 ∈ 𝑉) → (𝑦 ∈ (ℂ ↑m 𝑆) ↔ 𝑦:𝑆⟶ℂ))
36	33, 34, 35	sylancr 588	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)) → (𝑦 ∈ (ℂ ↑m 𝑆) ↔ 𝑦:𝑆⟶ℂ))
37	32, 36	mpbird 257	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)) → 𝑦 ∈ (ℂ ↑m 𝑆))
38	31, 37	jca 511	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)) → (𝑓 ∈ ((ℂ ↑m 𝑆) ↑pm ℤ) ∧ 𝑦 ∈ (ℂ ↑m 𝑆)))
39	38	ex 412	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝑉 → ((𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥) → (𝑓 ∈ ((ℂ ↑m 𝑆) ↑pm ℤ) ∧ 𝑦 ∈ (ℂ ↑m 𝑆))))
40	39	rexlimdvw 3144	. . . . . . . . 9 ⊢ (𝑆 ∈ 𝑉 → (∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥) → (𝑓 ∈ ((ℂ ↑m 𝑆) ↑pm ℤ) ∧ 𝑦 ∈ (ℂ ↑m 𝑆))))
41	40	ssopab2dv 5507	. . . . . . . 8 ⊢ (𝑆 ∈ 𝑉 → {⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)} ⊆ {⟨𝑓, 𝑦⟩ ∣ (𝑓 ∈ ((ℂ ↑m 𝑆) ↑pm ℤ) ∧ 𝑦 ∈ (ℂ ↑m 𝑆))})
42		df-xp 5638	. . . . . . . 8 ⊢ (((ℂ ↑m 𝑆) ↑pm ℤ) × (ℂ ↑m 𝑆)) = {⟨𝑓, 𝑦⟩ ∣ (𝑓 ∈ ((ℂ ↑m 𝑆) ↑pm ℤ) ∧ 𝑦 ∈ (ℂ ↑m 𝑆))}
43	41, 42	sseqtrrdi 3977	. . . . . . 7 ⊢ (𝑆 ∈ 𝑉 → {⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)} ⊆ (((ℂ ↑m 𝑆) ↑pm ℤ) × (ℂ ↑m 𝑆)))
44		ovex 7401	. . . . . . . . 9 ⊢ ((ℂ ↑m 𝑆) ↑pm ℤ) ∈ V
45	44, 27	xpex 7708	. . . . . . . 8 ⊢ (((ℂ ↑m 𝑆) ↑pm ℤ) × (ℂ ↑m 𝑆)) ∈ V
46	45	ssex 5268	. . . . . . 7 ⊢ ({⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)} ⊆ (((ℂ ↑m 𝑆) ↑pm ℤ) × (ℂ ↑m 𝑆)) → {⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)} ∈ V)
47	43, 46	syl 17	. . . . . 6 ⊢ (𝑆 ∈ 𝑉 → {⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)} ∈ V)
48	14, 23, 24, 47	fvmptd3 6973	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → (⇝𝑢‘𝑆) = {⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)})
49	48	breqd 5111	. . . 4 ⊢ (𝑆 ∈ 𝑉 → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ 𝐹{⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)}𝐺))
50		simpl 482	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → 𝑓 = 𝐹)
51	50	feq1d 6652	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ↔ 𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆)))
52		simpr 484	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → 𝑦 = 𝐺)
53	52	feq1d 6652	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (𝑦:𝑆⟶ℂ ↔ 𝐺:𝑆⟶ℂ))
54	50	fveq1d 6844	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (𝑓‘𝑘) = (𝐹‘𝑘))
55	54	fveq1d 6844	. . . . . . . . . . . . 13 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → ((𝑓‘𝑘)‘𝑧) = ((𝐹‘𝑘)‘𝑧))
56	52	fveq1d 6844	. . . . . . . . . . . . 13 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (𝑦‘𝑧) = (𝐺‘𝑧))
57	55, 56	oveq12d 7386	. . . . . . . . . . . 12 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧)) = (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))
58	57	fveq2d 6846	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) = (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))))
59	58	breq1d 5110	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → ((abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥 ↔ (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥))
60	59	ralbidv 3161	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥 ↔ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥))
61	60	rexralbidv 3204	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥 ↔ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥))
62	61	ralbidv 3161	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥))
63	51, 53, 62	3anbi123d 1439	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → ((𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥) ↔ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)))
64	63	rexbidv 3162	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥) ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)))
65		eqid 2737	. . . . 5 ⊢ {⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)} = {⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)}
66	64, 65	brabga 5490	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹{⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)}𝐺 ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)))
67	49, 66	sylan9bb 509	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ V ∧ 𝐺 ∈ V)) → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)))
68	67	ex 412	. 2 ⊢ (𝑆 ∈ 𝑉 → ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥))))
69	3, 13, 68	pm5.21ndd 379	1 ⊢ (𝑆 ∈ 𝑉 → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  Vcvv 3442   ⊆ wss 3903   class class class wbr 5100  {copab 5162   × cxp 5630  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368   ↑_m cmap 8775   ↑_pm cpm 8776  ℂcc 11036   < clt 11178   − cmin 11376  ℤcz 12500  ℤ_≥cuz 12763  ℝ⁺crp 12917  abscabs 15169  ⇝_𝑢culm 26353

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-map 8777  df-pm 8778  df-neg 11379  df-z 12501  df-uz 12764  df-ulm 26354

This theorem is referenced by:  ulmcl  26358  ulmf  26359  ulm2  26362