Theorem ulmval 25539

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 25537	. . . 4 ⊢ Rel (⇝𝑢‘𝑆)
2	1	brrelex12i 5642	. . 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 1147	. . . 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 6787	. . . . . . 7 ⊢ (ℤ≥‘𝑛) ∈ V
6		fex 7102	. . . . . . 7 ⊢ ((𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ (ℤ≥‘𝑛) ∈ V) → 𝐹 ∈ V)
7	5, 6	mpan2 688	. . . . . 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 7102	. . . . . 6 ⊢ ((𝐺:𝑆⟶ℂ ∧ 𝑆 ∈ 𝑉) → 𝐺 ∈ V)
10	9	expcom 414	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → (𝐺:𝑆⟶ℂ → 𝐺 ∈ V))
11	8, 10	anim12d 609	. . . 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 3219	. 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 25536	. . . . . 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 7283	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (ℂ ↑m 𝑠) = (ℂ ↑m 𝑆))
16	15	feq3d 6587	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑠) ↔ 𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆)))
17		feq2 6582	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (𝑦:𝑠⟶ℂ ↔ 𝑦:𝑆⟶ℂ))
18		raleq 3342	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (∀𝑧 ∈ 𝑠 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥 ↔ ∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥))
19	18	rexralbidv 3230	. . . . . . . . . 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 3112	. . . . . . . . 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 1435	. . . . . . . 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 3226	. . . . . . 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 5140	. . . . . 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 3450	. . . . . 6 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
25		simpr1 1193	. . . . . . . . . . . . 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 12603	. . . . . . . . . . . . 13 ⊢ (ℤ≥‘𝑛) ⊆ ℤ
27		ovex 7308	. . . . . . . . . . . . . 14 ⊢ (ℂ ↑m 𝑆) ∈ V
28		zex 12328	. . . . . . . . . . . . . 14 ⊢ ℤ ∈ V
29		elpm2r 8633	. . . . . . . . . . . . . 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 699	. . . . . . . . . . . . 13 ⊢ ((𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ (ℤ≥‘𝑛) ⊆ ℤ) → 𝑓 ∈ ((ℂ ↑m 𝑆) ↑pm ℤ))
31	25, 26, 30	sylancl 586	. . . . . . . . . . . 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 1194	. . . . . . . . . . . . 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 10952	. . . . . . . . . . . . . 14 ⊢ ℂ ∈ V
34		simpl 483	. . . . . . . . . . . . . 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 8628	. . . . . . . . . . . . . 14 ⊢ ((ℂ ∈ V ∧ 𝑆 ∈ 𝑉) → (𝑦 ∈ (ℂ ↑m 𝑆) ↔ 𝑦:𝑆⟶ℂ))
36	33, 34, 35	sylancr 587	. . . . . . . . . . . . 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 256	. . . . . . . . . . . 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 512	. . . . . . . . . . 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 413	. . . . . . . . . 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 3219	. . . . . . . . 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 5464	. . . . . . . 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 5595	. . . . . . . 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 3972	. . . . . . 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 7308	. . . . . . . . 9 ⊢ ((ℂ ↑m 𝑆) ↑pm ℤ) ∈ V
45	44, 27	xpex 7603	. . . . . . . 8 ⊢ (((ℂ ↑m 𝑆) ↑pm ℤ) × (ℂ ↑m 𝑆)) ∈ V
46	45	ssex 5245	. . . . . . 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 6898	. . . . 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 5085	. . . 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 483	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → 𝑓 = 𝐹)
51	50	feq1d 6585	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ↔ 𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆)))
52		simpr 485	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → 𝑦 = 𝐺)
53	52	feq1d 6585	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (𝑦:𝑆⟶ℂ ↔ 𝐺:𝑆⟶ℂ))
54	50	fveq1d 6776	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (𝑓‘𝑘) = (𝐹‘𝑘))
55	54	fveq1d 6776	. . . . . . . . . . . . 13 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → ((𝑓‘𝑘)‘𝑧) = ((𝐹‘𝑘)‘𝑧))
56	52	fveq1d 6776	. . . . . . . . . . . . 13 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (𝑦‘𝑧) = (𝐺‘𝑧))
57	55, 56	oveq12d 7293	. . . . . . . . . . . 12 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧)) = (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))
58	57	fveq2d 6778	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) = (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))))
59	58	breq1d 5084	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → ((abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥 ↔ (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥))
60	59	ralbidv 3112	. . . . . . . . 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 3230	. . . . . . . 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 3112	. . . . . . 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 1435	. . . . . 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 3226	. . . . 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 2738	. . . . 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 5447	. . . 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 510	. . 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 413	. 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 381	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 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ⊆ wss 3887   class class class wbr 5074  {copab 5136   × cxp 5587  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ↑_m cmap 8615   ↑_pm cpm 8616  ℂcc 10869   < clt 11009   − cmin 11205  ℤcz 12319  ℤ_≥cuz 12582  ℝ⁺crp 12730  abscabs 14945  ⇝_𝑢culm 25535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-pm 8618  df-neg 11208  df-z 12320  df-uz 12583  df-ulm 25536

This theorem is referenced by:  ulmcl  25540  ulmf  25541  ulm2  25544