Theorem ulmval 26623

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 26621	. . . 4 ⊢ Rel (⇝𝑢‘𝑆)
2	1	brrelex12i 5714	. . 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 1166	. . . 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 6895	. . . . . . 7 ⊢ (ℤ≥‘𝑛) ∈ V
6		fex 7229	. . . . . . 7 ⊢ ((𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ (ℤ≥‘𝑛) ∈ V) → 𝐹 ∈ V)
7	5, 6	mpan2 704	. . . . . 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 7229	. . . . . 6 ⊢ ((𝐺:𝑆⟶ℂ ∧ 𝑆 ∈ 𝑉) → 𝐺 ∈ V)
10	9	expcom 419	. . . . 5 ⊢ (𝑆 ∈ 𝑉 → (𝐺:𝑆⟶ℂ → 𝐺 ∈ V))
11	8, 10	anim12d 621	. . . 4 ⊢ (𝑆 ∈ 𝑉 → ((𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ) → (𝐹 ∈ V ∧ 𝐺 ∈ V)))
12	4, 11	syl5 35	. . 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 3170	. 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 26620	. . . . . 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 7425	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (ℂ ↑m 𝑠) = (ℂ ↑m 𝑆))
16	15	feq3d 6691	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑠) ↔ 𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆)))
17		feq2 6685	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (𝑦:𝑠⟶ℂ ↔ 𝑦:𝑆⟶ℂ))
18		raleq 3318	. . . . . . . . . . 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 3187	. . . . . . . . 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 1464	. . . . . . . 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 3188	. . . . . . 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 5175	. . . . . 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 3474	. . . . . 6 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
25		simpr1 1213	. . . . . . . . . . . . 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 12912	. . . . . . . . . . . . 13 ⊢ (ℤ≥‘𝑛) ⊆ ℤ
27		ovex 7450	. . . . . . . . . . . . . 14 ⊢ (ℂ ↑m 𝑆) ∈ V
28		zex 12628	. . . . . . . . . . . . . 14 ⊢ ℤ ∈ V
29		elpm2r 8848	. . . . . . . . . . . . . 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 715	. . . . . . . . . . . . 13 ⊢ ((𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ (ℤ≥‘𝑛) ⊆ ℤ) → 𝑓 ∈ ((ℂ ↑m 𝑆) ↑pm ℤ))
31	25, 26, 30	sylancl 598	. . . . . . . . . . . 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 1214	. . . . . . . . . . . . 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 11209	. . . . . . . . . . . . . 14 ⊢ ℂ ∈ V
34		simpl 488	. . . . . . . . . . . . . 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 8842	. . . . . . . . . . . . . 14 ⊢ ((ℂ ∈ V ∧ 𝑆 ∈ 𝑉) → (𝑦 ∈ (ℂ ↑m 𝑆) ↔ 𝑦:𝑆⟶ℂ))
36	33, 34, 35	sylancr 599	. . . . . . . . . . . . 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 260	. . . . . . . . . . . 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 521	. . . . . . . . . . 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 418	. . . . . . . . . 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 3170	. . . . . . . . 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 5534	. . . . . . . 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 5665	. . . . . . . 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 3975	. . . . . . 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 7450	. . . . . . . . 9 ⊢ ((ℂ ↑m 𝑆) ↑pm ℤ) ∈ V
45	44, 27	xpex 7756	. . . . . . . 8 ⊢ (((ℂ ↑m 𝑆) ↑pm ℤ) × (ℂ ↑m 𝑆)) ∈ V
46	45	ssex 5289	. . . . . . 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 18	. . . . . 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 7014	. . . . 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 5118	. . . 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 488	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → 𝑓 = 𝐹)
51	50	feq1d 6688	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (𝑓:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ↔ 𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆)))
52		simpr 490	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → 𝑦 = 𝐺)
53	52	feq1d 6688	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (𝑦:𝑆⟶ℂ ↔ 𝐺:𝑆⟶ℂ))
54	50	fveq1d 6884	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (𝑓‘𝑘) = (𝐹‘𝑘))
55	54	fveq1d 6884	. . . . . . . . . . . . 13 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → ((𝑓‘𝑘)‘𝑧) = ((𝐹‘𝑘)‘𝑧))
56	52	fveq1d 6884	. . . . . . . . . . . . 13 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (𝑦‘𝑧) = (𝐺‘𝑧))
57	55, 56	oveq12d 7435	. . . . . . . . . . . 12 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧)) = (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))
58	57	fveq2d 6886	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) = (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))))
59	58	breq1d 5117	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → ((abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥 ↔ (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥))
60	59	ralbidv 3187	. . . . . . . . 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 3187	. . . . . . 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 1464	. . . . . 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 3188	. . . . 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 2762	. . . . 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 5516	. . . 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 519	. . 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 418	. 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 382	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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ∃wrex 3088  Vcvv 3453   ⊆ wss 3902   class class class wbr 5107  {copab 5171   × cxp 5657  ⟶wf 6533  ‘cfv 6537  (class class class)co 7417   ↑_m cmap 8830   ↑_pm cpm 8831  ℂcc 11126   < clt 11271   − cmin 11469  ℤcz 12619  ℤ_≥cuz 12891  ℝ⁺crp 13046  abscabs 15325  ⇝_𝑢culm 26619

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-map 8832  df-pm 8833  df-neg 11472  df-z 12620  df-uz 12892  df-ulm 26620

This theorem is used by:  ulmcl  26624  ulmf  26625  ulm2  26628