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Theorem ulmcau 24985

Description: A sequence of functions converges uniformly iff it is uniformly Cauchy, which is to say that for every 0 < 𝑥 there is a 𝑗 such that for all 𝑗 ≤ 𝑘 the functions 𝐹(𝑘) and 𝐹(𝑗) are uniformly within 𝑥 of each other on 𝑆. This is the four-quantifier version, see ulmcau2 24986 for the more conventional five-quantifier version. (Contributed by Mario Carneiro, 1-Mar-2015.)

**Hypotheses**
Ref	Expression
ulmcau.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
ulmcau.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
ulmcau.s	⊢ (𝜑 → 𝑆 ∈ 𝑉)
ulmcau.f	⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑_m 𝑆))

**Assertion**
Ref	Expression
ulmcau	⊢ (𝜑 → (𝐹 ∈ dom (⇝_𝑢‘𝑆) ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥))

Distinct variable groups: 𝑗,𝑘,𝑥,𝑧,𝐹 𝜑,𝑗,𝑘,𝑥,𝑧 𝑆,𝑗,𝑘,𝑥,𝑧 𝑗,𝑍,𝑘,𝑥,𝑧 𝑗,𝑀,𝑘,𝑧 Allowed substitution hints: 𝑀(𝑥) 𝑉(𝑥,𝑧,𝑗,𝑘)

Proof of Theorem ulmcau Dummy variables 𝑔 𝑚 𝑛 𝑝 𝑞 𝑟 𝑣 𝑤 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldmg 5769	. . . 4 ⊢ (𝐹 ∈ dom (⇝_𝑢‘𝑆) → (𝐹 ∈ dom (⇝_𝑢‘𝑆) ↔ ∃𝑔 𝐹(⇝_𝑢‘𝑆)𝑔))
2	1	ibi 269	. . 3 ⊢ (𝐹 ∈ dom (⇝_𝑢‘𝑆) → ∃𝑔 𝐹(⇝_𝑢‘𝑆)𝑔)
3		ulmcau.z	. . . . . . . 8 ⊢ 𝑍 = (ℤ_≥‘𝑀)
4		ulmcau.m	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
5	4	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) → 𝑀 ∈ ℤ)
6		ulmcau.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑_m 𝑆))
7	6	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) → 𝐹:𝑍⟶(ℂ ↑_m 𝑆))
8		eqidd 2824	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑘)‘𝑧))
9		eqidd 2824	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑧 ∈ 𝑆) → (𝑔‘𝑧) = (𝑔‘𝑧))
10		simplr 767	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) → 𝐹(⇝_𝑢‘𝑆)𝑔)
11		rphalfcl 12419	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 / 2) ∈ ℝ⁺)
12	11	adantl 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) → (𝑥 / 2) ∈ ℝ⁺)
13	3, 5, 7, 8, 9, 10, 12	ulmi 24976	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2))
14		simpr 487	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍)
15	14, 3	eleqtrdi 2925	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ_≥‘𝑀))
16		eluzelz 12256	. . . . . . . . . 10 ⊢ (𝑗 ∈ (ℤ_≥‘𝑀) → 𝑗 ∈ ℤ)
17		uzid 12261	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ_≥‘𝑗))
18		fveq2 6672	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
19	18	fveq1d 6674	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑗)‘𝑧))
20	19	fvoveq1d 7180	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → (abs‘(((𝐹‘𝑘)‘𝑧) − (𝑔‘𝑧))) = (abs‘(((𝐹‘𝑗)‘𝑧) − (𝑔‘𝑧))))
21	20	breq1d 5078	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → ((abs‘(((𝐹‘𝑘)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2) ↔ (abs‘(((𝐹‘𝑗)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2)))
22	21	ralbidv 3199	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2) ↔ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑗)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2)))
23	22	rspcv 3620	. . . . . . . . . 10 ⊢ (𝑗 ∈ (ℤ_≥‘𝑗) → (∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2) → ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑗)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2)))
24	15, 16, 17, 23	4syl 19	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2) → ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑗)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2)))
25		r19.26 3172	. . . . . . . . . . . . . . 15 ⊢ (∀𝑧 ∈ 𝑆 ((abs‘(((𝐹‘𝑗)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2) ∧ (abs‘(((𝐹‘𝑘)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2)) ↔ (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑗)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2) ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2)))
26	7	ffvelrnda 6853	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ (ℂ ↑_m 𝑆))
27	26	adantr 483	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑗) ∈ (ℂ ↑_m 𝑆))
28		elmapi 8430	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹‘𝑗) ∈ (ℂ ↑_m 𝑆) → (𝐹‘𝑗):𝑆⟶ℂ)
29	27, 28	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑗):𝑆⟶ℂ)
30	29	ffvelrnda 6853	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑗)‘𝑧) ∈ ℂ)
31		ulmcl 24971	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹(⇝_𝑢‘𝑆)𝑔 → 𝑔:𝑆⟶ℂ)
32	31	ad4antlr 731	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑔:𝑆⟶ℂ)
33	32	ffvelrnda 6853	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → (𝑔‘𝑧) ∈ ℂ)
34	30, 33	abssubd 14815	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → (abs‘(((𝐹‘𝑗)‘𝑧) − (𝑔‘𝑧))) = (abs‘((𝑔‘𝑧) − ((𝐹‘𝑗)‘𝑧))))
35	34	breq1d 5078	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → ((abs‘(((𝐹‘𝑗)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2) ↔ (abs‘((𝑔‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)))
36	35	biimpd 231	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → ((abs‘(((𝐹‘𝑗)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2) → (abs‘((𝑔‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)))
37	3	uztrn2 12265	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
38		ffvelrn 6851	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹:𝑍⟶(ℂ ↑_m 𝑆) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ (ℂ ↑_m 𝑆))
39	7, 37, 38	syl2an 597	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑘) ∈ (ℂ ↑_m 𝑆))
40	39	anassrs 470	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ (ℂ ↑_m 𝑆))
41		elmapi 8430	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹‘𝑘) ∈ (ℂ ↑_m 𝑆) → (𝐹‘𝑘):𝑆⟶ℂ)
42	40, 41	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘):𝑆⟶ℂ)
43	42	ffvelrnda 6853	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑘)‘𝑧) ∈ ℂ)
44		rpre 12400	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
45	44	ad4antlr 731	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → 𝑥 ∈ ℝ)
46		abs3lem 14700	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐹‘𝑘)‘𝑧) ∈ ℂ ∧ ((𝐹‘𝑗)‘𝑧) ∈ ℂ) ∧ ((𝑔‘𝑧) ∈ ℂ ∧ 𝑥 ∈ ℝ)) → (((abs‘(((𝐹‘𝑘)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2) ∧ (abs‘((𝑔‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)) → (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥))
47	43, 30, 33, 45, 46	syl22anc 836	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → (((abs‘(((𝐹‘𝑘)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2) ∧ (abs‘((𝑔‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)) → (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥))
48	36, 47	sylan2d 606	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → (((abs‘(((𝐹‘𝑘)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2) ∧ (abs‘(((𝐹‘𝑗)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2)) → (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥))
49	48	ancomsd 468	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → (((abs‘(((𝐹‘𝑗)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2) ∧ (abs‘(((𝐹‘𝑘)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2)) → (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥))
50	49	ralimdva 3179	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (∀𝑧 ∈ 𝑆 ((abs‘(((𝐹‘𝑗)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2) ∧ (abs‘(((𝐹‘𝑘)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2)) → ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥))
51	25, 50	syl5bir 245	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑗)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2) ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2)) → ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥))
52	51	expdimp 455	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑗)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2)) → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2) → ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥))
53	52	an32s 650	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑗)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2) → ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥))
54	53	ralimdva 3179	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑗)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2)) → (∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2) → ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥))
55	54	ex 415	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑗)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2) → (∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2) → ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥)))
56	55	com23 86	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2) → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑗)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2) → ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥)))
57	24, 56	mpdd 43	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2) → ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥))
58	57	reximdva 3276	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝑔‘𝑧))) < (𝑥 / 2) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥))
59	13, 58	mpd 15	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) ∧ 𝑥 ∈ ℝ⁺) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥)
60	59	ralrimiva 3184	. . . . 5 ⊢ ((𝜑 ∧ 𝐹(⇝_𝑢‘𝑆)𝑔) → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥)
61	60	ex 415	. . . 4 ⊢ (𝜑 → (𝐹(⇝_𝑢‘𝑆)𝑔 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥))
62	61	exlimdv 1934	. . 3 ⊢ (𝜑 → (∃𝑔 𝐹(⇝_𝑢‘𝑆)𝑔 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥))
63	2, 62	syl5 34	. 2 ⊢ (𝜑 → (𝐹 ∈ dom (⇝_𝑢‘𝑆) → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥))
64		ulmrel 24968	. . . 4 ⊢ Rel (⇝_𝑢‘𝑆)
65		ulmcau.s	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
66	3, 4, 65, 6	ulmcaulem 24984	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥 ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥))
67	66	biimpa 479	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥)
68		rphalfcl 12419	. . . . . . . 8 ⊢ (𝑟 ∈ ℝ⁺ → (𝑟 / 2) ∈ ℝ⁺)
69		breq2 5072	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑟 / 2) → ((abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 ↔ (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < (𝑟 / 2)))
70	69	ralbidv 3199	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑟 / 2) → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 ↔ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < (𝑟 / 2)))
71	70	2ralbidv 3201	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑟 / 2) → (∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < (𝑟 / 2)))
72	71	rexbidv 3299	. . . . . . . . . 10 ⊢ (𝑥 = (𝑟 / 2) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < (𝑟 / 2)))
73		ralcom 3356	. . . . . . . . . . . . . 14 ⊢ (∀𝑤 ∈ 𝑆 ∀𝑚 ∈ (ℤ_≥‘𝑞)(abs‘(((𝐹‘𝑞)‘𝑤) − ((𝐹‘𝑚)‘𝑤))) < (𝑟 / 2) ↔ ∀𝑚 ∈ (ℤ_≥‘𝑞)∀𝑤 ∈ 𝑆 (abs‘(((𝐹‘𝑞)‘𝑤) − ((𝐹‘𝑚)‘𝑤))) < (𝑟 / 2))
74		fveq2 6672	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = 𝑘 → (ℤ_≥‘𝑞) = (ℤ_≥‘𝑘))
75		fveq2 6672	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = 𝑧 → ((𝐹‘𝑞)‘𝑤) = ((𝐹‘𝑞)‘𝑧))
76		fveq2 6672	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = 𝑧 → ((𝐹‘𝑚)‘𝑤) = ((𝐹‘𝑚)‘𝑧))
77	75, 76	oveq12d 7176	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = 𝑧 → (((𝐹‘𝑞)‘𝑤) − ((𝐹‘𝑚)‘𝑤)) = (((𝐹‘𝑞)‘𝑧) − ((𝐹‘𝑚)‘𝑧)))
78	77	fveq2d 6676	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑧 → (abs‘(((𝐹‘𝑞)‘𝑤) − ((𝐹‘𝑚)‘𝑤))) = (abs‘(((𝐹‘𝑞)‘𝑧) − ((𝐹‘𝑚)‘𝑧))))
79	78	breq1d 5078	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑧 → ((abs‘(((𝐹‘𝑞)‘𝑤) − ((𝐹‘𝑚)‘𝑤))) < (𝑟 / 2) ↔ (abs‘(((𝐹‘𝑞)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < (𝑟 / 2)))
80	79	cbvralvw 3451	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑤 ∈ 𝑆 (abs‘(((𝐹‘𝑞)‘𝑤) − ((𝐹‘𝑚)‘𝑤))) < (𝑟 / 2) ↔ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑞)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < (𝑟 / 2))
81		fveq2 6672	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑞 = 𝑘 → (𝐹‘𝑞) = (𝐹‘𝑘))
82	81	fveq1d 6674	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑞 = 𝑘 → ((𝐹‘𝑞)‘𝑧) = ((𝐹‘𝑘)‘𝑧))
83	82	fvoveq1d 7180	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑞 = 𝑘 → (abs‘(((𝐹‘𝑞)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) = (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))))
84	83	breq1d 5078	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞 = 𝑘 → ((abs‘(((𝐹‘𝑞)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < (𝑟 / 2) ↔ (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < (𝑟 / 2)))
85	84	ralbidv 3199	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 = 𝑘 → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑞)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < (𝑟 / 2) ↔ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < (𝑟 / 2)))
86	80, 85	syl5bb 285	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = 𝑘 → (∀𝑤 ∈ 𝑆 (abs‘(((𝐹‘𝑞)‘𝑤) − ((𝐹‘𝑚)‘𝑤))) < (𝑟 / 2) ↔ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < (𝑟 / 2)))
87	74, 86	raleqbidv 3403	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑘 → (∀𝑚 ∈ (ℤ_≥‘𝑞)∀𝑤 ∈ 𝑆 (abs‘(((𝐹‘𝑞)‘𝑤) − ((𝐹‘𝑚)‘𝑤))) < (𝑟 / 2) ↔ ∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < (𝑟 / 2)))
88	73, 87	syl5bb 285	. . . . . . . . . . . . 13 ⊢ (𝑞 = 𝑘 → (∀𝑤 ∈ 𝑆 ∀𝑚 ∈ (ℤ_≥‘𝑞)(abs‘(((𝐹‘𝑞)‘𝑤) − ((𝐹‘𝑚)‘𝑤))) < (𝑟 / 2) ↔ ∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < (𝑟 / 2)))
89	88	cbvralvw 3451	. . . . . . . . . . . 12 ⊢ (∀𝑞 ∈ (ℤ_≥‘𝑝)∀𝑤 ∈ 𝑆 ∀𝑚 ∈ (ℤ_≥‘𝑞)(abs‘(((𝐹‘𝑞)‘𝑤) − ((𝐹‘𝑚)‘𝑤))) < (𝑟 / 2) ↔ ∀𝑘 ∈ (ℤ_≥‘𝑝)∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < (𝑟 / 2))
90		fveq2 6672	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑗 → (ℤ_≥‘𝑝) = (ℤ_≥‘𝑗))
91	90	raleqdv 3417	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑗 → (∀𝑘 ∈ (ℤ_≥‘𝑝)∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < (𝑟 / 2) ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < (𝑟 / 2)))
92	89, 91	syl5bb 285	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑗 → (∀𝑞 ∈ (ℤ_≥‘𝑝)∀𝑤 ∈ 𝑆 ∀𝑚 ∈ (ℤ_≥‘𝑞)(abs‘(((𝐹‘𝑞)‘𝑤) − ((𝐹‘𝑚)‘𝑤))) < (𝑟 / 2) ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < (𝑟 / 2)))
93	92	cbvrexvw 3452	. . . . . . . . . 10 ⊢ (∃𝑝 ∈ 𝑍 ∀𝑞 ∈ (ℤ_≥‘𝑝)∀𝑤 ∈ 𝑆 ∀𝑚 ∈ (ℤ_≥‘𝑞)(abs‘(((𝐹‘𝑞)‘𝑤) − ((𝐹‘𝑚)‘𝑤))) < (𝑟 / 2) ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < (𝑟 / 2))
94	72, 93	syl6bbr 291	. . . . . . . . 9 ⊢ (𝑥 = (𝑟 / 2) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 ↔ ∃𝑝 ∈ 𝑍 ∀𝑞 ∈ (ℤ_≥‘𝑝)∀𝑤 ∈ 𝑆 ∀𝑚 ∈ (ℤ_≥‘𝑞)(abs‘(((𝐹‘𝑞)‘𝑤) − ((𝐹‘𝑚)‘𝑤))) < (𝑟 / 2)))
95	94	rspccva 3624	. . . . . . . 8 ⊢ ((∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 ∧ (𝑟 / 2) ∈ ℝ⁺) → ∃𝑝 ∈ 𝑍 ∀𝑞 ∈ (ℤ_≥‘𝑝)∀𝑤 ∈ 𝑆 ∀𝑚 ∈ (ℤ_≥‘𝑞)(abs‘(((𝐹‘𝑞)‘𝑤) − ((𝐹‘𝑚)‘𝑤))) < (𝑟 / 2))
96	67, 68, 95	syl2an 597	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) → ∃𝑝 ∈ 𝑍 ∀𝑞 ∈ (ℤ_≥‘𝑝)∀𝑤 ∈ 𝑆 ∀𝑚 ∈ (ℤ_≥‘𝑞)(abs‘(((𝐹‘𝑞)‘𝑤) − ((𝐹‘𝑚)‘𝑤))) < (𝑟 / 2))
97	3	uztrn2 12265	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ 𝑍 ∧ 𝑞 ∈ (ℤ_≥‘𝑝)) → 𝑞 ∈ 𝑍)
98		eqid 2823	. . . . . . . . . . . . . 14 ⊢ (ℤ_≥‘𝑞) = (ℤ_≥‘𝑞)
99		eluzelz 12256	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 ∈ (ℤ_≥‘𝑀) → 𝑞 ∈ ℤ)
100	99, 3	eleq2s 2933	. . . . . . . . . . . . . . 15 ⊢ (𝑞 ∈ 𝑍 → 𝑞 ∈ ℤ)
101	100	ad2antlr 725	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑞 ∈ 𝑍) ∧ 𝑤 ∈ 𝑆) → 𝑞 ∈ ℤ)
102	68	adantl 484	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) → (𝑟 / 2) ∈ ℝ⁺)
103	102	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑞 ∈ 𝑍) ∧ 𝑤 ∈ 𝑆) → (𝑟 / 2) ∈ ℝ⁺)
104		simplr 767	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑞 ∈ 𝑍) ∧ 𝑤 ∈ 𝑆) → 𝑞 ∈ 𝑍)
105	3	uztrn2 12265	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 ∈ 𝑍 ∧ 𝑚 ∈ (ℤ_≥‘𝑞)) → 𝑚 ∈ 𝑍)
106	104, 105	sylan 582	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑞 ∈ 𝑍) ∧ 𝑤 ∈ 𝑆) ∧ 𝑚 ∈ (ℤ_≥‘𝑞)) → 𝑚 ∈ 𝑍)
107		fveq2 6672	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → (𝐹‘𝑛) = (𝐹‘𝑚))
108	107	fveq1d 6674	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → ((𝐹‘𝑛)‘𝑤) = ((𝐹‘𝑚)‘𝑤))
109		eqid 2823	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤)) = (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))
110		fvex 6685	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑚)‘𝑤) ∈ V
111	108, 109, 110	fvmpt 6770	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ 𝑍 → ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))‘𝑚) = ((𝐹‘𝑚)‘𝑤))
112	106, 111	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑞 ∈ 𝑍) ∧ 𝑤 ∈ 𝑆) ∧ 𝑚 ∈ (ℤ_≥‘𝑞)) → ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))‘𝑚) = ((𝐹‘𝑚)‘𝑤))
113	6	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) → 𝐹:𝑍⟶(ℂ ↑_m 𝑆))
114	113	ffvelrnda 6853	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ (ℂ ↑_m 𝑆))
115		elmapi 8430	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹‘𝑛) ∈ (ℂ ↑_m 𝑆) → (𝐹‘𝑛):𝑆⟶ℂ)
116	114, 115	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):𝑆⟶ℂ)
117	116	ffvelrnda 6853	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑛 ∈ 𝑍) ∧ 𝑦 ∈ 𝑆) → ((𝐹‘𝑛)‘𝑦) ∈ ℂ)
118	117	an32s 650	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑦 ∈ 𝑆) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑦) ∈ ℂ)
119	118	fmpttd 6881	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑦 ∈ 𝑆) → (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦)):𝑍⟶ℂ)
120	119	ffvelrnda 6853	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑦 ∈ 𝑆) ∧ 𝑞 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑞) ∈ ℂ)
121		fveq2 6672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 = 𝑦 → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑘)‘𝑦))
122		fveq2 6672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑧 = 𝑦 → ((𝐹‘𝑗)‘𝑧) = ((𝐹‘𝑗)‘𝑦))
123	121, 122	oveq12d 7176	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 = 𝑦 → (((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧)) = (((𝐹‘𝑘)‘𝑦) − ((𝐹‘𝑗)‘𝑦)))
124	123	fveq2d 6676	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑧 = 𝑦 → (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) = (abs‘(((𝐹‘𝑘)‘𝑦) − ((𝐹‘𝑗)‘𝑦))))
125	124	breq1d 5078	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 = 𝑦 → ((abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥 ↔ (abs‘(((𝐹‘𝑘)‘𝑦) − ((𝐹‘𝑗)‘𝑦))) < 𝑥))
126	125	rspcv 3620	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ 𝑆 → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥 → (abs‘(((𝐹‘𝑘)‘𝑦) − ((𝐹‘𝑗)‘𝑦))) < 𝑥))
127	126	ralimdv 3180	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ 𝑆 → (∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥 → ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(((𝐹‘𝑘)‘𝑦) − ((𝐹‘𝑗)‘𝑦))) < 𝑥))
128	127	reximdv 3275	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ 𝑆 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(((𝐹‘𝑘)‘𝑦) − ((𝐹‘𝑗)‘𝑦))) < 𝑥))
129	128	ralimdv 3180	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ 𝑆 → (∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(((𝐹‘𝑘)‘𝑦) − ((𝐹‘𝑗)‘𝑦))) < 𝑥))
130	129	impcom 410	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥 ∧ 𝑦 ∈ 𝑆) → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(((𝐹‘𝑘)‘𝑦) − ((𝐹‘𝑗)‘𝑦))) < 𝑥)
131	130	adantll 712	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑦 ∈ 𝑆) → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(((𝐹‘𝑘)‘𝑦) − ((𝐹‘𝑗)‘𝑦))) < 𝑥)
132		fveq2 6672	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑞 = 𝑘 → ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑞) = ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑘))
133	132	fvoveq1d 7180	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑞 = 𝑘 → (abs‘(((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑞) − ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑝))) = (abs‘(((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑘) − ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑝))))
134	133	breq1d 5078	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑞 = 𝑘 → ((abs‘(((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑞) − ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑝))) < 𝑟 ↔ (abs‘(((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑘) − ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑝))) < 𝑟))
135	134	cbvralvw 3451	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (∀𝑞 ∈ (ℤ_≥‘𝑝)(abs‘(((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑞) − ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑝))) < 𝑟 ↔ ∀𝑘 ∈ (ℤ_≥‘𝑝)(abs‘(((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑘) − ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑝))) < 𝑟)
136		fveq2 6672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑝 = 𝑗 → ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑝) = ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑗))
137	136	oveq2d 7174	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑝 = 𝑗 → (((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑘) − ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑝)) = (((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑘) − ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑗)))
138	137	fveq2d 6676	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑝 = 𝑗 → (abs‘(((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑘) − ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑝))) = (abs‘(((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑘) − ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑗))))
139	138	breq1d 5078	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑝 = 𝑗 → ((abs‘(((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑘) − ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑝))) < 𝑟 ↔ (abs‘(((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑘) − ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑗))) < 𝑟))
140	90, 139	raleqbidv 3403	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑝 = 𝑗 → (∀𝑘 ∈ (ℤ_≥‘𝑝)(abs‘(((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑘) − ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑝))) < 𝑟 ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑘) − ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑗))) < 𝑟))
141	135, 140	syl5bb 285	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑝 = 𝑗 → (∀𝑞 ∈ (ℤ_≥‘𝑝)(abs‘(((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑞) − ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑝))) < 𝑟 ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑘) − ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑗))) < 𝑟))
142	141	cbvrexvw 3452	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∃𝑝 ∈ 𝑍 ∀𝑞 ∈ (ℤ_≥‘𝑝)(abs‘(((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑞) − ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑝))) < 𝑟 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑘) − ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑗))) < 𝑟)
143		fveq2 6672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
144	143	fveq1d 6674	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑛 = 𝑘 → ((𝐹‘𝑛)‘𝑦) = ((𝐹‘𝑘)‘𝑦))
145		eqid 2823	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦)) = (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))
146		fvex 6685	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐹‘𝑘)‘𝑦) ∈ V
147	144, 145, 146	fvmpt 6770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑘 ∈ 𝑍 → ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑘) = ((𝐹‘𝑘)‘𝑦))
148	37, 147	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑘) = ((𝐹‘𝑘)‘𝑦))
149		fveq2 6672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑛 = 𝑗 → (𝐹‘𝑛) = (𝐹‘𝑗))
150	149	fveq1d 6674	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑛 = 𝑗 → ((𝐹‘𝑛)‘𝑦) = ((𝐹‘𝑗)‘𝑦))
151		fvex 6685	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐹‘𝑗)‘𝑦) ∈ V
152	150, 145, 151	fvmpt 6770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ 𝑍 → ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑗) = ((𝐹‘𝑗)‘𝑦))
153	152	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑗) = ((𝐹‘𝑗)‘𝑦))
154	148, 153	oveq12d 7176	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑘) − ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑗)) = (((𝐹‘𝑘)‘𝑦) − ((𝐹‘𝑗)‘𝑦)))
155	154	fveq2d 6676	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (abs‘(((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑘) − ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑗))) = (abs‘(((𝐹‘𝑘)‘𝑦) − ((𝐹‘𝑗)‘𝑦))))
156	155	breq1d 5078	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((abs‘(((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑘) − ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑗))) < 𝑟 ↔ (abs‘(((𝐹‘𝑘)‘𝑦) − ((𝐹‘𝑗)‘𝑦))) < 𝑟))
157	156	ralbidva 3198	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ 𝑍 → (∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑘) − ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑗))) < 𝑟 ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(((𝐹‘𝑘)‘𝑦) − ((𝐹‘𝑗)‘𝑦))) < 𝑟))
158	157	rexbiia 3248	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑘) − ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑗))) < 𝑟 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(((𝐹‘𝑘)‘𝑦) − ((𝐹‘𝑗)‘𝑦))) < 𝑟)
159	142, 158	bitri 277	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃𝑝 ∈ 𝑍 ∀𝑞 ∈ (ℤ_≥‘𝑝)(abs‘(((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑞) − ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑝))) < 𝑟 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(((𝐹‘𝑘)‘𝑦) − ((𝐹‘𝑗)‘𝑦))) < 𝑟)
160		breq2 5072	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑟 = 𝑥 → ((abs‘(((𝐹‘𝑘)‘𝑦) − ((𝐹‘𝑗)‘𝑦))) < 𝑟 ↔ (abs‘(((𝐹‘𝑘)‘𝑦) − ((𝐹‘𝑗)‘𝑦))) < 𝑥))
161	160	ralbidv 3199	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑟 = 𝑥 → (∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(((𝐹‘𝑘)‘𝑦) − ((𝐹‘𝑗)‘𝑦))) < 𝑟 ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(((𝐹‘𝑘)‘𝑦) − ((𝐹‘𝑗)‘𝑦))) < 𝑥))
162	161	rexbidv 3299	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑟 = 𝑥 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(((𝐹‘𝑘)‘𝑦) − ((𝐹‘𝑗)‘𝑦))) < 𝑟 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(((𝐹‘𝑘)‘𝑦) − ((𝐹‘𝑗)‘𝑦))) < 𝑥))
163	159, 162	syl5bb 285	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑟 = 𝑥 → (∃𝑝 ∈ 𝑍 ∀𝑞 ∈ (ℤ_≥‘𝑝)(abs‘(((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑞) − ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑝))) < 𝑟 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(((𝐹‘𝑘)‘𝑦) − ((𝐹‘𝑗)‘𝑦))) < 𝑥))
164	163	cbvralvw 3451	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑟 ∈ ℝ⁺ ∃𝑝 ∈ 𝑍 ∀𝑞 ∈ (ℤ_≥‘𝑝)(abs‘(((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑞) − ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑝))) < 𝑟 ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘(((𝐹‘𝑘)‘𝑦) − ((𝐹‘𝑗)‘𝑦))) < 𝑥)
165	131, 164	sylibr 236	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑦 ∈ 𝑆) → ∀𝑟 ∈ ℝ⁺ ∃𝑝 ∈ 𝑍 ∀𝑞 ∈ (ℤ_≥‘𝑝)(abs‘(((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑞) − ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))‘𝑝))) < 𝑟)
166	3	fvexi 6686	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑍 ∈ V
167	166	mptex 6988	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦)) ∈ V
168	167	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑦 ∈ 𝑆) → (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦)) ∈ V)
169	3, 120, 165, 168	caucvg 15037	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑦 ∈ 𝑆) → (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦)) ∈ dom ⇝ )
170	169	ralrimiva 3184	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) → ∀𝑦 ∈ 𝑆 (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦)) ∈ dom ⇝ )
171	170	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑞 ∈ 𝑍) → ∀𝑦 ∈ 𝑆 (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦)) ∈ dom ⇝ )
172		fveq2 6672	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑛)‘𝑦) = ((𝐹‘𝑛)‘𝑤))
173	172	mpteq2dv 5164	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑤 → (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦)) = (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤)))
174	173	eleq1d 2899	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑤 → ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦)) ∈ dom ⇝ ↔ (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤)) ∈ dom ⇝ ))
175	174	rspccva 3624	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑦 ∈ 𝑆 (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦)) ∈ dom ⇝ ∧ 𝑤 ∈ 𝑆) → (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤)) ∈ dom ⇝ )
176	171, 175	sylan 582	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑞 ∈ 𝑍) ∧ 𝑤 ∈ 𝑆) → (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤)) ∈ dom ⇝ )
177		climdm 14913	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤)) ∈ dom ⇝ ↔ (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤)) ⇝ ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))))
178	176, 177	sylib 220	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑞 ∈ 𝑍) ∧ 𝑤 ∈ 𝑆) → (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤)) ⇝ ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))))
179	98, 101, 103, 112, 178	climi2 14870	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑞 ∈ 𝑍) ∧ 𝑤 ∈ 𝑆) → ∃𝑣 ∈ (ℤ_≥‘𝑞)∀𝑚 ∈ (ℤ_≥‘𝑣)(abs‘(((𝐹‘𝑚)‘𝑤) − ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))))) < (𝑟 / 2))
180	98	r19.29uz 14712	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑚 ∈ (ℤ_≥‘𝑞)(abs‘(((𝐹‘𝑞)‘𝑤) − ((𝐹‘𝑚)‘𝑤))) < (𝑟 / 2) ∧ ∃𝑣 ∈ (ℤ_≥‘𝑞)∀𝑚 ∈ (ℤ_≥‘𝑣)(abs‘(((𝐹‘𝑚)‘𝑤) − ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))))) < (𝑟 / 2)) → ∃𝑣 ∈ (ℤ_≥‘𝑞)∀𝑚 ∈ (ℤ_≥‘𝑣)((abs‘(((𝐹‘𝑞)‘𝑤) − ((𝐹‘𝑚)‘𝑤))) < (𝑟 / 2) ∧ (abs‘(((𝐹‘𝑚)‘𝑤) − ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))))) < (𝑟 / 2)))
181	98	r19.2uz 14713	. . . . . . . . . . . . . . 15 ⊢ (∃𝑣 ∈ (ℤ_≥‘𝑞)∀𝑚 ∈ (ℤ_≥‘𝑣)((abs‘(((𝐹‘𝑞)‘𝑤) − ((𝐹‘𝑚)‘𝑤))) < (𝑟 / 2) ∧ (abs‘(((𝐹‘𝑚)‘𝑤) − ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))))) < (𝑟 / 2)) → ∃𝑚 ∈ (ℤ_≥‘𝑞)((abs‘(((𝐹‘𝑞)‘𝑤) − ((𝐹‘𝑚)‘𝑤))) < (𝑟 / 2) ∧ (abs‘(((𝐹‘𝑚)‘𝑤) − ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))))) < (𝑟 / 2)))
182	180, 181	syl 17	. . . . . . . . . . . . . 14 ⊢ ((∀𝑚 ∈ (ℤ_≥‘𝑞)(abs‘(((𝐹‘𝑞)‘𝑤) − ((𝐹‘𝑚)‘𝑤))) < (𝑟 / 2) ∧ ∃𝑣 ∈ (ℤ_≥‘𝑞)∀𝑚 ∈ (ℤ_≥‘𝑣)(abs‘(((𝐹‘𝑚)‘𝑤) − ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))))) < (𝑟 / 2)) → ∃𝑚 ∈ (ℤ_≥‘𝑞)((abs‘(((𝐹‘𝑞)‘𝑤) − ((𝐹‘𝑚)‘𝑤))) < (𝑟 / 2) ∧ (abs‘(((𝐹‘𝑚)‘𝑤) − ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))))) < (𝑟 / 2)))
183	6	ad2antrr 724	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) → 𝐹:𝑍⟶(ℂ ↑_m 𝑆))
184	183	ffvelrnda 6853	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑞 ∈ 𝑍) → (𝐹‘𝑞) ∈ (ℂ ↑_m 𝑆))
185		elmapi 8430	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑞) ∈ (ℂ ↑_m 𝑆) → (𝐹‘𝑞):𝑆⟶ℂ)
186	184, 185	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑞 ∈ 𝑍) → (𝐹‘𝑞):𝑆⟶ℂ)
187	186	ffvelrnda 6853	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑞 ∈ 𝑍) ∧ 𝑤 ∈ 𝑆) → ((𝐹‘𝑞)‘𝑤) ∈ ℂ)
188	187	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑞 ∈ 𝑍) ∧ 𝑤 ∈ 𝑆) ∧ 𝑚 ∈ (ℤ_≥‘𝑞)) → ((𝐹‘𝑞)‘𝑤) ∈ ℂ)
189		climcl 14858	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤)) ⇝ ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))) → ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))) ∈ ℂ)
190	178, 189	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑞 ∈ 𝑍) ∧ 𝑤 ∈ 𝑆) → ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))) ∈ ℂ)
191	190	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑞 ∈ 𝑍) ∧ 𝑤 ∈ 𝑆) ∧ 𝑚 ∈ (ℤ_≥‘𝑞)) → ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))) ∈ ℂ)
192	6	ad5antr 732	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑞 ∈ 𝑍) ∧ 𝑤 ∈ 𝑆) ∧ 𝑚 ∈ (ℤ_≥‘𝑞)) → 𝐹:𝑍⟶(ℂ ↑_m 𝑆))
193	192, 106	ffvelrnd 6854	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑞 ∈ 𝑍) ∧ 𝑤 ∈ 𝑆) ∧ 𝑚 ∈ (ℤ_≥‘𝑞)) → (𝐹‘𝑚) ∈ (ℂ ↑_m 𝑆))
194		elmapi 8430	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑚) ∈ (ℂ ↑_m 𝑆) → (𝐹‘𝑚):𝑆⟶ℂ)
195	193, 194	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑞 ∈ 𝑍) ∧ 𝑤 ∈ 𝑆) ∧ 𝑚 ∈ (ℤ_≥‘𝑞)) → (𝐹‘𝑚):𝑆⟶ℂ)
196		simplr 767	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑞 ∈ 𝑍) ∧ 𝑤 ∈ 𝑆) ∧ 𝑚 ∈ (ℤ_≥‘𝑞)) → 𝑤 ∈ 𝑆)
197	195, 196	ffvelrnd 6854	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑞 ∈ 𝑍) ∧ 𝑤 ∈ 𝑆) ∧ 𝑚 ∈ (ℤ_≥‘𝑞)) → ((𝐹‘𝑚)‘𝑤) ∈ ℂ)
198		rpre 12400	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ)
199	198	ad4antlr 731	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑞 ∈ 𝑍) ∧ 𝑤 ∈ 𝑆) ∧ 𝑚 ∈ (ℤ_≥‘𝑞)) → 𝑟 ∈ ℝ)
200		abs3lem 14700	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐹‘𝑞)‘𝑤) ∈ ℂ ∧ ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))) ∈ ℂ) ∧ (((𝐹‘𝑚)‘𝑤) ∈ ℂ ∧ 𝑟 ∈ ℝ)) → (((abs‘(((𝐹‘𝑞)‘𝑤) − ((𝐹‘𝑚)‘𝑤))) < (𝑟 / 2) ∧ (abs‘(((𝐹‘𝑚)‘𝑤) − ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))))) < (𝑟 / 2)) → (abs‘(((𝐹‘𝑞)‘𝑤) − ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))))) < 𝑟))
201	188, 191, 197, 199, 200	syl22anc 836	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑞 ∈ 𝑍) ∧ 𝑤 ∈ 𝑆) ∧ 𝑚 ∈ (ℤ_≥‘𝑞)) → (((abs‘(((𝐹‘𝑞)‘𝑤) − ((𝐹‘𝑚)‘𝑤))) < (𝑟 / 2) ∧ (abs‘(((𝐹‘𝑚)‘𝑤) − ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))))) < (𝑟 / 2)) → (abs‘(((𝐹‘𝑞)‘𝑤) − ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))))) < 𝑟))
202	201	rexlimdva 3286	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑞 ∈ 𝑍) ∧ 𝑤 ∈ 𝑆) → (∃𝑚 ∈ (ℤ_≥‘𝑞)((abs‘(((𝐹‘𝑞)‘𝑤) − ((𝐹‘𝑚)‘𝑤))) < (𝑟 / 2) ∧ (abs‘(((𝐹‘𝑚)‘𝑤) − ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))))) < (𝑟 / 2)) → (abs‘(((𝐹‘𝑞)‘𝑤) − ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))))) < 𝑟))
203	182, 202	syl5 34	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑞 ∈ 𝑍) ∧ 𝑤 ∈ 𝑆) → ((∀𝑚 ∈ (ℤ_≥‘𝑞)(abs‘(((𝐹‘𝑞)‘𝑤) − ((𝐹‘𝑚)‘𝑤))) < (𝑟 / 2) ∧ ∃𝑣 ∈ (ℤ_≥‘𝑞)∀𝑚 ∈ (ℤ_≥‘𝑣)(abs‘(((𝐹‘𝑚)‘𝑤) − ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))))) < (𝑟 / 2)) → (abs‘(((𝐹‘𝑞)‘𝑤) − ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))))) < 𝑟))
204	179, 203	mpan2d 692	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑞 ∈ 𝑍) ∧ 𝑤 ∈ 𝑆) → (∀𝑚 ∈ (ℤ_≥‘𝑞)(abs‘(((𝐹‘𝑞)‘𝑤) − ((𝐹‘𝑚)‘𝑤))) < (𝑟 / 2) → (abs‘(((𝐹‘𝑞)‘𝑤) − ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))))) < 𝑟))
205	204	ralimdva 3179	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑞 ∈ 𝑍) → (∀𝑤 ∈ 𝑆 ∀𝑚 ∈ (ℤ_≥‘𝑞)(abs‘(((𝐹‘𝑞)‘𝑤) − ((𝐹‘𝑚)‘𝑤))) < (𝑟 / 2) → ∀𝑤 ∈ 𝑆 (abs‘(((𝐹‘𝑞)‘𝑤) − ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))))) < 𝑟))
206	97, 205	sylan2 594	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑝 ∈ 𝑍 ∧ 𝑞 ∈ (ℤ_≥‘𝑝))) → (∀𝑤 ∈ 𝑆 ∀𝑚 ∈ (ℤ_≥‘𝑞)(abs‘(((𝐹‘𝑞)‘𝑤) − ((𝐹‘𝑚)‘𝑤))) < (𝑟 / 2) → ∀𝑤 ∈ 𝑆 (abs‘(((𝐹‘𝑞)‘𝑤) − ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))))) < 𝑟))
207	206	anassrs 470	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑝 ∈ 𝑍) ∧ 𝑞 ∈ (ℤ_≥‘𝑝)) → (∀𝑤 ∈ 𝑆 ∀𝑚 ∈ (ℤ_≥‘𝑞)(abs‘(((𝐹‘𝑞)‘𝑤) − ((𝐹‘𝑚)‘𝑤))) < (𝑟 / 2) → ∀𝑤 ∈ 𝑆 (abs‘(((𝐹‘𝑞)‘𝑤) − ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))))) < 𝑟))
208	207	ralimdva 3179	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑝 ∈ 𝑍) → (∀𝑞 ∈ (ℤ_≥‘𝑝)∀𝑤 ∈ 𝑆 ∀𝑚 ∈ (ℤ_≥‘𝑞)(abs‘(((𝐹‘𝑞)‘𝑤) − ((𝐹‘𝑚)‘𝑤))) < (𝑟 / 2) → ∀𝑞 ∈ (ℤ_≥‘𝑝)∀𝑤 ∈ 𝑆 (abs‘(((𝐹‘𝑞)‘𝑤) − ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))))) < 𝑟))
209	208	reximdva 3276	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) → (∃𝑝 ∈ 𝑍 ∀𝑞 ∈ (ℤ_≥‘𝑝)∀𝑤 ∈ 𝑆 ∀𝑚 ∈ (ℤ_≥‘𝑞)(abs‘(((𝐹‘𝑞)‘𝑤) − ((𝐹‘𝑚)‘𝑤))) < (𝑟 / 2) → ∃𝑝 ∈ 𝑍 ∀𝑞 ∈ (ℤ_≥‘𝑝)∀𝑤 ∈ 𝑆 (abs‘(((𝐹‘𝑞)‘𝑤) − ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))))) < 𝑟))
210	96, 209	mpd 15	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑟 ∈ ℝ⁺) → ∃𝑝 ∈ 𝑍 ∀𝑞 ∈ (ℤ_≥‘𝑝)∀𝑤 ∈ 𝑆 (abs‘(((𝐹‘𝑞)‘𝑤) − ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))))) < 𝑟)
211	210	ralrimiva 3184	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) → ∀𝑟 ∈ ℝ⁺ ∃𝑝 ∈ 𝑍 ∀𝑞 ∈ (ℤ_≥‘𝑝)∀𝑤 ∈ 𝑆 (abs‘(((𝐹‘𝑞)‘𝑤) − ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))))) < 𝑟)
212	4	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) → 𝑀 ∈ ℤ)
213		eqidd 2824	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ (𝑞 ∈ 𝑍 ∧ 𝑤 ∈ 𝑆)) → ((𝐹‘𝑞)‘𝑤) = ((𝐹‘𝑞)‘𝑤))
214	173	fveq2d 6676	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))) = ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))))
215		eqid 2823	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑆 ↦ ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦)))) = (𝑦 ∈ 𝑆 ↦ ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))))
216		fvex 6685	. . . . . . . 8 ⊢ ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))) ∈ V
217	214, 215, 216	fvmpt 6770	. . . . . . 7 ⊢ (𝑤 ∈ 𝑆 → ((𝑦 ∈ 𝑆 ↦ ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))))‘𝑤) = ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))))
218	217	adantl 484	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑤 ∈ 𝑆) → ((𝑦 ∈ 𝑆 ↦ ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))))‘𝑤) = ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))))
219		climdm 14913	. . . . . . . . 9 ⊢ ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦)) ∈ dom ⇝ ↔ (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦)) ⇝ ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))))
220	169, 219	sylib 220	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑦 ∈ 𝑆) → (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦)) ⇝ ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))))
221		climcl 14858	. . . . . . . 8 ⊢ ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦)) ⇝ ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))) → ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))) ∈ ℂ)
222	220, 221	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) ∧ 𝑦 ∈ 𝑆) → ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))) ∈ ℂ)
223	222	fmpttd 6881	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) → (𝑦 ∈ 𝑆 ↦ ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦)))):𝑆⟶ℂ)
224	65	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) → 𝑆 ∈ 𝑉)
225	3, 212, 113, 213, 218, 223, 224	ulm2 24975	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) → (𝐹(⇝_𝑢‘𝑆)(𝑦 ∈ 𝑆 ↦ ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦)))) ↔ ∀𝑟 ∈ ℝ⁺ ∃𝑝 ∈ 𝑍 ∀𝑞 ∈ (ℤ_≥‘𝑝)∀𝑤 ∈ 𝑆 (abs‘(((𝐹‘𝑞)‘𝑤) − ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑤))))) < 𝑟))
226	211, 225	mpbird 259	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) → 𝐹(⇝_𝑢‘𝑆)(𝑦 ∈ 𝑆 ↦ ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦)))))
227		releldm 5816	. . . 4 ⊢ ((Rel (⇝_𝑢‘𝑆) ∧ 𝐹(⇝_𝑢‘𝑆)(𝑦 ∈ 𝑆 ↦ ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑦))))) → 𝐹 ∈ dom (⇝_𝑢‘𝑆))
228	64, 226, 227	sylancr 589	. . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥) → 𝐹 ∈ dom (⇝_𝑢‘𝑆))
229	228	ex 415	. 2 ⊢ (𝜑 → (∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥 → 𝐹 ∈ dom (⇝_𝑢‘𝑆)))
230	63, 229	impbid 214	1 ⊢ (𝜑 → (𝐹 ∈ dom (⇝_𝑢‘𝑆) ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 Vcvv 3496 class class class wbr 5068 ↦ cmpt 5148 dom cdm 5557 Rel wrel 5562 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ↑_m cmap 8408 ℂcc 10537 ℝcr 10538 < clt 10677 − cmin 10872 / cdiv 11299 2c2 11695 ℤcz 11984 ℤ_≥cuz 12246 ℝ⁺crp 12392 abscabs 14595 ⇝ cli 14843 ⇝_𝑢culm 24966

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-ico 12747 df-fl 13165 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-limsup 14830 df-clim 14847 df-rlim 14848 df-ulm 24967

This theorem is referenced by: ulmcau2 24986 mtest 24994

W3C validator