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Theorem ulmcaulem 25553

Description: Lemma for ulmcau 25554 and ulmcau2 25555: show the equivalence of the four- and five-quantifier forms of the Cauchy convergence condition. Compare cau3 15067. (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
ulmcaulem	⊢ (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))

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

Proof of Theorem ulmcaulem Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 5078	. . . . . 6 ⊢ (𝑥 = 𝑤 → ((abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥 ↔ (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑤))
2	1	ralbidv 3112	. . . . 5 ⊢ (𝑥 = 𝑤 → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥 ↔ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑤))
3	2	rexralbidv 3230	. . . 4 ⊢ (𝑥 = 𝑤 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑤))
4	3	cbvralvw 3383	. . 3 ⊢ (∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥 ↔ ∀𝑤 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑤)
5		rphalfcl 12757	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 / 2) ∈ ℝ⁺)
6		breq2 5078	. . . . . . . . . 10 ⊢ (𝑤 = (𝑥 / 2) → ((abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑤 ↔ (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)))
7	6	ralbidv 3112	. . . . . . . . 9 ⊢ (𝑤 = (𝑥 / 2) → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑤 ↔ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)))
8	7	rexralbidv 3230	. . . . . . . 8 ⊢ (𝑤 = (𝑥 / 2) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑤 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)))
9	8	rspcv 3557	. . . . . . 7 ⊢ ((𝑥 / 2) ∈ ℝ⁺ → (∀𝑤 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)))
10	5, 9	syl 17	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (∀𝑤 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)))
11	10	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (∀𝑤 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)))
12		fveq2 6774	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚))
13	12	fveq1d 6776	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑚 → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑚)‘𝑧))
14	13	fvoveq1d 7297	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) = (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))))
15	14	breq1d 5084	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → ((abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) ↔ (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)))
16	15	ralbidv 3112	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) ↔ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)))
17	16	cbvralvw 3383	. . . . . . . 8 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) ↔ ∀𝑚 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2))
18	17	biimpi 215	. . . . . . 7 ⊢ (∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → ∀𝑚 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2))
19		uzss 12605	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (ℤ_≥‘𝑗) → (ℤ_≥‘𝑘) ⊆ (ℤ_≥‘𝑗))
20	19	ad2antlr 724	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)) → (ℤ_≥‘𝑘) ⊆ (ℤ_≥‘𝑗))
21		ssralv 3987	. . . . . . . . . . . . . 14 ⊢ ((ℤ_≥‘𝑘) ⊆ (ℤ_≥‘𝑗) → (∀𝑚 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → ∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)))
22	20, 21	syl 17	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)) → (∀𝑚 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → ∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)))
23		r19.26 3095	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑧 ∈ 𝑆 ((abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) ∧ (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)) ↔ (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)))
24		ulmcau.f	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑_m 𝑆))
25	24	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐹:𝑍⟶(ℂ ↑_m 𝑆))
26	25	ad3antrrr 727	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) → 𝐹:𝑍⟶(ℂ ↑_m 𝑆))
27		ulmcau.z	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 𝑍 = (ℤ_≥‘𝑀)
28	27	uztrn2 12601	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
29	28	adantll 711	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
30	27	uztrn2 12601	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑘 ∈ 𝑍 ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) → 𝑚 ∈ 𝑍)
31	29, 30	sylan 580	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) → 𝑚 ∈ 𝑍)
32	26, 31	ffvelrnd 6962	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) → (𝐹‘𝑚) ∈ (ℂ ↑_m 𝑆))
33		elmapi 8637	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹‘𝑚) ∈ (ℂ ↑_m 𝑆) → (𝐹‘𝑚):𝑆⟶ℂ)
34	32, 33	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) → (𝐹‘𝑚):𝑆⟶ℂ)
35	34	ffvelrnda 6961	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑚)‘𝑧) ∈ ℂ)
36	25	ffvelrnda 6961	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ (ℂ ↑_m 𝑆))
37	36	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) → (𝐹‘𝑗) ∈ (ℂ ↑_m 𝑆))
38		elmapi 8637	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹‘𝑗) ∈ (ℂ ↑_m 𝑆) → (𝐹‘𝑗):𝑆⟶ℂ)
39	37, 38	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) → (𝐹‘𝑗):𝑆⟶ℂ)
40	39	ffvelrnda 6961	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑗)‘𝑧) ∈ ℂ)
41	35, 40	abssubd 15165	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) ∧ 𝑧 ∈ 𝑆) → (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) = (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))))
42	41	breq1d 5084	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) ∧ 𝑧 ∈ 𝑆) → ((abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) ↔ (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < (𝑥 / 2)))
43	42	biimpd 228	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) ∧ 𝑧 ∈ 𝑆) → ((abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < (𝑥 / 2)))
44		ffvelrn 6959	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐹:𝑍⟶(ℂ ↑_m 𝑆) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ (ℂ ↑_m 𝑆))
45	25, 28, 44	syl2an 596	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑘) ∈ (ℂ ↑_m 𝑆))
46	45	anassrs 468	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ (ℂ ↑_m 𝑆))
47	46	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) → (𝐹‘𝑘) ∈ (ℂ ↑_m 𝑆))
48		elmapi 8637	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹‘𝑘) ∈ (ℂ ↑_m 𝑆) → (𝐹‘𝑘):𝑆⟶ℂ)
49	47, 48	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) → (𝐹‘𝑘):𝑆⟶ℂ)
50	49	ffvelrnda 6961	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑘)‘𝑧) ∈ ℂ)
51		rpre 12738	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
52	51	ad2antlr 724	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → 𝑥 ∈ ℝ)
53	52	ad3antrrr 727	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) ∧ 𝑧 ∈ 𝑆) → 𝑥 ∈ ℝ)
54		abs3lem 15050	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐹‘𝑘)‘𝑧) ∈ ℂ ∧ ((𝐹‘𝑚)‘𝑧) ∈ ℂ) ∧ (((𝐹‘𝑗)‘𝑧) ∈ ℂ ∧ 𝑥 ∈ ℝ)) → (((abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) ∧ (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < (𝑥 / 2)) → (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥))
55	50, 35, 40, 53, 54	syl22anc 836	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) ∧ 𝑧 ∈ 𝑆) → (((abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) ∧ (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < (𝑥 / 2)) → (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥))
56	43, 55	sylan2d 605	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) ∧ 𝑧 ∈ 𝑆) → (((abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) ∧ (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)) → (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥))
57	56	ralimdva 3108	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) → (∀𝑧 ∈ 𝑆 ((abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) ∧ (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)) → ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥))
58	23, 57	syl5bir 242	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) → ((∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)) → ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥))
59	58	expdimp 453	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)) → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥))
60	59	an32s 649	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥))
61	60	ralimdva 3108	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)) → (∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → ∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥))
62	22, 61	syld 47	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)) → (∀𝑚 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → ∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥))
63	62	impancom 452	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)) → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → ∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥))
64	63	an32s 649	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → ∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥))
65	64	ralimdva 3108	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)) → (∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥))
66	65	ex 413	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → (∀𝑚 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → (∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥)))
67	66	com23 86	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → (∀𝑚 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥)))
68	18, 67	mpdi 45	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥))
69	68	reximdva 3203	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥))
70	11, 69	syld 47	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (∀𝑤 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑤 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥))
71	70	ralrimdva 3106	. . 3 ⊢ (𝜑 → (∀𝑤 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑤 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥))
72	4, 71	syl5bi 241	. 2 ⊢ (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))
73		eluzelz 12592	. . . . . . . . 9 ⊢ (𝑗 ∈ (ℤ_≥‘𝑀) → 𝑗 ∈ ℤ)
74	73, 27	eleq2s 2857	. . . . . . . 8 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ)
75		uzid 12597	. . . . . . . 8 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ_≥‘𝑗))
76	74, 75	syl 17	. . . . . . 7 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ (ℤ_≥‘𝑗))
77	76	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ_≥‘𝑗))
78		fveq2 6774	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → (ℤ_≥‘𝑘) = (ℤ_≥‘𝑗))
79		fveq2 6774	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
80	79	fveq1d 6776	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑗)‘𝑧))
81	80	fvoveq1d 7297	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) = (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))))
82	81	breq1d 5084	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 ↔ (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥))
83	82	ralbidv 3112	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 ↔ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥))
84	78, 83	raleqbidv 3336	. . . . . . 7 ⊢ (𝑘 = 𝑗 → (∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 ↔ ∀𝑚 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥))
85	84	rspcv 3557	. . . . . 6 ⊢ (𝑗 ∈ (ℤ_≥‘𝑗) → (∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 → ∀𝑚 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥))
86	77, 85	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 → ∀𝑚 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥))
87		fveq2 6774	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑘 → (𝐹‘𝑚) = (𝐹‘𝑘))
88	87	fveq1d 6776	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑘 → ((𝐹‘𝑚)‘𝑧) = ((𝐹‘𝑘)‘𝑧))
89	88	oveq2d 7291	. . . . . . . . . 10 ⊢ (𝑚 = 𝑘 → (((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧)) = (((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑘)‘𝑧)))
90	89	fveq2d 6778	. . . . . . . . 9 ⊢ (𝑚 = 𝑘 → (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) = (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑘)‘𝑧))))
91	90	breq1d 5084	. . . . . . . 8 ⊢ (𝑚 = 𝑘 → ((abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 ↔ (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑘)‘𝑧))) < 𝑥))
92	91	ralbidv 3112	. . . . . . 7 ⊢ (𝑚 = 𝑘 → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 ↔ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑘)‘𝑧))) < 𝑥))
93	92	cbvralvw 3383	. . . . . 6 ⊢ (∀𝑚 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑘)‘𝑧))) < 𝑥)
94	24	ffvelrnda 6961	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ (ℂ ↑_m 𝑆))
95	94	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑗) ∈ (ℂ ↑_m 𝑆))
96	95, 38	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑗):𝑆⟶ℂ)
97	96	ffvelrnda 6961	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑗)‘𝑧) ∈ ℂ)
98	24, 28, 44	syl2an 596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑘) ∈ (ℂ ↑_m 𝑆))
99	98	anassrs 468	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ (ℂ ↑_m 𝑆))
100	99, 48	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘):𝑆⟶ℂ)
101	100	ffvelrnda 6961	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑘)‘𝑧) ∈ ℂ)
102	97, 101	abssubd 15165	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑘)‘𝑧))) = (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))))
103	102	breq1d 5084	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → ((abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑘)‘𝑧))) < 𝑥 ↔ (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥))
104	103	ralbidva 3111	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑘)‘𝑧))) < 𝑥 ↔ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥))
105	104	ralbidva 3111	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑘)‘𝑧))) < 𝑥 ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥))
106	93, 105	bitrid 282	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑚 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥))
107	86, 106	sylibd 238	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 → ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥))
108	107	reximdva 3203	. . 3 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥))
109	108	ralimdv 3109	. 2 ⊢ (𝜑 → (∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑘)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥))
110	72, 109	impbid 211	1 ⊢ (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))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 ⊆ wss 3887 class class class wbr 5074 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ↑_m cmap 8615 ℂcc 10869 ℝcr 10870 < clt 11009 − cmin 11205 / cdiv 11632 2c2 12028 ℤcz 12319 ℤ_≥cuz 12582 ℝ⁺crp 12730 abscabs 14945

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-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

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-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 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-pss 3906 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-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 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-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 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-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-sup 9201 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-seq 13722 df-exp 13783 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947

This theorem is referenced by: ulmcau 25554 ulmcau2 25555

W3C validator