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Theorem ulmcaulem 26778

Description: Lemma for ulmcau 26779 and ulmcau2 26780: show the equivalence of the four- and five-quantifier forms of the Cauchy convergence condition. Compare cau3 15567. (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 5112	. . . . . 6 ⊢ (𝑥 = 𝑤 → ((abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥 ↔ (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑤))
2	1	ralbidv 3189	. . . . 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 3232	. . . 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 3244	. . 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 13189	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 / 2) ∈ ℝ⁺)
6		breq2 5112	. . . . . . . . . 10 ⊢ (𝑤 = (𝑥 / 2) → ((abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑤 ↔ (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)))
7	6	ralbidv 3189	. . . . . . . . 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 3232	. . . . . . . 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 3576	. . . . . . 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 18	. . . . . 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 487	. . . . 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 6916	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚))
13	12	fveq1d 6918	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑚 → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑚)‘𝑧))
14	13	fvoveq1d 7473	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) = (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))))
15	14	breq1d 5118	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → ((abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) ↔ (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)))
16	15	ralbidv 3189	. . . . . . . . 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 3244	. . . . . . . 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 219	. . . . . . 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 13028	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (ℤ_≥‘𝑗) → (ℤ_≥‘𝑘) ⊆ (ℤ_≥‘𝑗))
20	19	ad2antlr 740	. . . . . . . . . . . . . 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 3982	. . . . . . . . . . . . . 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 18	. . . . . . . . . . . . 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 3126	. . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐹:𝑍⟶(ℂ ↑_m 𝑆))
26	25	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . . . . . 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 13024	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
29	28	adantll 727	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
30	27	uztrn2 13024	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑘 ∈ 𝑍 ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) → 𝑚 ∈ 𝑍)
31	29, 30	sylan 592	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) → 𝑚 ∈ 𝑍)
32	26, 31	ffvelcdmd 7117	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) → (𝐹‘𝑚) ∈ (ℂ ↑_m 𝑆))
33		elmapi 8907	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹‘𝑚) ∈ (ℂ ↑_m 𝑆) → (𝐹‘𝑚):𝑆⟶ℂ)
34	32, 33	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) → (𝐹‘𝑚):𝑆⟶ℂ)
35	34	ffvelcdmda 7116	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑚)‘𝑧) ∈ ℂ)
36	25	ffvelcdmda 7116	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ (ℂ ↑_m 𝑆))
37	36	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) → (𝐹‘𝑗) ∈ (ℂ ↑_m 𝑆))
38		elmapi 8907	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹‘𝑗) ∈ (ℂ ↑_m 𝑆) → (𝐹‘𝑗):𝑆⟶ℂ)
39	37, 38	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) → (𝐹‘𝑗):𝑆⟶ℂ)
40	39	ffvelcdmda 7116	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑗)‘𝑧) ∈ ℂ)
41	35, 40	abssubd 15667	. . . . . . . . . . . . . . . . . . . . 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 5118	. . . . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . . . 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		ffvelcdm 7113	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐹:𝑍⟶(ℂ ↑_m 𝑆) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ (ℂ ↑_m 𝑆))
45	25, 28, 44	syl2an 608	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑘) ∈ (ℂ ↑_m 𝑆))
46	45	anassrs 473	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ (ℂ ↑_m 𝑆))
47	46	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) → (𝐹‘𝑘) ∈ (ℂ ↑_m 𝑆))
48		elmapi 8907	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹‘𝑘) ∈ (ℂ ↑_m 𝑆) → (𝐹‘𝑘):𝑆⟶ℂ)
49	47, 48	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) → (𝐹‘𝑘):𝑆⟶ℂ)
50	49	ffvelcdmda 7116	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑘)‘𝑧) ∈ ℂ)
51		rpre 13169	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
52	51	ad2antlr 740	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → 𝑥 ∈ ℝ)
53	52	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) ∧ 𝑧 ∈ 𝑆) → 𝑥 ∈ ℝ)
54		abs3lem 15550	. . . . . . . . . . . . . . . . . . . 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 852	. . . . . . . . . . . . . . . . . . 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 617	. . . . . . . . . . . . . . . . . 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 3178	. . . . . . . . . . . . . . . . 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	biimtrrid 246	. . . . . . . . . . . . . . . 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 458	. . . . . . . . . . . . . . 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 665	. . . . . . . . . . . . . 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 3178	. . . . . . . . . . . . 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 48	. . . . . . . . . . . 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 457	. . . . . . . . . . 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 665	. . . . . . . . . 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 3178	. . . . . . . . 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 418	. . . . . . . 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 87	. . . . . . 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 46	. . . . . 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 3179	. . . . 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 48	. . . 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 3166	. . 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	biimtrid 245	. 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 13015	. . . . . . . . 9 ⊢ (𝑗 ∈ (ℤ_≥‘𝑀) → 𝑗 ∈ ℤ)
74	73, 27	eleq2s 2882	. . . . . . . 8 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ)
75		uzid 13020	. . . . . . . 8 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ_≥‘𝑗))
76	74, 75	syl 18	. . . . . . 7 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ (ℤ_≥‘𝑗))
77	76	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ_≥‘𝑗))
78		fveq2 6916	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → (ℤ_≥‘𝑘) = (ℤ_≥‘𝑗))
79		fveq2 6916	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
80	79	fveq1d 6918	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑗)‘𝑧))
81	80	fvoveq1d 7473	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) = (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))))
82	81	breq1d 5118	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 ↔ (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥))
83	82	ralbidv 3189	. . . . . . . 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 3338	. . . . . . 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 3576	. . . . . 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 18	. . . . 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 6916	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑘 → (𝐹‘𝑚) = (𝐹‘𝑘))
88	87	fveq1d 6918	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑘 → ((𝐹‘𝑚)‘𝑧) = ((𝐹‘𝑘)‘𝑧))
89	88	oveq2d 7467	. . . . . . . . . 10 ⊢ (𝑚 = 𝑘 → (((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧)) = (((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑘)‘𝑧)))
90	89	fveq2d 6920	. . . . . . . . 9 ⊢ (𝑚 = 𝑘 → (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) = (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑘)‘𝑧))))
91	90	breq1d 5118	. . . . . . . 8 ⊢ (𝑚 = 𝑘 → ((abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 ↔ (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑘)‘𝑧))) < 𝑥))
92	91	ralbidv 3189	. . . . . . 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 3244	. . . . . 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	ffvelcdmda 7116	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ (ℂ ↑_m 𝑆))
95	94	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑗) ∈ (ℂ ↑_m 𝑆))
96	95, 38	syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑗):𝑆⟶ℂ)
97	96	ffvelcdmda 7116	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑗)‘𝑧) ∈ ℂ)
98	24, 28, 44	syl2an 608	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑘) ∈ (ℂ ↑_m 𝑆))
99	98	anassrs 473	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ (ℂ ↑_m 𝑆))
100	99, 48	syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘):𝑆⟶ℂ)
101	100	ffvelcdmda 7116	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑘)‘𝑧) ∈ ℂ)
102	97, 101	abssubd 15667	. . . . . . . . 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 5118	. . . . . . . 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 3187	. . . . . . 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 3187	. . . . . 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 286	. . . . 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 242	. . . 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 3179	. . 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 3180	. 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 215	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ∀wral 3080 ∃wrex 3090 ⊆ wss 3900 class class class wbr 5108 ⟶wf 6562 ‘cfv 6566 (class class class)co 7451 ↑_m cmap 8885 ℂcc 11237 ℝcr 11238 < clt 11382 − cmin 11580 / cdiv 12013 2c2 12437 ℤcz 12733 ℤ_≥cuz 13005 ℝ⁺crp 13160 abscabs 15444

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-1st 8035 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-er 8755 df-map 8887 df-en 9012 df-dom 9013 df-sdom 9014 df-sup 9471 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-n0 12647 df-z 12734 df-uz 13006 df-rp 13161 df-seq 14185 df-exp 14245 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446

This theorem is used by: ulmcau 26779 ulmcau2 26780

W3C validator