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Theorem ulmcaulem 26457

Description: Lemma for ulmcau 26458 and ulmcau2 26459: show the equivalence of the four- and five-quantifier forms of the Cauchy convergence condition. Compare cau3 15383. (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 5104	. . . . . 6 ⊢ (𝑥 = 𝑤 → ((abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥 ↔ (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑤))
2	1	ralbidv 3185	. . . . 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 3228	. . . 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 3240	. . 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 13022	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 / 2) ∈ ℝ⁺)
6		breq2 5104	. . . . . . . . . 10 ⊢ (𝑤 = (𝑥 / 2) → ((abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑤 ↔ (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)))
7	6	ralbidv 3185	. . . . . . . . 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 3228	. . . . . . . 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 3577	. . . . . . 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 485	. . . . 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 6867	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚))
13	12	fveq1d 6869	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑚 → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑚)‘𝑧))
14	13	fvoveq1d 7418	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) = (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))))
15	14	breq1d 5110	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → ((abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) ↔ (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)))
16	15	ralbidv 3185	. . . . . . . . 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 3240	. . . . . . . 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 218	. . . . . . 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 12862	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (ℤ_≥‘𝑗) → (ℤ_≥‘𝑘) ⊆ (ℤ_≥‘𝑗))
20	19	ad2antlr 737	. . . . . . . . . . . . . 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 4005	. . . . . . . . . . . . . 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 3122	. . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐹:𝑍⟶(ℂ ↑_m 𝑆))
26	25	ad3antrrr 740	. . . . . . . . . . . . . . . . . . . . . . . . 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 12858	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
29	28	adantll 724	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
30	27	uztrn2 12858	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑘 ∈ 𝑍 ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) → 𝑚 ∈ 𝑍)
31	29, 30	sylan 589	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) → 𝑚 ∈ 𝑍)
32	26, 31	ffvelcdmd 7066	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) → (𝐹‘𝑚) ∈ (ℂ ↑_m 𝑆))
33		elmapi 8830	. . . . . . . . . . . . . . . . . . . . . . . 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	ffvelcdmda 7065	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑚)‘𝑧) ∈ ℂ)
36	25	ffvelcdmda 7065	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ (ℂ ↑_m 𝑆))
37	36	ad2antrr 736	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) → (𝐹‘𝑗) ∈ (ℂ ↑_m 𝑆))
38		elmapi 8830	. . . . . . . . . . . . . . . . . . . . . . . 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	ffvelcdmda 7065	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑗)‘𝑧) ∈ ℂ)
41	35, 40	abssubd 15483	. . . . . . . . . . . . . . . . . . . . 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 5110	. . . . . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . . . . 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 7062	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐹:𝑍⟶(ℂ ↑_m 𝑆) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ (ℂ ↑_m 𝑆))
45	25, 28, 44	syl2an 605	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑘) ∈ (ℂ ↑_m 𝑆))
46	45	anassrs 471	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ (ℂ ↑_m 𝑆))
47	46	adantr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) → (𝐹‘𝑘) ∈ (ℂ ↑_m 𝑆))
48		elmapi 8830	. . . . . . . . . . . . . . . . . . . . . 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	ffvelcdmda 7065	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑘)‘𝑧) ∈ ℂ)
51		rpre 13002	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
52	51	ad2antlr 737	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → 𝑥 ∈ ℝ)
53	52	ad3antrrr 740	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) ∧ 𝑧 ∈ 𝑆) → 𝑥 ∈ ℝ)
54		abs3lem 15366	. . . . . . . . . . . . . . . . . . . 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 849	. . . . . . . . . . . . . . . . . . 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 614	. . . . . . . . . . . . . . . . . 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 3174	. . . . . . . . . . . . . . . . 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 245	. . . . . . . . . . . . . . . 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 456	. . . . . . . . . . . . . . 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 662	. . . . . . . . . . . . . 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 3174	. . . . . . . . . . . . 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 455	. . . . . . . . . . 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 662	. . . . . . . . . 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 3174	. . . . . . . . 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 416	. . . . . . . 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 3175	. . . . 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 3162	. . 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 244	. 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 12849	. . . . . . . . 9 ⊢ (𝑗 ∈ (ℤ_≥‘𝑀) → 𝑗 ∈ ℤ)
74	73, 27	eleq2s 2880	. . . . . . . 8 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ)
75		uzid 12854	. . . . . . . 8 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ_≥‘𝑗))
76	74, 75	syl 17	. . . . . . 7 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ (ℤ_≥‘𝑗))
77	76	adantl 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ_≥‘𝑗))
78		fveq2 6867	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → (ℤ_≥‘𝑘) = (ℤ_≥‘𝑗))
79		fveq2 6867	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
80	79	fveq1d 6869	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑗)‘𝑧))
81	80	fvoveq1d 7418	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) = (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))))
82	81	breq1d 5110	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 ↔ (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥))
83	82	ralbidv 3185	. . . . . . . 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 3577	. . . . . 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 6867	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑘 → (𝐹‘𝑚) = (𝐹‘𝑘))
88	87	fveq1d 6869	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑘 → ((𝐹‘𝑚)‘𝑧) = ((𝐹‘𝑘)‘𝑧))
89	88	oveq2d 7412	. . . . . . . . . 10 ⊢ (𝑚 = 𝑘 → (((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧)) = (((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑘)‘𝑧)))
90	89	fveq2d 6871	. . . . . . . . 9 ⊢ (𝑚 = 𝑘 → (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) = (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑘)‘𝑧))))
91	90	breq1d 5110	. . . . . . . 8 ⊢ (𝑚 = 𝑘 → ((abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 ↔ (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑘)‘𝑧))) < 𝑥))
92	91	ralbidv 3185	. . . . . . 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 3240	. . . . . 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 7065	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ (ℂ ↑_m 𝑆))
95	94	adantr 484	. . . . . . . . . . . 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	ffvelcdmda 7065	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑗)‘𝑧) ∈ ℂ)
98	24, 28, 44	syl2an 605	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑘) ∈ (ℂ ↑_m 𝑆))
99	98	anassrs 471	. . . . . . . . . . . 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	ffvelcdmda 7065	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑘)‘𝑧) ∈ ℂ)
102	97, 101	abssubd 15483	. . . . . . . . 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 5110	. . . . . . . 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 3183	. . . . . . 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 3183	. . . . . 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 285	. . . . 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 241	. . . 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 3175	. . 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 3176	. 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 214	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 208 ∧ wa 399 = wceq 1560 ∈ wcel 2142 ∀wral 3076 ∃wrex 3086 ⊆ wss 3904 class class class wbr 5100 ⟶wf 6517 ‘cfv 6521 (class class class)co 7396 ↑_m cmap 8808 ℂcc 11071 ℝcr 11072 < clt 11216 − cmin 11414 / cdiv 11844 2c2 12272 ℤcz 12568 ℤ_≥cuz 12839 ℝ⁺crp 12993 abscabs 15261

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150 ax-pre-sup 11151

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-er 8678 df-map 8810 df-en 8928 df-dom 8929 df-sdom 8930 df-sup 9388 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-div 11845 df-nn 12211 df-2 12280 df-3 12281 df-n0 12482 df-z 12569 df-uz 12840 df-rp 12994 df-seq 14015 df-exp 14075 df-cj 15126 df-re 15127 df-im 15128 df-sqrt 15262 df-abs 15263

This theorem is referenced by: ulmcau 26458 ulmcau2 26459

W3C validator