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Theorem ulmcaulem 26357

Description: Lemma for ulmcau 26358 and ulmcau2 26359: show the equivalence of the four- and five-quantifier forms of the Cauchy convergence condition. Compare cau3 15277. (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 5100	. . . . . 6 ⊢ (𝑥 = 𝑤 → ((abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑥 ↔ (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑤))
2	1	ralbidv 3157	. . . . 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 3200	. . . 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 3212	. . 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 12932	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 / 2) ∈ ℝ⁺)
6		breq2 5100	. . . . . . . . . 10 ⊢ (𝑤 = (𝑥 / 2) → ((abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < 𝑤 ↔ (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)))
7	6	ralbidv 3157	. . . . . . . . 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 3200	. . . . . . . 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 3570	. . . . . . 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 481	. . . . 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 6832	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚))
13	12	fveq1d 6834	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑚 → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑚)‘𝑧))
14	13	fvoveq1d 7378	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) = (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))))
15	14	breq1d 5106	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → ((abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2) ↔ (abs‘(((𝐹‘𝑚)‘𝑧) − ((𝐹‘𝑗)‘𝑧))) < (𝑥 / 2)))
16	15	ralbidv 3157	. . . . . . . . 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 3212	. . . . . . . 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 216	. . . . . . 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 12772	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (ℤ_≥‘𝑗) → (ℤ_≥‘𝑘) ⊆ (ℤ_≥‘𝑗))
20	19	ad2antlr 727	. . . . . . . . . . . . . 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 4000	. . . . . . . . . . . . . 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 3094	. . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐹:𝑍⟶(ℂ ↑_m 𝑆))
26	25	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . . . 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 12768	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
29	28	adantll 714	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
30	27	uztrn2 12768	. . . . . . . . . . . . . . . . . . . . . . . . . 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	ffvelcdmd 7028	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) → (𝐹‘𝑚) ∈ (ℂ ↑_m 𝑆))
33		elmapi 8784	. . . . . . . . . . . . . . . . . . . . . . . 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 7027	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑚)‘𝑧) ∈ ℂ)
36	25	ffvelcdmda 7027	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ (ℂ ↑_m 𝑆))
37	36	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) → (𝐹‘𝑗) ∈ (ℂ ↑_m 𝑆))
38		elmapi 8784	. . . . . . . . . . . . . . . . . . . . . . . 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 7027	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑗)‘𝑧) ∈ ℂ)
41	35, 40	abssubd 15377	. . . . . . . . . . . . . . . . . . . . 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 5106	. . . . . . . . . . . . . . . . . . . 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 229	. . . . . . . . . . . . . . . . . . 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 7024	. . . . . . . . . . . . . . . . . . . . . . . . 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 467	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ (ℂ ↑_m 𝑆))
47	46	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) → (𝐹‘𝑘) ∈ (ℂ ↑_m 𝑆))
48		elmapi 8784	. . . . . . . . . . . . . . . . . . . . . 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 7027	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑘)‘𝑧) ∈ ℂ)
51		rpre 12912	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
52	51	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → 𝑥 ∈ ℝ)
53	52	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) ∧ 𝑧 ∈ 𝑆) → 𝑥 ∈ ℝ)
54		abs3lem 15260	. . . . . . . . . . . . . . . . . . . 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 838	. . . . . . . . . . . . . . . . . . 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 3146	. . . . . . . . . . . . . . . . 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 243	. . . . . . . . . . . . . . . 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 452	. . . . . . . . . . . . . . 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 652	. . . . . . . . . . . . . 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 3146	. . . . . . . . . . . . 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 451	. . . . . . . . . . 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 652	. . . . . . . . . 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 3146	. . . . . . . . 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 412	. . . . . . . 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 3147	. . . . 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 3134	. . 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 242	. 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 12759	. . . . . . . . 9 ⊢ (𝑗 ∈ (ℤ_≥‘𝑀) → 𝑗 ∈ ℤ)
74	73, 27	eleq2s 2852	. . . . . . . 8 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ)
75		uzid 12764	. . . . . . . 8 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ_≥‘𝑗))
76	74, 75	syl 17	. . . . . . 7 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ (ℤ_≥‘𝑗))
77	76	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ_≥‘𝑗))
78		fveq2 6832	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → (ℤ_≥‘𝑘) = (ℤ_≥‘𝑗))
79		fveq2 6832	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
80	79	fveq1d 6834	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑗)‘𝑧))
81	80	fvoveq1d 7378	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) = (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))))
82	81	breq1d 5106	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((abs‘(((𝐹‘𝑘)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 ↔ (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥))
83	82	ralbidv 3157	. . . . . . . 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 3314	. . . . . . 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 3570	. . . . . 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 6832	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑘 → (𝐹‘𝑚) = (𝐹‘𝑘))
88	87	fveq1d 6834	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑘 → ((𝐹‘𝑚)‘𝑧) = ((𝐹‘𝑘)‘𝑧))
89	88	oveq2d 7372	. . . . . . . . . 10 ⊢ (𝑚 = 𝑘 → (((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧)) = (((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑘)‘𝑧)))
90	89	fveq2d 6836	. . . . . . . . 9 ⊢ (𝑚 = 𝑘 → (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) = (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑘)‘𝑧))))
91	90	breq1d 5106	. . . . . . . 8 ⊢ (𝑚 = 𝑘 → ((abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑚)‘𝑧))) < 𝑥 ↔ (abs‘(((𝐹‘𝑗)‘𝑧) − ((𝐹‘𝑘)‘𝑧))) < 𝑥))
92	91	ralbidv 3157	. . . . . . 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 3212	. . . . . 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 7027	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ (ℂ ↑_m 𝑆))
95	94	adantr 480	. . . . . . . . . . . 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 7027	. . . . . . . . . 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 467	. . . . . . . . . . . 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 7027	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑘)‘𝑧) ∈ ℂ)
102	97, 101	abssubd 15377	. . . . . . . . 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 5106	. . . . . . . 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 3155	. . . . . . 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 3155	. . . . . 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 283	. . . . 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 239	. . . 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 3147	. . 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 3148	. 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 212	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 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∀wral 3049 ∃wrex 3058 ⊆ wss 3899 class class class wbr 5096 ⟶wf 6486 ‘cfv 6490 (class class class)co 7356 ↑_m cmap 8761 ℂcc 11022 ℝcr 11023 < clt 11164 − cmin 11362 / cdiv 11792 2c2 12198 ℤcz 12486 ℤ_≥cuz 12749 ℝ⁺crp 12903 abscabs 15155

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8633 df-map 8763 df-en 8882 df-dom 8883 df-sdom 8884 df-sup 9343 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-n0 12400 df-z 12487 df-uz 12750 df-rp 12904 df-seq 13923 df-exp 13983 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157

This theorem is referenced by: ulmcau 26358 ulmcau2 26359

W3C validator