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Theorem mtest 25563

Description: The Weierstrass M-test. If 𝐹 is a sequence of functions which are uniformly bounded by the convergent sequence 𝑀(𝑘), then the series generated by the sequence 𝐹 converges uniformly. (Contributed by Mario Carneiro, 3-Mar-2015.)

**Hypotheses**
Ref	Expression
mtest.z	⊢ 𝑍 = (ℤ_≥‘𝑁)
mtest.n	⊢ (𝜑 → 𝑁 ∈ ℤ)
mtest.s	⊢ (𝜑 → 𝑆 ∈ 𝑉)
mtest.f	⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑_m 𝑆))
mtest.m	⊢ (𝜑 → 𝑀 ∈ 𝑊)
mtest.c	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑀‘𝑘) ∈ ℝ)
mtest.l	⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → (abs‘((𝐹‘𝑘)‘𝑧)) ≤ (𝑀‘𝑘))
mtest.d	⊢ (𝜑 → seq𝑁( + , 𝑀) ∈ dom ⇝ )

**Assertion**
Ref	Expression
mtest	⊢ (𝜑 → seq𝑁( ∘_f + , 𝐹) ∈ dom (⇝_𝑢‘𝑆))

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

Proof of Theorem mtest Dummy variables 𝑖 𝑗 𝑛 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mtest.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ)
2		mtest.d	. . . 4 ⊢ (𝜑 → seq𝑁( + , 𝑀) ∈ dom ⇝ )
3		mtest.z	. . . . 5 ⊢ 𝑍 = (ℤ_≥‘𝑁)
4	3	climcau 15382	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ seq𝑁( + , 𝑀) ∈ dom ⇝ ) → ∀𝑟 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑗)(abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))) < 𝑟)
5	1, 2, 4	syl2anc 584	. . 3 ⊢ (𝜑 → ∀𝑟 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑗)(abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))) < 𝑟)
6		seqfn 13733	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℤ → seq𝑁( ∘_f + , 𝐹) Fn (ℤ_≥‘𝑁))
7	1, 6	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → seq𝑁( ∘_f + , 𝐹) Fn (ℤ_≥‘𝑁))
8	3	fneq2i 6531	. . . . . . . . . . . . . . . . . 18 ⊢ (seq𝑁( ∘_f + , 𝐹) Fn 𝑍 ↔ seq𝑁( ∘_f + , 𝐹) Fn (ℤ_≥‘𝑁))
9	7, 8	sylibr 233	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → seq𝑁( ∘_f + , 𝐹) Fn 𝑍)
10		mtest.s	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
11	10	elexd 3452	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑆 ∈ V)
12	11	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝑆 ∈ V)
13		simpr 485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ 𝑍)
14	13, 3	eleqtrdi 2849	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ (ℤ_≥‘𝑁))
15		mtest.f	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑_m 𝑆))
16	15	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝐹:𝑍⟶(ℂ ↑_m 𝑆))
17		elfzuz 13252	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ (𝑁...𝑖) → 𝑘 ∈ (ℤ_≥‘𝑁))
18	17, 3	eleqtrrdi 2850	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ (𝑁...𝑖) → 𝑘 ∈ 𝑍)
19		ffvelrn 6959	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹:𝑍⟶(ℂ ↑_m 𝑆) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ (ℂ ↑_m 𝑆))
20	16, 18, 19	syl2an 596	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ (𝑁...𝑖)) → (𝐹‘𝑘) ∈ (ℂ ↑_m 𝑆))
21		elmapi 8637	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹‘𝑘) ∈ (ℂ ↑_m 𝑆) → (𝐹‘𝑘):𝑆⟶ℂ)
22	20, 21	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ (𝑁...𝑖)) → (𝐹‘𝑘):𝑆⟶ℂ)
23	22	feqmptd 6837	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ (𝑁...𝑖)) → (𝐹‘𝑘) = (𝑧 ∈ 𝑆 ↦ ((𝐹‘𝑘)‘𝑧)))
24	18	adantl 482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ (𝑁...𝑖)) → 𝑘 ∈ 𝑍)
25		fveq2 6774	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
26	25	fveq1d 6776	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝑘 → ((𝐹‘𝑛)‘𝑧) = ((𝐹‘𝑘)‘𝑧))
27		eqid 2738	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)) = (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧))
28		fvex 6787	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹‘𝑘)‘𝑧) ∈ V
29	26, 27, 28	fvmpt 6875	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ 𝑍 → ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧))‘𝑘) = ((𝐹‘𝑘)‘𝑧))
30	24, 29	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ (𝑁...𝑖)) → ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧))‘𝑘) = ((𝐹‘𝑘)‘𝑧))
31	30	mpteq2dv 5176	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ (𝑁...𝑖)) → (𝑧 ∈ 𝑆 ↦ ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧))‘𝑘)) = (𝑧 ∈ 𝑆 ↦ ((𝐹‘𝑘)‘𝑧)))
32	23, 31	eqtr4d 2781	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ (𝑁...𝑖)) → (𝐹‘𝑘) = (𝑧 ∈ 𝑆 ↦ ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧))‘𝑘)))
33	12, 14, 32	seqof 13780	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (seq𝑁( ∘_f + , 𝐹)‘𝑖) = (𝑧 ∈ 𝑆 ↦ (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑖)))
34	1	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑁 ∈ ℤ)
35	15	ffvelrnda 6961	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ (ℂ ↑_m 𝑆))
36		elmapi 8637	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐹‘𝑛) ∈ (ℂ ↑_m 𝑆) → (𝐹‘𝑛):𝑆⟶ℂ)
37	35, 36	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):𝑆⟶ℂ)
38	37	ffvelrnda 6961	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑛)‘𝑧) ∈ ℂ)
39	38	an32s 649	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑧) ∈ ℂ)
40	39	fmpttd 6989	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)):𝑍⟶ℂ)
41	40	ffvelrnda 6961	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ 𝑖 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧))‘𝑖) ∈ ℂ)
42	3, 34, 41	serf 13751	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧))):𝑍⟶ℂ)
43	42	ffvelrnda 6961	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ 𝑖 ∈ 𝑍) → (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑖) ∈ ℂ)
44	43	an32s 649	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑧 ∈ 𝑆) → (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑖) ∈ ℂ)
45	44	fmpttd 6989	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝑧 ∈ 𝑆 ↦ (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑖)):𝑆⟶ℂ)
46		cnex 10952	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℂ ∈ V
47		elmapg 8628	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℂ ∈ V ∧ 𝑆 ∈ V) → ((𝑧 ∈ 𝑆 ↦ (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑖)) ∈ (ℂ ↑_m 𝑆) ↔ (𝑧 ∈ 𝑆 ↦ (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑖)):𝑆⟶ℂ))
48	46, 12, 47	sylancr 587	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((𝑧 ∈ 𝑆 ↦ (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑖)) ∈ (ℂ ↑_m 𝑆) ↔ (𝑧 ∈ 𝑆 ↦ (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑖)):𝑆⟶ℂ))
49	45, 48	mpbird 256	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝑧 ∈ 𝑆 ↦ (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑖)) ∈ (ℂ ↑_m 𝑆))
50	33, 49	eqeltrd 2839	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (seq𝑁( ∘_f + , 𝐹)‘𝑖) ∈ (ℂ ↑_m 𝑆))
51	50	ralrimiva 3103	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑖 ∈ 𝑍 (seq𝑁( ∘_f + , 𝐹)‘𝑖) ∈ (ℂ ↑_m 𝑆))
52		ffnfv 6992	. . . . . . . . . . . . . . . . 17 ⊢ (seq𝑁( ∘_f + , 𝐹):𝑍⟶(ℂ ↑_m 𝑆) ↔ (seq𝑁( ∘_f + , 𝐹) Fn 𝑍 ∧ ∀𝑖 ∈ 𝑍 (seq𝑁( ∘_f + , 𝐹)‘𝑖) ∈ (ℂ ↑_m 𝑆)))
53	9, 51, 52	sylanbrc 583	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → seq𝑁( ∘_f + , 𝐹):𝑍⟶(ℂ ↑_m 𝑆))
54	53	ad2antrr 723	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → seq𝑁( ∘_f + , 𝐹):𝑍⟶(ℂ ↑_m 𝑆))
55	3	uztrn2 12601	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → 𝑖 ∈ 𝑍)
56	55	adantl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → 𝑖 ∈ 𝑍)
57	54, 56	ffvelrnd 6962	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → (seq𝑁( ∘_f + , 𝐹)‘𝑖) ∈ (ℂ ↑_m 𝑆))
58		elmapi 8637	. . . . . . . . . . . . . 14 ⊢ ((seq𝑁( ∘_f + , 𝐹)‘𝑖) ∈ (ℂ ↑_m 𝑆) → (seq𝑁( ∘_f + , 𝐹)‘𝑖):𝑆⟶ℂ)
59	57, 58	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → (seq𝑁( ∘_f + , 𝐹)‘𝑖):𝑆⟶ℂ)
60	59	ffvelrnda 6961	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → ((seq𝑁( ∘_f + , 𝐹)‘𝑖)‘𝑧) ∈ ℂ)
61		simprl 768	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → 𝑗 ∈ 𝑍)
62	54, 61	ffvelrnd 6962	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → (seq𝑁( ∘_f + , 𝐹)‘𝑗) ∈ (ℂ ↑_m 𝑆))
63		elmapi 8637	. . . . . . . . . . . . . 14 ⊢ ((seq𝑁( ∘_f + , 𝐹)‘𝑗) ∈ (ℂ ↑_m 𝑆) → (seq𝑁( ∘_f + , 𝐹)‘𝑗):𝑆⟶ℂ)
64	62, 63	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → (seq𝑁( ∘_f + , 𝐹)‘𝑗):𝑆⟶ℂ)
65	64	ffvelrnda 6961	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → ((seq𝑁( ∘_f + , 𝐹)‘𝑗)‘𝑧) ∈ ℂ)
66	60, 65	subcld 11332	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → (((seq𝑁( ∘_f + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘_f + , 𝐹)‘𝑗)‘𝑧)) ∈ ℂ)
67	66	abscld 15148	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → (abs‘(((seq𝑁( ∘_f + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘_f + , 𝐹)‘𝑗)‘𝑧))) ∈ ℝ)
68		fzfid 13693	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → ((𝑗 + 1)...𝑖) ∈ Fin)
69		ssun2 4107	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 + 1)...𝑖) ⊆ ((𝑁...𝑗) ∪ ((𝑗 + 1)...𝑖))
70	61, 3	eleqtrdi 2849	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → 𝑗 ∈ (ℤ_≥‘𝑁))
71		simprr 770	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → 𝑖 ∈ (ℤ_≥‘𝑗))
72		elfzuzb 13250	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (𝑁...𝑖) ↔ (𝑗 ∈ (ℤ_≥‘𝑁) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)))
73	70, 71, 72	sylanbrc 583	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → 𝑗 ∈ (𝑁...𝑖))
74		fzsplit 13282	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (𝑁...𝑖) → (𝑁...𝑖) = ((𝑁...𝑗) ∪ ((𝑗 + 1)...𝑖)))
75	73, 74	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → (𝑁...𝑖) = ((𝑁...𝑗) ∪ ((𝑗 + 1)...𝑖)))
76	69, 75	sseqtrrid 3974	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → ((𝑗 + 1)...𝑖) ⊆ (𝑁...𝑖))
77	76	sselda 3921	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ ((𝑗 + 1)...𝑖)) → 𝑘 ∈ (𝑁...𝑖))
78	77	adantlr 712	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑘 ∈ ((𝑗 + 1)...𝑖)) → 𝑘 ∈ (𝑁...𝑖))
79	15	ad2antrr 723	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → 𝐹:𝑍⟶(ℂ ↑_m 𝑆))
80	79, 18, 19	syl2an 596	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (𝑁...𝑖)) → (𝐹‘𝑘) ∈ (ℂ ↑_m 𝑆))
81	80, 21	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (𝑁...𝑖)) → (𝐹‘𝑘):𝑆⟶ℂ)
82	81	ffvelrnda 6961	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (𝑁...𝑖)) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑘)‘𝑧) ∈ ℂ)
83	82	an32s 649	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑘 ∈ (𝑁...𝑖)) → ((𝐹‘𝑘)‘𝑧) ∈ ℂ)
84	78, 83	syldan 591	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑘 ∈ ((𝑗 + 1)...𝑖)) → ((𝐹‘𝑘)‘𝑧) ∈ ℂ)
85	84	abscld 15148	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑘 ∈ ((𝑗 + 1)...𝑖)) → (abs‘((𝐹‘𝑘)‘𝑧)) ∈ ℝ)
86	68, 85	fsumrecl 15446	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(abs‘((𝐹‘𝑘)‘𝑧)) ∈ ℝ)
87		mtest.c	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑀‘𝑘) ∈ ℝ)
88	3, 1, 87	serfre 13752	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → seq𝑁( + , 𝑀):𝑍⟶ℝ)
89	88	ad2antrr 723	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → seq𝑁( + , 𝑀):𝑍⟶ℝ)
90	89, 56	ffvelrnd 6962	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → (seq𝑁( + , 𝑀)‘𝑖) ∈ ℝ)
91	89, 61	ffvelrnd 6962	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → (seq𝑁( + , 𝑀)‘𝑗) ∈ ℝ)
92	90, 91	resubcld 11403	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → ((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗)) ∈ ℝ)
93	92	recnd 11003	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → ((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗)) ∈ ℂ)
94	93	abscld 15148	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → (abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))) ∈ ℝ)
95	94	adantr 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → (abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))) ∈ ℝ)
96	55, 33	sylan2 593	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → (seq𝑁( ∘_f + , 𝐹)‘𝑖) = (𝑧 ∈ 𝑆 ↦ (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑖)))
97	96	adantlr 712	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → (seq𝑁( ∘_f + , 𝐹)‘𝑖) = (𝑧 ∈ 𝑆 ↦ (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑖)))
98	97	fveq1d 6776	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → ((seq𝑁( ∘_f + , 𝐹)‘𝑖)‘𝑧) = ((𝑧 ∈ 𝑆 ↦ (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑖))‘𝑧))
99		fvex 6787	. . . . . . . . . . . . . . . 16 ⊢ (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑖) ∈ V
100		eqid 2738	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ 𝑆 ↦ (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑖)) = (𝑧 ∈ 𝑆 ↦ (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑖))
101	100	fvmpt2 6886	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝑆 ∧ (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑖) ∈ V) → ((𝑧 ∈ 𝑆 ↦ (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑖))‘𝑧) = (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑖))
102	99, 101	mpan2 688	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑆 → ((𝑧 ∈ 𝑆 ↦ (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑖))‘𝑧) = (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑖))
103	98, 102	sylan9eq 2798	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → ((seq𝑁( ∘_f + , 𝐹)‘𝑖)‘𝑧) = (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑖))
104		fveq2 6774	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑗 → (seq𝑁( ∘_f + , 𝐹)‘𝑖) = (seq𝑁( ∘_f + , 𝐹)‘𝑗))
105		fveq2 6774	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑗 → (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑖) = (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑗))
106	105	mpteq2dv 5176	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑗 → (𝑧 ∈ 𝑆 ↦ (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑖)) = (𝑧 ∈ 𝑆 ↦ (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑗)))
107	104, 106	eqeq12d 2754	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑗 → ((seq𝑁( ∘_f + , 𝐹)‘𝑖) = (𝑧 ∈ 𝑆 ↦ (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑖)) ↔ (seq𝑁( ∘_f + , 𝐹)‘𝑗) = (𝑧 ∈ 𝑆 ↦ (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑗))))
108	33	ralrimiva 3103	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑖 ∈ 𝑍 (seq𝑁( ∘_f + , 𝐹)‘𝑖) = (𝑧 ∈ 𝑆 ↦ (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑖)))
109	108	ad2antrr 723	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → ∀𝑖 ∈ 𝑍 (seq𝑁( ∘_f + , 𝐹)‘𝑖) = (𝑧 ∈ 𝑆 ↦ (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑖)))
110	107, 109, 61	rspcdva 3562	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → (seq𝑁( ∘_f + , 𝐹)‘𝑗) = (𝑧 ∈ 𝑆 ↦ (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑗)))
111	110	fveq1d 6776	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → ((seq𝑁( ∘_f + , 𝐹)‘𝑗)‘𝑧) = ((𝑧 ∈ 𝑆 ↦ (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑗))‘𝑧))
112		fvex 6787	. . . . . . . . . . . . . . . 16 ⊢ (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑗) ∈ V
113		eqid 2738	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ 𝑆 ↦ (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑗)) = (𝑧 ∈ 𝑆 ↦ (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑗))
114	113	fvmpt2 6886	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝑆 ∧ (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑗) ∈ V) → ((𝑧 ∈ 𝑆 ↦ (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑗))‘𝑧) = (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑗))
115	112, 114	mpan2 688	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑆 → ((𝑧 ∈ 𝑆 ↦ (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑗))‘𝑧) = (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑗))
116	111, 115	sylan9eq 2798	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → ((seq𝑁( ∘_f + , 𝐹)‘𝑗)‘𝑧) = (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑗))
117	103, 116	oveq12d 7293	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → (((seq𝑁( ∘_f + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘_f + , 𝐹)‘𝑗)‘𝑧)) = ((seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑖) − (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑗)))
118	18	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑘 ∈ (𝑁...𝑖)) → 𝑘 ∈ 𝑍)
119	118, 29	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑘 ∈ (𝑁...𝑖)) → ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧))‘𝑘) = ((𝐹‘𝑘)‘𝑧))
120	56	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → 𝑖 ∈ 𝑍)
121	120, 3	eleqtrdi 2849	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → 𝑖 ∈ (ℤ_≥‘𝑁))
122	119, 121, 83	fsumser 15442	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → Σ𝑘 ∈ (𝑁...𝑖)((𝐹‘𝑘)‘𝑧) = (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑖))
123		elfzuz 13252	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (𝑁...𝑗) → 𝑘 ∈ (ℤ_≥‘𝑁))
124	123, 3	eleqtrrdi 2850	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (𝑁...𝑗) → 𝑘 ∈ 𝑍)
125	124	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑘 ∈ (𝑁...𝑗)) → 𝑘 ∈ 𝑍)
126	125, 29	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑘 ∈ (𝑁...𝑗)) → ((𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧))‘𝑘) = ((𝐹‘𝑘)‘𝑧))
127	61	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → 𝑗 ∈ 𝑍)
128	127, 3	eleqtrdi 2849	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → 𝑗 ∈ (ℤ_≥‘𝑁))
129	79, 124, 19	syl2an 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (𝑁...𝑗)) → (𝐹‘𝑘) ∈ (ℂ ↑_m 𝑆))
130	129, 21	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (𝑁...𝑗)) → (𝐹‘𝑘):𝑆⟶ℂ)
131	130	ffvelrnda 6961	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (𝑁...𝑗)) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑘)‘𝑧) ∈ ℂ)
132	131	an32s 649	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑘 ∈ (𝑁...𝑗)) → ((𝐹‘𝑘)‘𝑧) ∈ ℂ)
133	126, 128, 132	fsumser 15442	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → Σ𝑘 ∈ (𝑁...𝑗)((𝐹‘𝑘)‘𝑧) = (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑗))
134	122, 133	oveq12d 7293	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → (Σ𝑘 ∈ (𝑁...𝑖)((𝐹‘𝑘)‘𝑧) − Σ𝑘 ∈ (𝑁...𝑗)((𝐹‘𝑘)‘𝑧)) = ((seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑖) − (seq𝑁( + , (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑧)))‘𝑗)))
135		fzfid 13693	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → (𝑁...𝑗) ∈ Fin)
136	135, 132	fsumcl 15445	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → Σ𝑘 ∈ (𝑁...𝑗)((𝐹‘𝑘)‘𝑧) ∈ ℂ)
137	68, 84	fsumcl 15445	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → Σ𝑘 ∈ ((𝑗 + 1)...𝑖)((𝐹‘𝑘)‘𝑧) ∈ ℂ)
138		eluzelre 12593	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ (ℤ_≥‘𝑁) → 𝑗 ∈ ℝ)
139	70, 138	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → 𝑗 ∈ ℝ)
140	139	ltp1d 11905	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → 𝑗 < (𝑗 + 1))
141		fzdisj 13283	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 < (𝑗 + 1) → ((𝑁...𝑗) ∩ ((𝑗 + 1)...𝑖)) = ∅)
142	140, 141	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → ((𝑁...𝑗) ∩ ((𝑗 + 1)...𝑖)) = ∅)
143	142	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → ((𝑁...𝑗) ∩ ((𝑗 + 1)...𝑖)) = ∅)
144	75	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → (𝑁...𝑖) = ((𝑁...𝑗) ∪ ((𝑗 + 1)...𝑖)))
145		fzfid 13693	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → (𝑁...𝑖) ∈ Fin)
146	143, 144, 145, 83	fsumsplit 15453	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → Σ𝑘 ∈ (𝑁...𝑖)((𝐹‘𝑘)‘𝑧) = (Σ𝑘 ∈ (𝑁...𝑗)((𝐹‘𝑘)‘𝑧) + Σ𝑘 ∈ ((𝑗 + 1)...𝑖)((𝐹‘𝑘)‘𝑧)))
147	136, 137, 146	mvrladdd 11388	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → (Σ𝑘 ∈ (𝑁...𝑖)((𝐹‘𝑘)‘𝑧) − Σ𝑘 ∈ (𝑁...𝑗)((𝐹‘𝑘)‘𝑧)) = Σ𝑘 ∈ ((𝑗 + 1)...𝑖)((𝐹‘𝑘)‘𝑧))
148	117, 134, 147	3eqtr2d 2784	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → (((seq𝑁( ∘_f + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘_f + , 𝐹)‘𝑗)‘𝑧)) = Σ𝑘 ∈ ((𝑗 + 1)...𝑖)((𝐹‘𝑘)‘𝑧))
149	148	fveq2d 6778	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → (abs‘(((seq𝑁( ∘_f + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘_f + , 𝐹)‘𝑗)‘𝑧))) = (abs‘Σ𝑘 ∈ ((𝑗 + 1)...𝑖)((𝐹‘𝑘)‘𝑧)))
150	68, 84	fsumabs 15513	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → (abs‘Σ𝑘 ∈ ((𝑗 + 1)...𝑖)((𝐹‘𝑘)‘𝑧)) ≤ Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(abs‘((𝐹‘𝑘)‘𝑧)))
151	149, 150	eqbrtrd 5096	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → (abs‘(((seq𝑁( ∘_f + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘_f + , 𝐹)‘𝑗)‘𝑧))) ≤ Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(abs‘((𝐹‘𝑘)‘𝑧)))
152		simpll 764	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → 𝜑)
153	152, 18, 87	syl2an 596	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (𝑁...𝑖)) → (𝑀‘𝑘) ∈ ℝ)
154	77, 153	syldan 591	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ ((𝑗 + 1)...𝑖)) → (𝑀‘𝑘) ∈ ℝ)
155	154	adantlr 712	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑘 ∈ ((𝑗 + 1)...𝑖)) → (𝑀‘𝑘) ∈ ℝ)
156	78, 18	syl 17	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑘 ∈ ((𝑗 + 1)...𝑖)) → 𝑘 ∈ 𝑍)
157		mtest.l	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → (abs‘((𝐹‘𝑘)‘𝑧)) ≤ (𝑀‘𝑘))
158	157	ad4ant14 749	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → (abs‘((𝐹‘𝑘)‘𝑧)) ≤ (𝑀‘𝑘))
159	158	anass1rs 652	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑘 ∈ 𝑍) → (abs‘((𝐹‘𝑘)‘𝑧)) ≤ (𝑀‘𝑘))
160	156, 159	syldan 591	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑘 ∈ ((𝑗 + 1)...𝑖)) → (abs‘((𝐹‘𝑘)‘𝑧)) ≤ (𝑀‘𝑘))
161	68, 85, 155, 160	fsumle 15511	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(abs‘((𝐹‘𝑘)‘𝑧)) ≤ Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(𝑀‘𝑘))
162		eqidd 2739	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (𝑁...𝑖)) → (𝑀‘𝑘) = (𝑀‘𝑘))
163	56, 3	eleqtrdi 2849	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → 𝑖 ∈ (ℤ_≥‘𝑁))
164	153	recnd 11003	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (𝑁...𝑖)) → (𝑀‘𝑘) ∈ ℂ)
165	162, 163, 164	fsumser 15442	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → Σ𝑘 ∈ (𝑁...𝑖)(𝑀‘𝑘) = (seq𝑁( + , 𝑀)‘𝑖))
166		eqidd 2739	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (𝑁...𝑗)) → (𝑀‘𝑘) = (𝑀‘𝑘))
167	152, 124, 87	syl2an 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (𝑁...𝑗)) → (𝑀‘𝑘) ∈ ℝ)
168	167	recnd 11003	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ (𝑁...𝑗)) → (𝑀‘𝑘) ∈ ℂ)
169	166, 70, 168	fsumser 15442	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → Σ𝑘 ∈ (𝑁...𝑗)(𝑀‘𝑘) = (seq𝑁( + , 𝑀)‘𝑗))
170	165, 169	oveq12d 7293	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → (Σ𝑘 ∈ (𝑁...𝑖)(𝑀‘𝑘) − Σ𝑘 ∈ (𝑁...𝑗)(𝑀‘𝑘)) = ((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗)))
171		fzfid 13693	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → (𝑁...𝑗) ∈ Fin)
172	171, 168	fsumcl 15445	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → Σ𝑘 ∈ (𝑁...𝑗)(𝑀‘𝑘) ∈ ℂ)
173		fzfid 13693	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → ((𝑗 + 1)...𝑖) ∈ Fin)
174	77, 164	syldan 591	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑘 ∈ ((𝑗 + 1)...𝑖)) → (𝑀‘𝑘) ∈ ℂ)
175	173, 174	fsumcl 15445	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(𝑀‘𝑘) ∈ ℂ)
176		fzfid 13693	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → (𝑁...𝑖) ∈ Fin)
177	142, 75, 176, 164	fsumsplit 15453	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → Σ𝑘 ∈ (𝑁...𝑖)(𝑀‘𝑘) = (Σ𝑘 ∈ (𝑁...𝑗)(𝑀‘𝑘) + Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(𝑀‘𝑘)))
178	172, 175, 177	mvrladdd 11388	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → (Σ𝑘 ∈ (𝑁...𝑖)(𝑀‘𝑘) − Σ𝑘 ∈ (𝑁...𝑗)(𝑀‘𝑘)) = Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(𝑀‘𝑘))
179	170, 178	eqtr3d 2780	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → ((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗)) = Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(𝑀‘𝑘))
180	179	fveq2d 6778	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → (abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))) = (abs‘Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(𝑀‘𝑘)))
181	180	adantr 481	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → (abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))) = (abs‘Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(𝑀‘𝑘)))
182	179, 92	eqeltrrd 2840	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(𝑀‘𝑘) ∈ ℝ)
183	182	adantr 481	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(𝑀‘𝑘) ∈ ℝ)
184		0red 10978	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑘 ∈ ((𝑗 + 1)...𝑖)) → 0 ∈ ℝ)
185	84	absge0d 15156	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑘 ∈ ((𝑗 + 1)...𝑖)) → 0 ≤ (abs‘((𝐹‘𝑘)‘𝑧)))
186	184, 85, 155, 185, 160	letrd 11132	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) ∧ 𝑘 ∈ ((𝑗 + 1)...𝑖)) → 0 ≤ (𝑀‘𝑘))
187	68, 155, 186	fsumge0 15507	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → 0 ≤ Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(𝑀‘𝑘))
188	183, 187	absidd 15134	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → (abs‘Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(𝑀‘𝑘)) = Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(𝑀‘𝑘))
189	181, 188	eqtrd 2778	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → (abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))) = Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(𝑀‘𝑘))
190	161, 189	breqtrrd 5102	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(abs‘((𝐹‘𝑘)‘𝑧)) ≤ (abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))))
191	67, 86, 95, 151, 190	letrd 11132	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → (abs‘(((seq𝑁( ∘_f + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘_f + , 𝐹)‘𝑗)‘𝑧))) ≤ (abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))))
192		simpllr 773	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → 𝑟 ∈ ℝ⁺)
193	192	rpred 12772	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → 𝑟 ∈ ℝ)
194		lelttr 11065	. . . . . . . . . 10 ⊢ (((abs‘(((seq𝑁( ∘_f + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘_f + , 𝐹)‘𝑗)‘𝑧))) ∈ ℝ ∧ (abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))) ∈ ℝ ∧ 𝑟 ∈ ℝ) → (((abs‘(((seq𝑁( ∘_f + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘_f + , 𝐹)‘𝑗)‘𝑧))) ≤ (abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))) ∧ (abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))) < 𝑟) → (abs‘(((seq𝑁( ∘_f + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘_f + , 𝐹)‘𝑗)‘𝑧))) < 𝑟))
195	67, 95, 193, 194	syl3anc 1370	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → (((abs‘(((seq𝑁( ∘_f + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘_f + , 𝐹)‘𝑗)‘𝑧))) ≤ (abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))) ∧ (abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))) < 𝑟) → (abs‘(((seq𝑁( ∘_f + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘_f + , 𝐹)‘𝑗)‘𝑧))) < 𝑟))
196	191, 195	mpand 692	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) ∧ 𝑧 ∈ 𝑆) → ((abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))) < 𝑟 → (abs‘(((seq𝑁( ∘_f + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘_f + , 𝐹)‘𝑗)‘𝑧))) < 𝑟))
197	196	ralrimdva 3106	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑗))) → ((abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))) < 𝑟 → ∀𝑧 ∈ 𝑆 (abs‘(((seq𝑁( ∘_f + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘_f + , 𝐹)‘𝑗)‘𝑧))) < 𝑟))
198	197	anassrs 468	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑖 ∈ (ℤ_≥‘𝑗)) → ((abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))) < 𝑟 → ∀𝑧 ∈ 𝑆 (abs‘(((seq𝑁( ∘_f + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘_f + , 𝐹)‘𝑗)‘𝑧))) < 𝑟))
199	198	ralimdva 3108	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → (∀𝑖 ∈ (ℤ_≥‘𝑗)(abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))) < 𝑟 → ∀𝑖 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((seq𝑁( ∘_f + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘_f + , 𝐹)‘𝑗)‘𝑧))) < 𝑟))
200	199	reximdva 3203	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (∃𝑗 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑗)(abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))) < 𝑟 → ∃𝑗 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((seq𝑁( ∘_f + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘_f + , 𝐹)‘𝑗)‘𝑧))) < 𝑟))
201	200	ralimdva 3108	. . 3 ⊢ (𝜑 → (∀𝑟 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑗)(abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))) < 𝑟 → ∀𝑟 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((seq𝑁( ∘_f + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘_f + , 𝐹)‘𝑗)‘𝑧))) < 𝑟))
202	5, 201	mpd 15	. 2 ⊢ (𝜑 → ∀𝑟 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((seq𝑁( ∘_f + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘_f + , 𝐹)‘𝑗)‘𝑧))) < 𝑟)
203	3, 1, 10, 53	ulmcau 25554	. 2 ⊢ (𝜑 → (seq𝑁( ∘_f + , 𝐹) ∈ dom (⇝_𝑢‘𝑆) ↔ ∀𝑟 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((seq𝑁( ∘_f + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘_f + , 𝐹)‘𝑗)‘𝑧))) < 𝑟))
204	202, 203	mpbird 256	1 ⊢ (𝜑 → seq𝑁( ∘_f + , 𝐹) ∈ dom (⇝_𝑢‘𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 Vcvv 3432 ∪ cun 3885 ∩ cin 3886 ∅c0 4256 class class class wbr 5074 ↦ cmpt 5157 dom cdm 5589 Fn wfn 6428 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ∘_f cof 7531 ↑_m cmap 8615 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 < clt 11009 ≤ cle 11010 − cmin 11205 ℤcz 12319 ℤ_≥cuz 12582 ℝ⁺crp 12730 ...cfz 13239 seqcseq 13721 abscabs 14945 ⇝ cli 15193 Σcsu 15397 ⇝_𝑢culm 25535

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-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 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-int 4880 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-se 5545 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-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 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-ico 13085 df-fz 13240 df-fzo 13383 df-fl 13512 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-limsup 15180 df-clim 15197 df-rlim 15198 df-sum 15398 df-ulm 25536

This theorem is referenced by: pserulm 25581 lgamgulmlem6 26183 knoppcnlem6 34678

W3C validator