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Theorem mtest 23876
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 = (ℤ𝑁)
mtest.n (𝜑𝑁 ∈ ℤ)
mtest.s (𝜑𝑆𝑉)
mtest.f (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))
mtest.m (𝜑𝑀𝑊)
mtest.c ((𝜑𝑘𝑍) → (𝑀𝑘) ∈ ℝ)
mtest.l ((𝜑 ∧ (𝑘𝑍𝑧𝑆)) → (abs‘((𝐹𝑘)‘𝑧)) ≤ (𝑀𝑘))
mtest.d (𝜑 → seq𝑁( + , 𝑀) ∈ dom ⇝ )
Assertion
Ref Expression
mtest (𝜑 → seq𝑁( ∘𝑓 + , 𝐹) ∈ dom (⇝𝑢𝑆))
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.
StepHypRef Expression
1 mtest.n . . . 4 (𝜑𝑁 ∈ ℤ)
2 mtest.d . . . 4 (𝜑 → seq𝑁( + , 𝑀) ∈ dom ⇝ )
3 mtest.z . . . . 5 𝑍 = (ℤ𝑁)
43climcau 14192 . . . 4 ((𝑁 ∈ ℤ ∧ seq𝑁( + , 𝑀) ∈ dom ⇝ ) → ∀𝑟 ∈ ℝ+𝑗𝑍𝑖 ∈ (ℤ𝑗)(abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))) < 𝑟)
51, 2, 4syl2anc 690 . . 3 (𝜑 → ∀𝑟 ∈ ℝ+𝑗𝑍𝑖 ∈ (ℤ𝑗)(abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))) < 𝑟)
6 seqfn 12627 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℤ → seq𝑁( ∘𝑓 + , 𝐹) Fn (ℤ𝑁))
71, 6syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → seq𝑁( ∘𝑓 + , 𝐹) Fn (ℤ𝑁))
83fneq2i 5883 . . . . . . . . . . . . . . . . . 18 (seq𝑁( ∘𝑓 + , 𝐹) Fn 𝑍 ↔ seq𝑁( ∘𝑓 + , 𝐹) Fn (ℤ𝑁))
97, 8sylibr 222 . . . . . . . . . . . . . . . . 17 (𝜑 → seq𝑁( ∘𝑓 + , 𝐹) Fn 𝑍)
10 mtest.s . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑆𝑉)
11 elex 3181 . . . . . . . . . . . . . . . . . . . . . 22 (𝑆𝑉𝑆 ∈ V)
1210, 11syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑆 ∈ V)
1312adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑖𝑍) → 𝑆 ∈ V)
14 simpr 475 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑖𝑍) → 𝑖𝑍)
1514, 3syl6eleq 2694 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑖𝑍) → 𝑖 ∈ (ℤ𝑁))
16 mtest.f . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))
1716adantr 479 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑖𝑍) → 𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))
18 elfzuz 12161 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 ∈ (𝑁...𝑖) → 𝑘 ∈ (ℤ𝑁))
1918, 3syl6eleqr 2695 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ (𝑁...𝑖) → 𝑘𝑍)
20 ffvelrn 6247 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐹:𝑍⟶(ℂ ↑𝑚 𝑆) ∧ 𝑘𝑍) → (𝐹𝑘) ∈ (ℂ ↑𝑚 𝑆))
2117, 19, 20syl2an 492 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑖𝑍) ∧ 𝑘 ∈ (𝑁...𝑖)) → (𝐹𝑘) ∈ (ℂ ↑𝑚 𝑆))
22 elmapi 7739 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹𝑘) ∈ (ℂ ↑𝑚 𝑆) → (𝐹𝑘):𝑆⟶ℂ)
2321, 22syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖𝑍) ∧ 𝑘 ∈ (𝑁...𝑖)) → (𝐹𝑘):𝑆⟶ℂ)
2423feqmptd 6141 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖𝑍) ∧ 𝑘 ∈ (𝑁...𝑖)) → (𝐹𝑘) = (𝑧𝑆 ↦ ((𝐹𝑘)‘𝑧)))
2519adantl 480 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑖𝑍) ∧ 𝑘 ∈ (𝑁...𝑖)) → 𝑘𝑍)
26 fveq2 6085 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
2726fveq1d 6087 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑘 → ((𝐹𝑛)‘𝑧) = ((𝐹𝑘)‘𝑧))
28 eqid 2606 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)) = (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧))
29 fvex 6095 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐹𝑘)‘𝑧) ∈ V
3027, 28, 29fvmpt 6173 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘𝑍 → ((𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧))‘𝑘) = ((𝐹𝑘)‘𝑧))
3125, 30syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖𝑍) ∧ 𝑘 ∈ (𝑁...𝑖)) → ((𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧))‘𝑘) = ((𝐹𝑘)‘𝑧))
3231mpteq2dv 4664 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖𝑍) ∧ 𝑘 ∈ (𝑁...𝑖)) → (𝑧𝑆 ↦ ((𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧))‘𝑘)) = (𝑧𝑆 ↦ ((𝐹𝑘)‘𝑧)))
3324, 32eqtr4d 2643 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖𝑍) ∧ 𝑘 ∈ (𝑁...𝑖)) → (𝐹𝑘) = (𝑧𝑆 ↦ ((𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧))‘𝑘)))
3413, 15, 33seqof 12672 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖𝑍) → (seq𝑁( ∘𝑓 + , 𝐹)‘𝑖) = (𝑧𝑆 ↦ (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑖)))
351adantr 479 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑧𝑆) → 𝑁 ∈ ℤ)
3616ffvelrnda 6249 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑛𝑍) → (𝐹𝑛) ∈ (ℂ ↑𝑚 𝑆))
37 elmapi 7739 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐹𝑛) ∈ (ℂ ↑𝑚 𝑆) → (𝐹𝑛):𝑆⟶ℂ)
3836, 37syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑛𝑍) → (𝐹𝑛):𝑆⟶ℂ)
3938ffvelrnda 6249 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑛𝑍) ∧ 𝑧𝑆) → ((𝐹𝑛)‘𝑧) ∈ ℂ)
4039an32s 841 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑧𝑆) ∧ 𝑛𝑍) → ((𝐹𝑛)‘𝑧) ∈ ℂ)
4140, 28fmptd 6274 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑧𝑆) → (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)):𝑍⟶ℂ)
4241ffvelrnda 6249 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑧𝑆) ∧ 𝑖𝑍) → ((𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧))‘𝑖) ∈ ℂ)
433, 35, 42serf 12643 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑧𝑆) → seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧))):𝑍⟶ℂ)
4443ffvelrnda 6249 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑧𝑆) ∧ 𝑖𝑍) → (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑖) ∈ ℂ)
4544an32s 841 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖𝑍) ∧ 𝑧𝑆) → (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑖) ∈ ℂ)
46 eqid 2606 . . . . . . . . . . . . . . . . . . . . 21 (𝑧𝑆 ↦ (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑖)) = (𝑧𝑆 ↦ (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑖))
4745, 46fmptd 6274 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑖𝑍) → (𝑧𝑆 ↦ (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑖)):𝑆⟶ℂ)
48 cnex 9870 . . . . . . . . . . . . . . . . . . . . 21 ℂ ∈ V
49 elmapg 7731 . . . . . . . . . . . . . . . . . . . . 21 ((ℂ ∈ V ∧ 𝑆 ∈ V) → ((𝑧𝑆 ↦ (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑖)) ∈ (ℂ ↑𝑚 𝑆) ↔ (𝑧𝑆 ↦ (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑖)):𝑆⟶ℂ))
5048, 13, 49sylancr 693 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑖𝑍) → ((𝑧𝑆 ↦ (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑖)) ∈ (ℂ ↑𝑚 𝑆) ↔ (𝑧𝑆 ↦ (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑖)):𝑆⟶ℂ))
5147, 50mpbird 245 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖𝑍) → (𝑧𝑆 ↦ (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑖)) ∈ (ℂ ↑𝑚 𝑆))
5234, 51eqeltrd 2684 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖𝑍) → (seq𝑁( ∘𝑓 + , 𝐹)‘𝑖) ∈ (ℂ ↑𝑚 𝑆))
5352ralrimiva 2945 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑖𝑍 (seq𝑁( ∘𝑓 + , 𝐹)‘𝑖) ∈ (ℂ ↑𝑚 𝑆))
54 ffnfv 6277 . . . . . . . . . . . . . . . . 17 (seq𝑁( ∘𝑓 + , 𝐹):𝑍⟶(ℂ ↑𝑚 𝑆) ↔ (seq𝑁( ∘𝑓 + , 𝐹) Fn 𝑍 ∧ ∀𝑖𝑍 (seq𝑁( ∘𝑓 + , 𝐹)‘𝑖) ∈ (ℂ ↑𝑚 𝑆)))
559, 53, 54sylanbrc 694 . . . . . . . . . . . . . . . 16 (𝜑 → seq𝑁( ∘𝑓 + , 𝐹):𝑍⟶(ℂ ↑𝑚 𝑆))
5655ad2antrr 757 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → seq𝑁( ∘𝑓 + , 𝐹):𝑍⟶(ℂ ↑𝑚 𝑆))
573uztrn2 11534 . . . . . . . . . . . . . . . 16 ((𝑗𝑍𝑖 ∈ (ℤ𝑗)) → 𝑖𝑍)
5857adantl 480 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → 𝑖𝑍)
5956, 58ffvelrnd 6250 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → (seq𝑁( ∘𝑓 + , 𝐹)‘𝑖) ∈ (ℂ ↑𝑚 𝑆))
60 elmapi 7739 . . . . . . . . . . . . . 14 ((seq𝑁( ∘𝑓 + , 𝐹)‘𝑖) ∈ (ℂ ↑𝑚 𝑆) → (seq𝑁( ∘𝑓 + , 𝐹)‘𝑖):𝑆⟶ℂ)
6159, 60syl 17 . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → (seq𝑁( ∘𝑓 + , 𝐹)‘𝑖):𝑆⟶ℂ)
6261ffvelrnda 6249 . . . . . . . . . . . 12 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → ((seq𝑁( ∘𝑓 + , 𝐹)‘𝑖)‘𝑧) ∈ ℂ)
63 simprl 789 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → 𝑗𝑍)
6456, 63ffvelrnd 6250 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → (seq𝑁( ∘𝑓 + , 𝐹)‘𝑗) ∈ (ℂ ↑𝑚 𝑆))
65 elmapi 7739 . . . . . . . . . . . . . 14 ((seq𝑁( ∘𝑓 + , 𝐹)‘𝑗) ∈ (ℂ ↑𝑚 𝑆) → (seq𝑁( ∘𝑓 + , 𝐹)‘𝑗):𝑆⟶ℂ)
6664, 65syl 17 . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → (seq𝑁( ∘𝑓 + , 𝐹)‘𝑗):𝑆⟶ℂ)
6766ffvelrnda 6249 . . . . . . . . . . . 12 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → ((seq𝑁( ∘𝑓 + , 𝐹)‘𝑗)‘𝑧) ∈ ℂ)
6862, 67subcld 10240 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → (((seq𝑁( ∘𝑓 + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘𝑓 + , 𝐹)‘𝑗)‘𝑧)) ∈ ℂ)
6968abscld 13966 . . . . . . . . . 10 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → (abs‘(((seq𝑁( ∘𝑓 + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘𝑓 + , 𝐹)‘𝑗)‘𝑧))) ∈ ℝ)
70 fzfid 12586 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → ((𝑗 + 1)...𝑖) ∈ Fin)
71 ssun2 3735 . . . . . . . . . . . . . . . 16 ((𝑗 + 1)...𝑖) ⊆ ((𝑁...𝑗) ∪ ((𝑗 + 1)...𝑖))
7263, 3syl6eleq 2694 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → 𝑗 ∈ (ℤ𝑁))
73 simprr 791 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → 𝑖 ∈ (ℤ𝑗))
74 elfzuzb 12159 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (𝑁...𝑖) ↔ (𝑗 ∈ (ℤ𝑁) ∧ 𝑖 ∈ (ℤ𝑗)))
7572, 73, 74sylanbrc 694 . . . . . . . . . . . . . . . . 17 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → 𝑗 ∈ (𝑁...𝑖))
76 fzsplit 12190 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (𝑁...𝑖) → (𝑁...𝑖) = ((𝑁...𝑗) ∪ ((𝑗 + 1)...𝑖)))
7775, 76syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → (𝑁...𝑖) = ((𝑁...𝑗) ∪ ((𝑗 + 1)...𝑖)))
7871, 77syl5sseqr 3613 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → ((𝑗 + 1)...𝑖) ⊆ (𝑁...𝑖))
7978sselda 3564 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ ((𝑗 + 1)...𝑖)) → 𝑘 ∈ (𝑁...𝑖))
8079adantlr 746 . . . . . . . . . . . . 13 (((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) ∧ 𝑘 ∈ ((𝑗 + 1)...𝑖)) → 𝑘 ∈ (𝑁...𝑖))
8116ad2antrr 757 . . . . . . . . . . . . . . . . 17 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → 𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))
8281, 19, 20syl2an 492 . . . . . . . . . . . . . . . 16 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑁...𝑖)) → (𝐹𝑘) ∈ (ℂ ↑𝑚 𝑆))
8382, 22syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑁...𝑖)) → (𝐹𝑘):𝑆⟶ℂ)
8483ffvelrnda 6249 . . . . . . . . . . . . . 14 (((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑁...𝑖)) ∧ 𝑧𝑆) → ((𝐹𝑘)‘𝑧) ∈ ℂ)
8584an32s 841 . . . . . . . . . . . . 13 (((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑖)) → ((𝐹𝑘)‘𝑧) ∈ ℂ)
8680, 85syldan 485 . . . . . . . . . . . 12 (((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) ∧ 𝑘 ∈ ((𝑗 + 1)...𝑖)) → ((𝐹𝑘)‘𝑧) ∈ ℂ)
8786abscld 13966 . . . . . . . . . . 11 (((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) ∧ 𝑘 ∈ ((𝑗 + 1)...𝑖)) → (abs‘((𝐹𝑘)‘𝑧)) ∈ ℝ)
8870, 87fsumrecl 14255 . . . . . . . . . 10 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(abs‘((𝐹𝑘)‘𝑧)) ∈ ℝ)
89 mtest.c . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑍) → (𝑀𝑘) ∈ ℝ)
903, 1, 89serfre 12644 . . . . . . . . . . . . . . . 16 (𝜑 → seq𝑁( + , 𝑀):𝑍⟶ℝ)
9190ad2antrr 757 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → seq𝑁( + , 𝑀):𝑍⟶ℝ)
9291, 58ffvelrnd 6250 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → (seq𝑁( + , 𝑀)‘𝑖) ∈ ℝ)
9391, 63ffvelrnd 6250 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → (seq𝑁( + , 𝑀)‘𝑗) ∈ ℝ)
9492, 93resubcld 10306 . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → ((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗)) ∈ ℝ)
9594recnd 9921 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → ((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗)) ∈ ℂ)
9695abscld 13966 . . . . . . . . . . 11 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → (abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))) ∈ ℝ)
9796adantr 479 . . . . . . . . . 10 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → (abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))) ∈ ℝ)
9857, 34sylan2 489 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → (seq𝑁( ∘𝑓 + , 𝐹)‘𝑖) = (𝑧𝑆 ↦ (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑖)))
9998adantlr 746 . . . . . . . . . . . . . . . 16 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → (seq𝑁( ∘𝑓 + , 𝐹)‘𝑖) = (𝑧𝑆 ↦ (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑖)))
10099fveq1d 6087 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → ((seq𝑁( ∘𝑓 + , 𝐹)‘𝑖)‘𝑧) = ((𝑧𝑆 ↦ (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑖))‘𝑧))
101 fvex 6095 . . . . . . . . . . . . . . . 16 (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑖) ∈ V
10246fvmpt2 6182 . . . . . . . . . . . . . . . 16 ((𝑧𝑆 ∧ (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑖) ∈ V) → ((𝑧𝑆 ↦ (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑖))‘𝑧) = (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑖))
103101, 102mpan2 702 . . . . . . . . . . . . . . 15 (𝑧𝑆 → ((𝑧𝑆 ↦ (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑖))‘𝑧) = (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑖))
104100, 103sylan9eq 2660 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → ((seq𝑁( ∘𝑓 + , 𝐹)‘𝑖)‘𝑧) = (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑖))
10534ralrimiva 2945 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑖𝑍 (seq𝑁( ∘𝑓 + , 𝐹)‘𝑖) = (𝑧𝑆 ↦ (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑖)))
106105ad2antrr 757 . . . . . . . . . . . . . . . . 17 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → ∀𝑖𝑍 (seq𝑁( ∘𝑓 + , 𝐹)‘𝑖) = (𝑧𝑆 ↦ (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑖)))
107 fveq2 6085 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 𝑗 → (seq𝑁( ∘𝑓 + , 𝐹)‘𝑖) = (seq𝑁( ∘𝑓 + , 𝐹)‘𝑗))
108 fveq2 6085 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 𝑗 → (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑖) = (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑗))
109108mpteq2dv 4664 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 𝑗 → (𝑧𝑆 ↦ (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑖)) = (𝑧𝑆 ↦ (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑗)))
110107, 109eqeq12d 2621 . . . . . . . . . . . . . . . . . 18 (𝑖 = 𝑗 → ((seq𝑁( ∘𝑓 + , 𝐹)‘𝑖) = (𝑧𝑆 ↦ (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑖)) ↔ (seq𝑁( ∘𝑓 + , 𝐹)‘𝑗) = (𝑧𝑆 ↦ (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑗))))
111110rspccv 3275 . . . . . . . . . . . . . . . . 17 (∀𝑖𝑍 (seq𝑁( ∘𝑓 + , 𝐹)‘𝑖) = (𝑧𝑆 ↦ (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑖)) → (𝑗𝑍 → (seq𝑁( ∘𝑓 + , 𝐹)‘𝑗) = (𝑧𝑆 ↦ (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑗))))
112106, 63, 111sylc 62 . . . . . . . . . . . . . . . 16 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → (seq𝑁( ∘𝑓 + , 𝐹)‘𝑗) = (𝑧𝑆 ↦ (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑗)))
113112fveq1d 6087 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → ((seq𝑁( ∘𝑓 + , 𝐹)‘𝑗)‘𝑧) = ((𝑧𝑆 ↦ (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑗))‘𝑧))
114 fvex 6095 . . . . . . . . . . . . . . . 16 (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑗) ∈ V
115 eqid 2606 . . . . . . . . . . . . . . . . 17 (𝑧𝑆 ↦ (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑗)) = (𝑧𝑆 ↦ (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑗))
116115fvmpt2 6182 . . . . . . . . . . . . . . . 16 ((𝑧𝑆 ∧ (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑗) ∈ V) → ((𝑧𝑆 ↦ (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑗))‘𝑧) = (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑗))
117114, 116mpan2 702 . . . . . . . . . . . . . . 15 (𝑧𝑆 → ((𝑧𝑆 ↦ (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑗))‘𝑧) = (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑗))
118113, 117sylan9eq 2660 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → ((seq𝑁( ∘𝑓 + , 𝐹)‘𝑗)‘𝑧) = (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑗))
119104, 118oveq12d 6542 . . . . . . . . . . . . 13 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → (((seq𝑁( ∘𝑓 + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘𝑓 + , 𝐹)‘𝑗)‘𝑧)) = ((seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑖) − (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑗)))
12019adantl 480 . . . . . . . . . . . . . . . 16 (((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑖)) → 𝑘𝑍)
121120, 30syl 17 . . . . . . . . . . . . . . 15 (((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑖)) → ((𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧))‘𝑘) = ((𝐹𝑘)‘𝑧))
12258adantr 479 . . . . . . . . . . . . . . . 16 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → 𝑖𝑍)
123122, 3syl6eleq 2694 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → 𝑖 ∈ (ℤ𝑁))
124121, 123, 85fsumser 14251 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑖)((𝐹𝑘)‘𝑧) = (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑖))
125 elfzuz 12161 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (𝑁...𝑗) → 𝑘 ∈ (ℤ𝑁))
126125, 3syl6eleqr 2695 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (𝑁...𝑗) → 𝑘𝑍)
127126adantl 480 . . . . . . . . . . . . . . . 16 (((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑗)) → 𝑘𝑍)
128127, 30syl 17 . . . . . . . . . . . . . . 15 (((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑗)) → ((𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧))‘𝑘) = ((𝐹𝑘)‘𝑧))
12963adantr 479 . . . . . . . . . . . . . . . 16 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → 𝑗𝑍)
130129, 3syl6eleq 2694 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → 𝑗 ∈ (ℤ𝑁))
13181, 126, 20syl2an 492 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑁...𝑗)) → (𝐹𝑘) ∈ (ℂ ↑𝑚 𝑆))
132131, 22syl 17 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑁...𝑗)) → (𝐹𝑘):𝑆⟶ℂ)
133132ffvelrnda 6249 . . . . . . . . . . . . . . . 16 (((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑁...𝑗)) ∧ 𝑧𝑆) → ((𝐹𝑘)‘𝑧) ∈ ℂ)
134133an32s 841 . . . . . . . . . . . . . . 15 (((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑗)) → ((𝐹𝑘)‘𝑧) ∈ ℂ)
135128, 130, 134fsumser 14251 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑗)((𝐹𝑘)‘𝑧) = (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑗))
136124, 135oveq12d 6542 . . . . . . . . . . . . 13 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → (Σ𝑘 ∈ (𝑁...𝑖)((𝐹𝑘)‘𝑧) − Σ𝑘 ∈ (𝑁...𝑗)((𝐹𝑘)‘𝑧)) = ((seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑖) − (seq𝑁( + , (𝑛𝑍 ↦ ((𝐹𝑛)‘𝑧)))‘𝑗)))
137 eluzelre 11527 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (ℤ𝑁) → 𝑗 ∈ ℝ)
13872, 137syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → 𝑗 ∈ ℝ)
139138ltp1d 10800 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → 𝑗 < (𝑗 + 1))
140 fzdisj 12191 . . . . . . . . . . . . . . . . . 18 (𝑗 < (𝑗 + 1) → ((𝑁...𝑗) ∩ ((𝑗 + 1)...𝑖)) = ∅)
141139, 140syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → ((𝑁...𝑗) ∩ ((𝑗 + 1)...𝑖)) = ∅)
142141adantr 479 . . . . . . . . . . . . . . . 16 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → ((𝑁...𝑗) ∩ ((𝑗 + 1)...𝑖)) = ∅)
14377adantr 479 . . . . . . . . . . . . . . . 16 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → (𝑁...𝑖) = ((𝑁...𝑗) ∪ ((𝑗 + 1)...𝑖)))
144 fzfid 12586 . . . . . . . . . . . . . . . 16 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → (𝑁...𝑖) ∈ Fin)
145142, 143, 144, 85fsumsplit 14261 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑖)((𝐹𝑘)‘𝑧) = (Σ𝑘 ∈ (𝑁...𝑗)((𝐹𝑘)‘𝑧) + Σ𝑘 ∈ ((𝑗 + 1)...𝑖)((𝐹𝑘)‘𝑧)))
146145eqcomd 2612 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → (Σ𝑘 ∈ (𝑁...𝑗)((𝐹𝑘)‘𝑧) + Σ𝑘 ∈ ((𝑗 + 1)...𝑖)((𝐹𝑘)‘𝑧)) = Σ𝑘 ∈ (𝑁...𝑖)((𝐹𝑘)‘𝑧))
147144, 85fsumcl 14254 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑖)((𝐹𝑘)‘𝑧) ∈ ℂ)
148 fzfid 12586 . . . . . . . . . . . . . . . 16 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → (𝑁...𝑗) ∈ Fin)
149148, 134fsumcl 14254 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑗)((𝐹𝑘)‘𝑧) ∈ ℂ)
15070, 86fsumcl 14254 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → Σ𝑘 ∈ ((𝑗 + 1)...𝑖)((𝐹𝑘)‘𝑧) ∈ ℂ)
151147, 149, 150subaddd 10258 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → ((Σ𝑘 ∈ (𝑁...𝑖)((𝐹𝑘)‘𝑧) − Σ𝑘 ∈ (𝑁...𝑗)((𝐹𝑘)‘𝑧)) = Σ𝑘 ∈ ((𝑗 + 1)...𝑖)((𝐹𝑘)‘𝑧) ↔ (Σ𝑘 ∈ (𝑁...𝑗)((𝐹𝑘)‘𝑧) + Σ𝑘 ∈ ((𝑗 + 1)...𝑖)((𝐹𝑘)‘𝑧)) = Σ𝑘 ∈ (𝑁...𝑖)((𝐹𝑘)‘𝑧)))
152146, 151mpbird 245 . . . . . . . . . . . . 13 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → (Σ𝑘 ∈ (𝑁...𝑖)((𝐹𝑘)‘𝑧) − Σ𝑘 ∈ (𝑁...𝑗)((𝐹𝑘)‘𝑧)) = Σ𝑘 ∈ ((𝑗 + 1)...𝑖)((𝐹𝑘)‘𝑧))
153119, 136, 1523eqtr2d 2646 . . . . . . . . . . . 12 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → (((seq𝑁( ∘𝑓 + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘𝑓 + , 𝐹)‘𝑗)‘𝑧)) = Σ𝑘 ∈ ((𝑗 + 1)...𝑖)((𝐹𝑘)‘𝑧))
154153fveq2d 6089 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → (abs‘(((seq𝑁( ∘𝑓 + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘𝑓 + , 𝐹)‘𝑗)‘𝑧))) = (abs‘Σ𝑘 ∈ ((𝑗 + 1)...𝑖)((𝐹𝑘)‘𝑧)))
15570, 86fsumabs 14317 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → (abs‘Σ𝑘 ∈ ((𝑗 + 1)...𝑖)((𝐹𝑘)‘𝑧)) ≤ Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(abs‘((𝐹𝑘)‘𝑧)))
156154, 155eqbrtrd 4596 . . . . . . . . . 10 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → (abs‘(((seq𝑁( ∘𝑓 + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘𝑓 + , 𝐹)‘𝑗)‘𝑧))) ≤ Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(abs‘((𝐹𝑘)‘𝑧)))
157 simpll 785 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → 𝜑)
158157, 19, 89syl2an 492 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑁...𝑖)) → (𝑀𝑘) ∈ ℝ)
15979, 158syldan 485 . . . . . . . . . . . . 13 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ ((𝑗 + 1)...𝑖)) → (𝑀𝑘) ∈ ℝ)
160159adantlr 746 . . . . . . . . . . . 12 (((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) ∧ 𝑘 ∈ ((𝑗 + 1)...𝑖)) → (𝑀𝑘) ∈ ℝ)
16180, 19syl 17 . . . . . . . . . . . . 13 (((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) ∧ 𝑘 ∈ ((𝑗 + 1)...𝑖)) → 𝑘𝑍)
162 mtest.l . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘𝑍𝑧𝑆)) → (abs‘((𝐹𝑘)‘𝑧)) ≤ (𝑀𝑘))
163162adantlr 746 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑘𝑍𝑧𝑆)) → (abs‘((𝐹𝑘)‘𝑧)) ≤ (𝑀𝑘))
164163adantlr 746 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ (𝑘𝑍𝑧𝑆)) → (abs‘((𝐹𝑘)‘𝑧)) ≤ (𝑀𝑘))
165164anass1rs 844 . . . . . . . . . . . . 13 (((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) ∧ 𝑘𝑍) → (abs‘((𝐹𝑘)‘𝑧)) ≤ (𝑀𝑘))
166161, 165syldan 485 . . . . . . . . . . . 12 (((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) ∧ 𝑘 ∈ ((𝑗 + 1)...𝑖)) → (abs‘((𝐹𝑘)‘𝑧)) ≤ (𝑀𝑘))
16770, 87, 160, 166fsumle 14315 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(abs‘((𝐹𝑘)‘𝑧)) ≤ Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(𝑀𝑘))
168 eqidd 2607 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑁...𝑖)) → (𝑀𝑘) = (𝑀𝑘))
16958, 3syl6eleq 2694 . . . . . . . . . . . . . . . . 17 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → 𝑖 ∈ (ℤ𝑁))
170158recnd 9921 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑁...𝑖)) → (𝑀𝑘) ∈ ℂ)
171168, 169, 170fsumser 14251 . . . . . . . . . . . . . . . 16 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑁...𝑖)(𝑀𝑘) = (seq𝑁( + , 𝑀)‘𝑖))
172 eqidd 2607 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑁...𝑗)) → (𝑀𝑘) = (𝑀𝑘))
173157, 126, 89syl2an 492 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑁...𝑗)) → (𝑀𝑘) ∈ ℝ)
174173recnd 9921 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑁...𝑗)) → (𝑀𝑘) ∈ ℂ)
175172, 72, 174fsumser 14251 . . . . . . . . . . . . . . . 16 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑁...𝑗)(𝑀𝑘) = (seq𝑁( + , 𝑀)‘𝑗))
176171, 175oveq12d 6542 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → (Σ𝑘 ∈ (𝑁...𝑖)(𝑀𝑘) − Σ𝑘 ∈ (𝑁...𝑗)(𝑀𝑘)) = ((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗)))
177 fzfid 12586 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → (𝑁...𝑖) ∈ Fin)
178141, 77, 177, 170fsumsplit 14261 . . . . . . . . . . . . . . . . 17 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑁...𝑖)(𝑀𝑘) = (Σ𝑘 ∈ (𝑁...𝑗)(𝑀𝑘) + Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(𝑀𝑘)))
179178eqcomd 2612 . . . . . . . . . . . . . . . 16 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → (Σ𝑘 ∈ (𝑁...𝑗)(𝑀𝑘) + Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(𝑀𝑘)) = Σ𝑘 ∈ (𝑁...𝑖)(𝑀𝑘))
180177, 170fsumcl 14254 . . . . . . . . . . . . . . . . 17 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑁...𝑖)(𝑀𝑘) ∈ ℂ)
181 fzfid 12586 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → (𝑁...𝑗) ∈ Fin)
182181, 174fsumcl 14254 . . . . . . . . . . . . . . . . 17 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑁...𝑗)(𝑀𝑘) ∈ ℂ)
183 fzfid 12586 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → ((𝑗 + 1)...𝑖) ∈ Fin)
18479, 170syldan 485 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ ((𝑗 + 1)...𝑖)) → (𝑀𝑘) ∈ ℂ)
185183, 184fsumcl 14254 . . . . . . . . . . . . . . . . 17 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(𝑀𝑘) ∈ ℂ)
186180, 182, 185subaddd 10258 . . . . . . . . . . . . . . . 16 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → ((Σ𝑘 ∈ (𝑁...𝑖)(𝑀𝑘) − Σ𝑘 ∈ (𝑁...𝑗)(𝑀𝑘)) = Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(𝑀𝑘) ↔ (Σ𝑘 ∈ (𝑁...𝑗)(𝑀𝑘) + Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(𝑀𝑘)) = Σ𝑘 ∈ (𝑁...𝑖)(𝑀𝑘)))
187179, 186mpbird 245 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → (Σ𝑘 ∈ (𝑁...𝑖)(𝑀𝑘) − Σ𝑘 ∈ (𝑁...𝑗)(𝑀𝑘)) = Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(𝑀𝑘))
188176, 187eqtr3d 2642 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → ((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗)) = Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(𝑀𝑘))
189188fveq2d 6089 . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → (abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))) = (abs‘Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(𝑀𝑘)))
190189adantr 479 . . . . . . . . . . . 12 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → (abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))) = (abs‘Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(𝑀𝑘)))
191188, 94eqeltrrd 2685 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(𝑀𝑘) ∈ ℝ)
192191adantr 479 . . . . . . . . . . . . 13 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(𝑀𝑘) ∈ ℝ)
193 0red 9894 . . . . . . . . . . . . . . 15 (((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) ∧ 𝑘 ∈ ((𝑗 + 1)...𝑖)) → 0 ∈ ℝ)
19486absge0d 13974 . . . . . . . . . . . . . . 15 (((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) ∧ 𝑘 ∈ ((𝑗 + 1)...𝑖)) → 0 ≤ (abs‘((𝐹𝑘)‘𝑧)))
195193, 87, 160, 194, 166letrd 10042 . . . . . . . . . . . . . 14 (((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) ∧ 𝑘 ∈ ((𝑗 + 1)...𝑖)) → 0 ≤ (𝑀𝑘))
19670, 160, 195fsumge0 14311 . . . . . . . . . . . . 13 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → 0 ≤ Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(𝑀𝑘))
197192, 196absidd 13952 . . . . . . . . . . . 12 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → (abs‘Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(𝑀𝑘)) = Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(𝑀𝑘))
198190, 197eqtrd 2640 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → (abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))) = Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(𝑀𝑘))
199167, 198breqtrrd 4602 . . . . . . . . . 10 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → Σ𝑘 ∈ ((𝑗 + 1)...𝑖)(abs‘((𝐹𝑘)‘𝑧)) ≤ (abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))))
20069, 88, 97, 156, 199letrd 10042 . . . . . . . . 9 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → (abs‘(((seq𝑁( ∘𝑓 + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘𝑓 + , 𝐹)‘𝑗)‘𝑧))) ≤ (abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))))
201 simpllr 794 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → 𝑟 ∈ ℝ+)
202201rpred 11701 . . . . . . . . . 10 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → 𝑟 ∈ ℝ)
203 lelttr 9976 . . . . . . . . . 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))
20469, 97, 202, 203syl3anc 1317 . . . . . . . . 9 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → (((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))
205200, 204mpand 706 . . . . . . . 8 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) ∧ 𝑧𝑆) → ((abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))) < 𝑟 → (abs‘(((seq𝑁( ∘𝑓 + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘𝑓 + , 𝐹)‘𝑗)‘𝑧))) < 𝑟))
206205ralrimdva 2948 . . . . . . 7 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑗𝑍𝑖 ∈ (ℤ𝑗))) → ((abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))) < 𝑟 → ∀𝑧𝑆 (abs‘(((seq𝑁( ∘𝑓 + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘𝑓 + , 𝐹)‘𝑗)‘𝑧))) < 𝑟))
207206anassrs 677 . . . . . 6 ((((𝜑𝑟 ∈ ℝ+) ∧ 𝑗𝑍) ∧ 𝑖 ∈ (ℤ𝑗)) → ((abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))) < 𝑟 → ∀𝑧𝑆 (abs‘(((seq𝑁( ∘𝑓 + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘𝑓 + , 𝐹)‘𝑗)‘𝑧))) < 𝑟))
208207ralimdva 2941 . . . . 5 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑗𝑍) → (∀𝑖 ∈ (ℤ𝑗)(abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))) < 𝑟 → ∀𝑖 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((seq𝑁( ∘𝑓 + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘𝑓 + , 𝐹)‘𝑗)‘𝑧))) < 𝑟))
209208reximdva 2996 . . . 4 ((𝜑𝑟 ∈ ℝ+) → (∃𝑗𝑍𝑖 ∈ (ℤ𝑗)(abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))) < 𝑟 → ∃𝑗𝑍𝑖 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((seq𝑁( ∘𝑓 + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘𝑓 + , 𝐹)‘𝑗)‘𝑧))) < 𝑟))
210209ralimdva 2941 . . 3 (𝜑 → (∀𝑟 ∈ ℝ+𝑗𝑍𝑖 ∈ (ℤ𝑗)(abs‘((seq𝑁( + , 𝑀)‘𝑖) − (seq𝑁( + , 𝑀)‘𝑗))) < 𝑟 → ∀𝑟 ∈ ℝ+𝑗𝑍𝑖 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((seq𝑁( ∘𝑓 + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘𝑓 + , 𝐹)‘𝑗)‘𝑧))) < 𝑟))
2115, 210mpd 15 . 2 (𝜑 → ∀𝑟 ∈ ℝ+𝑗𝑍𝑖 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((seq𝑁( ∘𝑓 + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘𝑓 + , 𝐹)‘𝑗)‘𝑧))) < 𝑟)
2123, 1, 10, 55ulmcau 23867 . 2 (𝜑 → (seq𝑁( ∘𝑓 + , 𝐹) ∈ dom (⇝𝑢𝑆) ↔ ∀𝑟 ∈ ℝ+𝑗𝑍𝑖 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((seq𝑁( ∘𝑓 + , 𝐹)‘𝑖)‘𝑧) − ((seq𝑁( ∘𝑓 + , 𝐹)‘𝑗)‘𝑧))) < 𝑟))
213211, 212mpbird 245 1 (𝜑 → seq𝑁( ∘𝑓 + , 𝐹) ∈ dom (⇝𝑢𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wral 2892  wrex 2893  Vcvv 3169  cun 3534  cin 3535  c0 3870   class class class wbr 4574  cmpt 4634  dom cdm 5025   Fn wfn 5782  wf 5783  cfv 5787  (class class class)co 6524  𝑓 cof 6767  𝑚 cmap 7718  cc 9787  cr 9788  0cc0 9789  1c1 9790   + caddc 9792   < clt 9927  cle 9928  cmin 10114  cz 11207  cuz 11516  +crp 11661  ...cfz 12149  seqcseq 12615  abscabs 13765  cli 14006  Σcsu 14207  𝑢culm 23848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-inf2 8395  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865  ax-pre-mulgt0 9866  ax-pre-sup 9867  ax-addf 9868  ax-mulf 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-se 4985  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-isom 5796  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-of 6769  df-om 6932  df-1st 7033  df-2nd 7034  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-1o 7421  df-oadd 7425  df-er 7603  df-map 7720  df-pm 7721  df-en 7816  df-dom 7817  df-sdom 7818  df-fin 7819  df-sup 8205  df-inf 8206  df-oi 8272  df-card 8622  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-sub 10116  df-neg 10117  df-div 10531  df-nn 10865  df-2 10923  df-3 10924  df-n0 11137  df-z 11208  df-uz 11517  df-rp 11662  df-ico 12005  df-fz 12150  df-fzo 12287  df-fl 12407  df-seq 12616  df-exp 12675  df-hash 12932  df-cj 13630  df-re 13631  df-im 13632  df-sqrt 13766  df-abs 13767  df-limsup 13993  df-clim 14010  df-rlim 14011  df-sum 14208  df-ulm 23849
This theorem is referenced by:  pserulm  23894  lgamgulmlem6  24474  knoppcnlem6  31461
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