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Theorem mtestbdd 24379
Description: Given the hypotheses of the Weierstrass M-test, the convergent function of the sequence is uniformly bounded. (Contributed by Mario Carneiro, 9-Jul-2017.)
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 ⇝ )
mtest.t (𝜑 → seq𝑁( ∘𝑓 + , 𝐹)(⇝𝑢𝑆)𝑇)
Assertion
Ref Expression
mtestbdd (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧𝑆 (abs‘(𝑇𝑧)) ≤ 𝑥)
Distinct variable groups:   𝑥,𝑘,𝑧,𝐹   𝑘,𝑀,𝑥,𝑧   𝑘,𝑁,𝑥,𝑧   𝜑,𝑘,𝑥,𝑧   𝑥,𝑇,𝑧   𝑘,𝑍,𝑥,𝑧   𝑆,𝑘,𝑥,𝑧
Allowed substitution hints:   𝑇(𝑘)   𝑉(𝑥,𝑧,𝑘)   𝑊(𝑥,𝑧,𝑘)

Proof of Theorem mtestbdd
Dummy variables 𝑗 𝑛 𝑚 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mtest.n . . 3 (𝜑𝑁 ∈ ℤ)
2 mtest.d . . 3 (𝜑 → seq𝑁( + , 𝑀) ∈ dom ⇝ )
3 mtest.z . . . . . 6 𝑍 = (ℤ𝑁)
4 mtest.c . . . . . . 7 ((𝜑𝑘𝑍) → (𝑀𝑘) ∈ ℝ)
54recnd 10270 . . . . . 6 ((𝜑𝑘𝑍) → (𝑀𝑘) ∈ ℂ)
63, 1, 5serf 13036 . . . . 5 (𝜑 → seq𝑁( + , 𝑀):𝑍⟶ℂ)
76ffvelrnda 6502 . . . 4 ((𝜑𝑚𝑍) → (seq𝑁( + , 𝑀)‘𝑚) ∈ ℂ)
87ralrimiva 3115 . . 3 (𝜑 → ∀𝑚𝑍 (seq𝑁( + , 𝑀)‘𝑚) ∈ ℂ)
93climbdd 14610 . . 3 ((𝑁 ∈ ℤ ∧ seq𝑁( + , 𝑀) ∈ dom ⇝ ∧ ∀𝑚𝑍 (seq𝑁( + , 𝑀)‘𝑚) ∈ ℂ) → ∃𝑦 ∈ ℝ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)
101, 2, 8, 9syl3anc 1476 . 2 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)
111adantr 466 . . 3 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → 𝑁 ∈ ℤ)
12 seqfn 13020 . . . . . . 7 (𝑁 ∈ ℤ → seq𝑁( ∘𝑓 + , 𝐹) Fn (ℤ𝑁))
131, 12syl 17 . . . . . 6 (𝜑 → seq𝑁( ∘𝑓 + , 𝐹) Fn (ℤ𝑁))
143fneq2i 6126 . . . . . 6 (seq𝑁( ∘𝑓 + , 𝐹) Fn 𝑍 ↔ seq𝑁( ∘𝑓 + , 𝐹) Fn (ℤ𝑁))
1513, 14sylibr 224 . . . . 5 (𝜑 → seq𝑁( ∘𝑓 + , 𝐹) Fn 𝑍)
16 mtest.t . . . . 5 (𝜑 → seq𝑁( ∘𝑓 + , 𝐹)(⇝𝑢𝑆)𝑇)
17 ulmf2 24358 . . . . 5 ((seq𝑁( ∘𝑓 + , 𝐹) Fn 𝑍 ∧ seq𝑁( ∘𝑓 + , 𝐹)(⇝𝑢𝑆)𝑇) → seq𝑁( ∘𝑓 + , 𝐹):𝑍⟶(ℂ ↑𝑚 𝑆))
1815, 16, 17syl2anc 573 . . . 4 (𝜑 → seq𝑁( ∘𝑓 + , 𝐹):𝑍⟶(ℂ ↑𝑚 𝑆))
1918adantr 466 . . 3 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → seq𝑁( ∘𝑓 + , 𝐹):𝑍⟶(ℂ ↑𝑚 𝑆))
20 simplrl 762 . . . 4 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) → 𝑦 ∈ ℝ)
21 fveq2 6332 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → ((𝐹𝑗)‘𝑥) = ((𝐹𝑗)‘𝑧))
2221mpteq2dv 4879 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑥)) = (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧)))
2322seqeq3d 13016 . . . . . . . . . . 11 (𝑥 = 𝑧 → seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑥))) = seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧))))
2423fveq1d 6334 . . . . . . . . . 10 (𝑥 = 𝑧 → (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑥)))‘𝑛) = (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧)))‘𝑛))
25 eqid 2771 . . . . . . . . . 10 (𝑥𝑆 ↦ (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑥)))‘𝑛)) = (𝑥𝑆 ↦ (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑥)))‘𝑛))
26 fvex 6342 . . . . . . . . . 10 (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧)))‘𝑛) ∈ V
2724, 25, 26fvmpt 6424 . . . . . . . . 9 (𝑧𝑆 → ((𝑥𝑆 ↦ (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑥)))‘𝑛))‘𝑧) = (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧)))‘𝑛))
2827adantl 467 . . . . . . . 8 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → ((𝑥𝑆 ↦ (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑥)))‘𝑛))‘𝑧) = (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧)))‘𝑛))
29 mtest.f . . . . . . . . . . . . . . 15 (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))
3029ad3antrrr 709 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → 𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))
3130feqmptd 6391 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → 𝐹 = (𝑗𝑍 ↦ (𝐹𝑗)))
3230ffvelrnda 6502 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑗𝑍) → (𝐹𝑗) ∈ (ℂ ↑𝑚 𝑆))
33 elmapi 8031 . . . . . . . . . . . . . . . 16 ((𝐹𝑗) ∈ (ℂ ↑𝑚 𝑆) → (𝐹𝑗):𝑆⟶ℂ)
3432, 33syl 17 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑗𝑍) → (𝐹𝑗):𝑆⟶ℂ)
3534feqmptd 6391 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑗𝑍) → (𝐹𝑗) = (𝑥𝑆 ↦ ((𝐹𝑗)‘𝑥)))
3635mpteq2dva 4878 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (𝑗𝑍 ↦ (𝐹𝑗)) = (𝑗𝑍 ↦ (𝑥𝑆 ↦ ((𝐹𝑗)‘𝑥))))
3731, 36eqtrd 2805 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → 𝐹 = (𝑗𝑍 ↦ (𝑥𝑆 ↦ ((𝐹𝑗)‘𝑥))))
3837seqeq3d 13016 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → seq𝑁( ∘𝑓 + , 𝐹) = seq𝑁( ∘𝑓 + , (𝑗𝑍 ↦ (𝑥𝑆 ↦ ((𝐹𝑗)‘𝑥)))))
3938fveq1d 6334 . . . . . . . . . 10 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (seq𝑁( ∘𝑓 + , 𝐹)‘𝑛) = (seq𝑁( ∘𝑓 + , (𝑗𝑍 ↦ (𝑥𝑆 ↦ ((𝐹𝑗)‘𝑥))))‘𝑛))
40 mtest.s . . . . . . . . . . . 12 (𝜑𝑆𝑉)
4140ad3antrrr 709 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → 𝑆𝑉)
42 simplr 752 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → 𝑛𝑍)
4342, 3syl6eleq 2860 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → 𝑛 ∈ (ℤ𝑁))
44 elfzuz 12545 . . . . . . . . . . . . . 14 (𝑘 ∈ (𝑁...𝑛) → 𝑘 ∈ (ℤ𝑁))
4544, 3syl6eleqr 2861 . . . . . . . . . . . . 13 (𝑘 ∈ (𝑁...𝑛) → 𝑘𝑍)
4645ssriv 3756 . . . . . . . . . . . 12 (𝑁...𝑛) ⊆ 𝑍
4746a1i 11 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (𝑁...𝑛) ⊆ 𝑍)
4834ffvelrnda 6502 . . . . . . . . . . . 12 ((((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑗𝑍) ∧ 𝑥𝑆) → ((𝐹𝑗)‘𝑥) ∈ ℂ)
4948anasss 457 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ (𝑗𝑍𝑥𝑆)) → ((𝐹𝑗)‘𝑥) ∈ ℂ)
5041, 43, 47, 49seqof2 13066 . . . . . . . . . 10 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (seq𝑁( ∘𝑓 + , (𝑗𝑍 ↦ (𝑥𝑆 ↦ ((𝐹𝑗)‘𝑥))))‘𝑛) = (𝑥𝑆 ↦ (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑥)))‘𝑛)))
5139, 50eqtrd 2805 . . . . . . . . 9 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (seq𝑁( ∘𝑓 + , 𝐹)‘𝑛) = (𝑥𝑆 ↦ (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑥)))‘𝑛)))
5251fveq1d 6334 . . . . . . . 8 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → ((seq𝑁( ∘𝑓 + , 𝐹)‘𝑛)‘𝑧) = ((𝑥𝑆 ↦ (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑥)))‘𝑛))‘𝑧))
5345adantl 467 . . . . . . . . . 10 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → 𝑘𝑍)
54 fveq2 6332 . . . . . . . . . . . 12 (𝑗 = 𝑘 → (𝐹𝑗) = (𝐹𝑘))
5554fveq1d 6334 . . . . . . . . . . 11 (𝑗 = 𝑘 → ((𝐹𝑗)‘𝑧) = ((𝐹𝑘)‘𝑧))
56 eqid 2771 . . . . . . . . . . 11 (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧)) = (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧))
57 fvex 6342 . . . . . . . . . . 11 ((𝐹𝑘)‘𝑧) ∈ V
5855, 56, 57fvmpt 6424 . . . . . . . . . 10 (𝑘𝑍 → ((𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧))‘𝑘) = ((𝐹𝑘)‘𝑧))
5953, 58syl 17 . . . . . . . . 9 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → ((𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧))‘𝑘) = ((𝐹𝑘)‘𝑧))
60 simplr 752 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑗𝑍) → 𝑧𝑆)
6134, 60ffvelrnd 6503 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑗𝑍) → ((𝐹𝑗)‘𝑧) ∈ ℂ)
6261, 56fmptd 6527 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧)):𝑍⟶ℂ)
6362ffvelrnda 6502 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘𝑍) → ((𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧))‘𝑘) ∈ ℂ)
6445, 63sylan2 580 . . . . . . . . . 10 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → ((𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧))‘𝑘) ∈ ℂ)
6559, 64eqeltrrd 2851 . . . . . . . . 9 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → ((𝐹𝑘)‘𝑧) ∈ ℂ)
6659, 43, 65fsumser 14669 . . . . . . . 8 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)((𝐹𝑘)‘𝑧) = (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧)))‘𝑛))
6728, 52, 663eqtr4d 2815 . . . . . . 7 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → ((seq𝑁( ∘𝑓 + , 𝐹)‘𝑛)‘𝑧) = Σ𝑘 ∈ (𝑁...𝑛)((𝐹𝑘)‘𝑧))
6867fveq2d 6336 . . . . . 6 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘((seq𝑁( ∘𝑓 + , 𝐹)‘𝑛)‘𝑧)) = (abs‘Σ𝑘 ∈ (𝑁...𝑛)((𝐹𝑘)‘𝑧)))
69 fzfid 12980 . . . . . . . . 9 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (𝑁...𝑛) ∈ Fin)
7069, 65fsumcl 14672 . . . . . . . 8 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)((𝐹𝑘)‘𝑧) ∈ ℂ)
7170abscld 14383 . . . . . . 7 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘Σ𝑘 ∈ (𝑁...𝑛)((𝐹𝑘)‘𝑧)) ∈ ℝ)
7265abscld 14383 . . . . . . . 8 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → (abs‘((𝐹𝑘)‘𝑧)) ∈ ℝ)
7369, 72fsumrecl 14673 . . . . . . 7 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(abs‘((𝐹𝑘)‘𝑧)) ∈ ℝ)
7420adantr 466 . . . . . . 7 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → 𝑦 ∈ ℝ)
7569, 65fsumabs 14740 . . . . . . 7 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘Σ𝑘 ∈ (𝑁...𝑛)((𝐹𝑘)‘𝑧)) ≤ Σ𝑘 ∈ (𝑁...𝑛)(abs‘((𝐹𝑘)‘𝑧)))
76 simp-4l 768 . . . . . . . . . 10 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → 𝜑)
7776, 53, 4syl2anc 573 . . . . . . . . 9 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → (𝑀𝑘) ∈ ℝ)
7869, 77fsumrecl 14673 . . . . . . . 8 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘) ∈ ℝ)
79 simplr 752 . . . . . . . . . 10 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → 𝑧𝑆)
80 mtest.l . . . . . . . . . 10 ((𝜑 ∧ (𝑘𝑍𝑧𝑆)) → (abs‘((𝐹𝑘)‘𝑧)) ≤ (𝑀𝑘))
8176, 53, 79, 80syl12anc 1474 . . . . . . . . 9 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → (abs‘((𝐹𝑘)‘𝑧)) ≤ (𝑀𝑘))
8269, 72, 77, 81fsumle 14738 . . . . . . . 8 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(abs‘((𝐹𝑘)‘𝑧)) ≤ Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘))
8378recnd 10270 . . . . . . . . . 10 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘) ∈ ℂ)
8483abscld 14383 . . . . . . . . 9 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘)) ∈ ℝ)
8578leabsd 14361 . . . . . . . . 9 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘) ≤ (abs‘Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘)))
86 eqidd 2772 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → (𝑀𝑘) = (𝑀𝑘))
8776, 53, 5syl2anc 573 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → (𝑀𝑘) ∈ ℂ)
8886, 43, 87fsumser 14669 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘) = (seq𝑁( + , 𝑀)‘𝑛))
8988fveq2d 6336 . . . . . . . . . 10 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘)) = (abs‘(seq𝑁( + , 𝑀)‘𝑛)))
90 simprr 756 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)
91 fveq2 6332 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (seq𝑁( + , 𝑀)‘𝑚) = (seq𝑁( + , 𝑀)‘𝑛))
9291fveq2d 6336 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (abs‘(seq𝑁( + , 𝑀)‘𝑚)) = (abs‘(seq𝑁( + , 𝑀)‘𝑛)))
9392breq1d 4796 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → ((abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦 ↔ (abs‘(seq𝑁( + , 𝑀)‘𝑛)) ≤ 𝑦))
9493rspccva 3459 . . . . . . . . . . . 12 ((∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦𝑛𝑍) → (abs‘(seq𝑁( + , 𝑀)‘𝑛)) ≤ 𝑦)
9590, 94sylan 569 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) → (abs‘(seq𝑁( + , 𝑀)‘𝑛)) ≤ 𝑦)
9695adantr 466 . . . . . . . . . 10 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘(seq𝑁( + , 𝑀)‘𝑛)) ≤ 𝑦)
9789, 96eqbrtrd 4808 . . . . . . . . 9 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘)) ≤ 𝑦)
9878, 84, 74, 85, 97letrd 10396 . . . . . . . 8 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘) ≤ 𝑦)
9973, 78, 74, 82, 98letrd 10396 . . . . . . 7 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(abs‘((𝐹𝑘)‘𝑧)) ≤ 𝑦)
10071, 73, 74, 75, 99letrd 10396 . . . . . 6 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘Σ𝑘 ∈ (𝑁...𝑛)((𝐹𝑘)‘𝑧)) ≤ 𝑦)
10168, 100eqbrtrd 4808 . . . . 5 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘((seq𝑁( ∘𝑓 + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑦)
102101ralrimiva 3115 . . . 4 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) → ∀𝑧𝑆 (abs‘((seq𝑁( ∘𝑓 + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑦)
103 breq2 4790 . . . . . 6 (𝑥 = 𝑦 → ((abs‘((seq𝑁( ∘𝑓 + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑥 ↔ (abs‘((seq𝑁( ∘𝑓 + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑦))
104103ralbidv 3135 . . . . 5 (𝑥 = 𝑦 → (∀𝑧𝑆 (abs‘((seq𝑁( ∘𝑓 + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑥 ↔ ∀𝑧𝑆 (abs‘((seq𝑁( ∘𝑓 + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑦))
105104rspcev 3460 . . . 4 ((𝑦 ∈ ℝ ∧ ∀𝑧𝑆 (abs‘((seq𝑁( ∘𝑓 + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑧𝑆 (abs‘((seq𝑁( ∘𝑓 + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑥)
10620, 102, 105syl2anc 573 . . 3 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) → ∃𝑥 ∈ ℝ ∀𝑧𝑆 (abs‘((seq𝑁( ∘𝑓 + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑥)
10716adantr 466 . . 3 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → seq𝑁( ∘𝑓 + , 𝐹)(⇝𝑢𝑆)𝑇)
1083, 11, 19, 106, 107ulmbdd 24372 . 2 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → ∃𝑥 ∈ ℝ ∀𝑧𝑆 (abs‘(𝑇𝑧)) ≤ 𝑥)
10910, 108rexlimddv 3183 1 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧𝑆 (abs‘(𝑇𝑧)) ≤ 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  wrex 3062  wss 3723   class class class wbr 4786  cmpt 4863  dom cdm 5249   Fn wfn 6026  wf 6027  cfv 6031  (class class class)co 6793  𝑓 cof 7042  𝑚 cmap 8009  cc 10136  cr 10137   + caddc 10141  cle 10277  cz 11579  cuz 11888  ...cfz 12533  seqcseq 13008  abscabs 14182  cli 14423  Σcsu 14624  𝑢culm 24350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-of 7044  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-map 8011  df-pm 8012  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-sup 8504  df-oi 8571  df-card 8965  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-nn 11223  df-2 11281  df-3 11282  df-n0 11495  df-z 11580  df-uz 11889  df-rp 12036  df-ico 12386  df-fz 12534  df-fzo 12674  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-clim 14427  df-sum 14625  df-ulm 24351
This theorem is referenced by:  lgamgulmlem6  24981
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