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Theorem mtestbdd 24920
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 10657 . . . . . 6 ((𝜑𝑘𝑍) → (𝑀𝑘) ∈ ℂ)
63, 1, 5serf 13386 . . . . 5 (𝜑 → seq𝑁( + , 𝑀):𝑍⟶ℂ)
76ffvelrnda 6843 . . . 4 ((𝜑𝑚𝑍) → (seq𝑁( + , 𝑀)‘𝑚) ∈ ℂ)
87ralrimiva 3179 . . 3 (𝜑 → ∀𝑚𝑍 (seq𝑁( + , 𝑀)‘𝑚) ∈ ℂ)
93climbdd 15016 . . 3 ((𝑁 ∈ ℤ ∧ seq𝑁( + , 𝑀) ∈ dom ⇝ ∧ ∀𝑚𝑍 (seq𝑁( + , 𝑀)‘𝑚) ∈ ℂ) → ∃𝑦 ∈ ℝ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)
101, 2, 8, 9syl3anc 1363 . 2 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)
111adantr 481 . . 3 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → 𝑁 ∈ ℤ)
12 seqfn 13369 . . . . . . 7 (𝑁 ∈ ℤ → seq𝑁( ∘f + , 𝐹) Fn (ℤ𝑁))
131, 12syl 17 . . . . . 6 (𝜑 → seq𝑁( ∘f + , 𝐹) Fn (ℤ𝑁))
143fneq2i 6444 . . . . . 6 (seq𝑁( ∘f + , 𝐹) Fn 𝑍 ↔ seq𝑁( ∘f + , 𝐹) Fn (ℤ𝑁))
1513, 14sylibr 235 . . . . 5 (𝜑 → seq𝑁( ∘f + , 𝐹) Fn 𝑍)
16 mtest.t . . . . 5 (𝜑 → seq𝑁( ∘f + , 𝐹)(⇝𝑢𝑆)𝑇)
17 ulmf2 24899 . . . . 5 ((seq𝑁( ∘f + , 𝐹) Fn 𝑍 ∧ seq𝑁( ∘f + , 𝐹)(⇝𝑢𝑆)𝑇) → seq𝑁( ∘f + , 𝐹):𝑍⟶(ℂ ↑m 𝑆))
1815, 16, 17syl2anc 584 . . . 4 (𝜑 → seq𝑁( ∘f + , 𝐹):𝑍⟶(ℂ ↑m 𝑆))
1918adantr 481 . . 3 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → seq𝑁( ∘f + , 𝐹):𝑍⟶(ℂ ↑m 𝑆))
20 simplrl 773 . . . 4 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) → 𝑦 ∈ ℝ)
21 fveq2 6663 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → ((𝐹𝑗)‘𝑥) = ((𝐹𝑗)‘𝑧))
2221mpteq2dv 5153 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑥)) = (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧)))
2322seqeq3d 13365 . . . . . . . . . . 11 (𝑥 = 𝑧 → seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑥))) = seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧))))
2423fveq1d 6665 . . . . . . . . . 10 (𝑥 = 𝑧 → (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑥)))‘𝑛) = (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧)))‘𝑛))
25 eqid 2818 . . . . . . . . . 10 (𝑥𝑆 ↦ (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑥)))‘𝑛)) = (𝑥𝑆 ↦ (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑥)))‘𝑛))
26 fvex 6676 . . . . . . . . . 10 (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧)))‘𝑛) ∈ V
2724, 25, 26fvmpt 6761 . . . . . . . . 9 (𝑧𝑆 → ((𝑥𝑆 ↦ (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑥)))‘𝑛))‘𝑧) = (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧)))‘𝑛))
2827adantl 482 . . . . . . . 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 726 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → 𝐹:𝑍⟶(ℂ ↑m 𝑆))
3130feqmptd 6726 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → 𝐹 = (𝑗𝑍 ↦ (𝐹𝑗)))
3230ffvelrnda 6843 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑗𝑍) → (𝐹𝑗) ∈ (ℂ ↑m 𝑆))
33 elmapi 8417 . . . . . . . . . . . . . . . 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 6726 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑗𝑍) → (𝐹𝑗) = (𝑥𝑆 ↦ ((𝐹𝑗)‘𝑥)))
3635mpteq2dva 5152 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (𝑗𝑍 ↦ (𝐹𝑗)) = (𝑗𝑍 ↦ (𝑥𝑆 ↦ ((𝐹𝑗)‘𝑥))))
3731, 36eqtrd 2853 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → 𝐹 = (𝑗𝑍 ↦ (𝑥𝑆 ↦ ((𝐹𝑗)‘𝑥))))
3837seqeq3d 13365 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → seq𝑁( ∘f + , 𝐹) = seq𝑁( ∘f + , (𝑗𝑍 ↦ (𝑥𝑆 ↦ ((𝐹𝑗)‘𝑥)))))
3938fveq1d 6665 . . . . . . . . . 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 726 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → 𝑆𝑉)
42 simplr 765 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → 𝑛𝑍)
4342, 3eleqtrdi 2920 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → 𝑛 ∈ (ℤ𝑁))
44 elfzuz 12892 . . . . . . . . . . . . . 14 (𝑘 ∈ (𝑁...𝑛) → 𝑘 ∈ (ℤ𝑁))
4544, 3eleqtrrdi 2921 . . . . . . . . . . . . 13 (𝑘 ∈ (𝑁...𝑛) → 𝑘𝑍)
4645ssriv 3968 . . . . . . . . . . . 12 (𝑁...𝑛) ⊆ 𝑍
4746a1i 11 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (𝑁...𝑛) ⊆ 𝑍)
4834ffvelrnda 6843 . . . . . . . . . . . 12 ((((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑗𝑍) ∧ 𝑥𝑆) → ((𝐹𝑗)‘𝑥) ∈ ℂ)
4948anasss 467 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ (𝑗𝑍𝑥𝑆)) → ((𝐹𝑗)‘𝑥) ∈ ℂ)
5041, 43, 47, 49seqof2 13416 . . . . . . . . . 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 2853 . . . . . . . . 9 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (seq𝑁( ∘f + , 𝐹)‘𝑛) = (𝑥𝑆 ↦ (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑥)))‘𝑛)))
5251fveq1d 6665 . . . . . . . 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 482 . . . . . . . . . 10 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → 𝑘𝑍)
54 fveq2 6663 . . . . . . . . . . . 12 (𝑗 = 𝑘 → (𝐹𝑗) = (𝐹𝑘))
5554fveq1d 6665 . . . . . . . . . . 11 (𝑗 = 𝑘 → ((𝐹𝑗)‘𝑧) = ((𝐹𝑘)‘𝑧))
56 eqid 2818 . . . . . . . . . . 11 (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧)) = (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧))
57 fvex 6676 . . . . . . . . . . 11 ((𝐹𝑘)‘𝑧) ∈ V
5855, 56, 57fvmpt 6761 . . . . . . . . . 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 765 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑗𝑍) → 𝑧𝑆)
6134, 60ffvelrnd 6844 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑗𝑍) → ((𝐹𝑗)‘𝑧) ∈ ℂ)
6261fmpttd 6871 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧)):𝑍⟶ℂ)
6362ffvelrnda 6843 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘𝑍) → ((𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧))‘𝑘) ∈ ℂ)
6445, 63sylan2 592 . . . . . . . . . 10 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → ((𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧))‘𝑘) ∈ ℂ)
6559, 64eqeltrrd 2911 . . . . . . . . 9 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → ((𝐹𝑘)‘𝑧) ∈ ℂ)
6659, 43, 65fsumser 15075 . . . . . . . 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 2863 . . . . . . 7 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → ((seq𝑁( ∘f + , 𝐹)‘𝑛)‘𝑧) = Σ𝑘 ∈ (𝑁...𝑛)((𝐹𝑘)‘𝑧))
6867fveq2d 6667 . . . . . 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 13329 . . . . . . . . 9 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (𝑁...𝑛) ∈ Fin)
7069, 65fsumcl 15078 . . . . . . . 8 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)((𝐹𝑘)‘𝑧) ∈ ℂ)
7170abscld 14784 . . . . . . 7 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘Σ𝑘 ∈ (𝑁...𝑛)((𝐹𝑘)‘𝑧)) ∈ ℝ)
7265abscld 14784 . . . . . . . 8 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → (abs‘((𝐹𝑘)‘𝑧)) ∈ ℝ)
7369, 72fsumrecl 15079 . . . . . . 7 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(abs‘((𝐹𝑘)‘𝑧)) ∈ ℝ)
7420adantr 481 . . . . . . 7 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → 𝑦 ∈ ℝ)
7569, 65fsumabs 15144 . . . . . . 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 779 . . . . . . . . . 10 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → 𝜑)
7776, 53, 4syl2anc 584 . . . . . . . . 9 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → (𝑀𝑘) ∈ ℝ)
7869, 77fsumrecl 15079 . . . . . . . 8 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘) ∈ ℝ)
79 simplr 765 . . . . . . . . . 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 832 . . . . . . . . 9 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → (abs‘((𝐹𝑘)‘𝑧)) ≤ (𝑀𝑘))
8269, 72, 77, 81fsumle 15142 . . . . . . . 8 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(abs‘((𝐹𝑘)‘𝑧)) ≤ Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘))
8378recnd 10657 . . . . . . . . . 10 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘) ∈ ℂ)
8483abscld 14784 . . . . . . . . 9 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘)) ∈ ℝ)
8578leabsd 14762 . . . . . . . . 9 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘) ≤ (abs‘Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘)))
86 eqidd 2819 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → (𝑀𝑘) = (𝑀𝑘))
8776, 53, 5syl2anc 584 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → (𝑀𝑘) ∈ ℂ)
8886, 43, 87fsumser 15075 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘) = (seq𝑁( + , 𝑀)‘𝑛))
8988fveq2d 6667 . . . . . . . . . 10 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘)) = (abs‘(seq𝑁( + , 𝑀)‘𝑛)))
90 simprr 769 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)
91 fveq2 6663 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (seq𝑁( + , 𝑀)‘𝑚) = (seq𝑁( + , 𝑀)‘𝑛))
9291fveq2d 6667 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (abs‘(seq𝑁( + , 𝑀)‘𝑚)) = (abs‘(seq𝑁( + , 𝑀)‘𝑛)))
9392breq1d 5067 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → ((abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦 ↔ (abs‘(seq𝑁( + , 𝑀)‘𝑛)) ≤ 𝑦))
9493rspccva 3619 . . . . . . . . . . . 12 ((∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦𝑛𝑍) → (abs‘(seq𝑁( + , 𝑀)‘𝑛)) ≤ 𝑦)
9590, 94sylan 580 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) → (abs‘(seq𝑁( + , 𝑀)‘𝑛)) ≤ 𝑦)
9695adantr 481 . . . . . . . . . 10 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘(seq𝑁( + , 𝑀)‘𝑛)) ≤ 𝑦)
9789, 96eqbrtrd 5079 . . . . . . . . 9 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘)) ≤ 𝑦)
9878, 84, 74, 85, 97letrd 10785 . . . . . . . 8 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘) ≤ 𝑦)
9973, 78, 74, 82, 98letrd 10785 . . . . . . 7 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(abs‘((𝐹𝑘)‘𝑧)) ≤ 𝑦)
10071, 73, 74, 75, 99letrd 10785 . . . . . 6 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘Σ𝑘 ∈ (𝑁...𝑛)((𝐹𝑘)‘𝑧)) ≤ 𝑦)
10168, 100eqbrtrd 5079 . . . . 5 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘((seq𝑁( ∘f + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑦)
102101ralrimiva 3179 . . . 4 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) → ∀𝑧𝑆 (abs‘((seq𝑁( ∘f + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑦)
103 brralrspcev 5117 . . . 4 ((𝑦 ∈ ℝ ∧ ∀𝑧𝑆 (abs‘((seq𝑁( ∘f + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑧𝑆 (abs‘((seq𝑁( ∘f + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑥)
10420, 102, 103syl2anc 584 . . 3 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) → ∃𝑥 ∈ ℝ ∀𝑧𝑆 (abs‘((seq𝑁( ∘f + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑥)
10516adantr 481 . . 3 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → seq𝑁( ∘f + , 𝐹)(⇝𝑢𝑆)𝑇)
1063, 11, 19, 104, 105ulmbdd 24913 . 2 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → ∃𝑥 ∈ ℝ ∀𝑧𝑆 (abs‘(𝑇𝑧)) ≤ 𝑥)
10710, 106rexlimddv 3288 1 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧𝑆 (abs‘(𝑇𝑧)) ≤ 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136  wss 3933   class class class wbr 5057  cmpt 5137  dom cdm 5548   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145  f cof 7396  m cmap 8395  cc 10523  cr 10524   + caddc 10528  cle 10664  cz 11969  cuz 12231  ...cfz 12880  seqcseq 13357  abscabs 14581  cli 14829  Σcsu 15030  𝑢culm 24891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-ico 12732  df-fz 12881  df-fzo 13022  df-seq 13358  df-exp 13418  df-hash 13679  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-clim 14833  df-sum 15031  df-ulm 24892
This theorem is referenced by:  lgamgulmlem6  25538
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