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Theorem mtestbdd 25764
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 11183 . . . . . 6 ((𝜑𝑘𝑍) → (𝑀𝑘) ∈ ℂ)
63, 1, 5serf 13936 . . . . 5 (𝜑 → seq𝑁( + , 𝑀):𝑍⟶ℂ)
76ffvelcdmda 7035 . . . 4 ((𝜑𝑚𝑍) → (seq𝑁( + , 𝑀)‘𝑚) ∈ ℂ)
87ralrimiva 3143 . . 3 (𝜑 → ∀𝑚𝑍 (seq𝑁( + , 𝑀)‘𝑚) ∈ ℂ)
93climbdd 15556 . . 3 ((𝑁 ∈ ℤ ∧ seq𝑁( + , 𝑀) ∈ dom ⇝ ∧ ∀𝑚𝑍 (seq𝑁( + , 𝑀)‘𝑚) ∈ ℂ) → ∃𝑦 ∈ ℝ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)
101, 2, 8, 9syl3anc 1371 . 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 13918 . . . . . . 7 (𝑁 ∈ ℤ → seq𝑁( ∘f + , 𝐹) Fn (ℤ𝑁))
131, 12syl 17 . . . . . 6 (𝜑 → seq𝑁( ∘f + , 𝐹) Fn (ℤ𝑁))
143fneq2i 6600 . . . . . 6 (seq𝑁( ∘f + , 𝐹) Fn 𝑍 ↔ seq𝑁( ∘f + , 𝐹) Fn (ℤ𝑁))
1513, 14sylibr 233 . . . . 5 (𝜑 → seq𝑁( ∘f + , 𝐹) Fn 𝑍)
16 mtest.t . . . . 5 (𝜑 → seq𝑁( ∘f + , 𝐹)(⇝𝑢𝑆)𝑇)
17 ulmf2 25743 . . . . 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 775 . . . 4 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) → 𝑦 ∈ ℝ)
21 fveq2 6842 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → ((𝐹𝑗)‘𝑥) = ((𝐹𝑗)‘𝑧))
2221mpteq2dv 5207 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑥)) = (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧)))
2322seqeq3d 13914 . . . . . . . . . . 11 (𝑥 = 𝑧 → seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑥))) = seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧))))
2423fveq1d 6844 . . . . . . . . . 10 (𝑥 = 𝑧 → (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑥)))‘𝑛) = (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧)))‘𝑛))
25 eqid 2736 . . . . . . . . . 10 (𝑥𝑆 ↦ (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑥)))‘𝑛)) = (𝑥𝑆 ↦ (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑥)))‘𝑛))
26 fvex 6855 . . . . . . . . . 10 (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧)))‘𝑛) ∈ V
2724, 25, 26fvmpt 6948 . . . . . . . . 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 728 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → 𝐹:𝑍⟶(ℂ ↑m 𝑆))
3130feqmptd 6910 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → 𝐹 = (𝑗𝑍 ↦ (𝐹𝑗)))
3230ffvelcdmda 7035 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑗𝑍) → (𝐹𝑗) ∈ (ℂ ↑m 𝑆))
33 elmapi 8787 . . . . . . . . . . . . . . . 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 6910 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑗𝑍) → (𝐹𝑗) = (𝑥𝑆 ↦ ((𝐹𝑗)‘𝑥)))
3635mpteq2dva 5205 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (𝑗𝑍 ↦ (𝐹𝑗)) = (𝑗𝑍 ↦ (𝑥𝑆 ↦ ((𝐹𝑗)‘𝑥))))
3731, 36eqtrd 2776 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → 𝐹 = (𝑗𝑍 ↦ (𝑥𝑆 ↦ ((𝐹𝑗)‘𝑥))))
3837seqeq3d 13914 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → seq𝑁( ∘f + , 𝐹) = seq𝑁( ∘f + , (𝑗𝑍 ↦ (𝑥𝑆 ↦ ((𝐹𝑗)‘𝑥)))))
3938fveq1d 6844 . . . . . . . . . 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 728 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → 𝑆𝑉)
42 simplr 767 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → 𝑛𝑍)
4342, 3eleqtrdi 2848 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → 𝑛 ∈ (ℤ𝑁))
44 elfzuz 13437 . . . . . . . . . . . . . 14 (𝑘 ∈ (𝑁...𝑛) → 𝑘 ∈ (ℤ𝑁))
4544, 3eleqtrrdi 2849 . . . . . . . . . . . . 13 (𝑘 ∈ (𝑁...𝑛) → 𝑘𝑍)
4645ssriv 3948 . . . . . . . . . . . 12 (𝑁...𝑛) ⊆ 𝑍
4746a1i 11 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (𝑁...𝑛) ⊆ 𝑍)
4834ffvelcdmda 7035 . . . . . . . . . . . 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 13966 . . . . . . . . . 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 2776 . . . . . . . . 9 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (seq𝑁( ∘f + , 𝐹)‘𝑛) = (𝑥𝑆 ↦ (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑥)))‘𝑛)))
5251fveq1d 6844 . . . . . . . 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 6842 . . . . . . . . . . . 12 (𝑗 = 𝑘 → (𝐹𝑗) = (𝐹𝑘))
5554fveq1d 6844 . . . . . . . . . . 11 (𝑗 = 𝑘 → ((𝐹𝑗)‘𝑧) = ((𝐹𝑘)‘𝑧))
56 eqid 2736 . . . . . . . . . . 11 (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧)) = (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧))
57 fvex 6855 . . . . . . . . . . 11 ((𝐹𝑘)‘𝑧) ∈ V
5855, 56, 57fvmpt 6948 . . . . . . . . . 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 767 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑗𝑍) → 𝑧𝑆)
6134, 60ffvelcdmd 7036 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑗𝑍) → ((𝐹𝑗)‘𝑧) ∈ ℂ)
6261fmpttd 7063 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧)):𝑍⟶ℂ)
6362ffvelcdmda 7035 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘𝑍) → ((𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧))‘𝑘) ∈ ℂ)
6445, 63sylan2 593 . . . . . . . . . 10 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → ((𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧))‘𝑘) ∈ ℂ)
6559, 64eqeltrrd 2839 . . . . . . . . 9 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → ((𝐹𝑘)‘𝑧) ∈ ℂ)
6659, 43, 65fsumser 15615 . . . . . . . 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 2786 . . . . . . 7 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → ((seq𝑁( ∘f + , 𝐹)‘𝑛)‘𝑧) = Σ𝑘 ∈ (𝑁...𝑛)((𝐹𝑘)‘𝑧))
6867fveq2d 6846 . . . . . 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 13878 . . . . . . . . 9 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (𝑁...𝑛) ∈ Fin)
7069, 65fsumcl 15618 . . . . . . . 8 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)((𝐹𝑘)‘𝑧) ∈ ℂ)
7170abscld 15321 . . . . . . 7 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘Σ𝑘 ∈ (𝑁...𝑛)((𝐹𝑘)‘𝑧)) ∈ ℝ)
7265abscld 15321 . . . . . . . 8 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → (abs‘((𝐹𝑘)‘𝑧)) ∈ ℝ)
7369, 72fsumrecl 15619 . . . . . . 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 15686 . . . . . . 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 781 . . . . . . . . . 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 15619 . . . . . . . 8 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘) ∈ ℝ)
79 simplr 767 . . . . . . . . . 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 835 . . . . . . . . 9 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → (abs‘((𝐹𝑘)‘𝑧)) ≤ (𝑀𝑘))
8269, 72, 77, 81fsumle 15684 . . . . . . . 8 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(abs‘((𝐹𝑘)‘𝑧)) ≤ Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘))
8378recnd 11183 . . . . . . . . . 10 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘) ∈ ℂ)
8483abscld 15321 . . . . . . . . 9 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘)) ∈ ℝ)
8578leabsd 15299 . . . . . . . . 9 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘) ≤ (abs‘Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘)))
86 eqidd 2737 . . . . . . . . . . . 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 15615 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘) = (seq𝑁( + , 𝑀)‘𝑛))
8988fveq2d 6846 . . . . . . . . . 10 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘)) = (abs‘(seq𝑁( + , 𝑀)‘𝑛)))
90 simprr 771 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)
91 fveq2 6842 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (seq𝑁( + , 𝑀)‘𝑚) = (seq𝑁( + , 𝑀)‘𝑛))
9291fveq2d 6846 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (abs‘(seq𝑁( + , 𝑀)‘𝑚)) = (abs‘(seq𝑁( + , 𝑀)‘𝑛)))
9392breq1d 5115 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → ((abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦 ↔ (abs‘(seq𝑁( + , 𝑀)‘𝑛)) ≤ 𝑦))
9493rspccva 3580 . . . . . . . . . . . 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 5127 . . . . . . . . 9 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘)) ≤ 𝑦)
9878, 84, 74, 85, 97letrd 11312 . . . . . . . 8 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘) ≤ 𝑦)
9973, 78, 74, 82, 98letrd 11312 . . . . . . 7 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(abs‘((𝐹𝑘)‘𝑧)) ≤ 𝑦)
10071, 73, 74, 75, 99letrd 11312 . . . . . 6 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘Σ𝑘 ∈ (𝑁...𝑛)((𝐹𝑘)‘𝑧)) ≤ 𝑦)
10168, 100eqbrtrd 5127 . . . . 5 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘((seq𝑁( ∘f + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑦)
102101ralrimiva 3143 . . . 4 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) → ∀𝑧𝑆 (abs‘((seq𝑁( ∘f + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑦)
103 brralrspcev 5165 . . . 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 25757 . 2 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → ∃𝑥 ∈ ℝ ∀𝑧𝑆 (abs‘(𝑇𝑧)) ≤ 𝑥)
10710, 106rexlimddv 3158 1 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧𝑆 (abs‘(𝑇𝑧)) ≤ 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3064  wrex 3073  wss 3910   class class class wbr 5105  cmpt 5188  dom cdm 5633   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  f cof 7615  m cmap 8765  cc 11049  cr 11050   + caddc 11054  cle 11190  cz 12499  cuz 12763  ...cfz 13424  seqcseq 13906  abscabs 15119  cli 15366  Σcsu 15570  𝑢culm 25735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-ico 13270  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571  df-ulm 25736
This theorem is referenced by:  lgamgulmlem6  26383
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