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Theorem mtestbdd 25670
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 11104 . . . . . 6 ((𝜑𝑘𝑍) → (𝑀𝑘) ∈ ℂ)
63, 1, 5serf 13852 . . . . 5 (𝜑 → seq𝑁( + , 𝑀):𝑍⟶ℂ)
76ffvelcdmda 7017 . . . 4 ((𝜑𝑚𝑍) → (seq𝑁( + , 𝑀)‘𝑚) ∈ ℂ)
87ralrimiva 3139 . . 3 (𝜑 → ∀𝑚𝑍 (seq𝑁( + , 𝑀)‘𝑚) ∈ ℂ)
93climbdd 15482 . . 3 ((𝑁 ∈ ℤ ∧ seq𝑁( + , 𝑀) ∈ dom ⇝ ∧ ∀𝑚𝑍 (seq𝑁( + , 𝑀)‘𝑚) ∈ ℂ) → ∃𝑦 ∈ ℝ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)
101, 2, 8, 9syl3anc 1370 . 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 13834 . . . . . . 7 (𝑁 ∈ ℤ → seq𝑁( ∘f + , 𝐹) Fn (ℤ𝑁))
131, 12syl 17 . . . . . 6 (𝜑 → seq𝑁( ∘f + , 𝐹) Fn (ℤ𝑁))
143fneq2i 6583 . . . . . 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 25649 . . . . 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 774 . . . 4 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) → 𝑦 ∈ ℝ)
21 fveq2 6825 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → ((𝐹𝑗)‘𝑥) = ((𝐹𝑗)‘𝑧))
2221mpteq2dv 5194 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑥)) = (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧)))
2322seqeq3d 13830 . . . . . . . . . . 11 (𝑥 = 𝑧 → seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑥))) = seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧))))
2423fveq1d 6827 . . . . . . . . . 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 6838 . . . . . . . . . 10 (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧)))‘𝑛) ∈ V
2724, 25, 26fvmpt 6931 . . . . . . . . 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 727 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → 𝐹:𝑍⟶(ℂ ↑m 𝑆))
3130feqmptd 6893 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → 𝐹 = (𝑗𝑍 ↦ (𝐹𝑗)))
3230ffvelcdmda 7017 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑗𝑍) → (𝐹𝑗) ∈ (ℂ ↑m 𝑆))
33 elmapi 8708 . . . . . . . . . . . . . . . 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 6893 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑗𝑍) → (𝐹𝑗) = (𝑥𝑆 ↦ ((𝐹𝑗)‘𝑥)))
3635mpteq2dva 5192 . . . . . . . . . . . . 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 13830 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → seq𝑁( ∘f + , 𝐹) = seq𝑁( ∘f + , (𝑗𝑍 ↦ (𝑥𝑆 ↦ ((𝐹𝑗)‘𝑥)))))
3938fveq1d 6827 . . . . . . . . . 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 727 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → 𝑆𝑉)
42 simplr 766 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → 𝑛𝑍)
4342, 3eleqtrdi 2847 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → 𝑛 ∈ (ℤ𝑁))
44 elfzuz 13353 . . . . . . . . . . . . . 14 (𝑘 ∈ (𝑁...𝑛) → 𝑘 ∈ (ℤ𝑁))
4544, 3eleqtrrdi 2848 . . . . . . . . . . . . 13 (𝑘 ∈ (𝑁...𝑛) → 𝑘𝑍)
4645ssriv 3936 . . . . . . . . . . . 12 (𝑁...𝑛) ⊆ 𝑍
4746a1i 11 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (𝑁...𝑛) ⊆ 𝑍)
4834ffvelcdmda 7017 . . . . . . . . . . . 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 13882 . . . . . . . . . 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 6827 . . . . . . . 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 6825 . . . . . . . . . . . 12 (𝑗 = 𝑘 → (𝐹𝑗) = (𝐹𝑘))
5554fveq1d 6827 . . . . . . . . . . 11 (𝑗 = 𝑘 → ((𝐹𝑗)‘𝑧) = ((𝐹𝑘)‘𝑧))
56 eqid 2736 . . . . . . . . . . 11 (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧)) = (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧))
57 fvex 6838 . . . . . . . . . . 11 ((𝐹𝑘)‘𝑧) ∈ V
5855, 56, 57fvmpt 6931 . . . . . . . . . 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 766 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑗𝑍) → 𝑧𝑆)
6134, 60ffvelcdmd 7018 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑗𝑍) → ((𝐹𝑗)‘𝑧) ∈ ℂ)
6261fmpttd 7045 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧)):𝑍⟶ℂ)
6362ffvelcdmda 7017 . . . . . . . . . . 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 2838 . . . . . . . . 9 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → ((𝐹𝑘)‘𝑧) ∈ ℂ)
6659, 43, 65fsumser 15541 . . . . . . . 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 6829 . . . . . 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 13794 . . . . . . . . 9 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (𝑁...𝑛) ∈ Fin)
7069, 65fsumcl 15544 . . . . . . . 8 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)((𝐹𝑘)‘𝑧) ∈ ℂ)
7170abscld 15247 . . . . . . 7 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘Σ𝑘 ∈ (𝑁...𝑛)((𝐹𝑘)‘𝑧)) ∈ ℝ)
7265abscld 15247 . . . . . . . 8 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → (abs‘((𝐹𝑘)‘𝑧)) ∈ ℝ)
7369, 72fsumrecl 15545 . . . . . . 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 15612 . . . . . . 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 780 . . . . . . . . . 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 15545 . . . . . . . 8 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘) ∈ ℝ)
79 simplr 766 . . . . . . . . . 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 834 . . . . . . . . 9 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → (abs‘((𝐹𝑘)‘𝑧)) ≤ (𝑀𝑘))
8269, 72, 77, 81fsumle 15610 . . . . . . . 8 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(abs‘((𝐹𝑘)‘𝑧)) ≤ Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘))
8378recnd 11104 . . . . . . . . . 10 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘) ∈ ℂ)
8483abscld 15247 . . . . . . . . 9 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘)) ∈ ℝ)
8578leabsd 15225 . . . . . . . . 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 15541 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘) = (seq𝑁( + , 𝑀)‘𝑛))
8988fveq2d 6829 . . . . . . . . . 10 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘)) = (abs‘(seq𝑁( + , 𝑀)‘𝑛)))
90 simprr 770 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)
91 fveq2 6825 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (seq𝑁( + , 𝑀)‘𝑚) = (seq𝑁( + , 𝑀)‘𝑛))
9291fveq2d 6829 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (abs‘(seq𝑁( + , 𝑀)‘𝑚)) = (abs‘(seq𝑁( + , 𝑀)‘𝑛)))
9392breq1d 5102 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → ((abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦 ↔ (abs‘(seq𝑁( + , 𝑀)‘𝑛)) ≤ 𝑦))
9493rspccva 3569 . . . . . . . . . . . 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 5114 . . . . . . . . 9 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘)) ≤ 𝑦)
9878, 84, 74, 85, 97letrd 11233 . . . . . . . 8 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘) ≤ 𝑦)
9973, 78, 74, 82, 98letrd 11233 . . . . . . 7 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(abs‘((𝐹𝑘)‘𝑧)) ≤ 𝑦)
10071, 73, 74, 75, 99letrd 11233 . . . . . 6 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘Σ𝑘 ∈ (𝑁...𝑛)((𝐹𝑘)‘𝑧)) ≤ 𝑦)
10168, 100eqbrtrd 5114 . . . . 5 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘((seq𝑁( ∘f + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑦)
102101ralrimiva 3139 . . . 4 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) → ∀𝑧𝑆 (abs‘((seq𝑁( ∘f + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑦)
103 brralrspcev 5152 . . . 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 25663 . 2 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → ∃𝑥 ∈ ℝ ∀𝑧𝑆 (abs‘(𝑇𝑧)) ≤ 𝑥)
10710, 106rexlimddv 3154 1 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧𝑆 (abs‘(𝑇𝑧)) ≤ 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1540  wcel 2105  wral 3061  wrex 3070  wss 3898   class class class wbr 5092  cmpt 5175  dom cdm 5620   Fn wfn 6474  wf 6475  cfv 6479  (class class class)co 7337  f cof 7593  m cmap 8686  cc 10970  cr 10971   + caddc 10975  cle 11111  cz 12420  cuz 12683  ...cfz 13340  seqcseq 13822  abscabs 15044  cli 15292  Σcsu 15496  𝑢culm 25641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-inf2 9498  ax-cnex 11028  ax-resscn 11029  ax-1cn 11030  ax-icn 11031  ax-addcl 11032  ax-addrcl 11033  ax-mulcl 11034  ax-mulrcl 11035  ax-mulcom 11036  ax-addass 11037  ax-mulass 11038  ax-distr 11039  ax-i2m1 11040  ax-1ne0 11041  ax-1rid 11042  ax-rnegex 11043  ax-rrecex 11044  ax-cnre 11045  ax-pre-lttri 11046  ax-pre-lttrn 11047  ax-pre-ltadd 11048  ax-pre-mulgt0 11049  ax-pre-sup 11050
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-int 4895  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-se 5576  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-isom 6488  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-of 7595  df-om 7781  df-1st 7899  df-2nd 7900  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-rdg 8311  df-1o 8367  df-er 8569  df-map 8688  df-pm 8689  df-en 8805  df-dom 8806  df-sdom 8807  df-fin 8808  df-sup 9299  df-oi 9367  df-card 9796  df-pnf 11112  df-mnf 11113  df-xr 11114  df-ltxr 11115  df-le 11116  df-sub 11308  df-neg 11309  df-div 11734  df-nn 12075  df-2 12137  df-3 12138  df-n0 12335  df-z 12421  df-uz 12684  df-rp 12832  df-ico 13186  df-fz 13341  df-fzo 13484  df-seq 13823  df-exp 13884  df-hash 14146  df-cj 14909  df-re 14910  df-im 14911  df-sqrt 15045  df-abs 15046  df-clim 15296  df-sum 15497  df-ulm 25642
This theorem is referenced by:  lgamgulmlem6  26289
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