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Theorem mtestbdd 23963
 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 9947 . . . . . 6 ((𝜑𝑘𝑍) → (𝑀𝑘) ∈ ℂ)
63, 1, 5serf 12691 . . . . 5 (𝜑 → seq𝑁( + , 𝑀):𝑍⟶ℂ)
76ffvelrnda 6267 . . . 4 ((𝜑𝑚𝑍) → (seq𝑁( + , 𝑀)‘𝑚) ∈ ℂ)
87ralrimiva 2949 . . 3 (𝜑 → ∀𝑚𝑍 (seq𝑁( + , 𝑀)‘𝑚) ∈ ℂ)
93climbdd 14250 . . 3 ((𝑁 ∈ ℤ ∧ seq𝑁( + , 𝑀) ∈ dom ⇝ ∧ ∀𝑚𝑍 (seq𝑁( + , 𝑀)‘𝑚) ∈ ℂ) → ∃𝑦 ∈ ℝ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)
101, 2, 8, 9syl3anc 1318 . 2 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)
111adantr 480 . . 3 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → 𝑁 ∈ ℤ)
12 seqfn 12675 . . . . . . 7 (𝑁 ∈ ℤ → seq𝑁( ∘𝑓 + , 𝐹) Fn (ℤ𝑁))
131, 12syl 17 . . . . . 6 (𝜑 → seq𝑁( ∘𝑓 + , 𝐹) Fn (ℤ𝑁))
143fneq2i 5900 . . . . . 6 (seq𝑁( ∘𝑓 + , 𝐹) Fn 𝑍 ↔ seq𝑁( ∘𝑓 + , 𝐹) Fn (ℤ𝑁))
1513, 14sylibr 223 . . . . 5 (𝜑 → seq𝑁( ∘𝑓 + , 𝐹) Fn 𝑍)
16 mtest.t . . . . 5 (𝜑 → seq𝑁( ∘𝑓 + , 𝐹)(⇝𝑢𝑆)𝑇)
17 ulmf2 23942 . . . . 5 ((seq𝑁( ∘𝑓 + , 𝐹) Fn 𝑍 ∧ seq𝑁( ∘𝑓 + , 𝐹)(⇝𝑢𝑆)𝑇) → seq𝑁( ∘𝑓 + , 𝐹):𝑍⟶(ℂ ↑𝑚 𝑆))
1815, 16, 17syl2anc 691 . . . 4 (𝜑 → seq𝑁( ∘𝑓 + , 𝐹):𝑍⟶(ℂ ↑𝑚 𝑆))
1918adantr 480 . . 3 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → seq𝑁( ∘𝑓 + , 𝐹):𝑍⟶(ℂ ↑𝑚 𝑆))
20 simplrl 796 . . . 4 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) → 𝑦 ∈ ℝ)
21 fveq2 6103 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → ((𝐹𝑗)‘𝑥) = ((𝐹𝑗)‘𝑧))
2221mpteq2dv 4673 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑥)) = (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧)))
2322seqeq3d 12671 . . . . . . . . . . 11 (𝑥 = 𝑧 → seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑥))) = seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧))))
2423fveq1d 6105 . . . . . . . . . 10 (𝑥 = 𝑧 → (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑥)))‘𝑛) = (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧)))‘𝑛))
25 eqid 2610 . . . . . . . . . 10 (𝑥𝑆 ↦ (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑥)))‘𝑛)) = (𝑥𝑆 ↦ (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑥)))‘𝑛))
26 fvex 6113 . . . . . . . . . 10 (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧)))‘𝑛) ∈ V
2724, 25, 26fvmpt 6191 . . . . . . . . 9 (𝑧𝑆 → ((𝑥𝑆 ↦ (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑥)))‘𝑛))‘𝑧) = (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧)))‘𝑛))
2827adantl 481 . . . . . . . 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 762 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → 𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))
3130feqmptd 6159 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → 𝐹 = (𝑗𝑍 ↦ (𝐹𝑗)))
3230ffvelrnda 6267 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑗𝑍) → (𝐹𝑗) ∈ (ℂ ↑𝑚 𝑆))
33 elmapi 7765 . . . . . . . . . . . . . . . 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 6159 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑗𝑍) → (𝐹𝑗) = (𝑥𝑆 ↦ ((𝐹𝑗)‘𝑥)))
3635mpteq2dva 4672 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (𝑗𝑍 ↦ (𝐹𝑗)) = (𝑗𝑍 ↦ (𝑥𝑆 ↦ ((𝐹𝑗)‘𝑥))))
3731, 36eqtrd 2644 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → 𝐹 = (𝑗𝑍 ↦ (𝑥𝑆 ↦ ((𝐹𝑗)‘𝑥))))
3837seqeq3d 12671 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → seq𝑁( ∘𝑓 + , 𝐹) = seq𝑁( ∘𝑓 + , (𝑗𝑍 ↦ (𝑥𝑆 ↦ ((𝐹𝑗)‘𝑥)))))
3938fveq1d 6105 . . . . . . . . . 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 762 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → 𝑆𝑉)
42 simplr 788 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → 𝑛𝑍)
4342, 3syl6eleq 2698 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → 𝑛 ∈ (ℤ𝑁))
44 elfzuz 12209 . . . . . . . . . . . . . 14 (𝑘 ∈ (𝑁...𝑛) → 𝑘 ∈ (ℤ𝑁))
4544, 3syl6eleqr 2699 . . . . . . . . . . . . 13 (𝑘 ∈ (𝑁...𝑛) → 𝑘𝑍)
4645ssriv 3572 . . . . . . . . . . . 12 (𝑁...𝑛) ⊆ 𝑍
4746a1i 11 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (𝑁...𝑛) ⊆ 𝑍)
4834ffvelrnda 6267 . . . . . . . . . . . 12 ((((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑗𝑍) ∧ 𝑥𝑆) → ((𝐹𝑗)‘𝑥) ∈ ℂ)
4948anasss 677 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ (𝑗𝑍𝑥𝑆)) → ((𝐹𝑗)‘𝑥) ∈ ℂ)
5041, 43, 47, 49seqof2 12721 . . . . . . . . . 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 2644 . . . . . . . . 9 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (seq𝑁( ∘𝑓 + , 𝐹)‘𝑛) = (𝑥𝑆 ↦ (seq𝑁( + , (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑥)))‘𝑛)))
5251fveq1d 6105 . . . . . . . 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 481 . . . . . . . . . 10 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → 𝑘𝑍)
54 fveq2 6103 . . . . . . . . . . . 12 (𝑗 = 𝑘 → (𝐹𝑗) = (𝐹𝑘))
5554fveq1d 6105 . . . . . . . . . . 11 (𝑗 = 𝑘 → ((𝐹𝑗)‘𝑧) = ((𝐹𝑘)‘𝑧))
56 eqid 2610 . . . . . . . . . . 11 (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧)) = (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧))
57 fvex 6113 . . . . . . . . . . 11 ((𝐹𝑘)‘𝑧) ∈ V
5855, 56, 57fvmpt 6191 . . . . . . . . . 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 788 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑗𝑍) → 𝑧𝑆)
6134, 60ffvelrnd 6268 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑗𝑍) → ((𝐹𝑗)‘𝑧) ∈ ℂ)
6261, 56fmptd 6292 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧)):𝑍⟶ℂ)
6362ffvelrnda 6267 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘𝑍) → ((𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧))‘𝑘) ∈ ℂ)
6445, 63sylan2 490 . . . . . . . . . 10 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → ((𝑗𝑍 ↦ ((𝐹𝑗)‘𝑧))‘𝑘) ∈ ℂ)
6559, 64eqeltrrd 2689 . . . . . . . . 9 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → ((𝐹𝑘)‘𝑧) ∈ ℂ)
6659, 43, 65fsumser 14308 . . . . . . . 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 2654 . . . . . . 7 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → ((seq𝑁( ∘𝑓 + , 𝐹)‘𝑛)‘𝑧) = Σ𝑘 ∈ (𝑁...𝑛)((𝐹𝑘)‘𝑧))
6867fveq2d 6107 . . . . . 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 12634 . . . . . . . . 9 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (𝑁...𝑛) ∈ Fin)
7069, 65fsumcl 14311 . . . . . . . 8 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)((𝐹𝑘)‘𝑧) ∈ ℂ)
7170abscld 14023 . . . . . . 7 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘Σ𝑘 ∈ (𝑁...𝑛)((𝐹𝑘)‘𝑧)) ∈ ℝ)
7265abscld 14023 . . . . . . . 8 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → (abs‘((𝐹𝑘)‘𝑧)) ∈ ℝ)
7369, 72fsumrecl 14312 . . . . . . 7 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(abs‘((𝐹𝑘)‘𝑧)) ∈ ℝ)
7420adantr 480 . . . . . . 7 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → 𝑦 ∈ ℝ)
7569, 65fsumabs 14374 . . . . . . 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 802 . . . . . . . . . 10 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → 𝜑)
7776, 53, 4syl2anc 691 . . . . . . . . 9 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → (𝑀𝑘) ∈ ℝ)
7869, 77fsumrecl 14312 . . . . . . . 8 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘) ∈ ℝ)
79 simplr 788 . . . . . . . . . 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 1316 . . . . . . . . 9 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → (abs‘((𝐹𝑘)‘𝑧)) ≤ (𝑀𝑘))
8269, 72, 77, 81fsumle 14372 . . . . . . . 8 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(abs‘((𝐹𝑘)‘𝑧)) ≤ Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘))
8378recnd 9947 . . . . . . . . . 10 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘) ∈ ℂ)
8483abscld 14023 . . . . . . . . 9 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘)) ∈ ℝ)
8578leabsd 14001 . . . . . . . . 9 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘) ≤ (abs‘Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘)))
86 eqidd 2611 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → (𝑀𝑘) = (𝑀𝑘))
8776, 53, 5syl2anc 691 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) ∧ 𝑘 ∈ (𝑁...𝑛)) → (𝑀𝑘) ∈ ℂ)
8886, 43, 87fsumser 14308 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘) = (seq𝑁( + , 𝑀)‘𝑛))
8988fveq2d 6107 . . . . . . . . . 10 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘)) = (abs‘(seq𝑁( + , 𝑀)‘𝑛)))
90 simprr 792 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)
91 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (seq𝑁( + , 𝑀)‘𝑚) = (seq𝑁( + , 𝑀)‘𝑛))
9291fveq2d 6107 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (abs‘(seq𝑁( + , 𝑀)‘𝑚)) = (abs‘(seq𝑁( + , 𝑀)‘𝑛)))
9392breq1d 4593 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → ((abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦 ↔ (abs‘(seq𝑁( + , 𝑀)‘𝑛)) ≤ 𝑦))
9493rspccva 3281 . . . . . . . . . . . 12 ((∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦𝑛𝑍) → (abs‘(seq𝑁( + , 𝑀)‘𝑛)) ≤ 𝑦)
9590, 94sylan 487 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) → (abs‘(seq𝑁( + , 𝑀)‘𝑛)) ≤ 𝑦)
9695adantr 480 . . . . . . . . . 10 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘(seq𝑁( + , 𝑀)‘𝑛)) ≤ 𝑦)
9789, 96eqbrtrd 4605 . . . . . . . . 9 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘)) ≤ 𝑦)
9878, 84, 74, 85, 97letrd 10073 . . . . . . . 8 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(𝑀𝑘) ≤ 𝑦)
9973, 78, 74, 82, 98letrd 10073 . . . . . . 7 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → Σ𝑘 ∈ (𝑁...𝑛)(abs‘((𝐹𝑘)‘𝑧)) ≤ 𝑦)
10071, 73, 74, 75, 99letrd 10073 . . . . . 6 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘Σ𝑘 ∈ (𝑁...𝑛)((𝐹𝑘)‘𝑧)) ≤ 𝑦)
10168, 100eqbrtrd 4605 . . . . 5 ((((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) ∧ 𝑧𝑆) → (abs‘((seq𝑁( ∘𝑓 + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑦)
102101ralrimiva 2949 . . . 4 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) → ∀𝑧𝑆 (abs‘((seq𝑁( ∘𝑓 + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑦)
103 breq2 4587 . . . . . 6 (𝑥 = 𝑦 → ((abs‘((seq𝑁( ∘𝑓 + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑥 ↔ (abs‘((seq𝑁( ∘𝑓 + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑦))
104103ralbidv 2969 . . . . 5 (𝑥 = 𝑦 → (∀𝑧𝑆 (abs‘((seq𝑁( ∘𝑓 + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑥 ↔ ∀𝑧𝑆 (abs‘((seq𝑁( ∘𝑓 + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑦))
105104rspcev 3282 . . . 4 ((𝑦 ∈ ℝ ∧ ∀𝑧𝑆 (abs‘((seq𝑁( ∘𝑓 + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑧𝑆 (abs‘((seq𝑁( ∘𝑓 + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑥)
10620, 102, 105syl2anc 691 . . 3 (((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) ∧ 𝑛𝑍) → ∃𝑥 ∈ ℝ ∀𝑧𝑆 (abs‘((seq𝑁( ∘𝑓 + , 𝐹)‘𝑛)‘𝑧)) ≤ 𝑥)
10716adantr 480 . . 3 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → seq𝑁( ∘𝑓 + , 𝐹)(⇝𝑢𝑆)𝑇)
1083, 11, 19, 106, 107ulmbdd 23956 . 2 ((𝜑 ∧ (𝑦 ∈ ℝ ∧ ∀𝑚𝑍 (abs‘(seq𝑁( + , 𝑀)‘𝑚)) ≤ 𝑦)) → ∃𝑥 ∈ ℝ ∀𝑧𝑆 (abs‘(𝑇𝑧)) ≤ 𝑥)
10910, 108rexlimddv 3017 1 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧𝑆 (abs‘(𝑇𝑧)) ≤ 𝑥)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540   class class class wbr 4583   ↦ cmpt 4643  dom cdm 5038   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793   ↑𝑚 cmap 7744  ℂcc 9813  ℝcr 9814   + caddc 9818   ≤ cle 9954  ℤcz 11254  ℤ≥cuz 11563  ...cfz 12197  seqcseq 12663  abscabs 13822   ⇝ cli 14063  Σcsu 14264  ⇝𝑢culm 23934 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-ico 12052  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-ulm 23935 This theorem is referenced by:  lgamgulmlem6  24560
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