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Theorem fsumabs 15837
Description: Generalized triangle inequality: the absolute value of a finite sum is less than or equal to the sum of absolute values. (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
Hypotheses
Ref Expression
fsumabs.1 (𝜑𝐴 ∈ Fin)
fsumabs.2 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
Assertion
Ref Expression
fsumabs (𝜑 → (abs‘Σ𝑘𝐴 𝐵) ≤ Σ𝑘𝐴 (abs‘𝐵))
Distinct variable groups:   𝐴,𝑘   𝜑,𝑘
Allowed substitution hint:   𝐵(𝑘)

Proof of Theorem fsumabs
Dummy variables 𝑤 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 4006 . 2 𝐴𝐴
2 fsumabs.1 . . 3 (𝜑𝐴 ∈ Fin)
3 sseq1 4009 . . . . . 6 (𝑤 = ∅ → (𝑤𝐴 ↔ ∅ ⊆ 𝐴))
4 sumeq1 15725 . . . . . . . 8 (𝑤 = ∅ → Σ𝑘𝑤 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
54fveq2d 6910 . . . . . . 7 (𝑤 = ∅ → (abs‘Σ𝑘𝑤 𝐵) = (abs‘Σ𝑘 ∈ ∅ 𝐵))
6 sumeq1 15725 . . . . . . 7 (𝑤 = ∅ → Σ𝑘𝑤 (abs‘𝐵) = Σ𝑘 ∈ ∅ (abs‘𝐵))
75, 6breq12d 5156 . . . . . 6 (𝑤 = ∅ → ((abs‘Σ𝑘𝑤 𝐵) ≤ Σ𝑘𝑤 (abs‘𝐵) ↔ (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵)))
83, 7imbi12d 344 . . . . 5 (𝑤 = ∅ → ((𝑤𝐴 → (abs‘Σ𝑘𝑤 𝐵) ≤ Σ𝑘𝑤 (abs‘𝐵)) ↔ (∅ ⊆ 𝐴 → (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵))))
98imbi2d 340 . . . 4 (𝑤 = ∅ → ((𝜑 → (𝑤𝐴 → (abs‘Σ𝑘𝑤 𝐵) ≤ Σ𝑘𝑤 (abs‘𝐵))) ↔ (𝜑 → (∅ ⊆ 𝐴 → (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵)))))
10 sseq1 4009 . . . . . 6 (𝑤 = 𝑥 → (𝑤𝐴𝑥𝐴))
11 sumeq1 15725 . . . . . . . 8 (𝑤 = 𝑥 → Σ𝑘𝑤 𝐵 = Σ𝑘𝑥 𝐵)
1211fveq2d 6910 . . . . . . 7 (𝑤 = 𝑥 → (abs‘Σ𝑘𝑤 𝐵) = (abs‘Σ𝑘𝑥 𝐵))
13 sumeq1 15725 . . . . . . 7 (𝑤 = 𝑥 → Σ𝑘𝑤 (abs‘𝐵) = Σ𝑘𝑥 (abs‘𝐵))
1412, 13breq12d 5156 . . . . . 6 (𝑤 = 𝑥 → ((abs‘Σ𝑘𝑤 𝐵) ≤ Σ𝑘𝑤 (abs‘𝐵) ↔ (abs‘Σ𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (abs‘𝐵)))
1510, 14imbi12d 344 . . . . 5 (𝑤 = 𝑥 → ((𝑤𝐴 → (abs‘Σ𝑘𝑤 𝐵) ≤ Σ𝑘𝑤 (abs‘𝐵)) ↔ (𝑥𝐴 → (abs‘Σ𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (abs‘𝐵))))
1615imbi2d 340 . . . 4 (𝑤 = 𝑥 → ((𝜑 → (𝑤𝐴 → (abs‘Σ𝑘𝑤 𝐵) ≤ Σ𝑘𝑤 (abs‘𝐵))) ↔ (𝜑 → (𝑥𝐴 → (abs‘Σ𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (abs‘𝐵)))))
17 sseq1 4009 . . . . . 6 (𝑤 = (𝑥 ∪ {𝑦}) → (𝑤𝐴 ↔ (𝑥 ∪ {𝑦}) ⊆ 𝐴))
18 sumeq1 15725 . . . . . . . 8 (𝑤 = (𝑥 ∪ {𝑦}) → Σ𝑘𝑤 𝐵 = Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵)
1918fveq2d 6910 . . . . . . 7 (𝑤 = (𝑥 ∪ {𝑦}) → (abs‘Σ𝑘𝑤 𝐵) = (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵))
20 sumeq1 15725 . . . . . . 7 (𝑤 = (𝑥 ∪ {𝑦}) → Σ𝑘𝑤 (abs‘𝐵) = Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵))
2119, 20breq12d 5156 . . . . . 6 (𝑤 = (𝑥 ∪ {𝑦}) → ((abs‘Σ𝑘𝑤 𝐵) ≤ Σ𝑘𝑤 (abs‘𝐵) ↔ (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵)))
2217, 21imbi12d 344 . . . . 5 (𝑤 = (𝑥 ∪ {𝑦}) → ((𝑤𝐴 → (abs‘Σ𝑘𝑤 𝐵) ≤ Σ𝑘𝑤 (abs‘𝐵)) ↔ ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵))))
2322imbi2d 340 . . . 4 (𝑤 = (𝑥 ∪ {𝑦}) → ((𝜑 → (𝑤𝐴 → (abs‘Σ𝑘𝑤 𝐵) ≤ Σ𝑘𝑤 (abs‘𝐵))) ↔ (𝜑 → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵)))))
24 sseq1 4009 . . . . . 6 (𝑤 = 𝐴 → (𝑤𝐴𝐴𝐴))
25 sumeq1 15725 . . . . . . . 8 (𝑤 = 𝐴 → Σ𝑘𝑤 𝐵 = Σ𝑘𝐴 𝐵)
2625fveq2d 6910 . . . . . . 7 (𝑤 = 𝐴 → (abs‘Σ𝑘𝑤 𝐵) = (abs‘Σ𝑘𝐴 𝐵))
27 sumeq1 15725 . . . . . . 7 (𝑤 = 𝐴 → Σ𝑘𝑤 (abs‘𝐵) = Σ𝑘𝐴 (abs‘𝐵))
2826, 27breq12d 5156 . . . . . 6 (𝑤 = 𝐴 → ((abs‘Σ𝑘𝑤 𝐵) ≤ Σ𝑘𝑤 (abs‘𝐵) ↔ (abs‘Σ𝑘𝐴 𝐵) ≤ Σ𝑘𝐴 (abs‘𝐵)))
2924, 28imbi12d 344 . . . . 5 (𝑤 = 𝐴 → ((𝑤𝐴 → (abs‘Σ𝑘𝑤 𝐵) ≤ Σ𝑘𝑤 (abs‘𝐵)) ↔ (𝐴𝐴 → (abs‘Σ𝑘𝐴 𝐵) ≤ Σ𝑘𝐴 (abs‘𝐵))))
3029imbi2d 340 . . . 4 (𝑤 = 𝐴 → ((𝜑 → (𝑤𝐴 → (abs‘Σ𝑘𝑤 𝐵) ≤ Σ𝑘𝑤 (abs‘𝐵))) ↔ (𝜑 → (𝐴𝐴 → (abs‘Σ𝑘𝐴 𝐵) ≤ Σ𝑘𝐴 (abs‘𝐵)))))
31 0le0 12367 . . . . . 6 0 ≤ 0
32 sum0 15757 . . . . . . . 8 Σ𝑘 ∈ ∅ 𝐵 = 0
3332fveq2i 6909 . . . . . . 7 (abs‘Σ𝑘 ∈ ∅ 𝐵) = (abs‘0)
34 abs0 15324 . . . . . . 7 (abs‘0) = 0
3533, 34eqtri 2765 . . . . . 6 (abs‘Σ𝑘 ∈ ∅ 𝐵) = 0
36 sum0 15757 . . . . . 6 Σ𝑘 ∈ ∅ (abs‘𝐵) = 0
3731, 35, 363brtr4i 5173 . . . . 5 (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵)
38372a1i 12 . . . 4 (𝜑 → (∅ ⊆ 𝐴 → (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵)))
39 ssun1 4178 . . . . . . . . . 10 𝑥 ⊆ (𝑥 ∪ {𝑦})
40 sstr 3992 . . . . . . . . . 10 ((𝑥 ⊆ (𝑥 ∪ {𝑦}) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑥𝐴)
4139, 40mpan 690 . . . . . . . . 9 ((𝑥 ∪ {𝑦}) ⊆ 𝐴𝑥𝐴)
4241imim1i 63 . . . . . . . 8 ((𝑥𝐴 → (abs‘Σ𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (abs‘𝐵)) → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → (abs‘Σ𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (abs‘𝐵)))
43 simpll 767 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝜑)
4443, 2syl 17 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝐴 ∈ Fin)
45 simpr 484 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∪ {𝑦}) ⊆ 𝐴)
4645unssad 4193 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑥𝐴)
4744, 46ssfid 9301 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑥 ∈ Fin)
4846sselda 3983 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘𝑥) → 𝑘𝐴)
49 fsumabs.2 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
5043, 48, 49syl2an2r 685 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘𝑥) → 𝐵 ∈ ℂ)
5147, 50fsumcl 15769 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘𝑥 𝐵 ∈ ℂ)
5251abscld 15475 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘Σ𝑘𝑥 𝐵) ∈ ℝ)
5350abscld 15475 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘𝑥) → (abs‘𝐵) ∈ ℝ)
5447, 53fsumrecl 15770 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘𝑥 (abs‘𝐵) ∈ ℝ)
5545unssbd 4194 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → {𝑦} ⊆ 𝐴)
56 vex 3484 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
5756snss 4785 . . . . . . . . . . . . . . . 16 (𝑦𝐴 ↔ {𝑦} ⊆ 𝐴)
5855, 57sylibr 234 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑦𝐴)
5949ralrimiva 3146 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
6043, 59syl 17 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ∀𝑘𝐴 𝐵 ∈ ℂ)
61 nfcsb1v 3923 . . . . . . . . . . . . . . . . 17 𝑘𝑦 / 𝑘𝐵
6261nfel1 2922 . . . . . . . . . . . . . . . 16 𝑘𝑦 / 𝑘𝐵 ∈ ℂ
63 csbeq1a 3913 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑦𝐵 = 𝑦 / 𝑘𝐵)
6463eleq1d 2826 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑦 → (𝐵 ∈ ℂ ↔ 𝑦 / 𝑘𝐵 ∈ ℂ))
6562, 64rspc 3610 . . . . . . . . . . . . . . 15 (𝑦𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → 𝑦 / 𝑘𝐵 ∈ ℂ))
6658, 60, 65sylc 65 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑦 / 𝑘𝐵 ∈ ℂ)
6766abscld 15475 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘𝑦 / 𝑘𝐵) ∈ ℝ)
6852, 54, 67leadd1d 11857 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ((abs‘Σ𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (abs‘𝐵) ↔ ((abs‘Σ𝑘𝑥 𝐵) + (abs‘𝑦 / 𝑘𝐵)) ≤ (Σ𝑘𝑥 (abs‘𝐵) + (abs‘𝑦 / 𝑘𝐵))))
69 simplr 769 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ¬ 𝑦𝑥)
70 disjsn 4711 . . . . . . . . . . . . . . . 16 ((𝑥 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦𝑥)
7169, 70sylibr 234 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∩ {𝑦}) = ∅)
72 eqidd 2738 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∪ {𝑦}) = (𝑥 ∪ {𝑦}))
7344, 45ssfid 9301 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∪ {𝑦}) ∈ Fin)
7445sselda 3983 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ (𝑥 ∪ {𝑦})) → 𝑘𝐴)
7543, 74, 49syl2an2r 685 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ (𝑥 ∪ {𝑦})) → 𝐵 ∈ ℂ)
7675abscld 15475 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ (𝑥 ∪ {𝑦})) → (abs‘𝐵) ∈ ℝ)
7776recnd 11289 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ (𝑥 ∪ {𝑦})) → (abs‘𝐵) ∈ ℂ)
7871, 72, 73, 77fsumsplit 15777 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵) = (Σ𝑘𝑥 (abs‘𝐵) + Σ𝑘 ∈ {𝑦} (abs‘𝐵)))
79 csbfv2g 6955 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ V → 𝑦 / 𝑘(abs‘𝐵) = (abs‘𝑦 / 𝑘𝐵))
8079elv 3485 . . . . . . . . . . . . . . . . . 18 𝑦 / 𝑘(abs‘𝐵) = (abs‘𝑦 / 𝑘𝐵)
8167recnd 11289 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘𝑦 / 𝑘𝐵) ∈ ℂ)
8280, 81eqeltrid 2845 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑦 / 𝑘(abs‘𝐵) ∈ ℂ)
83 sumsns 15786 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ V ∧ 𝑦 / 𝑘(abs‘𝐵) ∈ ℂ) → Σ𝑘 ∈ {𝑦} (abs‘𝐵) = 𝑦 / 𝑘(abs‘𝐵))
8456, 82, 83sylancr 587 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ {𝑦} (abs‘𝐵) = 𝑦 / 𝑘(abs‘𝐵))
8584, 80eqtrdi 2793 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ {𝑦} (abs‘𝐵) = (abs‘𝑦 / 𝑘𝐵))
8685oveq2d 7447 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (Σ𝑘𝑥 (abs‘𝐵) + Σ𝑘 ∈ {𝑦} (abs‘𝐵)) = (Σ𝑘𝑥 (abs‘𝐵) + (abs‘𝑦 / 𝑘𝐵)))
8778, 86eqtrd 2777 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵) = (Σ𝑘𝑥 (abs‘𝐵) + (abs‘𝑦 / 𝑘𝐵)))
8887breq2d 5155 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (((abs‘Σ𝑘𝑥 𝐵) + (abs‘𝑦 / 𝑘𝐵)) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵) ↔ ((abs‘Σ𝑘𝑥 𝐵) + (abs‘𝑦 / 𝑘𝐵)) ≤ (Σ𝑘𝑥 (abs‘𝐵) + (abs‘𝑦 / 𝑘𝐵))))
8968, 88bitr4d 282 . . . . . . . . . . 11 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ((abs‘Σ𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (abs‘𝐵) ↔ ((abs‘Σ𝑘𝑥 𝐵) + (abs‘𝑦 / 𝑘𝐵)) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵)))
9071, 72, 73, 75fsumsplit 15777 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵 = (Σ𝑘𝑥 𝐵 + Σ𝑘 ∈ {𝑦}𝐵))
91 sumsns 15786 . . . . . . . . . . . . . . . . 17 ((𝑦𝐴𝑦 / 𝑘𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑦}𝐵 = 𝑦 / 𝑘𝐵)
9258, 66, 91syl2anc 584 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ {𝑦}𝐵 = 𝑦 / 𝑘𝐵)
9392oveq2d 7447 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (Σ𝑘𝑥 𝐵 + Σ𝑘 ∈ {𝑦}𝐵) = (Σ𝑘𝑥 𝐵 + 𝑦 / 𝑘𝐵))
9490, 93eqtrd 2777 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵 = (Σ𝑘𝑥 𝐵 + 𝑦 / 𝑘𝐵))
9594fveq2d 6910 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) = (abs‘(Σ𝑘𝑥 𝐵 + 𝑦 / 𝑘𝐵)))
9651, 66abstrid 15495 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘(Σ𝑘𝑥 𝐵 + 𝑦 / 𝑘𝐵)) ≤ ((abs‘Σ𝑘𝑥 𝐵) + (abs‘𝑦 / 𝑘𝐵)))
9795, 96eqbrtrd 5165 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ ((abs‘Σ𝑘𝑥 𝐵) + (abs‘𝑦 / 𝑘𝐵)))
9873, 75fsumcl 15769 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵 ∈ ℂ)
9998abscld 15475 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ∈ ℝ)
10052, 67readdcld 11290 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ((abs‘Σ𝑘𝑥 𝐵) + (abs‘𝑦 / 𝑘𝐵)) ∈ ℝ)
10173, 76fsumrecl 15770 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵) ∈ ℝ)
102 letr 11355 . . . . . . . . . . . . 13 (((abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ∈ ℝ ∧ ((abs‘Σ𝑘𝑥 𝐵) + (abs‘𝑦 / 𝑘𝐵)) ∈ ℝ ∧ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵) ∈ ℝ) → (((abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ ((abs‘Σ𝑘𝑥 𝐵) + (abs‘𝑦 / 𝑘𝐵)) ∧ ((abs‘Σ𝑘𝑥 𝐵) + (abs‘𝑦 / 𝑘𝐵)) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵)) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵)))
10399, 100, 101, 102syl3anc 1373 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (((abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ ((abs‘Σ𝑘𝑥 𝐵) + (abs‘𝑦 / 𝑘𝐵)) ∧ ((abs‘Σ𝑘𝑥 𝐵) + (abs‘𝑦 / 𝑘𝐵)) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵)) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵)))
10497, 103mpand 695 . . . . . . . . . . 11 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (((abs‘Σ𝑘𝑥 𝐵) + (abs‘𝑦 / 𝑘𝐵)) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵)))
10589, 104sylbid 240 . . . . . . . . . 10 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ((abs‘Σ𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (abs‘𝐵) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵)))
106105ex 412 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝑦𝑥) → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → ((abs‘Σ𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (abs‘𝐵) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵))))
107106a2d 29 . . . . . . . 8 ((𝜑 ∧ ¬ 𝑦𝑥) → (((𝑥 ∪ {𝑦}) ⊆ 𝐴 → (abs‘Σ𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (abs‘𝐵)) → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵))))
10842, 107syl5 34 . . . . . . 7 ((𝜑 ∧ ¬ 𝑦𝑥) → ((𝑥𝐴 → (abs‘Σ𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (abs‘𝐵)) → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵))))
109108expcom 413 . . . . . 6 𝑦𝑥 → (𝜑 → ((𝑥𝐴 → (abs‘Σ𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (abs‘𝐵)) → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵)))))
110109a2d 29 . . . . 5 𝑦𝑥 → ((𝜑 → (𝑥𝐴 → (abs‘Σ𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (abs‘𝐵))) → (𝜑 → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵)))))
111110adantl 481 . . . 4 ((𝑥 ∈ Fin ∧ ¬ 𝑦𝑥) → ((𝜑 → (𝑥𝐴 → (abs‘Σ𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (abs‘𝐵))) → (𝜑 → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵)))))
1129, 16, 23, 30, 38, 111findcard2s 9205 . . 3 (𝐴 ∈ Fin → (𝜑 → (𝐴𝐴 → (abs‘Σ𝑘𝐴 𝐵) ≤ Σ𝑘𝐴 (abs‘𝐵))))
1132, 112mpcom 38 . 2 (𝜑 → (𝐴𝐴 → (abs‘Σ𝑘𝐴 𝐵) ≤ Σ𝑘𝐴 (abs‘𝐵)))
1141, 113mpi 20 1 (𝜑 → (abs‘Σ𝑘𝐴 𝐵) ≤ Σ𝑘𝐴 (abs‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  csb 3899  cun 3949  cin 3950  wss 3951  c0 4333  {csn 4626   class class class wbr 5143  cfv 6561  (class class class)co 7431  Fincfn 8985  cc 11153  cr 11154  0cc0 11155   + caddc 11158  cle 11296  abscabs 15273  Σcsu 15722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-fz 13548  df-fzo 13695  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723
This theorem is referenced by:  o1fsum  15849  seqabs  15850  cvgcmpce  15854  mertenslem1  15920  dvfsumabs  26063  mtest  26447  mtestbdd  26448  abelthlem7  26482  fsumharmonic  27055  ftalem1  27116  ftalem5  27120  dchrisumlem2  27534  dchrmusum2  27538  dchrvmasumlem3  27543  dchrvmasumiflem1  27545  dchrisum0lem1  27560  dchrisum0lem2a  27561  mudivsum  27574  mulogsumlem  27575  2vmadivsumlem  27584  selberglem2  27590  selberg3lem1  27601  selberg4lem1  27604  pntrsumbnd  27610  pntrlog2bndlem1  27621  pntrlog2bndlem3  27623  knoppndvlem11  36523  fourierdlem73  46194  etransclem23  46272
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