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Theorem fsumabs 11487
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 3187 . 2 𝐴𝐴
2 fsumabs.1 . . 3 (𝜑𝐴 ∈ Fin)
3 sseq1 3190 . . . . . 6 (𝑤 = ∅ → (𝑤𝐴 ↔ ∅ ⊆ 𝐴))
4 sumeq1 11377 . . . . . . . 8 (𝑤 = ∅ → Σ𝑘𝑤 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
54fveq2d 5531 . . . . . . 7 (𝑤 = ∅ → (abs‘Σ𝑘𝑤 𝐵) = (abs‘Σ𝑘 ∈ ∅ 𝐵))
6 sumeq1 11377 . . . . . . 7 (𝑤 = ∅ → Σ𝑘𝑤 (abs‘𝐵) = Σ𝑘 ∈ ∅ (abs‘𝐵))
75, 6breq12d 4028 . . . . . 6 (𝑤 = ∅ → ((abs‘Σ𝑘𝑤 𝐵) ≤ Σ𝑘𝑤 (abs‘𝐵) ↔ (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵)))
83, 7imbi12d 234 . . . . 5 (𝑤 = ∅ → ((𝑤𝐴 → (abs‘Σ𝑘𝑤 𝐵) ≤ Σ𝑘𝑤 (abs‘𝐵)) ↔ (∅ ⊆ 𝐴 → (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵))))
98imbi2d 230 . . . 4 (𝑤 = ∅ → ((𝜑 → (𝑤𝐴 → (abs‘Σ𝑘𝑤 𝐵) ≤ Σ𝑘𝑤 (abs‘𝐵))) ↔ (𝜑 → (∅ ⊆ 𝐴 → (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵)))))
10 sseq1 3190 . . . . . 6 (𝑤 = 𝑥 → (𝑤𝐴𝑥𝐴))
11 sumeq1 11377 . . . . . . . 8 (𝑤 = 𝑥 → Σ𝑘𝑤 𝐵 = Σ𝑘𝑥 𝐵)
1211fveq2d 5531 . . . . . . 7 (𝑤 = 𝑥 → (abs‘Σ𝑘𝑤 𝐵) = (abs‘Σ𝑘𝑥 𝐵))
13 sumeq1 11377 . . . . . . 7 (𝑤 = 𝑥 → Σ𝑘𝑤 (abs‘𝐵) = Σ𝑘𝑥 (abs‘𝐵))
1412, 13breq12d 4028 . . . . . 6 (𝑤 = 𝑥 → ((abs‘Σ𝑘𝑤 𝐵) ≤ Σ𝑘𝑤 (abs‘𝐵) ↔ (abs‘Σ𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (abs‘𝐵)))
1510, 14imbi12d 234 . . . . 5 (𝑤 = 𝑥 → ((𝑤𝐴 → (abs‘Σ𝑘𝑤 𝐵) ≤ Σ𝑘𝑤 (abs‘𝐵)) ↔ (𝑥𝐴 → (abs‘Σ𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (abs‘𝐵))))
1615imbi2d 230 . . . 4 (𝑤 = 𝑥 → ((𝜑 → (𝑤𝐴 → (abs‘Σ𝑘𝑤 𝐵) ≤ Σ𝑘𝑤 (abs‘𝐵))) ↔ (𝜑 → (𝑥𝐴 → (abs‘Σ𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (abs‘𝐵)))))
17 sseq1 3190 . . . . . 6 (𝑤 = (𝑥 ∪ {𝑦}) → (𝑤𝐴 ↔ (𝑥 ∪ {𝑦}) ⊆ 𝐴))
18 sumeq1 11377 . . . . . . . 8 (𝑤 = (𝑥 ∪ {𝑦}) → Σ𝑘𝑤 𝐵 = Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵)
1918fveq2d 5531 . . . . . . 7 (𝑤 = (𝑥 ∪ {𝑦}) → (abs‘Σ𝑘𝑤 𝐵) = (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵))
20 sumeq1 11377 . . . . . . 7 (𝑤 = (𝑥 ∪ {𝑦}) → Σ𝑘𝑤 (abs‘𝐵) = Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵))
2119, 20breq12d 4028 . . . . . 6 (𝑤 = (𝑥 ∪ {𝑦}) → ((abs‘Σ𝑘𝑤 𝐵) ≤ Σ𝑘𝑤 (abs‘𝐵) ↔ (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵)))
2217, 21imbi12d 234 . . . . 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 230 . . . 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 3190 . . . . . 6 (𝑤 = 𝐴 → (𝑤𝐴𝐴𝐴))
25 sumeq1 11377 . . . . . . . 8 (𝑤 = 𝐴 → Σ𝑘𝑤 𝐵 = Σ𝑘𝐴 𝐵)
2625fveq2d 5531 . . . . . . 7 (𝑤 = 𝐴 → (abs‘Σ𝑘𝑤 𝐵) = (abs‘Σ𝑘𝐴 𝐵))
27 sumeq1 11377 . . . . . . 7 (𝑤 = 𝐴 → Σ𝑘𝑤 (abs‘𝐵) = Σ𝑘𝐴 (abs‘𝐵))
2826, 27breq12d 4028 . . . . . 6 (𝑤 = 𝐴 → ((abs‘Σ𝑘𝑤 𝐵) ≤ Σ𝑘𝑤 (abs‘𝐵) ↔ (abs‘Σ𝑘𝐴 𝐵) ≤ Σ𝑘𝐴 (abs‘𝐵)))
2924, 28imbi12d 234 . . . . 5 (𝑤 = 𝐴 → ((𝑤𝐴 → (abs‘Σ𝑘𝑤 𝐵) ≤ Σ𝑘𝑤 (abs‘𝐵)) ↔ (𝐴𝐴 → (abs‘Σ𝑘𝐴 𝐵) ≤ Σ𝑘𝐴 (abs‘𝐵))))
3029imbi2d 230 . . . 4 (𝑤 = 𝐴 → ((𝜑 → (𝑤𝐴 → (abs‘Σ𝑘𝑤 𝐵) ≤ Σ𝑘𝑤 (abs‘𝐵))) ↔ (𝜑 → (𝐴𝐴 → (abs‘Σ𝑘𝐴 𝐵) ≤ Σ𝑘𝐴 (abs‘𝐵)))))
31 0le0 9022 . . . . . 6 0 ≤ 0
32 sum0 11410 . . . . . . . 8 Σ𝑘 ∈ ∅ 𝐵 = 0
3332fveq2i 5530 . . . . . . 7 (abs‘Σ𝑘 ∈ ∅ 𝐵) = (abs‘0)
34 abs0 11081 . . . . . . 7 (abs‘0) = 0
3533, 34eqtri 2208 . . . . . 6 (abs‘Σ𝑘 ∈ ∅ 𝐵) = 0
36 sum0 11410 . . . . . 6 Σ𝑘 ∈ ∅ (abs‘𝐵) = 0
3731, 35, 363brtr4i 4045 . . . . 5 (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵)
38372a1i 27 . . . 4 (𝜑 → (∅ ⊆ 𝐴 → (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵)))
39 ssun1 3310 . . . . . . . . 9 𝑥 ⊆ (𝑥 ∪ {𝑦})
40 sstr 3175 . . . . . . . . 9 ((𝑥 ⊆ (𝑥 ∪ {𝑦}) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑥𝐴)
4139, 40mpan 424 . . . . . . . 8 ((𝑥 ∪ {𝑦}) ⊆ 𝐴𝑥𝐴)
4241imim1i 60 . . . . . . 7 ((𝑥𝐴 → (abs‘Σ𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (abs‘𝐵)) → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → (abs‘Σ𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (abs‘𝐵)))
43 simplrl 535 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑥 ∈ Fin)
44 simpll 527 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝜑)
45 simpr 110 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∪ {𝑦}) ⊆ 𝐴)
4645unssad 3324 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑥𝐴)
4746sselda 3167 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘𝑥) → 𝑘𝐴)
48 fsumabs.2 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
4944, 47, 48syl2an2r 595 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘𝑥) → 𝐵 ∈ ℂ)
5043, 49fsumcl 11422 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘𝑥 𝐵 ∈ ℂ)
5150abscld 11204 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘Σ𝑘𝑥 𝐵) ∈ ℝ)
5249abscld 11204 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘𝑥) → (abs‘𝐵) ∈ ℝ)
5343, 52fsumrecl 11423 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘𝑥 (abs‘𝐵) ∈ ℝ)
54 simpr 110 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∪ {𝑦}) ⊆ 𝐴)
5554unssbd 3325 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → {𝑦} ⊆ 𝐴)
56 vex 2752 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
5756snss 3739 . . . . . . . . . . . . . . . 16 (𝑦𝐴 ↔ {𝑦} ⊆ 𝐴)
5855, 57sylibr 134 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑦𝐴)
5958adantlrl 482 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑦𝐴)
6048ralrimiva 2560 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
6160ad2antrr 488 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ∀𝑘𝐴 𝐵 ∈ ℂ)
62 nfcsb1v 3102 . . . . . . . . . . . . . . . 16 𝑘𝑦 / 𝑘𝐵
6362nfel1 2340 . . . . . . . . . . . . . . 15 𝑘𝑦 / 𝑘𝐵 ∈ ℂ
64 csbeq1a 3078 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑦𝐵 = 𝑦 / 𝑘𝐵)
6564eleq1d 2256 . . . . . . . . . . . . . . 15 (𝑘 = 𝑦 → (𝐵 ∈ ℂ ↔ 𝑦 / 𝑘𝐵 ∈ ℂ))
6663, 65rspc 2847 . . . . . . . . . . . . . 14 (𝑦𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → 𝑦 / 𝑘𝐵 ∈ ℂ))
6759, 61, 66sylc 62 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑦 / 𝑘𝐵 ∈ ℂ)
6867abscld 11204 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘𝑦 / 𝑘𝐵) ∈ ℝ)
6951, 53, 68leadd1d 8510 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ((abs‘Σ𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (abs‘𝐵) ↔ ((abs‘Σ𝑘𝑥 𝐵) + (abs‘𝑦 / 𝑘𝐵)) ≤ (Σ𝑘𝑥 (abs‘𝐵) + (abs‘𝑦 / 𝑘𝐵))))
70 simplr 528 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ¬ 𝑦𝑥)
7170adantlrl 482 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ¬ 𝑦𝑥)
72 disjsn 3666 . . . . . . . . . . . . . . 15 ((𝑥 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦𝑥)
7371, 72sylibr 134 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∩ {𝑦}) = ∅)
74 eqidd 2188 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∪ {𝑦}) = (𝑥 ∪ {𝑦}))
7556a1i 9 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑦 ∈ V)
76 unsnfi 6932 . . . . . . . . . . . . . . 15 ((𝑥 ∈ Fin ∧ 𝑦 ∈ V ∧ ¬ 𝑦𝑥) → (𝑥 ∪ {𝑦}) ∈ Fin)
7743, 75, 71, 76syl3anc 1248 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∪ {𝑦}) ∈ Fin)
7845sselda 3167 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ (𝑥 ∪ {𝑦})) → 𝑘𝐴)
7944, 78, 48syl2an2r 595 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ (𝑥 ∪ {𝑦})) → 𝐵 ∈ ℂ)
8079abscld 11204 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ (𝑥 ∪ {𝑦})) → (abs‘𝐵) ∈ ℝ)
8180recnd 8000 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ (𝑥 ∪ {𝑦})) → (abs‘𝐵) ∈ ℂ)
8273, 74, 77, 81fsumsplit 11429 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵) = (Σ𝑘𝑥 (abs‘𝐵) + Σ𝑘 ∈ {𝑦} (abs‘𝐵)))
83 csbfv2g 5565 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ V → 𝑦 / 𝑘(abs‘𝐵) = (abs‘𝑦 / 𝑘𝐵))
8483elv 2753 . . . . . . . . . . . . . . . . . 18 𝑦 / 𝑘(abs‘𝐵) = (abs‘𝑦 / 𝑘𝐵)
8560ad2antrr 488 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ∀𝑘𝐴 𝐵 ∈ ℂ)
8658, 85, 66sylc 62 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑦 / 𝑘𝐵 ∈ ℂ)
8786abscld 11204 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘𝑦 / 𝑘𝐵) ∈ ℝ)
8887recnd 8000 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘𝑦 / 𝑘𝐵) ∈ ℂ)
8984, 88eqeltrid 2274 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑦 / 𝑘(abs‘𝐵) ∈ ℂ)
90 sumsns 11437 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ V ∧ 𝑦 / 𝑘(abs‘𝐵) ∈ ℂ) → Σ𝑘 ∈ {𝑦} (abs‘𝐵) = 𝑦 / 𝑘(abs‘𝐵))
9156, 89, 90sylancr 414 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ {𝑦} (abs‘𝐵) = 𝑦 / 𝑘(abs‘𝐵))
9291, 84eqtrdi 2236 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ {𝑦} (abs‘𝐵) = (abs‘𝑦 / 𝑘𝐵))
9392oveq2d 5904 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (Σ𝑘𝑥 (abs‘𝐵) + Σ𝑘 ∈ {𝑦} (abs‘𝐵)) = (Σ𝑘𝑥 (abs‘𝐵) + (abs‘𝑦 / 𝑘𝐵)))
9493adantlrl 482 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (Σ𝑘𝑥 (abs‘𝐵) + Σ𝑘 ∈ {𝑦} (abs‘𝐵)) = (Σ𝑘𝑥 (abs‘𝐵) + (abs‘𝑦 / 𝑘𝐵)))
9582, 94eqtrd 2220 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵) = (Σ𝑘𝑥 (abs‘𝐵) + (abs‘𝑦 / 𝑘𝐵)))
9695breq2d 4027 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 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))))
9769, 96bitr4d 191 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ((abs‘Σ𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (abs‘𝐵) ↔ ((abs‘Σ𝑘𝑥 𝐵) + (abs‘𝑦 / 𝑘𝐵)) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵)))
9870, 72sylibr 134 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∩ {𝑦}) = ∅)
9998adantlrl 482 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∩ {𝑦}) = ∅)
10099, 74, 77, 79fsumsplit 11429 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵 = (Σ𝑘𝑥 𝐵 + Σ𝑘 ∈ {𝑦}𝐵))
101 sumsns 11437 . . . . . . . . . . . . . . . . 17 ((𝑦𝐴𝑦 / 𝑘𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑦}𝐵 = 𝑦 / 𝑘𝐵)
10258, 86, 101syl2anc 411 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ {𝑦}𝐵 = 𝑦 / 𝑘𝐵)
103102oveq2d 5904 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (Σ𝑘𝑥 𝐵 + Σ𝑘 ∈ {𝑦}𝐵) = (Σ𝑘𝑥 𝐵 + 𝑦 / 𝑘𝐵))
104103adantlrl 482 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (Σ𝑘𝑥 𝐵 + Σ𝑘 ∈ {𝑦}𝐵) = (Σ𝑘𝑥 𝐵 + 𝑦 / 𝑘𝐵))
105100, 104eqtrd 2220 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵 = (Σ𝑘𝑥 𝐵 + 𝑦 / 𝑘𝐵))
106105fveq2d 5531 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) = (abs‘(Σ𝑘𝑥 𝐵 + 𝑦 / 𝑘𝐵)))
10786adantlrl 482 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑦 / 𝑘𝐵 ∈ ℂ)
10850, 107abstrid 11219 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘(Σ𝑘𝑥 𝐵 + 𝑦 / 𝑘𝐵)) ≤ ((abs‘Σ𝑘𝑥 𝐵) + (abs‘𝑦 / 𝑘𝐵)))
109106, 108eqbrtrd 4037 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ ((abs‘Σ𝑘𝑥 𝐵) + (abs‘𝑦 / 𝑘𝐵)))
11077, 79fsumcl 11422 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵 ∈ ℂ)
111110abscld 11204 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ∈ ℝ)
11251, 68readdcld 8001 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ((abs‘Σ𝑘𝑥 𝐵) + (abs‘𝑦 / 𝑘𝐵)) ∈ ℝ)
11377, 80fsumrecl 11423 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵) ∈ ℝ)
114 letr 8054 . . . . . . . . . . . 12 (((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)))
115111, 112, 113, 114syl3anc 1248 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 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)))
116109, 115mpand 429 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 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)))
11797, 116sylbid 150 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ((abs‘Σ𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (abs‘𝐵) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵)))
118117ex 115 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → ((abs‘Σ𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (abs‘𝐵) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵))))
119118a2d 26 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) → (((𝑥 ∪ {𝑦}) ⊆ 𝐴 → (abs‘Σ𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (abs‘𝐵)) → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵))))
12042, 119syl5 32 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) → ((𝑥𝐴 → (abs‘Σ𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (abs‘𝐵)) → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵))))
121120expcom 116 . . . . 5 ((𝑥 ∈ Fin ∧ ¬ 𝑦𝑥) → (𝜑 → ((𝑥𝐴 → (abs‘Σ𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (abs‘𝐵)) → ((𝑥 ∪ {𝑦}) ⊆ 𝐴 → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵)))))
122121a2d 26 . . . 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)))))
1239, 16, 23, 30, 38, 122findcard2s 6904 . . 3 (𝐴 ∈ Fin → (𝜑 → (𝐴𝐴 → (abs‘Σ𝑘𝐴 𝐵) ≤ Σ𝑘𝐴 (abs‘𝐵))))
1242, 123mpcom 36 . 2 (𝜑 → (𝐴𝐴 → (abs‘Σ𝑘𝐴 𝐵) ≤ Σ𝑘𝐴 (abs‘𝐵)))
1251, 124mpi 15 1 (𝜑 → (abs‘Σ𝑘𝐴 𝐵) ≤ Σ𝑘𝐴 (abs‘𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104   = wceq 1363  wcel 2158  wral 2465  Vcvv 2749  csb 3069  cun 3139  cin 3140  wss 3141  c0 3434  {csn 3604   class class class wbr 4015  cfv 5228  (class class class)co 5888  Fincfn 6754  cc 7823  cr 7824  0cc0 7825   + caddc 7828  cle 8007  abscabs 11020  Σcsu 11375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-pre-mulext 7943  ax-arch 7944  ax-caucvg 7945
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-isom 5237  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-irdg 6385  df-frec 6406  df-1o 6431  df-oadd 6435  df-er 6549  df-en 6755  df-dom 6756  df-fin 6757  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553  df-div 8644  df-inn 8934  df-2 8992  df-3 8993  df-4 8994  df-n0 9191  df-z 9268  df-uz 9543  df-q 9634  df-rp 9668  df-fz 10023  df-fzo 10157  df-seqfrec 10460  df-exp 10534  df-ihash 10770  df-cj 10865  df-re 10866  df-im 10867  df-rsqrt 11021  df-abs 11022  df-clim 11301  df-sumdc 11376
This theorem is referenced by:  iserabs  11497  cvgratnnlemabsle  11549  mertenslemi1  11557
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