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Theorem fsumabs 11174
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 3085 . 2 𝐴𝐴
2 fsumabs.1 . . 3 (𝜑𝐴 ∈ Fin)
3 sseq1 3088 . . . . . 6 (𝑤 = ∅ → (𝑤𝐴 ↔ ∅ ⊆ 𝐴))
4 sumeq1 11064 . . . . . . . 8 (𝑤 = ∅ → Σ𝑘𝑤 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
54fveq2d 5391 . . . . . . 7 (𝑤 = ∅ → (abs‘Σ𝑘𝑤 𝐵) = (abs‘Σ𝑘 ∈ ∅ 𝐵))
6 sumeq1 11064 . . . . . . 7 (𝑤 = ∅ → Σ𝑘𝑤 (abs‘𝐵) = Σ𝑘 ∈ ∅ (abs‘𝐵))
75, 6breq12d 3910 . . . . . 6 (𝑤 = ∅ → ((abs‘Σ𝑘𝑤 𝐵) ≤ Σ𝑘𝑤 (abs‘𝐵) ↔ (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵)))
83, 7imbi12d 233 . . . . 5 (𝑤 = ∅ → ((𝑤𝐴 → (abs‘Σ𝑘𝑤 𝐵) ≤ Σ𝑘𝑤 (abs‘𝐵)) ↔ (∅ ⊆ 𝐴 → (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵))))
98imbi2d 229 . . . 4 (𝑤 = ∅ → ((𝜑 → (𝑤𝐴 → (abs‘Σ𝑘𝑤 𝐵) ≤ Σ𝑘𝑤 (abs‘𝐵))) ↔ (𝜑 → (∅ ⊆ 𝐴 → (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵)))))
10 sseq1 3088 . . . . . 6 (𝑤 = 𝑥 → (𝑤𝐴𝑥𝐴))
11 sumeq1 11064 . . . . . . . 8 (𝑤 = 𝑥 → Σ𝑘𝑤 𝐵 = Σ𝑘𝑥 𝐵)
1211fveq2d 5391 . . . . . . 7 (𝑤 = 𝑥 → (abs‘Σ𝑘𝑤 𝐵) = (abs‘Σ𝑘𝑥 𝐵))
13 sumeq1 11064 . . . . . . 7 (𝑤 = 𝑥 → Σ𝑘𝑤 (abs‘𝐵) = Σ𝑘𝑥 (abs‘𝐵))
1412, 13breq12d 3910 . . . . . 6 (𝑤 = 𝑥 → ((abs‘Σ𝑘𝑤 𝐵) ≤ Σ𝑘𝑤 (abs‘𝐵) ↔ (abs‘Σ𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (abs‘𝐵)))
1510, 14imbi12d 233 . . . . 5 (𝑤 = 𝑥 → ((𝑤𝐴 → (abs‘Σ𝑘𝑤 𝐵) ≤ Σ𝑘𝑤 (abs‘𝐵)) ↔ (𝑥𝐴 → (abs‘Σ𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (abs‘𝐵))))
1615imbi2d 229 . . . 4 (𝑤 = 𝑥 → ((𝜑 → (𝑤𝐴 → (abs‘Σ𝑘𝑤 𝐵) ≤ Σ𝑘𝑤 (abs‘𝐵))) ↔ (𝜑 → (𝑥𝐴 → (abs‘Σ𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (abs‘𝐵)))))
17 sseq1 3088 . . . . . 6 (𝑤 = (𝑥 ∪ {𝑦}) → (𝑤𝐴 ↔ (𝑥 ∪ {𝑦}) ⊆ 𝐴))
18 sumeq1 11064 . . . . . . . 8 (𝑤 = (𝑥 ∪ {𝑦}) → Σ𝑘𝑤 𝐵 = Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵)
1918fveq2d 5391 . . . . . . 7 (𝑤 = (𝑥 ∪ {𝑦}) → (abs‘Σ𝑘𝑤 𝐵) = (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵))
20 sumeq1 11064 . . . . . . 7 (𝑤 = (𝑥 ∪ {𝑦}) → Σ𝑘𝑤 (abs‘𝐵) = Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵))
2119, 20breq12d 3910 . . . . . 6 (𝑤 = (𝑥 ∪ {𝑦}) → ((abs‘Σ𝑘𝑤 𝐵) ≤ Σ𝑘𝑤 (abs‘𝐵) ↔ (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵)))
2217, 21imbi12d 233 . . . . 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 229 . . . 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 3088 . . . . . 6 (𝑤 = 𝐴 → (𝑤𝐴𝐴𝐴))
25 sumeq1 11064 . . . . . . . 8 (𝑤 = 𝐴 → Σ𝑘𝑤 𝐵 = Σ𝑘𝐴 𝐵)
2625fveq2d 5391 . . . . . . 7 (𝑤 = 𝐴 → (abs‘Σ𝑘𝑤 𝐵) = (abs‘Σ𝑘𝐴 𝐵))
27 sumeq1 11064 . . . . . . 7 (𝑤 = 𝐴 → Σ𝑘𝑤 (abs‘𝐵) = Σ𝑘𝐴 (abs‘𝐵))
2826, 27breq12d 3910 . . . . . 6 (𝑤 = 𝐴 → ((abs‘Σ𝑘𝑤 𝐵) ≤ Σ𝑘𝑤 (abs‘𝐵) ↔ (abs‘Σ𝑘𝐴 𝐵) ≤ Σ𝑘𝐴 (abs‘𝐵)))
2924, 28imbi12d 233 . . . . 5 (𝑤 = 𝐴 → ((𝑤𝐴 → (abs‘Σ𝑘𝑤 𝐵) ≤ Σ𝑘𝑤 (abs‘𝐵)) ↔ (𝐴𝐴 → (abs‘Σ𝑘𝐴 𝐵) ≤ Σ𝑘𝐴 (abs‘𝐵))))
3029imbi2d 229 . . . 4 (𝑤 = 𝐴 → ((𝜑 → (𝑤𝐴 → (abs‘Σ𝑘𝑤 𝐵) ≤ Σ𝑘𝑤 (abs‘𝐵))) ↔ (𝜑 → (𝐴𝐴 → (abs‘Σ𝑘𝐴 𝐵) ≤ Σ𝑘𝐴 (abs‘𝐵)))))
31 0le0 8766 . . . . . 6 0 ≤ 0
32 sum0 11097 . . . . . . . 8 Σ𝑘 ∈ ∅ 𝐵 = 0
3332fveq2i 5390 . . . . . . 7 (abs‘Σ𝑘 ∈ ∅ 𝐵) = (abs‘0)
34 abs0 10770 . . . . . . 7 (abs‘0) = 0
3533, 34eqtri 2136 . . . . . 6 (abs‘Σ𝑘 ∈ ∅ 𝐵) = 0
36 sum0 11097 . . . . . 6 Σ𝑘 ∈ ∅ (abs‘𝐵) = 0
3731, 35, 363brtr4i 3926 . . . . 5 (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵)
38372a1i 27 . . . 4 (𝜑 → (∅ ⊆ 𝐴 → (abs‘Σ𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (abs‘𝐵)))
39 ssun1 3207 . . . . . . . . 9 𝑥 ⊆ (𝑥 ∪ {𝑦})
40 sstr 3073 . . . . . . . . 9 ((𝑥 ⊆ (𝑥 ∪ {𝑦}) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑥𝐴)
4139, 40mpan 418 . . . . . . . 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 507 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑥 ∈ Fin)
44 simpll 501 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝜑)
45 simpr 109 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∪ {𝑦}) ⊆ 𝐴)
4645unssad 3221 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑥𝐴)
4746sselda 3065 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘𝑥) → 𝑘𝐴)
48 fsumabs.2 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
4944, 47, 48syl2an2r 567 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘𝑥) → 𝐵 ∈ ℂ)
5043, 49fsumcl 11109 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘𝑥 𝐵 ∈ ℂ)
5150abscld 10893 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘Σ𝑘𝑥 𝐵) ∈ ℝ)
5249abscld 10893 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘𝑥) → (abs‘𝐵) ∈ ℝ)
5343, 52fsumrecl 11110 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘𝑥 (abs‘𝐵) ∈ ℝ)
54 simpr 109 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∪ {𝑦}) ⊆ 𝐴)
5554unssbd 3222 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → {𝑦} ⊆ 𝐴)
56 vex 2661 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
5756snss 3617 . . . . . . . . . . . . . . . 16 (𝑦𝐴 ↔ {𝑦} ⊆ 𝐴)
5855, 57sylibr 133 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑦𝐴)
5958adantlrl 471 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑦𝐴)
6048ralrimiva 2480 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
6160ad2antrr 477 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ∀𝑘𝐴 𝐵 ∈ ℂ)
62 nfcsb1v 3003 . . . . . . . . . . . . . . . 16 𝑘𝑦 / 𝑘𝐵
6362nfel1 2267 . . . . . . . . . . . . . . 15 𝑘𝑦 / 𝑘𝐵 ∈ ℂ
64 csbeq1a 2981 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑦𝐵 = 𝑦 / 𝑘𝐵)
6564eleq1d 2184 . . . . . . . . . . . . . . 15 (𝑘 = 𝑦 → (𝐵 ∈ ℂ ↔ 𝑦 / 𝑘𝐵 ∈ ℂ))
6663, 65rspc 2755 . . . . . . . . . . . . . 14 (𝑦𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → 𝑦 / 𝑘𝐵 ∈ ℂ))
6759, 61, 66sylc 62 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑦 / 𝑘𝐵 ∈ ℂ)
6867abscld 10893 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘𝑦 / 𝑘𝐵) ∈ ℝ)
6951, 53, 68leadd1d 8264 . . . . . . . . . . 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 502 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ¬ 𝑦𝑥)
7170adantlrl 471 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ¬ 𝑦𝑥)
72 disjsn 3553 . . . . . . . . . . . . . . 15 ((𝑥 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦𝑥)
7371, 72sylibr 133 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∩ {𝑦}) = ∅)
74 eqidd 2116 . . . . . . . . . . . . . 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 6773 . . . . . . . . . . . . . . 15 ((𝑥 ∈ Fin ∧ 𝑦 ∈ V ∧ ¬ 𝑦𝑥) → (𝑥 ∪ {𝑦}) ∈ Fin)
7743, 75, 71, 76syl3anc 1199 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∪ {𝑦}) ∈ Fin)
7845sselda 3065 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ (𝑥 ∪ {𝑦})) → 𝑘𝐴)
7944, 78, 48syl2an2r 567 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ (𝑥 ∪ {𝑦})) → 𝐵 ∈ ℂ)
8079abscld 10893 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ (𝑥 ∪ {𝑦})) → (abs‘𝐵) ∈ ℝ)
8180recnd 7758 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑘 ∈ (𝑥 ∪ {𝑦})) → (abs‘𝐵) ∈ ℂ)
8273, 74, 77, 81fsumsplit 11116 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵) = (Σ𝑘𝑥 (abs‘𝐵) + Σ𝑘 ∈ {𝑦} (abs‘𝐵)))
83 csbfv2g 5424 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ V → 𝑦 / 𝑘(abs‘𝐵) = (abs‘𝑦 / 𝑘𝐵))
8483elv 2662 . . . . . . . . . . . . . . . . . 18 𝑦 / 𝑘(abs‘𝐵) = (abs‘𝑦 / 𝑘𝐵)
8560ad2antrr 477 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ∀𝑘𝐴 𝐵 ∈ ℂ)
8658, 85, 66sylc 62 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑦 / 𝑘𝐵 ∈ ℂ)
8786abscld 10893 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘𝑦 / 𝑘𝐵) ∈ ℝ)
8887recnd 7758 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘𝑦 / 𝑘𝐵) ∈ ℂ)
8984, 88eqeltrid 2202 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑦 / 𝑘(abs‘𝐵) ∈ ℂ)
90 sumsns 11124 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ V ∧ 𝑦 / 𝑘(abs‘𝐵) ∈ ℂ) → Σ𝑘 ∈ {𝑦} (abs‘𝐵) = 𝑦 / 𝑘(abs‘𝐵))
9156, 89, 90sylancr 408 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ {𝑦} (abs‘𝐵) = 𝑦 / 𝑘(abs‘𝐵))
9291, 84syl6eq 2164 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ {𝑦} (abs‘𝐵) = (abs‘𝑦 / 𝑘𝐵))
9392oveq2d 5756 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (Σ𝑘𝑥 (abs‘𝐵) + Σ𝑘 ∈ {𝑦} (abs‘𝐵)) = (Σ𝑘𝑥 (abs‘𝐵) + (abs‘𝑦 / 𝑘𝐵)))
9493adantlrl 471 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (Σ𝑘𝑥 (abs‘𝐵) + Σ𝑘 ∈ {𝑦} (abs‘𝐵)) = (Σ𝑘𝑥 (abs‘𝐵) + (abs‘𝑦 / 𝑘𝐵)))
9582, 94eqtrd 2148 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵) = (Σ𝑘𝑥 (abs‘𝐵) + (abs‘𝑦 / 𝑘𝐵)))
9695breq2d 3909 . . . . . . . . . . 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 190 . . . . . . . . . 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 133 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∩ {𝑦}) = ∅)
9998adantlrl 471 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (𝑥 ∩ {𝑦}) = ∅)
10099, 74, 77, 79fsumsplit 11116 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵 = (Σ𝑘𝑥 𝐵 + Σ𝑘 ∈ {𝑦}𝐵))
101 sumsns 11124 . . . . . . . . . . . . . . . . 17 ((𝑦𝐴𝑦 / 𝑘𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑦}𝐵 = 𝑦 / 𝑘𝐵)
10258, 86, 101syl2anc 406 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ {𝑦}𝐵 = 𝑦 / 𝑘𝐵)
103102oveq2d 5756 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ 𝑦𝑥) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (Σ𝑘𝑥 𝐵 + Σ𝑘 ∈ {𝑦}𝐵) = (Σ𝑘𝑥 𝐵 + 𝑦 / 𝑘𝐵))
104103adantlrl 471 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (Σ𝑘𝑥 𝐵 + Σ𝑘 ∈ {𝑦}𝐵) = (Σ𝑘𝑥 𝐵 + 𝑦 / 𝑘𝐵))
105100, 104eqtrd 2148 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵 = (Σ𝑘𝑥 𝐵 + 𝑦 / 𝑘𝐵))
106105fveq2d 5391 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) = (abs‘(Σ𝑘𝑥 𝐵 + 𝑦 / 𝑘𝐵)))
10786adantlrl 471 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → 𝑦 / 𝑘𝐵 ∈ ℂ)
10850, 107abstrid 10908 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘(Σ𝑘𝑥 𝐵 + 𝑦 / 𝑘𝐵)) ≤ ((abs‘Σ𝑘𝑥 𝐵) + (abs‘𝑦 / 𝑘𝐵)))
109106, 108eqbrtrd 3918 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ≤ ((abs‘Σ𝑘𝑥 𝐵) + (abs‘𝑦 / 𝑘𝐵)))
11077, 79fsumcl 11109 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵 ∈ ℂ)
111110abscld 10893 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → (abs‘Σ𝑘 ∈ (𝑥 ∪ {𝑦})𝐵) ∈ ℝ)
11251, 68readdcld 7759 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → ((abs‘Σ𝑘𝑥 𝐵) + (abs‘𝑦 / 𝑘𝐵)) ∈ ℝ)
11377, 80fsumrecl 11110 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ Fin ∧ ¬ 𝑦𝑥)) ∧ (𝑥 ∪ {𝑦}) ⊆ 𝐴) → Σ𝑘 ∈ (𝑥 ∪ {𝑦})(abs‘𝐵) ∈ ℝ)
114 letr 7811 . . . . . . . . . . . 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 1199 . . . . . . . . . . 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 423 . . . . . . . . . 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 149 . . . . . . . . 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 114 . . . . . . . 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 115 . . . . 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 6750 . . 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 103   = wceq 1314  wcel 1463  wral 2391  Vcvv 2658  csb 2973  cun 3037  cin 3038  wss 3039  c0 3331  {csn 3495   class class class wbr 3897  cfv 5091  (class class class)co 5740  Fincfn 6600  cc 7582  cr 7583  0cc0 7584   + caddc 7587  cle 7765  abscabs 10709  Σcsu 11062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702  ax-arch 7703  ax-caucvg 7704
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-isom 5100  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-frec 6254  df-1o 6279  df-oadd 6283  df-er 6395  df-en 6601  df-dom 6602  df-fin 6603  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8393  df-inn 8678  df-2 8736  df-3 8737  df-4 8738  df-n0 8929  df-z 9006  df-uz 9276  df-q 9361  df-rp 9391  df-fz 9731  df-fzo 9860  df-seqfrec 10159  df-exp 10233  df-ihash 10462  df-cj 10554  df-re 10555  df-im 10556  df-rsqrt 10710  df-abs 10711  df-clim 10988  df-sumdc 11063
This theorem is referenced by:  iserabs  11184  cvgratnnlemabsle  11236  mertenslemi1  11244
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