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Theorem fsumo1 14588
Description: The finite sum of eventually bounded functions (where the index set 𝐵 does not depend on 𝑥) is eventually bounded. (Contributed by Mario Carneiro, 30-Apr-2016.) (Proof shortened by Mario Carneiro, 22-May-2016.)
Hypotheses
Ref Expression
fsumo1.1 (𝜑𝐴 ⊆ ℝ)
fsumo1.2 (𝜑𝐵 ∈ Fin)
fsumo1.3 ((𝜑 ∧ (𝑥𝐴𝑘𝐵)) → 𝐶𝑉)
fsumo1.4 ((𝜑𝑘𝐵) → (𝑥𝐴𝐶) ∈ 𝑂(1))
Assertion
Ref Expression
fsumo1 (𝜑 → (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ∈ 𝑂(1))
Distinct variable groups:   𝑥,𝑘,𝐴   𝐵,𝑘,𝑥   𝜑,𝑘,𝑥
Allowed substitution hints:   𝐶(𝑥,𝑘)   𝑉(𝑥,𝑘)

Proof of Theorem fsumo1
Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 3657 . 2 𝐵𝐵
2 fsumo1.2 . . 3 (𝜑𝐵 ∈ Fin)
3 sseq1 3659 . . . . . 6 (𝑤 = ∅ → (𝑤𝐵 ↔ ∅ ⊆ 𝐵))
4 sumeq1 14463 . . . . . . . . 9 (𝑤 = ∅ → Σ𝑘𝑤 𝐶 = Σ𝑘 ∈ ∅ 𝐶)
5 sum0 14496 . . . . . . . . 9 Σ𝑘 ∈ ∅ 𝐶 = 0
64, 5syl6eq 2701 . . . . . . . 8 (𝑤 = ∅ → Σ𝑘𝑤 𝐶 = 0)
76mpteq2dv 4778 . . . . . . 7 (𝑤 = ∅ → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) = (𝑥𝐴 ↦ 0))
87eleq1d 2715 . . . . . 6 (𝑤 = ∅ → ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝑂(1) ↔ (𝑥𝐴 ↦ 0) ∈ 𝑂(1)))
93, 8imbi12d 333 . . . . 5 (𝑤 = ∅ → ((𝑤𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝑂(1)) ↔ (∅ ⊆ 𝐵 → (𝑥𝐴 ↦ 0) ∈ 𝑂(1))))
109imbi2d 329 . . . 4 (𝑤 = ∅ → ((𝜑 → (𝑤𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝑂(1))) ↔ (𝜑 → (∅ ⊆ 𝐵 → (𝑥𝐴 ↦ 0) ∈ 𝑂(1)))))
11 sseq1 3659 . . . . . 6 (𝑤 = 𝑦 → (𝑤𝐵𝑦𝐵))
12 sumeq1 14463 . . . . . . . 8 (𝑤 = 𝑦 → Σ𝑘𝑤 𝐶 = Σ𝑘𝑦 𝐶)
1312mpteq2dv 4778 . . . . . . 7 (𝑤 = 𝑦 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) = (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶))
1413eleq1d 2715 . . . . . 6 (𝑤 = 𝑦 → ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝑂(1) ↔ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1)))
1511, 14imbi12d 333 . . . . 5 (𝑤 = 𝑦 → ((𝑤𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝑂(1)) ↔ (𝑦𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1))))
1615imbi2d 329 . . . 4 (𝑤 = 𝑦 → ((𝜑 → (𝑤𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝑂(1))) ↔ (𝜑 → (𝑦𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1)))))
17 sseq1 3659 . . . . . 6 (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤𝐵 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝐵))
18 sumeq1 14463 . . . . . . . 8 (𝑤 = (𝑦 ∪ {𝑧}) → Σ𝑘𝑤 𝐶 = Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶)
1918mpteq2dv 4778 . . . . . . 7 (𝑤 = (𝑦 ∪ {𝑧}) → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) = (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶))
2019eleq1d 2715 . . . . . 6 (𝑤 = (𝑦 ∪ {𝑧}) → ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝑂(1) ↔ (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1)))
2117, 20imbi12d 333 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → ((𝑤𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝑂(1)) ↔ ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1))))
2221imbi2d 329 . . . 4 (𝑤 = (𝑦 ∪ {𝑧}) → ((𝜑 → (𝑤𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝑂(1))) ↔ (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1)))))
23 sseq1 3659 . . . . . 6 (𝑤 = 𝐵 → (𝑤𝐵𝐵𝐵))
24 sumeq1 14463 . . . . . . . 8 (𝑤 = 𝐵 → Σ𝑘𝑤 𝐶 = Σ𝑘𝐵 𝐶)
2524mpteq2dv 4778 . . . . . . 7 (𝑤 = 𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) = (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶))
2625eleq1d 2715 . . . . . 6 (𝑤 = 𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝑂(1) ↔ (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ∈ 𝑂(1)))
2723, 26imbi12d 333 . . . . 5 (𝑤 = 𝐵 → ((𝑤𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝑂(1)) ↔ (𝐵𝐵 → (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ∈ 𝑂(1))))
2827imbi2d 329 . . . 4 (𝑤 = 𝐵 → ((𝜑 → (𝑤𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝑂(1))) ↔ (𝜑 → (𝐵𝐵 → (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ∈ 𝑂(1)))))
29 fsumo1.1 . . . . . 6 (𝜑𝐴 ⊆ ℝ)
30 0cn 10070 . . . . . 6 0 ∈ ℂ
31 o1const 14394 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 0 ∈ ℂ) → (𝑥𝐴 ↦ 0) ∈ 𝑂(1))
3229, 30, 31sylancl 695 . . . . 5 (𝜑 → (𝑥𝐴 ↦ 0) ∈ 𝑂(1))
3332a1d 25 . . . 4 (𝜑 → (∅ ⊆ 𝐵 → (𝑥𝐴 ↦ 0) ∈ 𝑂(1)))
34 ssun1 3809 . . . . . . . . . 10 𝑦 ⊆ (𝑦 ∪ {𝑧})
35 sstr 3644 . . . . . . . . . 10 ((𝑦 ⊆ (𝑦 ∪ {𝑧}) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵) → 𝑦𝐵)
3634, 35mpan 706 . . . . . . . . 9 ((𝑦 ∪ {𝑧}) ⊆ 𝐵𝑦𝐵)
3736imim1i 63 . . . . . . . 8 ((𝑦𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1)) → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1)))
38 simprl 809 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ¬ 𝑧𝑦)
39 disjsn 4278 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑦)
4038, 39sylibr 224 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∩ {𝑧}) = ∅)
4140adantr 480 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → (𝑦 ∩ {𝑧}) = ∅)
42 eqidd 2652 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
432adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝐵 ∈ Fin)
44 simprr 811 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∪ {𝑧}) ⊆ 𝐵)
45 ssfi 8221 . . . . . . . . . . . . . . . . . . 19 ((𝐵 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵) → (𝑦 ∪ {𝑧}) ∈ Fin)
4643, 44, 45syl2anc 694 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∪ {𝑧}) ∈ Fin)
4746adantr 480 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → (𝑦 ∪ {𝑧}) ∈ Fin)
4844sselda 3636 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑘𝐵)
4948adantlr 751 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑘𝐵)
50 fsumo1.3 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑥𝐴𝑘𝐵)) → 𝐶𝑉)
5150anass1rs 866 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘𝐵) ∧ 𝑥𝐴) → 𝐶𝑉)
52 fsumo1.4 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑘𝐵) → (𝑥𝐴𝐶) ∈ 𝑂(1))
5351, 52o1mptrcl 14397 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑘𝐵) ∧ 𝑥𝐴) → 𝐶 ∈ ℂ)
5453an32s 863 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ)
5554adantllr 755 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ)
5649, 55syldan 486 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝐶 ∈ ℂ)
5741, 42, 47, 56fsumsplit 14515 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶 = (Σ𝑘𝑦 𝐶 + Σ𝑘 ∈ {𝑧}𝐶))
58 nfcv 2793 . . . . . . . . . . . . . . . . . . 19 𝑤𝐶
59 nfcsb1v 3582 . . . . . . . . . . . . . . . . . . 19 𝑘𝑤 / 𝑘𝐶
60 csbeq1a 3575 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑤𝐶 = 𝑤 / 𝑘𝐶)
6158, 59, 60cbvsumi 14471 . . . . . . . . . . . . . . . . . 18 Σ𝑘 ∈ {𝑧}𝐶 = Σ𝑤 ∈ {𝑧}𝑤 / 𝑘𝐶
6244unssbd 3824 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → {𝑧} ⊆ 𝐵)
63 vex 3234 . . . . . . . . . . . . . . . . . . . . . 22 𝑧 ∈ V
6463snss 4348 . . . . . . . . . . . . . . . . . . . . 21 (𝑧𝐵 ↔ {𝑧} ⊆ 𝐵)
6562, 64sylibr 224 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝑧𝐵)
6665adantr 480 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → 𝑧𝐵)
6755ralrimiva 2995 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → ∀𝑘𝐵 𝐶 ∈ ℂ)
68 nfcsb1v 3582 . . . . . . . . . . . . . . . . . . . . . 22 𝑘𝑧 / 𝑘𝐶
6968nfel1 2808 . . . . . . . . . . . . . . . . . . . . 21 𝑘𝑧 / 𝑘𝐶 ∈ ℂ
70 csbeq1a 3575 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑧𝐶 = 𝑧 / 𝑘𝐶)
7170eleq1d 2715 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑧 → (𝐶 ∈ ℂ ↔ 𝑧 / 𝑘𝐶 ∈ ℂ))
7269, 71rspc 3334 . . . . . . . . . . . . . . . . . . . 20 (𝑧𝐵 → (∀𝑘𝐵 𝐶 ∈ ℂ → 𝑧 / 𝑘𝐶 ∈ ℂ))
7366, 67, 72sylc 65 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → 𝑧 / 𝑘𝐶 ∈ ℂ)
74 csbeq1 3569 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑧𝑤 / 𝑘𝐶 = 𝑧 / 𝑘𝐶)
7574sumsn 14519 . . . . . . . . . . . . . . . . . . 19 ((𝑧𝐵𝑧 / 𝑘𝐶 ∈ ℂ) → Σ𝑤 ∈ {𝑧}𝑤 / 𝑘𝐶 = 𝑧 / 𝑘𝐶)
7666, 73, 75syl2anc 694 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → Σ𝑤 ∈ {𝑧}𝑤 / 𝑘𝐶 = 𝑧 / 𝑘𝐶)
7761, 76syl5eq 2697 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → Σ𝑘 ∈ {𝑧}𝐶 = 𝑧 / 𝑘𝐶)
7877oveq2d 6706 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → (Σ𝑘𝑦 𝐶 + Σ𝑘 ∈ {𝑧}𝐶) = (Σ𝑘𝑦 𝐶 + 𝑧 / 𝑘𝐶))
7957, 78eqtrd 2685 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶 = (Σ𝑘𝑦 𝐶 + 𝑧 / 𝑘𝐶))
8079mpteq2dva 4777 . . . . . . . . . . . . . 14 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) = (𝑥𝐴 ↦ (Σ𝑘𝑦 𝐶 + 𝑧 / 𝑘𝐶)))
8129adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝐴 ⊆ ℝ)
82 reex 10065 . . . . . . . . . . . . . . . . 17 ℝ ∈ V
8382ssex 4835 . . . . . . . . . . . . . . . 16 (𝐴 ⊆ ℝ → 𝐴 ∈ V)
8481, 83syl 17 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝐴 ∈ V)
85 sumex 14462 . . . . . . . . . . . . . . . 16 Σ𝑘𝑦 𝐶 ∈ V
8685a1i 11 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → Σ𝑘𝑦 𝐶 ∈ V)
87 eqidd 2652 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) = (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶))
88 eqidd 2652 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥𝐴𝑧 / 𝑘𝐶) = (𝑥𝐴𝑧 / 𝑘𝐶))
8984, 86, 73, 87, 88offval2 6956 . . . . . . . . . . . . . 14 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ((𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∘𝑓 + (𝑥𝐴𝑧 / 𝑘𝐶)) = (𝑥𝐴 ↦ (Σ𝑘𝑦 𝐶 + 𝑧 / 𝑘𝐶)))
9080, 89eqtr4d 2688 . . . . . . . . . . . . 13 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) = ((𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∘𝑓 + (𝑥𝐴𝑧 / 𝑘𝐶)))
9190adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1)) → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) = ((𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∘𝑓 + (𝑥𝐴𝑧 / 𝑘𝐶)))
92 id 22 . . . . . . . . . . . . 13 ((𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1) → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1))
9352ralrimiva 2995 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑘𝐵 (𝑥𝐴𝐶) ∈ 𝑂(1))
9493adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑘𝐵 (𝑥𝐴𝐶) ∈ 𝑂(1))
95 nfcv 2793 . . . . . . . . . . . . . . . . 17 𝑘𝐴
9695, 68nfmpt 4779 . . . . . . . . . . . . . . . 16 𝑘(𝑥𝐴𝑧 / 𝑘𝐶)
9796nfel1 2808 . . . . . . . . . . . . . . 15 𝑘(𝑥𝐴𝑧 / 𝑘𝐶) ∈ 𝑂(1)
9870mpteq2dv 4778 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑧 → (𝑥𝐴𝐶) = (𝑥𝐴𝑧 / 𝑘𝐶))
9998eleq1d 2715 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → ((𝑥𝐴𝐶) ∈ 𝑂(1) ↔ (𝑥𝐴𝑧 / 𝑘𝐶) ∈ 𝑂(1)))
10097, 99rspc 3334 . . . . . . . . . . . . . 14 (𝑧𝐵 → (∀𝑘𝐵 (𝑥𝐴𝐶) ∈ 𝑂(1) → (𝑥𝐴𝑧 / 𝑘𝐶) ∈ 𝑂(1)))
10165, 94, 100sylc 65 . . . . . . . . . . . . 13 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥𝐴𝑧 / 𝑘𝐶) ∈ 𝑂(1))
102 o1add 14388 . . . . . . . . . . . . 13 (((𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1) ∧ (𝑥𝐴𝑧 / 𝑘𝐶) ∈ 𝑂(1)) → ((𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∘𝑓 + (𝑥𝐴𝑧 / 𝑘𝐶)) ∈ 𝑂(1))
10392, 101, 102syl2anr 494 . . . . . . . . . . . 12 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1)) → ((𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∘𝑓 + (𝑥𝐴𝑧 / 𝑘𝐶)) ∈ 𝑂(1))
10491, 103eqeltrd 2730 . . . . . . . . . . 11 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1)) → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1))
105104ex 449 . . . . . . . . . 10 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ((𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1) → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1)))
106105expr 642 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝑧𝑦) → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1) → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1))))
107106a2d 29 . . . . . . . 8 ((𝜑 ∧ ¬ 𝑧𝑦) → (((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1)) → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1))))
10837, 107syl5 34 . . . . . . 7 ((𝜑 ∧ ¬ 𝑧𝑦) → ((𝑦𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1)) → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1))))
109108expcom 450 . . . . . 6 𝑧𝑦 → (𝜑 → ((𝑦𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1)) → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1)))))
110109a2d 29 . . . . 5 𝑧𝑦 → ((𝜑 → (𝑦𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1))) → (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1)))))
111110adantl 481 . . . 4 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → ((𝜑 → (𝑦𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1))) → (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1)))))
11210, 16, 22, 28, 33, 111findcard2s 8242 . . 3 (𝐵 ∈ Fin → (𝜑 → (𝐵𝐵 → (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ∈ 𝑂(1))))
1132, 112mpcom 38 . 2 (𝜑 → (𝐵𝐵 → (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ∈ 𝑂(1)))
1141, 113mpi 20 1 (𝜑 → (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ∈ 𝑂(1))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  csb 3566  cun 3605  cin 3606  wss 3607  c0 3948  {csn 4210  cmpt 4762  (class class class)co 6690  𝑓 cof 6937  Fincfn 7997  cc 9972  cr 9973  0cc0 9974   + caddc 9977  𝑂(1)co1 14261  Σcsu 14460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-ico 12219  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-o1 14265  df-sum 14461
This theorem is referenced by:  rpvmasum2  25246
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