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Theorem fsumo1 15159
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 3992 . 2 𝐵𝐵
2 fsumo1.2 . . 3 (𝜑𝐵 ∈ Fin)
3 sseq1 3995 . . . . . 6 (𝑤 = ∅ → (𝑤𝐵 ↔ ∅ ⊆ 𝐵))
4 sumeq1 15038 . . . . . . . . 9 (𝑤 = ∅ → Σ𝑘𝑤 𝐶 = Σ𝑘 ∈ ∅ 𝐶)
5 sum0 15070 . . . . . . . . 9 Σ𝑘 ∈ ∅ 𝐶 = 0
64, 5syl6eq 2876 . . . . . . . 8 (𝑤 = ∅ → Σ𝑘𝑤 𝐶 = 0)
76mpteq2dv 5158 . . . . . . 7 (𝑤 = ∅ → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) = (𝑥𝐴 ↦ 0))
87eleq1d 2901 . . . . . 6 (𝑤 = ∅ → ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝑂(1) ↔ (𝑥𝐴 ↦ 0) ∈ 𝑂(1)))
93, 8imbi12d 346 . . . . 5 (𝑤 = ∅ → ((𝑤𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝑂(1)) ↔ (∅ ⊆ 𝐵 → (𝑥𝐴 ↦ 0) ∈ 𝑂(1))))
109imbi2d 342 . . . 4 (𝑤 = ∅ → ((𝜑 → (𝑤𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝑂(1))) ↔ (𝜑 → (∅ ⊆ 𝐵 → (𝑥𝐴 ↦ 0) ∈ 𝑂(1)))))
11 sseq1 3995 . . . . . 6 (𝑤 = 𝑦 → (𝑤𝐵𝑦𝐵))
12 sumeq1 15038 . . . . . . . 8 (𝑤 = 𝑦 → Σ𝑘𝑤 𝐶 = Σ𝑘𝑦 𝐶)
1312mpteq2dv 5158 . . . . . . 7 (𝑤 = 𝑦 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) = (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶))
1413eleq1d 2901 . . . . . 6 (𝑤 = 𝑦 → ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝑂(1) ↔ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1)))
1511, 14imbi12d 346 . . . . 5 (𝑤 = 𝑦 → ((𝑤𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝑂(1)) ↔ (𝑦𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1))))
1615imbi2d 342 . . . 4 (𝑤 = 𝑦 → ((𝜑 → (𝑤𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝑂(1))) ↔ (𝜑 → (𝑦𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1)))))
17 sseq1 3995 . . . . . 6 (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤𝐵 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝐵))
18 sumeq1 15038 . . . . . . . 8 (𝑤 = (𝑦 ∪ {𝑧}) → Σ𝑘𝑤 𝐶 = Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶)
1918mpteq2dv 5158 . . . . . . 7 (𝑤 = (𝑦 ∪ {𝑧}) → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) = (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶))
2019eleq1d 2901 . . . . . 6 (𝑤 = (𝑦 ∪ {𝑧}) → ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝑂(1) ↔ (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1)))
2117, 20imbi12d 346 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → ((𝑤𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝑂(1)) ↔ ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1))))
2221imbi2d 342 . . . 4 (𝑤 = (𝑦 ∪ {𝑧}) → ((𝜑 → (𝑤𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝑂(1))) ↔ (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1)))))
23 sseq1 3995 . . . . . 6 (𝑤 = 𝐵 → (𝑤𝐵𝐵𝐵))
24 sumeq1 15038 . . . . . . . 8 (𝑤 = 𝐵 → Σ𝑘𝑤 𝐶 = Σ𝑘𝐵 𝐶)
2524mpteq2dv 5158 . . . . . . 7 (𝑤 = 𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) = (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶))
2625eleq1d 2901 . . . . . 6 (𝑤 = 𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝑂(1) ↔ (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ∈ 𝑂(1)))
2723, 26imbi12d 346 . . . . 5 (𝑤 = 𝐵 → ((𝑤𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ∈ 𝑂(1)) ↔ (𝐵𝐵 → (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ∈ 𝑂(1))))
2827imbi2d 342 . . . 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 10625 . . . . . 6 0 ∈ ℂ
31 o1const 14969 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 0 ∈ ℂ) → (𝑥𝐴 ↦ 0) ∈ 𝑂(1))
3229, 30, 31sylancl 586 . . . . 5 (𝜑 → (𝑥𝐴 ↦ 0) ∈ 𝑂(1))
3332a1d 25 . . . 4 (𝜑 → (∅ ⊆ 𝐵 → (𝑥𝐴 ↦ 0) ∈ 𝑂(1)))
34 ssun1 4151 . . . . . . . . . 10 𝑦 ⊆ (𝑦 ∪ {𝑧})
35 sstr 3978 . . . . . . . . . 10 ((𝑦 ⊆ (𝑦 ∪ {𝑧}) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵) → 𝑦𝐵)
3634, 35mpan 686 . . . . . . . . 9 ((𝑦 ∪ {𝑧}) ⊆ 𝐵𝑦𝐵)
3736imim1i 63 . . . . . . . 8 ((𝑦𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1)) → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1)))
38 simprl 767 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ¬ 𝑧𝑦)
39 disjsn 4645 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑦)
4038, 39sylibr 235 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∩ {𝑧}) = ∅)
4140adantr 481 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → (𝑦 ∩ {𝑧}) = ∅)
42 eqidd 2826 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
432adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝐵 ∈ Fin)
44 simprr 769 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∪ {𝑧}) ⊆ 𝐵)
4543, 44ssfid 8733 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∪ {𝑧}) ∈ Fin)
4645adantr 481 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → (𝑦 ∪ {𝑧}) ∈ Fin)
4744sselda 3970 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑘𝐵)
4847adantlr 711 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑘𝐵)
49 fsumo1.3 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑥𝐴𝑘𝐵)) → 𝐶𝑉)
5049anass1rs 651 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘𝐵) ∧ 𝑥𝐴) → 𝐶𝑉)
51 fsumo1.4 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑘𝐵) → (𝑥𝐴𝐶) ∈ 𝑂(1))
5250, 51o1mptrcl 14972 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑘𝐵) ∧ 𝑥𝐴) → 𝐶 ∈ ℂ)
5352an32s 648 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ)
5453adantllr 715 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ)
5548, 54syldan 591 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝐶 ∈ ℂ)
5641, 42, 46, 55fsumsplit 15089 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶 = (Σ𝑘𝑦 𝐶 + Σ𝑘 ∈ {𝑧}𝐶))
57 nfcv 2981 . . . . . . . . . . . . . . . . . . 19 𝑤𝐶
58 nfcsb1v 3910 . . . . . . . . . . . . . . . . . . 19 𝑘𝑤 / 𝑘𝐶
59 csbeq1a 3900 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑤𝐶 = 𝑤 / 𝑘𝐶)
6057, 58, 59cbvsumi 15046 . . . . . . . . . . . . . . . . . 18 Σ𝑘 ∈ {𝑧}𝐶 = Σ𝑤 ∈ {𝑧}𝑤 / 𝑘𝐶
6144unssbd 4167 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → {𝑧} ⊆ 𝐵)
62 vex 3502 . . . . . . . . . . . . . . . . . . . . . 22 𝑧 ∈ V
6362snss 4716 . . . . . . . . . . . . . . . . . . . . 21 (𝑧𝐵 ↔ {𝑧} ⊆ 𝐵)
6461, 63sylibr 235 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝑧𝐵)
6564adantr 481 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → 𝑧𝐵)
6654ralrimiva 3186 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → ∀𝑘𝐵 𝐶 ∈ ℂ)
67 nfcsb1v 3910 . . . . . . . . . . . . . . . . . . . . . 22 𝑘𝑧 / 𝑘𝐶
6867nfel1 2998 . . . . . . . . . . . . . . . . . . . . 21 𝑘𝑧 / 𝑘𝐶 ∈ ℂ
69 csbeq1a 3900 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑧𝐶 = 𝑧 / 𝑘𝐶)
7069eleq1d 2901 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑧 → (𝐶 ∈ ℂ ↔ 𝑧 / 𝑘𝐶 ∈ ℂ))
7168, 70rspc 3614 . . . . . . . . . . . . . . . . . . . 20 (𝑧𝐵 → (∀𝑘𝐵 𝐶 ∈ ℂ → 𝑧 / 𝑘𝐶 ∈ ℂ))
7265, 66, 71sylc 65 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → 𝑧 / 𝑘𝐶 ∈ ℂ)
73 csbeq1 3889 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑧𝑤 / 𝑘𝐶 = 𝑧 / 𝑘𝐶)
7473sumsn 15093 . . . . . . . . . . . . . . . . . . 19 ((𝑧𝐵𝑧 / 𝑘𝐶 ∈ ℂ) → Σ𝑤 ∈ {𝑧}𝑤 / 𝑘𝐶 = 𝑧 / 𝑘𝐶)
7565, 72, 74syl2anc 584 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → Σ𝑤 ∈ {𝑧}𝑤 / 𝑘𝐶 = 𝑧 / 𝑘𝐶)
7660, 75syl5eq 2872 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → Σ𝑘 ∈ {𝑧}𝐶 = 𝑧 / 𝑘𝐶)
7776oveq2d 7167 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → (Σ𝑘𝑦 𝐶 + Σ𝑘 ∈ {𝑧}𝐶) = (Σ𝑘𝑦 𝐶 + 𝑧 / 𝑘𝐶))
7856, 77eqtrd 2860 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶 = (Σ𝑘𝑦 𝐶 + 𝑧 / 𝑘𝐶))
7978mpteq2dva 5157 . . . . . . . . . . . . . 14 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) = (𝑥𝐴 ↦ (Σ𝑘𝑦 𝐶 + 𝑧 / 𝑘𝐶)))
8029adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝐴 ⊆ ℝ)
81 reex 10620 . . . . . . . . . . . . . . . . 17 ℝ ∈ V
8281ssex 5221 . . . . . . . . . . . . . . . 16 (𝐴 ⊆ ℝ → 𝐴 ∈ V)
8380, 82syl 17 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝐴 ∈ V)
84 sumex 15037 . . . . . . . . . . . . . . . 16 Σ𝑘𝑦 𝐶 ∈ V
8584a1i 11 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → Σ𝑘𝑦 𝐶 ∈ V)
86 eqidd 2826 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) = (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶))
87 eqidd 2826 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥𝐴𝑧 / 𝑘𝐶) = (𝑥𝐴𝑧 / 𝑘𝐶))
8883, 85, 72, 86, 87offval2 7419 . . . . . . . . . . . . . 14 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ((𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∘f + (𝑥𝐴𝑧 / 𝑘𝐶)) = (𝑥𝐴 ↦ (Σ𝑘𝑦 𝐶 + 𝑧 / 𝑘𝐶)))
8979, 88eqtr4d 2863 . . . . . . . . . . . . 13 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) = ((𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∘f + (𝑥𝐴𝑧 / 𝑘𝐶)))
9089adantr 481 . . . . . . . . . . . 12 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1)) → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) = ((𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∘f + (𝑥𝐴𝑧 / 𝑘𝐶)))
91 id 22 . . . . . . . . . . . . 13 ((𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1) → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1))
9251ralrimiva 3186 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑘𝐵 (𝑥𝐴𝐶) ∈ 𝑂(1))
9392adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑘𝐵 (𝑥𝐴𝐶) ∈ 𝑂(1))
94 nfcv 2981 . . . . . . . . . . . . . . . . 17 𝑘𝐴
9594, 67nfmpt 5159 . . . . . . . . . . . . . . . 16 𝑘(𝑥𝐴𝑧 / 𝑘𝐶)
9695nfel1 2998 . . . . . . . . . . . . . . 15 𝑘(𝑥𝐴𝑧 / 𝑘𝐶) ∈ 𝑂(1)
9769mpteq2dv 5158 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑧 → (𝑥𝐴𝐶) = (𝑥𝐴𝑧 / 𝑘𝐶))
9897eleq1d 2901 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → ((𝑥𝐴𝐶) ∈ 𝑂(1) ↔ (𝑥𝐴𝑧 / 𝑘𝐶) ∈ 𝑂(1)))
9996, 98rspc 3614 . . . . . . . . . . . . . 14 (𝑧𝐵 → (∀𝑘𝐵 (𝑥𝐴𝐶) ∈ 𝑂(1) → (𝑥𝐴𝑧 / 𝑘𝐶) ∈ 𝑂(1)))
10064, 93, 99sylc 65 . . . . . . . . . . . . 13 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥𝐴𝑧 / 𝑘𝐶) ∈ 𝑂(1))
101 o1add 14963 . . . . . . . . . . . . 13 (((𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1) ∧ (𝑥𝐴𝑧 / 𝑘𝐶) ∈ 𝑂(1)) → ((𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∘f + (𝑥𝐴𝑧 / 𝑘𝐶)) ∈ 𝑂(1))
10291, 100, 101syl2anr 596 . . . . . . . . . . . 12 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1)) → ((𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∘f + (𝑥𝐴𝑧 / 𝑘𝐶)) ∈ 𝑂(1))
10390, 102eqeltrd 2917 . . . . . . . . . . 11 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1)) → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1))
104103ex 413 . . . . . . . . . 10 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ((𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1) → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1)))
105104expr 457 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝑧𝑦) → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1) → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1))))
106105a2d 29 . . . . . . . 8 ((𝜑 ∧ ¬ 𝑧𝑦) → (((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1)) → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1))))
10737, 106syl5 34 . . . . . . 7 ((𝜑 ∧ ¬ 𝑧𝑦) → ((𝑦𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1)) → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1))))
108107expcom 414 . . . . . 6 𝑧𝑦 → (𝜑 → ((𝑦𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1)) → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1)))))
109108a2d 29 . . . . 5 𝑧𝑦 → ((𝜑 → (𝑦𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1))) → (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1)))))
110109adantl 482 . . . 4 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → ((𝜑 → (𝑦𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ∈ 𝑂(1))) → (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ∈ 𝑂(1)))))
11110, 16, 22, 28, 33, 110findcard2s 8751 . . 3 (𝐵 ∈ Fin → (𝜑 → (𝐵𝐵 → (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ∈ 𝑂(1))))
1122, 111mpcom 38 . 2 (𝜑 → (𝐵𝐵 → (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ∈ 𝑂(1)))
1131, 112mpi 20 1 (𝜑 → (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ∈ 𝑂(1))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1530  wcel 2107  wral 3142  Vcvv 3499  csb 3886  cun 3937  cin 3938  wss 3939  c0 4294  {csn 4563  cmpt 5142  (class class class)co 7151  f cof 7400  Fincfn 8501  cc 10527  cr 10528  0cc0 10529   + caddc 10532  𝑂(1)co1 14836  Σcsu 15035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-inf2 9096  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-se 5513  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-of 7402  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8282  df-pm 8402  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-sup 8898  df-oi 8966  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-rp 12383  df-ico 12737  df-fz 12886  df-fzo 13027  df-seq 13363  df-exp 13423  df-hash 13684  df-cj 14451  df-re 14452  df-im 14453  df-sqrt 14587  df-abs 14588  df-clim 14838  df-rlim 14839  df-o1 14840  df-sum 15036
This theorem is referenced by:  rpvmasum2  26004
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