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Theorem fsumrlim 14742
Description: Limit of a finite sum of converging sequences. Note that 𝐶(𝑘) is a collection of functions with implicit parameter 𝑘, each of which converges to 𝐷(𝑘) as 𝑛 ⇝ +∞. (Contributed by Mario Carneiro, 22-May-2016.)
Hypotheses
Ref Expression
fsumrlim.1 (𝜑𝐴 ⊆ ℝ)
fsumrlim.2 (𝜑𝐵 ∈ Fin)
fsumrlim.3 ((𝜑 ∧ (𝑥𝐴𝑘𝐵)) → 𝐶𝑉)
fsumrlim.4 ((𝜑𝑘𝐵) → (𝑥𝐴𝐶) ⇝𝑟 𝐷)
Assertion
Ref Expression
fsumrlim (𝜑 → (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ⇝𝑟 Σ𝑘𝐵 𝐷)
Distinct variable groups:   𝑥,𝑘,𝐴   𝐵,𝑘,𝑥   𝜑,𝑘,𝑥
Allowed substitution hints:   𝐶(𝑥,𝑘)   𝐷(𝑥,𝑘)   𝑉(𝑥,𝑘)

Proof of Theorem fsumrlim
Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 3765 . 2 𝐵𝐵
2 fsumrlim.2 . . 3 (𝜑𝐵 ∈ Fin)
3 sseq1 3767 . . . . . 6 (𝑤 = ∅ → (𝑤𝐵 ↔ ∅ ⊆ 𝐵))
4 sumeq1 14618 . . . . . . . . 9 (𝑤 = ∅ → Σ𝑘𝑤 𝐶 = Σ𝑘 ∈ ∅ 𝐶)
5 sum0 14651 . . . . . . . . 9 Σ𝑘 ∈ ∅ 𝐶 = 0
64, 5syl6eq 2810 . . . . . . . 8 (𝑤 = ∅ → Σ𝑘𝑤 𝐶 = 0)
76mpteq2dv 4897 . . . . . . 7 (𝑤 = ∅ → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) = (𝑥𝐴 ↦ 0))
8 sumeq1 14618 . . . . . . . 8 (𝑤 = ∅ → Σ𝑘𝑤 𝐷 = Σ𝑘 ∈ ∅ 𝐷)
9 sum0 14651 . . . . . . . 8 Σ𝑘 ∈ ∅ 𝐷 = 0
108, 9syl6eq 2810 . . . . . . 7 (𝑤 = ∅ → Σ𝑘𝑤 𝐷 = 0)
117, 10breq12d 4817 . . . . . 6 (𝑤 = ∅ → ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ⇝𝑟 Σ𝑘𝑤 𝐷 ↔ (𝑥𝐴 ↦ 0) ⇝𝑟 0))
123, 11imbi12d 333 . . . . 5 (𝑤 = ∅ → ((𝑤𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ⇝𝑟 Σ𝑘𝑤 𝐷) ↔ (∅ ⊆ 𝐵 → (𝑥𝐴 ↦ 0) ⇝𝑟 0)))
1312imbi2d 329 . . . 4 (𝑤 = ∅ → ((𝜑 → (𝑤𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ⇝𝑟 Σ𝑘𝑤 𝐷)) ↔ (𝜑 → (∅ ⊆ 𝐵 → (𝑥𝐴 ↦ 0) ⇝𝑟 0))))
14 sseq1 3767 . . . . . 6 (𝑤 = 𝑦 → (𝑤𝐵𝑦𝐵))
15 sumeq1 14618 . . . . . . . 8 (𝑤 = 𝑦 → Σ𝑘𝑤 𝐶 = Σ𝑘𝑦 𝐶)
1615mpteq2dv 4897 . . . . . . 7 (𝑤 = 𝑦 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) = (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶))
17 sumeq1 14618 . . . . . . 7 (𝑤 = 𝑦 → Σ𝑘𝑤 𝐷 = Σ𝑘𝑦 𝐷)
1816, 17breq12d 4817 . . . . . 6 (𝑤 = 𝑦 → ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ⇝𝑟 Σ𝑘𝑤 𝐷 ↔ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷))
1914, 18imbi12d 333 . . . . 5 (𝑤 = 𝑦 → ((𝑤𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ⇝𝑟 Σ𝑘𝑤 𝐷) ↔ (𝑦𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷)))
2019imbi2d 329 . . . 4 (𝑤 = 𝑦 → ((𝜑 → (𝑤𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ⇝𝑟 Σ𝑘𝑤 𝐷)) ↔ (𝜑 → (𝑦𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷))))
21 sseq1 3767 . . . . . 6 (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤𝐵 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝐵))
22 sumeq1 14618 . . . . . . . 8 (𝑤 = (𝑦 ∪ {𝑧}) → Σ𝑘𝑤 𝐶 = Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶)
2322mpteq2dv 4897 . . . . . . 7 (𝑤 = (𝑦 ∪ {𝑧}) → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) = (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶))
24 sumeq1 14618 . . . . . . 7 (𝑤 = (𝑦 ∪ {𝑧}) → Σ𝑘𝑤 𝐷 = Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷)
2523, 24breq12d 4817 . . . . . 6 (𝑤 = (𝑦 ∪ {𝑧}) → ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ⇝𝑟 Σ𝑘𝑤 𝐷 ↔ (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷))
2621, 25imbi12d 333 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → ((𝑤𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ⇝𝑟 Σ𝑘𝑤 𝐷) ↔ ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷)))
2726imbi2d 329 . . . 4 (𝑤 = (𝑦 ∪ {𝑧}) → ((𝜑 → (𝑤𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ⇝𝑟 Σ𝑘𝑤 𝐷)) ↔ (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷))))
28 sseq1 3767 . . . . . 6 (𝑤 = 𝐵 → (𝑤𝐵𝐵𝐵))
29 sumeq1 14618 . . . . . . . 8 (𝑤 = 𝐵 → Σ𝑘𝑤 𝐶 = Σ𝑘𝐵 𝐶)
3029mpteq2dv 4897 . . . . . . 7 (𝑤 = 𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) = (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶))
31 sumeq1 14618 . . . . . . 7 (𝑤 = 𝐵 → Σ𝑘𝑤 𝐷 = Σ𝑘𝐵 𝐷)
3230, 31breq12d 4817 . . . . . 6 (𝑤 = 𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ⇝𝑟 Σ𝑘𝑤 𝐷 ↔ (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ⇝𝑟 Σ𝑘𝐵 𝐷))
3328, 32imbi12d 333 . . . . 5 (𝑤 = 𝐵 → ((𝑤𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ⇝𝑟 Σ𝑘𝑤 𝐷) ↔ (𝐵𝐵 → (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ⇝𝑟 Σ𝑘𝐵 𝐷)))
3433imbi2d 329 . . . 4 (𝑤 = 𝐵 → ((𝜑 → (𝑤𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ⇝𝑟 Σ𝑘𝑤 𝐷)) ↔ (𝜑 → (𝐵𝐵 → (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ⇝𝑟 Σ𝑘𝐵 𝐷))))
35 fsumrlim.1 . . . . . 6 (𝜑𝐴 ⊆ ℝ)
36 0cn 10224 . . . . . 6 0 ∈ ℂ
37 rlimconst 14474 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 0 ∈ ℂ) → (𝑥𝐴 ↦ 0) ⇝𝑟 0)
3835, 36, 37sylancl 697 . . . . 5 (𝜑 → (𝑥𝐴 ↦ 0) ⇝𝑟 0)
3938a1d 25 . . . 4 (𝜑 → (∅ ⊆ 𝐵 → (𝑥𝐴 ↦ 0) ⇝𝑟 0))
40 ssun1 3919 . . . . . . . . . 10 𝑦 ⊆ (𝑦 ∪ {𝑧})
41 sstr 3752 . . . . . . . . . 10 ((𝑦 ⊆ (𝑦 ∪ {𝑧}) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵) → 𝑦𝐵)
4240, 41mpan 708 . . . . . . . . 9 ((𝑦 ∪ {𝑧}) ⊆ 𝐵𝑦𝐵)
4342imim1i 63 . . . . . . . 8 ((𝑦𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷))
44 sumex 14617 . . . . . . . . . . . . . 14 Σ𝑘𝑦 𝑤 / 𝑥𝐶 ∈ V
4544a1i 11 . . . . . . . . . . . . 13 ((((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) ∧ 𝑤𝐴) → Σ𝑘𝑦 𝑤 / 𝑥𝐶 ∈ V)
46 simprr 813 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∪ {𝑧}) ⊆ 𝐵)
4746unssbd 3934 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → {𝑧} ⊆ 𝐵)
48 vex 3343 . . . . . . . . . . . . . . . . . . . . 21 𝑧 ∈ V
4948snss 4460 . . . . . . . . . . . . . . . . . . . 20 (𝑧𝐵 ↔ {𝑧} ⊆ 𝐵)
5047, 49sylibr 224 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝑧𝐵)
5150adantr 472 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → 𝑧𝐵)
52 fsumrlim.3 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑥𝐴𝑘𝐵)) → 𝐶𝑉)
5352anass1rs 884 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘𝐵) ∧ 𝑥𝐴) → 𝐶𝑉)
54 fsumrlim.4 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘𝐵) → (𝑥𝐴𝐶) ⇝𝑟 𝐷)
5553, 54rlimmptrcl 14537 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘𝐵) ∧ 𝑥𝐴) → 𝐶 ∈ ℂ)
5655an32s 881 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ)
5756adantllr 757 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ)
5857ralrimiva 3104 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → ∀𝑘𝐵 𝐶 ∈ ℂ)
59 nfcsb1v 3690 . . . . . . . . . . . . . . . . . . . 20 𝑘𝑧 / 𝑘𝐶
6059nfel1 2917 . . . . . . . . . . . . . . . . . . 19 𝑘𝑧 / 𝑘𝐶 ∈ ℂ
61 csbeq1a 3683 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑧𝐶 = 𝑧 / 𝑘𝐶)
6261eleq1d 2824 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑧 → (𝐶 ∈ ℂ ↔ 𝑧 / 𝑘𝐶 ∈ ℂ))
6360, 62rspc 3443 . . . . . . . . . . . . . . . . . 18 (𝑧𝐵 → (∀𝑘𝐵 𝐶 ∈ ℂ → 𝑧 / 𝑘𝐶 ∈ ℂ))
6451, 58, 63sylc 65 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → 𝑧 / 𝑘𝐶 ∈ ℂ)
6564ralrimiva 3104 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑥𝐴 𝑧 / 𝑘𝐶 ∈ ℂ)
6665adantr 472 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → ∀𝑥𝐴 𝑧 / 𝑘𝐶 ∈ ℂ)
67 nfcsb1v 3690 . . . . . . . . . . . . . . . . 17 𝑥𝑤 / 𝑥𝑧 / 𝑘𝐶
6867nfel1 2917 . . . . . . . . . . . . . . . 16 𝑥𝑤 / 𝑥𝑧 / 𝑘𝐶 ∈ ℂ
69 csbeq1a 3683 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑤𝑧 / 𝑘𝐶 = 𝑤 / 𝑥𝑧 / 𝑘𝐶)
7069eleq1d 2824 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑤 → (𝑧 / 𝑘𝐶 ∈ ℂ ↔ 𝑤 / 𝑥𝑧 / 𝑘𝐶 ∈ ℂ))
7168, 70rspc 3443 . . . . . . . . . . . . . . 15 (𝑤𝐴 → (∀𝑥𝐴 𝑧 / 𝑘𝐶 ∈ ℂ → 𝑤 / 𝑥𝑧 / 𝑘𝐶 ∈ ℂ))
7266, 71mpan9 487 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) ∧ 𝑤𝐴) → 𝑤 / 𝑥𝑧 / 𝑘𝐶 ∈ ℂ)
73 elex 3352 . . . . . . . . . . . . . 14 (𝑤 / 𝑥𝑧 / 𝑘𝐶 ∈ ℂ → 𝑤 / 𝑥𝑧 / 𝑘𝐶 ∈ V)
7472, 73syl 17 . . . . . . . . . . . . 13 ((((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) ∧ 𝑤𝐴) → 𝑤 / 𝑥𝑧 / 𝑘𝐶 ∈ V)
75 nfcv 2902 . . . . . . . . . . . . . . 15 𝑤Σ𝑘𝑦 𝐶
76 nfcv 2902 . . . . . . . . . . . . . . . 16 𝑥𝑦
77 nfcsb1v 3690 . . . . . . . . . . . . . . . 16 𝑥𝑤 / 𝑥𝐶
7876, 77nfsum 14620 . . . . . . . . . . . . . . 15 𝑥Σ𝑘𝑦 𝑤 / 𝑥𝐶
79 csbeq1a 3683 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑤𝐶 = 𝑤 / 𝑥𝐶)
8079sumeq2sdv 14634 . . . . . . . . . . . . . . 15 (𝑥 = 𝑤 → Σ𝑘𝑦 𝐶 = Σ𝑘𝑦 𝑤 / 𝑥𝐶)
8175, 78, 80cbvmpt 4901 . . . . . . . . . . . . . 14 (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) = (𝑤𝐴 ↦ Σ𝑘𝑦 𝑤 / 𝑥𝐶)
82 simpr 479 . . . . . . . . . . . . . 14 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷)
8381, 82syl5eqbrr 4840 . . . . . . . . . . . . 13 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → (𝑤𝐴 ↦ Σ𝑘𝑦 𝑤 / 𝑥𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷)
84 nfcv 2902 . . . . . . . . . . . . . . 15 𝑤𝑧 / 𝑘𝐶
8584, 67, 69cbvmpt 4901 . . . . . . . . . . . . . 14 (𝑥𝐴𝑧 / 𝑘𝐶) = (𝑤𝐴𝑤 / 𝑥𝑧 / 𝑘𝐶)
8654ralrimiva 3104 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑘𝐵 (𝑥𝐴𝐶) ⇝𝑟 𝐷)
8786adantr 472 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑘𝐵 (𝑥𝐴𝐶) ⇝𝑟 𝐷)
88 nfcv 2902 . . . . . . . . . . . . . . . . . . 19 𝑘𝐴
8988, 59nfmpt 4898 . . . . . . . . . . . . . . . . . 18 𝑘(𝑥𝐴𝑧 / 𝑘𝐶)
90 nfcv 2902 . . . . . . . . . . . . . . . . . 18 𝑘𝑟
91 nfcsb1v 3690 . . . . . . . . . . . . . . . . . 18 𝑘𝑧 / 𝑘𝐷
9289, 90, 91nfbr 4851 . . . . . . . . . . . . . . . . 17 𝑘(𝑥𝐴𝑧 / 𝑘𝐶) ⇝𝑟 𝑧 / 𝑘𝐷
9361mpteq2dv 4897 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑧 → (𝑥𝐴𝐶) = (𝑥𝐴𝑧 / 𝑘𝐶))
94 csbeq1a 3683 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑧𝐷 = 𝑧 / 𝑘𝐷)
9593, 94breq12d 4817 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑧 → ((𝑥𝐴𝐶) ⇝𝑟 𝐷 ↔ (𝑥𝐴𝑧 / 𝑘𝐶) ⇝𝑟 𝑧 / 𝑘𝐷))
9692, 95rspc 3443 . . . . . . . . . . . . . . . 16 (𝑧𝐵 → (∀𝑘𝐵 (𝑥𝐴𝐶) ⇝𝑟 𝐷 → (𝑥𝐴𝑧 / 𝑘𝐶) ⇝𝑟 𝑧 / 𝑘𝐷))
9750, 87, 96sylc 65 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥𝐴𝑧 / 𝑘𝐶) ⇝𝑟 𝑧 / 𝑘𝐷)
9897adantr 472 . . . . . . . . . . . . . 14 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → (𝑥𝐴𝑧 / 𝑘𝐶) ⇝𝑟 𝑧 / 𝑘𝐷)
9985, 98syl5eqbrr 4840 . . . . . . . . . . . . 13 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → (𝑤𝐴𝑤 / 𝑥𝑧 / 𝑘𝐶) ⇝𝑟 𝑧 / 𝑘𝐷)
10045, 74, 83, 99rlimadd 14572 . . . . . . . . . . . 12 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → (𝑤𝐴 ↦ (Σ𝑘𝑦 𝑤 / 𝑥𝐶 + 𝑤 / 𝑥𝑧 / 𝑘𝐶)) ⇝𝑟𝑘𝑦 𝐷 + 𝑧 / 𝑘𝐷))
101 simprl 811 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ¬ 𝑧𝑦)
102 disjsn 4390 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑦)
103101, 102sylibr 224 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∩ {𝑧}) = ∅)
104103adantr 472 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → (𝑦 ∩ {𝑧}) = ∅)
105 eqidd 2761 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
1062adantr 472 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝐵 ∈ Fin)
107 ssfi 8345 . . . . . . . . . . . . . . . . . . 19 ((𝐵 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵) → (𝑦 ∪ {𝑧}) ∈ Fin)
108106, 46, 107syl2anc 696 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∪ {𝑧}) ∈ Fin)
109108adantr 472 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → (𝑦 ∪ {𝑧}) ∈ Fin)
11046sselda 3744 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑘𝐵)
111110adantlr 753 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑘𝐵)
112111, 57syldan 488 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝐶 ∈ ℂ)
113104, 105, 109, 112fsumsplit 14670 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶 = (Σ𝑘𝑦 𝐶 + Σ𝑘 ∈ {𝑧}𝐶))
114 nfcv 2902 . . . . . . . . . . . . . . . . . . 19 𝑤𝐶
115 nfcsb1v 3690 . . . . . . . . . . . . . . . . . . 19 𝑘𝑤 / 𝑘𝐶
116 csbeq1a 3683 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑤𝐶 = 𝑤 / 𝑘𝐶)
117114, 115, 116cbvsumi 14626 . . . . . . . . . . . . . . . . . 18 Σ𝑘 ∈ {𝑧}𝐶 = Σ𝑤 ∈ {𝑧}𝑤 / 𝑘𝐶
118 csbeq1 3677 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑧𝑤 / 𝑘𝐶 = 𝑧 / 𝑘𝐶)
119118sumsn 14674 . . . . . . . . . . . . . . . . . . 19 ((𝑧𝐵𝑧 / 𝑘𝐶 ∈ ℂ) → Σ𝑤 ∈ {𝑧}𝑤 / 𝑘𝐶 = 𝑧 / 𝑘𝐶)
12051, 64, 119syl2anc 696 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → Σ𝑤 ∈ {𝑧}𝑤 / 𝑘𝐶 = 𝑧 / 𝑘𝐶)
121117, 120syl5eq 2806 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → Σ𝑘 ∈ {𝑧}𝐶 = 𝑧 / 𝑘𝐶)
122121oveq2d 6829 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → (Σ𝑘𝑦 𝐶 + Σ𝑘 ∈ {𝑧}𝐶) = (Σ𝑘𝑦 𝐶 + 𝑧 / 𝑘𝐶))
123113, 122eqtrd 2794 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶 = (Σ𝑘𝑦 𝐶 + 𝑧 / 𝑘𝐶))
124123mpteq2dva 4896 . . . . . . . . . . . . . 14 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) = (𝑥𝐴 ↦ (Σ𝑘𝑦 𝐶 + 𝑧 / 𝑘𝐶)))
125124adantr 472 . . . . . . . . . . . . 13 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) = (𝑥𝐴 ↦ (Σ𝑘𝑦 𝐶 + 𝑧 / 𝑘𝐶)))
126 nfcv 2902 . . . . . . . . . . . . . 14 𝑤𝑘𝑦 𝐶 + 𝑧 / 𝑘𝐶)
127 nfcv 2902 . . . . . . . . . . . . . . 15 𝑥 +
12878, 127, 67nfov 6839 . . . . . . . . . . . . . 14 𝑥𝑘𝑦 𝑤 / 𝑥𝐶 + 𝑤 / 𝑥𝑧 / 𝑘𝐶)
12980, 69oveq12d 6831 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (Σ𝑘𝑦 𝐶 + 𝑧 / 𝑘𝐶) = (Σ𝑘𝑦 𝑤 / 𝑥𝐶 + 𝑤 / 𝑥𝑧 / 𝑘𝐶))
130126, 128, 129cbvmpt 4901 . . . . . . . . . . . . 13 (𝑥𝐴 ↦ (Σ𝑘𝑦 𝐶 + 𝑧 / 𝑘𝐶)) = (𝑤𝐴 ↦ (Σ𝑘𝑦 𝑤 / 𝑥𝐶 + 𝑤 / 𝑥𝑧 / 𝑘𝐶))
131125, 130syl6eq 2810 . . . . . . . . . . . 12 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) = (𝑤𝐴 ↦ (Σ𝑘𝑦 𝑤 / 𝑥𝐶 + 𝑤 / 𝑥𝑧 / 𝑘𝐶)))
132 eqidd 2761 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
133 rlimcl 14433 . . . . . . . . . . . . . . . . . 18 ((𝑥𝐴𝐶) ⇝𝑟 𝐷𝐷 ∈ ℂ)
13454, 133syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐵) → 𝐷 ∈ ℂ)
135134adantlr 753 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑘𝐵) → 𝐷 ∈ ℂ)
136110, 135syldan 488 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝐷 ∈ ℂ)
137103, 132, 108, 136fsumsplit 14670 . . . . . . . . . . . . . 14 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷 = (Σ𝑘𝑦 𝐷 + Σ𝑘 ∈ {𝑧}𝐷))
138 nfcv 2902 . . . . . . . . . . . . . . . . 17 𝑤𝐷
139 nfcsb1v 3690 . . . . . . . . . . . . . . . . 17 𝑘𝑤 / 𝑘𝐷
140 csbeq1a 3683 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑤𝐷 = 𝑤 / 𝑘𝐷)
141138, 139, 140cbvsumi 14626 . . . . . . . . . . . . . . . 16 Σ𝑘 ∈ {𝑧}𝐷 = Σ𝑤 ∈ {𝑧}𝑤 / 𝑘𝐷
142135ralrimiva 3104 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑘𝐵 𝐷 ∈ ℂ)
14391nfel1 2917 . . . . . . . . . . . . . . . . . . 19 𝑘𝑧 / 𝑘𝐷 ∈ ℂ
14494eleq1d 2824 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑧 → (𝐷 ∈ ℂ ↔ 𝑧 / 𝑘𝐷 ∈ ℂ))
145143, 144rspc 3443 . . . . . . . . . . . . . . . . . 18 (𝑧𝐵 → (∀𝑘𝐵 𝐷 ∈ ℂ → 𝑧 / 𝑘𝐷 ∈ ℂ))
14650, 142, 145sylc 65 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝑧 / 𝑘𝐷 ∈ ℂ)
147 csbeq1 3677 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑧𝑤 / 𝑘𝐷 = 𝑧 / 𝑘𝐷)
148147sumsn 14674 . . . . . . . . . . . . . . . . 17 ((𝑧𝐵𝑧 / 𝑘𝐷 ∈ ℂ) → Σ𝑤 ∈ {𝑧}𝑤 / 𝑘𝐷 = 𝑧 / 𝑘𝐷)
14950, 146, 148syl2anc 696 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → Σ𝑤 ∈ {𝑧}𝑤 / 𝑘𝐷 = 𝑧 / 𝑘𝐷)
150141, 149syl5eq 2806 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → Σ𝑘 ∈ {𝑧}𝐷 = 𝑧 / 𝑘𝐷)
151150oveq2d 6829 . . . . . . . . . . . . . 14 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (Σ𝑘𝑦 𝐷 + Σ𝑘 ∈ {𝑧}𝐷) = (Σ𝑘𝑦 𝐷 + 𝑧 / 𝑘𝐷))
152137, 151eqtrd 2794 . . . . . . . . . . . . 13 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷 = (Σ𝑘𝑦 𝐷 + 𝑧 / 𝑘𝐷))
153152adantr 472 . . . . . . . . . . . 12 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷 = (Σ𝑘𝑦 𝐷 + 𝑧 / 𝑘𝐷))
154100, 131, 1533brtr4d 4836 . . . . . . . . . . 11 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷)
155154ex 449 . . . . . . . . . 10 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ((𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷 → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷))
156155expr 644 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝑧𝑦) → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷 → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷)))
157156a2d 29 . . . . . . . 8 ((𝜑 ∧ ¬ 𝑧𝑦) → (((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷)))
15843, 157syl5 34 . . . . . . 7 ((𝜑 ∧ ¬ 𝑧𝑦) → ((𝑦𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷)))
159158expcom 450 . . . . . 6 𝑧𝑦 → (𝜑 → ((𝑦𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷))))
160159a2d 29 . . . . 5 𝑧𝑦 → ((𝜑 → (𝑦𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷)) → (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷))))
161160adantl 473 . . . 4 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → ((𝜑 → (𝑦𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷)) → (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷))))
16213, 20, 27, 34, 39, 161findcard2s 8366 . . 3 (𝐵 ∈ Fin → (𝜑 → (𝐵𝐵 → (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ⇝𝑟 Σ𝑘𝐵 𝐷)))
1632, 162mpcom 38 . 2 (𝜑 → (𝐵𝐵 → (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ⇝𝑟 Σ𝑘𝐵 𝐷))
1641, 163mpi 20 1 (𝜑 → (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ⇝𝑟 Σ𝑘𝐵 𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  csb 3674  cun 3713  cin 3714  wss 3715  c0 4058  {csn 4321   class class class wbr 4804  cmpt 4881  (class class class)co 6813  Fincfn 8121  cc 10126  cr 10127  0cc0 10128   + caddc 10131  𝑟 crli 14415  Σcsu 14615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-pre-sup 10206  ax-addf 10207
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-pm 8026  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-sup 8513  df-oi 8580  df-card 8955  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-nn 11213  df-2 11271  df-3 11272  df-n0 11485  df-z 11570  df-uz 11880  df-rp 12026  df-fz 12520  df-fzo 12660  df-seq 12996  df-exp 13055  df-hash 13312  df-cj 14038  df-re 14039  df-im 14040  df-sqrt 14174  df-abs 14175  df-clim 14418  df-rlim 14419  df-sum 14616
This theorem is referenced by:  climfsum  14751  logexprlim  25149  signsplypnf  30936
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