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Theorem fsumrlim 15157
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 3964 . 2 𝐵𝐵
2 fsumrlim.2 . . 3 (𝜑𝐵 ∈ Fin)
3 sseq1 3967 . . . . . 6 (𝑤 = ∅ → (𝑤𝐵 ↔ ∅ ⊆ 𝐵))
4 sumeq1 15036 . . . . . . . . 9 (𝑤 = ∅ → Σ𝑘𝑤 𝐶 = Σ𝑘 ∈ ∅ 𝐶)
5 sum0 15069 . . . . . . . . 9 Σ𝑘 ∈ ∅ 𝐶 = 0
64, 5syl6eq 2873 . . . . . . . 8 (𝑤 = ∅ → Σ𝑘𝑤 𝐶 = 0)
76mpteq2dv 5138 . . . . . . 7 (𝑤 = ∅ → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) = (𝑥𝐴 ↦ 0))
8 sumeq1 15036 . . . . . . . 8 (𝑤 = ∅ → Σ𝑘𝑤 𝐷 = Σ𝑘 ∈ ∅ 𝐷)
9 sum0 15069 . . . . . . . 8 Σ𝑘 ∈ ∅ 𝐷 = 0
108, 9syl6eq 2873 . . . . . . 7 (𝑤 = ∅ → Σ𝑘𝑤 𝐷 = 0)
117, 10breq12d 5055 . . . . . 6 (𝑤 = ∅ → ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ⇝𝑟 Σ𝑘𝑤 𝐷 ↔ (𝑥𝐴 ↦ 0) ⇝𝑟 0))
123, 11imbi12d 348 . . . . 5 (𝑤 = ∅ → ((𝑤𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ⇝𝑟 Σ𝑘𝑤 𝐷) ↔ (∅ ⊆ 𝐵 → (𝑥𝐴 ↦ 0) ⇝𝑟 0)))
1312imbi2d 344 . . . 4 (𝑤 = ∅ → ((𝜑 → (𝑤𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ⇝𝑟 Σ𝑘𝑤 𝐷)) ↔ (𝜑 → (∅ ⊆ 𝐵 → (𝑥𝐴 ↦ 0) ⇝𝑟 0))))
14 sseq1 3967 . . . . . 6 (𝑤 = 𝑦 → (𝑤𝐵𝑦𝐵))
15 sumeq1 15036 . . . . . . . 8 (𝑤 = 𝑦 → Σ𝑘𝑤 𝐶 = Σ𝑘𝑦 𝐶)
1615mpteq2dv 5138 . . . . . . 7 (𝑤 = 𝑦 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) = (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶))
17 sumeq1 15036 . . . . . . 7 (𝑤 = 𝑦 → Σ𝑘𝑤 𝐷 = Σ𝑘𝑦 𝐷)
1816, 17breq12d 5055 . . . . . 6 (𝑤 = 𝑦 → ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ⇝𝑟 Σ𝑘𝑤 𝐷 ↔ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷))
1914, 18imbi12d 348 . . . . 5 (𝑤 = 𝑦 → ((𝑤𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ⇝𝑟 Σ𝑘𝑤 𝐷) ↔ (𝑦𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷)))
2019imbi2d 344 . . . 4 (𝑤 = 𝑦 → ((𝜑 → (𝑤𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ⇝𝑟 Σ𝑘𝑤 𝐷)) ↔ (𝜑 → (𝑦𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷))))
21 sseq1 3967 . . . . . 6 (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤𝐵 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝐵))
22 sumeq1 15036 . . . . . . . 8 (𝑤 = (𝑦 ∪ {𝑧}) → Σ𝑘𝑤 𝐶 = Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶)
2322mpteq2dv 5138 . . . . . . 7 (𝑤 = (𝑦 ∪ {𝑧}) → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) = (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶))
24 sumeq1 15036 . . . . . . 7 (𝑤 = (𝑦 ∪ {𝑧}) → Σ𝑘𝑤 𝐷 = Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷)
2523, 24breq12d 5055 . . . . . 6 (𝑤 = (𝑦 ∪ {𝑧}) → ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ⇝𝑟 Σ𝑘𝑤 𝐷 ↔ (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷))
2621, 25imbi12d 348 . . . . 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 344 . . . 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 3967 . . . . . 6 (𝑤 = 𝐵 → (𝑤𝐵𝐵𝐵))
29 sumeq1 15036 . . . . . . . 8 (𝑤 = 𝐵 → Σ𝑘𝑤 𝐶 = Σ𝑘𝐵 𝐶)
3029mpteq2dv 5138 . . . . . . 7 (𝑤 = 𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) = (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶))
31 sumeq1 15036 . . . . . . 7 (𝑤 = 𝐵 → Σ𝑘𝑤 𝐷 = Σ𝑘𝐵 𝐷)
3230, 31breq12d 5055 . . . . . 6 (𝑤 = 𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ⇝𝑟 Σ𝑘𝑤 𝐷 ↔ (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ⇝𝑟 Σ𝑘𝐵 𝐷))
3328, 32imbi12d 348 . . . . 5 (𝑤 = 𝐵 → ((𝑤𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ⇝𝑟 Σ𝑘𝑤 𝐷) ↔ (𝐵𝐵 → (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ⇝𝑟 Σ𝑘𝐵 𝐷)))
3433imbi2d 344 . . . 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 10622 . . . . . 6 0 ∈ ℂ
37 rlimconst 14892 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 0 ∈ ℂ) → (𝑥𝐴 ↦ 0) ⇝𝑟 0)
3835, 36, 37sylancl 589 . . . . 5 (𝜑 → (𝑥𝐴 ↦ 0) ⇝𝑟 0)
3938a1d 25 . . . 4 (𝜑 → (∅ ⊆ 𝐵 → (𝑥𝐴 ↦ 0) ⇝𝑟 0))
40 ssun1 4123 . . . . . . . . . 10 𝑦 ⊆ (𝑦 ∪ {𝑧})
41 sstr 3950 . . . . . . . . . 10 ((𝑦 ⊆ (𝑦 ∪ {𝑧}) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵) → 𝑦𝐵)
4240, 41mpan 689 . . . . . . . . 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 15035 . . . . . . . . . . . . . 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 772 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∪ {𝑧}) ⊆ 𝐵)
4746unssbd 4139 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → {𝑧} ⊆ 𝐵)
48 vex 3472 . . . . . . . . . . . . . . . . . . . . 21 𝑧 ∈ V
4948snss 4692 . . . . . . . . . . . . . . . . . . . 20 (𝑧𝐵 ↔ {𝑧} ⊆ 𝐵)
5047, 49sylibr 237 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝑧𝐵)
5150adantr 484 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → 𝑧𝐵)
52 fsumrlim.3 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑥𝐴𝑘𝐵)) → 𝐶𝑉)
5352anass1rs 654 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘𝐵) ∧ 𝑥𝐴) → 𝐶𝑉)
54 fsumrlim.4 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘𝐵) → (𝑥𝐴𝐶) ⇝𝑟 𝐷)
5553, 54rlimmptrcl 14955 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘𝐵) ∧ 𝑥𝐴) → 𝐶 ∈ ℂ)
5655an32s 651 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ)
5756adantllr 718 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ)
5857ralrimiva 3174 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → ∀𝑘𝐵 𝐶 ∈ ℂ)
59 nfcsb1v 3879 . . . . . . . . . . . . . . . . . . . 20 𝑘𝑧 / 𝑘𝐶
6059nfel1 2995 . . . . . . . . . . . . . . . . . . 19 𝑘𝑧 / 𝑘𝐶 ∈ ℂ
61 csbeq1a 3869 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑧𝐶 = 𝑧 / 𝑘𝐶)
6261eleq1d 2898 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑧 → (𝐶 ∈ ℂ ↔ 𝑧 / 𝑘𝐶 ∈ ℂ))
6360, 62rspc 3586 . . . . . . . . . . . . . . . . . 18 (𝑧𝐵 → (∀𝑘𝐵 𝐶 ∈ ℂ → 𝑧 / 𝑘𝐶 ∈ ℂ))
6451, 58, 63sylc 65 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → 𝑧 / 𝑘𝐶 ∈ ℂ)
6564ralrimiva 3174 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑥𝐴 𝑧 / 𝑘𝐶 ∈ ℂ)
6665adantr 484 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → ∀𝑥𝐴 𝑧 / 𝑘𝐶 ∈ ℂ)
67 nfcsb1v 3879 . . . . . . . . . . . . . . . . 17 𝑥𝑤 / 𝑥𝑧 / 𝑘𝐶
6867nfel1 2995 . . . . . . . . . . . . . . . 16 𝑥𝑤 / 𝑥𝑧 / 𝑘𝐶 ∈ ℂ
69 csbeq1a 3869 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑤𝑧 / 𝑘𝐶 = 𝑤 / 𝑥𝑧 / 𝑘𝐶)
7069eleq1d 2898 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑤 → (𝑧 / 𝑘𝐶 ∈ ℂ ↔ 𝑤 / 𝑥𝑧 / 𝑘𝐶 ∈ ℂ))
7168, 70rspc 3586 . . . . . . . . . . . . . . 15 (𝑤𝐴 → (∀𝑥𝐴 𝑧 / 𝑘𝐶 ∈ ℂ → 𝑤 / 𝑥𝑧 / 𝑘𝐶 ∈ ℂ))
7266, 71mpan9 510 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) ∧ 𝑤𝐴) → 𝑤 / 𝑥𝑧 / 𝑘𝐶 ∈ ℂ)
7372elexd 3489 . . . . . . . . . . . . 13 ((((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) ∧ 𝑤𝐴) → 𝑤 / 𝑥𝑧 / 𝑘𝐶 ∈ V)
74 nfcv 2979 . . . . . . . . . . . . . . 15 𝑤Σ𝑘𝑦 𝐶
75 nfcv 2979 . . . . . . . . . . . . . . . 16 𝑥𝑦
76 nfcsb1v 3879 . . . . . . . . . . . . . . . 16 𝑥𝑤 / 𝑥𝐶
7775, 76nfsum 15038 . . . . . . . . . . . . . . 15 𝑥Σ𝑘𝑦 𝑤 / 𝑥𝐶
78 csbeq1a 3869 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑤𝐶 = 𝑤 / 𝑥𝐶)
7978sumeq2sdv 15052 . . . . . . . . . . . . . . 15 (𝑥 = 𝑤 → Σ𝑘𝑦 𝐶 = Σ𝑘𝑦 𝑤 / 𝑥𝐶)
8074, 77, 79cbvmpt 5143 . . . . . . . . . . . . . 14 (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) = (𝑤𝐴 ↦ Σ𝑘𝑦 𝑤 / 𝑥𝐶)
81 simpr 488 . . . . . . . . . . . . . 14 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷)
8280, 81eqbrtrrid 5078 . . . . . . . . . . . . 13 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → (𝑤𝐴 ↦ Σ𝑘𝑦 𝑤 / 𝑥𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷)
83 nfcv 2979 . . . . . . . . . . . . . . 15 𝑤𝑧 / 𝑘𝐶
8483, 67, 69cbvmpt 5143 . . . . . . . . . . . . . 14 (𝑥𝐴𝑧 / 𝑘𝐶) = (𝑤𝐴𝑤 / 𝑥𝑧 / 𝑘𝐶)
8554ralrimiva 3174 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑘𝐵 (𝑥𝐴𝐶) ⇝𝑟 𝐷)
8685adantr 484 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑘𝐵 (𝑥𝐴𝐶) ⇝𝑟 𝐷)
87 nfcv 2979 . . . . . . . . . . . . . . . . . . 19 𝑘𝐴
8887, 59nfmpt 5139 . . . . . . . . . . . . . . . . . 18 𝑘(𝑥𝐴𝑧 / 𝑘𝐶)
89 nfcv 2979 . . . . . . . . . . . . . . . . . 18 𝑘𝑟
90 nfcsb1v 3879 . . . . . . . . . . . . . . . . . 18 𝑘𝑧 / 𝑘𝐷
9188, 89, 90nfbr 5089 . . . . . . . . . . . . . . . . 17 𝑘(𝑥𝐴𝑧 / 𝑘𝐶) ⇝𝑟 𝑧 / 𝑘𝐷
9261mpteq2dv 5138 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑧 → (𝑥𝐴𝐶) = (𝑥𝐴𝑧 / 𝑘𝐶))
93 csbeq1a 3869 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑧𝐷 = 𝑧 / 𝑘𝐷)
9492, 93breq12d 5055 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑧 → ((𝑥𝐴𝐶) ⇝𝑟 𝐷 ↔ (𝑥𝐴𝑧 / 𝑘𝐶) ⇝𝑟 𝑧 / 𝑘𝐷))
9591, 94rspc 3586 . . . . . . . . . . . . . . . 16 (𝑧𝐵 → (∀𝑘𝐵 (𝑥𝐴𝐶) ⇝𝑟 𝐷 → (𝑥𝐴𝑧 / 𝑘𝐶) ⇝𝑟 𝑧 / 𝑘𝐷))
9650, 86, 95sylc 65 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥𝐴𝑧 / 𝑘𝐶) ⇝𝑟 𝑧 / 𝑘𝐷)
9796adantr 484 . . . . . . . . . . . . . 14 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → (𝑥𝐴𝑧 / 𝑘𝐶) ⇝𝑟 𝑧 / 𝑘𝐷)
9884, 97eqbrtrrid 5078 . . . . . . . . . . . . 13 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → (𝑤𝐴𝑤 / 𝑥𝑧 / 𝑘𝐶) ⇝𝑟 𝑧 / 𝑘𝐷)
9945, 73, 82, 98rlimadd 14990 . . . . . . . . . . . 12 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → (𝑤𝐴 ↦ (Σ𝑘𝑦 𝑤 / 𝑥𝐶 + 𝑤 / 𝑥𝑧 / 𝑘𝐶)) ⇝𝑟𝑘𝑦 𝐷 + 𝑧 / 𝑘𝐷))
100 simprl 770 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ¬ 𝑧𝑦)
101 disjsn 4621 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑦)
102100, 101sylibr 237 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∩ {𝑧}) = ∅)
103102adantr 484 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → (𝑦 ∩ {𝑧}) = ∅)
104 eqidd 2823 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
1052adantr 484 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝐵 ∈ Fin)
106105, 46ssfid 8729 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∪ {𝑧}) ∈ Fin)
107106adantr 484 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → (𝑦 ∪ {𝑧}) ∈ Fin)
10846sselda 3942 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑘𝐵)
109108adantlr 714 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑘𝐵)
110109, 57syldan 594 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝐶 ∈ ℂ)
111103, 104, 107, 110fsumsplit 15088 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶 = (Σ𝑘𝑦 𝐶 + Σ𝑘 ∈ {𝑧}𝐶))
112 nfcv 2979 . . . . . . . . . . . . . . . . . . 19 𝑤𝐶
113 nfcsb1v 3879 . . . . . . . . . . . . . . . . . . 19 𝑘𝑤 / 𝑘𝐶
114 csbeq1a 3869 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑤𝐶 = 𝑤 / 𝑘𝐶)
115112, 113, 114cbvsumi 15045 . . . . . . . . . . . . . . . . . 18 Σ𝑘 ∈ {𝑧}𝐶 = Σ𝑤 ∈ {𝑧}𝑤 / 𝑘𝐶
116 csbeq1 3858 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑧𝑤 / 𝑘𝐶 = 𝑧 / 𝑘𝐶)
117116sumsn 15092 . . . . . . . . . . . . . . . . . . 19 ((𝑧𝐵𝑧 / 𝑘𝐶 ∈ ℂ) → Σ𝑤 ∈ {𝑧}𝑤 / 𝑘𝐶 = 𝑧 / 𝑘𝐶)
11851, 64, 117syl2anc 587 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → Σ𝑤 ∈ {𝑧}𝑤 / 𝑘𝐶 = 𝑧 / 𝑘𝐶)
119115, 118syl5eq 2869 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → Σ𝑘 ∈ {𝑧}𝐶 = 𝑧 / 𝑘𝐶)
120119oveq2d 7156 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → (Σ𝑘𝑦 𝐶 + Σ𝑘 ∈ {𝑧}𝐶) = (Σ𝑘𝑦 𝐶 + 𝑧 / 𝑘𝐶))
121111, 120eqtrd 2857 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶 = (Σ𝑘𝑦 𝐶 + 𝑧 / 𝑘𝐶))
122121mpteq2dva 5137 . . . . . . . . . . . . . 14 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) = (𝑥𝐴 ↦ (Σ𝑘𝑦 𝐶 + 𝑧 / 𝑘𝐶)))
123122adantr 484 . . . . . . . . . . . . 13 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) = (𝑥𝐴 ↦ (Σ𝑘𝑦 𝐶 + 𝑧 / 𝑘𝐶)))
124 nfcv 2979 . . . . . . . . . . . . . 14 𝑤𝑘𝑦 𝐶 + 𝑧 / 𝑘𝐶)
125 nfcv 2979 . . . . . . . . . . . . . . 15 𝑥 +
12677, 125, 67nfov 7170 . . . . . . . . . . . . . 14 𝑥𝑘𝑦 𝑤 / 𝑥𝐶 + 𝑤 / 𝑥𝑧 / 𝑘𝐶)
12779, 69oveq12d 7158 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (Σ𝑘𝑦 𝐶 + 𝑧 / 𝑘𝐶) = (Σ𝑘𝑦 𝑤 / 𝑥𝐶 + 𝑤 / 𝑥𝑧 / 𝑘𝐶))
128124, 126, 127cbvmpt 5143 . . . . . . . . . . . . 13 (𝑥𝐴 ↦ (Σ𝑘𝑦 𝐶 + 𝑧 / 𝑘𝐶)) = (𝑤𝐴 ↦ (Σ𝑘𝑦 𝑤 / 𝑥𝐶 + 𝑤 / 𝑥𝑧 / 𝑘𝐶))
129123, 128syl6eq 2873 . . . . . . . . . . . 12 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) = (𝑤𝐴 ↦ (Σ𝑘𝑦 𝑤 / 𝑥𝐶 + 𝑤 / 𝑥𝑧 / 𝑘𝐶)))
130 eqidd 2823 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
131 rlimcl 14851 . . . . . . . . . . . . . . . . . 18 ((𝑥𝐴𝐶) ⇝𝑟 𝐷𝐷 ∈ ℂ)
13254, 131syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐵) → 𝐷 ∈ ℂ)
133132adantlr 714 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑘𝐵) → 𝐷 ∈ ℂ)
134108, 133syldan 594 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝐷 ∈ ℂ)
135102, 130, 106, 134fsumsplit 15088 . . . . . . . . . . . . . 14 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷 = (Σ𝑘𝑦 𝐷 + Σ𝑘 ∈ {𝑧}𝐷))
136 nfcv 2979 . . . . . . . . . . . . . . . . 17 𝑤𝐷
137 nfcsb1v 3879 . . . . . . . . . . . . . . . . 17 𝑘𝑤 / 𝑘𝐷
138 csbeq1a 3869 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑤𝐷 = 𝑤 / 𝑘𝐷)
139136, 137, 138cbvsumi 15045 . . . . . . . . . . . . . . . 16 Σ𝑘 ∈ {𝑧}𝐷 = Σ𝑤 ∈ {𝑧}𝑤 / 𝑘𝐷
140133ralrimiva 3174 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑘𝐵 𝐷 ∈ ℂ)
14190nfel1 2995 . . . . . . . . . . . . . . . . . . 19 𝑘𝑧 / 𝑘𝐷 ∈ ℂ
14293eleq1d 2898 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑧 → (𝐷 ∈ ℂ ↔ 𝑧 / 𝑘𝐷 ∈ ℂ))
143141, 142rspc 3586 . . . . . . . . . . . . . . . . . 18 (𝑧𝐵 → (∀𝑘𝐵 𝐷 ∈ ℂ → 𝑧 / 𝑘𝐷 ∈ ℂ))
14450, 140, 143sylc 65 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝑧 / 𝑘𝐷 ∈ ℂ)
145 csbeq1 3858 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑧𝑤 / 𝑘𝐷 = 𝑧 / 𝑘𝐷)
146145sumsn 15092 . . . . . . . . . . . . . . . . 17 ((𝑧𝐵𝑧 / 𝑘𝐷 ∈ ℂ) → Σ𝑤 ∈ {𝑧}𝑤 / 𝑘𝐷 = 𝑧 / 𝑘𝐷)
14750, 144, 146syl2anc 587 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → Σ𝑤 ∈ {𝑧}𝑤 / 𝑘𝐷 = 𝑧 / 𝑘𝐷)
148139, 147syl5eq 2869 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → Σ𝑘 ∈ {𝑧}𝐷 = 𝑧 / 𝑘𝐷)
149148oveq2d 7156 . . . . . . . . . . . . . 14 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (Σ𝑘𝑦 𝐷 + Σ𝑘 ∈ {𝑧}𝐷) = (Σ𝑘𝑦 𝐷 + 𝑧 / 𝑘𝐷))
150135, 149eqtrd 2857 . . . . . . . . . . . . 13 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷 = (Σ𝑘𝑦 𝐷 + 𝑧 / 𝑘𝐷))
151150adantr 484 . . . . . . . . . . . 12 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷 = (Σ𝑘𝑦 𝐷 + 𝑧 / 𝑘𝐷))
15299, 129, 1513brtr4d 5074 . . . . . . . . . . 11 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷)
153152ex 416 . . . . . . . . . 10 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ((𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷 → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷))
154153expr 460 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝑧𝑦) → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷 → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷)))
155154a2d 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)))
15643, 155syl5 34 . . . . . . 7 ((𝜑 ∧ ¬ 𝑧𝑦) → ((𝑦𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷)))
157156expcom 417 . . . . . 6 𝑧𝑦 → (𝜑 → ((𝑦𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷))))
158157a2d 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))))
159158adantl 485 . . . 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))))
16013, 20, 27, 34, 39, 159findcard2s 8747 . . 3 (𝐵 ∈ Fin → (𝜑 → (𝐵𝐵 → (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ⇝𝑟 Σ𝑘𝐵 𝐷)))
1612, 160mpcom 38 . 2 (𝜑 → (𝐵𝐵 → (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ⇝𝑟 Σ𝑘𝐵 𝐷))
1621, 161mpi 20 1 (𝜑 → (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ⇝𝑟 Σ𝑘𝐵 𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2114  wral 3130  Vcvv 3469  csb 3855  cun 3906  cin 3907  wss 3908  c0 4265  {csn 4539   class class class wbr 5042  cmpt 5122  (class class class)co 7140  Fincfn 8496  cc 10524  cr 10525  0cc0 10526   + caddc 10529  𝑟 crli 14833  Σcsu 15033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-inf2 9092  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604  ax-addf 10605
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-se 5492  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-isom 6343  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-pm 8396  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-sup 8894  df-oi 8962  df-card 9356  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12886  df-fzo 13029  df-seq 13365  df-exp 13426  df-hash 13687  df-cj 14449  df-re 14450  df-im 14451  df-sqrt 14585  df-abs 14586  df-clim 14836  df-rlim 14837  df-sum 15034
This theorem is referenced by:  climfsum  15166  logexprlim  25807  signsplypnf  31894
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