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Theorem fsumrlim 15762
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 4004 . 2 𝐵𝐵
2 fsumrlim.2 . . 3 (𝜑𝐵 ∈ Fin)
3 sseq1 4007 . . . . . 6 (𝑤 = ∅ → (𝑤𝐵 ↔ ∅ ⊆ 𝐵))
4 sumeq1 15640 . . . . . . . . 9 (𝑤 = ∅ → Σ𝑘𝑤 𝐶 = Σ𝑘 ∈ ∅ 𝐶)
5 sum0 15672 . . . . . . . . 9 Σ𝑘 ∈ ∅ 𝐶 = 0
64, 5eqtrdi 2787 . . . . . . . 8 (𝑤 = ∅ → Σ𝑘𝑤 𝐶 = 0)
76mpteq2dv 5250 . . . . . . 7 (𝑤 = ∅ → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) = (𝑥𝐴 ↦ 0))
8 sumeq1 15640 . . . . . . . 8 (𝑤 = ∅ → Σ𝑘𝑤 𝐷 = Σ𝑘 ∈ ∅ 𝐷)
9 sum0 15672 . . . . . . . 8 Σ𝑘 ∈ ∅ 𝐷 = 0
108, 9eqtrdi 2787 . . . . . . 7 (𝑤 = ∅ → Σ𝑘𝑤 𝐷 = 0)
117, 10breq12d 5161 . . . . . 6 (𝑤 = ∅ → ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ⇝𝑟 Σ𝑘𝑤 𝐷 ↔ (𝑥𝐴 ↦ 0) ⇝𝑟 0))
123, 11imbi12d 344 . . . . 5 (𝑤 = ∅ → ((𝑤𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ⇝𝑟 Σ𝑘𝑤 𝐷) ↔ (∅ ⊆ 𝐵 → (𝑥𝐴 ↦ 0) ⇝𝑟 0)))
1312imbi2d 340 . . . 4 (𝑤 = ∅ → ((𝜑 → (𝑤𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ⇝𝑟 Σ𝑘𝑤 𝐷)) ↔ (𝜑 → (∅ ⊆ 𝐵 → (𝑥𝐴 ↦ 0) ⇝𝑟 0))))
14 sseq1 4007 . . . . . 6 (𝑤 = 𝑦 → (𝑤𝐵𝑦𝐵))
15 sumeq1 15640 . . . . . . . 8 (𝑤 = 𝑦 → Σ𝑘𝑤 𝐶 = Σ𝑘𝑦 𝐶)
1615mpteq2dv 5250 . . . . . . 7 (𝑤 = 𝑦 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) = (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶))
17 sumeq1 15640 . . . . . . 7 (𝑤 = 𝑦 → Σ𝑘𝑤 𝐷 = Σ𝑘𝑦 𝐷)
1816, 17breq12d 5161 . . . . . 6 (𝑤 = 𝑦 → ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ⇝𝑟 Σ𝑘𝑤 𝐷 ↔ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷))
1914, 18imbi12d 344 . . . . 5 (𝑤 = 𝑦 → ((𝑤𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ⇝𝑟 Σ𝑘𝑤 𝐷) ↔ (𝑦𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷)))
2019imbi2d 340 . . . 4 (𝑤 = 𝑦 → ((𝜑 → (𝑤𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ⇝𝑟 Σ𝑘𝑤 𝐷)) ↔ (𝜑 → (𝑦𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷))))
21 sseq1 4007 . . . . . 6 (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤𝐵 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝐵))
22 sumeq1 15640 . . . . . . . 8 (𝑤 = (𝑦 ∪ {𝑧}) → Σ𝑘𝑤 𝐶 = Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶)
2322mpteq2dv 5250 . . . . . . 7 (𝑤 = (𝑦 ∪ {𝑧}) → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) = (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶))
24 sumeq1 15640 . . . . . . 7 (𝑤 = (𝑦 ∪ {𝑧}) → Σ𝑘𝑤 𝐷 = Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷)
2523, 24breq12d 5161 . . . . . 6 (𝑤 = (𝑦 ∪ {𝑧}) → ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ⇝𝑟 Σ𝑘𝑤 𝐷 ↔ (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷))
2621, 25imbi12d 344 . . . . 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 340 . . . 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 4007 . . . . . 6 (𝑤 = 𝐵 → (𝑤𝐵𝐵𝐵))
29 sumeq1 15640 . . . . . . . 8 (𝑤 = 𝐵 → Σ𝑘𝑤 𝐶 = Σ𝑘𝐵 𝐶)
3029mpteq2dv 5250 . . . . . . 7 (𝑤 = 𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) = (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶))
31 sumeq1 15640 . . . . . . 7 (𝑤 = 𝐵 → Σ𝑘𝑤 𝐷 = Σ𝑘𝐵 𝐷)
3230, 31breq12d 5161 . . . . . 6 (𝑤 = 𝐵 → ((𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ⇝𝑟 Σ𝑘𝑤 𝐷 ↔ (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ⇝𝑟 Σ𝑘𝐵 𝐷))
3328, 32imbi12d 344 . . . . 5 (𝑤 = 𝐵 → ((𝑤𝐵 → (𝑥𝐴 ↦ Σ𝑘𝑤 𝐶) ⇝𝑟 Σ𝑘𝑤 𝐷) ↔ (𝐵𝐵 → (𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ⇝𝑟 Σ𝑘𝐵 𝐷)))
3433imbi2d 340 . . . 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 11211 . . . . . 6 0 ∈ ℂ
37 rlimconst 15493 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 0 ∈ ℂ) → (𝑥𝐴 ↦ 0) ⇝𝑟 0)
3835, 36, 37sylancl 585 . . . . 5 (𝜑 → (𝑥𝐴 ↦ 0) ⇝𝑟 0)
3938a1d 25 . . . 4 (𝜑 → (∅ ⊆ 𝐵 → (𝑥𝐴 ↦ 0) ⇝𝑟 0))
40 ssun1 4172 . . . . . . . . . 10 𝑦 ⊆ (𝑦 ∪ {𝑧})
41 sstr 3990 . . . . . . . . . 10 ((𝑦 ⊆ (𝑦 ∪ {𝑧}) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵) → 𝑦𝐵)
4240, 41mpan 687 . . . . . . . . 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 15639 . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∪ {𝑧}) ⊆ 𝐵)
4746unssbd 4188 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → {𝑧} ⊆ 𝐵)
48 vex 3477 . . . . . . . . . . . . . . . . . . . . 21 𝑧 ∈ V
4948snss 4789 . . . . . . . . . . . . . . . . . . . 20 (𝑧𝐵 ↔ {𝑧} ⊆ 𝐵)
5047, 49sylibr 233 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝑧𝐵)
5150adantr 480 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → 𝑧𝐵)
52 fsumrlim.3 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑥𝐴𝑘𝐵)) → 𝐶𝑉)
5352anass1rs 652 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘𝐵) ∧ 𝑥𝐴) → 𝐶𝑉)
54 fsumrlim.4 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘𝐵) → (𝑥𝐴𝐶) ⇝𝑟 𝐷)
5553, 54rlimmptrcl 15557 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘𝐵) ∧ 𝑥𝐴) → 𝐶 ∈ ℂ)
5655an32s 649 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ)
5756adantllr 716 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ ℂ)
5857ralrimiva 3145 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → ∀𝑘𝐵 𝐶 ∈ ℂ)
59 nfcsb1v 3918 . . . . . . . . . . . . . . . . . . . 20 𝑘𝑧 / 𝑘𝐶
6059nfel1 2918 . . . . . . . . . . . . . . . . . . 19 𝑘𝑧 / 𝑘𝐶 ∈ ℂ
61 csbeq1a 3907 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑧𝐶 = 𝑧 / 𝑘𝐶)
6261eleq1d 2817 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑧 → (𝐶 ∈ ℂ ↔ 𝑧 / 𝑘𝐶 ∈ ℂ))
6360, 62rspc 3600 . . . . . . . . . . . . . . . . . 18 (𝑧𝐵 → (∀𝑘𝐵 𝐶 ∈ ℂ → 𝑧 / 𝑘𝐶 ∈ ℂ))
6451, 58, 63sylc 65 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → 𝑧 / 𝑘𝐶 ∈ ℂ)
6564ralrimiva 3145 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑥𝐴 𝑧 / 𝑘𝐶 ∈ ℂ)
6665adantr 480 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → ∀𝑥𝐴 𝑧 / 𝑘𝐶 ∈ ℂ)
67 nfcsb1v 3918 . . . . . . . . . . . . . . . . 17 𝑥𝑤 / 𝑥𝑧 / 𝑘𝐶
6867nfel1 2918 . . . . . . . . . . . . . . . 16 𝑥𝑤 / 𝑥𝑧 / 𝑘𝐶 ∈ ℂ
69 csbeq1a 3907 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑤𝑧 / 𝑘𝐶 = 𝑤 / 𝑥𝑧 / 𝑘𝐶)
7069eleq1d 2817 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑤 → (𝑧 / 𝑘𝐶 ∈ ℂ ↔ 𝑤 / 𝑥𝑧 / 𝑘𝐶 ∈ ℂ))
7168, 70rspc 3600 . . . . . . . . . . . . . . 15 (𝑤𝐴 → (∀𝑥𝐴 𝑧 / 𝑘𝐶 ∈ ℂ → 𝑤 / 𝑥𝑧 / 𝑘𝐶 ∈ ℂ))
7266, 71mpan9 506 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) ∧ 𝑤𝐴) → 𝑤 / 𝑥𝑧 / 𝑘𝐶 ∈ ℂ)
7372elexd 3494 . . . . . . . . . . . . 13 ((((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) ∧ 𝑤𝐴) → 𝑤 / 𝑥𝑧 / 𝑘𝐶 ∈ V)
74 nfcv 2902 . . . . . . . . . . . . . . 15 𝑤Σ𝑘𝑦 𝐶
75 nfcv 2902 . . . . . . . . . . . . . . . 16 𝑥𝑦
76 nfcsb1v 3918 . . . . . . . . . . . . . . . 16 𝑥𝑤 / 𝑥𝐶
7775, 76nfsum 15642 . . . . . . . . . . . . . . 15 𝑥Σ𝑘𝑦 𝑤 / 𝑥𝐶
78 csbeq1a 3907 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑤𝐶 = 𝑤 / 𝑥𝐶)
7978sumeq2sdv 15655 . . . . . . . . . . . . . . 15 (𝑥 = 𝑤 → Σ𝑘𝑦 𝐶 = Σ𝑘𝑦 𝑤 / 𝑥𝐶)
8074, 77, 79cbvmpt 5259 . . . . . . . . . . . . . 14 (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) = (𝑤𝐴 ↦ Σ𝑘𝑦 𝑤 / 𝑥𝐶)
81 simpr 484 . . . . . . . . . . . . . 14 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷)
8280, 81eqbrtrrid 5184 . . . . . . . . . . . . 13 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → (𝑤𝐴 ↦ Σ𝑘𝑦 𝑤 / 𝑥𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷)
83 nfcv 2902 . . . . . . . . . . . . . . 15 𝑤𝑧 / 𝑘𝐶
8483, 67, 69cbvmpt 5259 . . . . . . . . . . . . . 14 (𝑥𝐴𝑧 / 𝑘𝐶) = (𝑤𝐴𝑤 / 𝑥𝑧 / 𝑘𝐶)
8554ralrimiva 3145 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑘𝐵 (𝑥𝐴𝐶) ⇝𝑟 𝐷)
8685adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑘𝐵 (𝑥𝐴𝐶) ⇝𝑟 𝐷)
87 nfcv 2902 . . . . . . . . . . . . . . . . . . 19 𝑘𝐴
8887, 59nfmpt 5255 . . . . . . . . . . . . . . . . . 18 𝑘(𝑥𝐴𝑧 / 𝑘𝐶)
89 nfcv 2902 . . . . . . . . . . . . . . . . . 18 𝑘𝑟
90 nfcsb1v 3918 . . . . . . . . . . . . . . . . . 18 𝑘𝑧 / 𝑘𝐷
9188, 89, 90nfbr 5195 . . . . . . . . . . . . . . . . 17 𝑘(𝑥𝐴𝑧 / 𝑘𝐶) ⇝𝑟 𝑧 / 𝑘𝐷
9261mpteq2dv 5250 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑧 → (𝑥𝐴𝐶) = (𝑥𝐴𝑧 / 𝑘𝐶))
93 csbeq1a 3907 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑧𝐷 = 𝑧 / 𝑘𝐷)
9492, 93breq12d 5161 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑧 → ((𝑥𝐴𝐶) ⇝𝑟 𝐷 ↔ (𝑥𝐴𝑧 / 𝑘𝐶) ⇝𝑟 𝑧 / 𝑘𝐷))
9591, 94rspc 3600 . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . 14 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → (𝑥𝐴𝑧 / 𝑘𝐶) ⇝𝑟 𝑧 / 𝑘𝐷)
9884, 97eqbrtrrid 5184 . . . . . . . . . . . . 13 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → (𝑤𝐴𝑤 / 𝑥𝑧 / 𝑘𝐶) ⇝𝑟 𝑧 / 𝑘𝐷)
9945, 73, 82, 98rlimadd 15592 . . . . . . . . . . . 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 768 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ¬ 𝑧𝑦)
101 disjsn 4715 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑦)
102100, 101sylibr 233 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∩ {𝑧}) = ∅)
103102adantr 480 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → (𝑦 ∩ {𝑧}) = ∅)
104 eqidd 2732 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
1052adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝐵 ∈ Fin)
106105, 46ssfid 9270 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∪ {𝑧}) ∈ Fin)
107106adantr 480 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → (𝑦 ∪ {𝑧}) ∈ Fin)
10846sselda 3982 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑘𝐵)
109108adantlr 712 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑘𝐵)
110109, 57syldan 590 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝐶 ∈ ℂ)
111103, 104, 107, 110fsumsplit 15692 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶 = (Σ𝑘𝑦 𝐶 + Σ𝑘 ∈ {𝑧}𝐶))
112 nfcv 2902 . . . . . . . . . . . . . . . . . . 19 𝑤𝐶
113 nfcsb1v 3918 . . . . . . . . . . . . . . . . . . 19 𝑘𝑤 / 𝑘𝐶
114 csbeq1a 3907 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑤𝐶 = 𝑤 / 𝑘𝐶)
115112, 113, 114cbvsumi 15648 . . . . . . . . . . . . . . . . . 18 Σ𝑘 ∈ {𝑧}𝐶 = Σ𝑤 ∈ {𝑧}𝑤 / 𝑘𝐶
116 csbeq1 3896 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑧𝑤 / 𝑘𝐶 = 𝑧 / 𝑘𝐶)
117116sumsn 15697 . . . . . . . . . . . . . . . . . . 19 ((𝑧𝐵𝑧 / 𝑘𝐶 ∈ ℂ) → Σ𝑤 ∈ {𝑧}𝑤 / 𝑘𝐶 = 𝑧 / 𝑘𝐶)
11851, 64, 117syl2anc 583 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → Σ𝑤 ∈ {𝑧}𝑤 / 𝑘𝐶 = 𝑧 / 𝑘𝐶)
119115, 118eqtrid 2783 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → Σ𝑘 ∈ {𝑧}𝐶 = 𝑧 / 𝑘𝐶)
120119oveq2d 7428 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → (Σ𝑘𝑦 𝐶 + Σ𝑘 ∈ {𝑧}𝐶) = (Σ𝑘𝑦 𝐶 + 𝑧 / 𝑘𝐶))
121111, 120eqtrd 2771 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥𝐴) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶 = (Σ𝑘𝑦 𝐶 + 𝑧 / 𝑘𝐶))
122121mpteq2dva 5248 . . . . . . . . . . . . . 14 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) = (𝑥𝐴 ↦ (Σ𝑘𝑦 𝐶 + 𝑧 / 𝑘𝐶)))
123122adantr 480 . . . . . . . . . . . . 13 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) = (𝑥𝐴 ↦ (Σ𝑘𝑦 𝐶 + 𝑧 / 𝑘𝐶)))
124 nfcv 2902 . . . . . . . . . . . . . 14 𝑤𝑘𝑦 𝐶 + 𝑧 / 𝑘𝐶)
125 nfcv 2902 . . . . . . . . . . . . . . 15 𝑥 +
12677, 125, 67nfov 7442 . . . . . . . . . . . . . 14 𝑥𝑘𝑦 𝑤 / 𝑥𝐶 + 𝑤 / 𝑥𝑧 / 𝑘𝐶)
12779, 69oveq12d 7430 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (Σ𝑘𝑦 𝐶 + 𝑧 / 𝑘𝐶) = (Σ𝑘𝑦 𝑤 / 𝑥𝐶 + 𝑤 / 𝑥𝑧 / 𝑘𝐶))
128124, 126, 127cbvmpt 5259 . . . . . . . . . . . . 13 (𝑥𝐴 ↦ (Σ𝑘𝑦 𝐶 + 𝑧 / 𝑘𝐶)) = (𝑤𝐴 ↦ (Σ𝑘𝑦 𝑤 / 𝑥𝐶 + 𝑤 / 𝑥𝑧 / 𝑘𝐶))
129123, 128eqtrdi 2787 . . . . . . . . . . . 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 2732 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
131 rlimcl 15452 . . . . . . . . . . . . . . . . . 18 ((𝑥𝐴𝐶) ⇝𝑟 𝐷𝐷 ∈ ℂ)
13254, 131syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐵) → 𝐷 ∈ ℂ)
133132adantlr 712 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑘𝐵) → 𝐷 ∈ ℂ)
134108, 133syldan 590 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝐷 ∈ ℂ)
135102, 130, 106, 134fsumsplit 15692 . . . . . . . . . . . . . 14 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷 = (Σ𝑘𝑦 𝐷 + Σ𝑘 ∈ {𝑧}𝐷))
136 nfcv 2902 . . . . . . . . . . . . . . . . 17 𝑤𝐷
137 nfcsb1v 3918 . . . . . . . . . . . . . . . . 17 𝑘𝑤 / 𝑘𝐷
138 csbeq1a 3907 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑤𝐷 = 𝑤 / 𝑘𝐷)
139136, 137, 138cbvsumi 15648 . . . . . . . . . . . . . . . 16 Σ𝑘 ∈ {𝑧}𝐷 = Σ𝑤 ∈ {𝑧}𝑤 / 𝑘𝐷
140133ralrimiva 3145 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑘𝐵 𝐷 ∈ ℂ)
14190nfel1 2918 . . . . . . . . . . . . . . . . . . 19 𝑘𝑧 / 𝑘𝐷 ∈ ℂ
14293eleq1d 2817 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑧 → (𝐷 ∈ ℂ ↔ 𝑧 / 𝑘𝐷 ∈ ℂ))
143141, 142rspc 3600 . . . . . . . . . . . . . . . . . 18 (𝑧𝐵 → (∀𝑘𝐵 𝐷 ∈ ℂ → 𝑧 / 𝑘𝐷 ∈ ℂ))
14450, 140, 143sylc 65 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝑧 / 𝑘𝐷 ∈ ℂ)
145 csbeq1 3896 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑧𝑤 / 𝑘𝐷 = 𝑧 / 𝑘𝐷)
146145sumsn 15697 . . . . . . . . . . . . . . . . 17 ((𝑧𝐵𝑧 / 𝑘𝐷 ∈ ℂ) → Σ𝑤 ∈ {𝑧}𝑤 / 𝑘𝐷 = 𝑧 / 𝑘𝐷)
14750, 144, 146syl2anc 583 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → Σ𝑤 ∈ {𝑧}𝑤 / 𝑘𝐷 = 𝑧 / 𝑘𝐷)
148139, 147eqtrid 2783 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → Σ𝑘 ∈ {𝑧}𝐷 = 𝑧 / 𝑘𝐷)
149148oveq2d 7428 . . . . . . . . . . . . . 14 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (Σ𝑘𝑦 𝐷 + Σ𝑘 ∈ {𝑧}𝐷) = (Σ𝑘𝑦 𝐷 + 𝑧 / 𝑘𝐷))
150135, 149eqtrd 2771 . . . . . . . . . . . . 13 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷 = (Σ𝑘𝑦 𝐷 + 𝑧 / 𝑘𝐷))
151150adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷 = (Σ𝑘𝑦 𝐷 + 𝑧 / 𝑘𝐷))
15299, 129, 1513brtr4d 5180 . . . . . . . . . . 11 (((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷) → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷)
153152ex 412 . . . . . . . . . 10 ((𝜑 ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ((𝑥𝐴 ↦ Σ𝑘𝑦 𝐶) ⇝𝑟 Σ𝑘𝑦 𝐷 → (𝑥𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷))
154153expr 456 . . . . . . . . 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 413 . . . . . 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 481 . . . 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 9168 . . 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 395   = wceq 1540  wcel 2105  wral 3060  Vcvv 3473  csb 3893  cun 3946  cin 3947  wss 3948  c0 4322  {csn 4628   class class class wbr 5148  cmpt 5231  (class class class)co 7412  Fincfn 8942  cc 11111  cr 11112  0cc0 11113   + caddc 11116  𝑟 crli 15434  Σcsu 15637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7728  ax-inf2 9639  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190  ax-pre-sup 11191
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-1st 7978  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-1o 8469  df-er 8706  df-pm 8826  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-sup 9440  df-oi 9508  df-card 9937  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-div 11877  df-nn 12218  df-2 12280  df-3 12281  df-n0 12478  df-z 12564  df-uz 12828  df-rp 12980  df-fz 13490  df-fzo 13633  df-seq 13972  df-exp 14033  df-hash 14296  df-cj 15051  df-re 15052  df-im 15053  df-sqrt 15187  df-abs 15188  df-clim 15437  df-rlim 15438  df-sum 15638
This theorem is referenced by:  climfsum  15771  logexprlim  26965  signsplypnf  33860
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