Theorem fsumrlim 15696

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	⊢ (𝜑 → 𝐴 ⊆ ℝ)
fsumrlim.2	⊢ (𝜑 → 𝐵 ∈ Fin)
fsumrlim.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ 𝑉)
fsumrlim.4	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷)

Assertion

Ref	Expression
fsumrlim	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝐵 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.

Step	Hyp	Ref	Expression
1		ssid 3966	. 2 ⊢ 𝐵 ⊆ 𝐵
2		fsumrlim.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ Fin)
3		sseq1 3969	. . . . . 6 ⊢ (𝑤 = ∅ → (𝑤 ⊆ 𝐵 ↔ ∅ ⊆ 𝐵))
4		sumeq1 15573	. . . . . . . . 9 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 𝐶 = Σ𝑘 ∈ ∅ 𝐶)
5		sum0 15606	. . . . . . . . 9 ⊢ Σ𝑘 ∈ ∅ 𝐶 = 0
6	4, 5	eqtrdi 2792	. . . . . . . 8 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 𝐶 = 0)
7	6	mpteq2dv 5207	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) = (𝑥 ∈ 𝐴 ↦ 0))
8		sumeq1 15573	. . . . . . . 8 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 𝐷 = Σ𝑘 ∈ ∅ 𝐷)
9		sum0 15606	. . . . . . . 8 ⊢ Σ𝑘 ∈ ∅ 𝐷 = 0
10	8, 9	eqtrdi 2792	. . . . . . 7 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 𝐷 = 0)
11	7, 10	breq12d 5118	. . . . . 6 ⊢ (𝑤 = ∅ → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑤 𝐷 ↔ (𝑥 ∈ 𝐴 ↦ 0) ⇝𝑟 0))
12	3, 11	imbi12d 344	. . . . 5 ⊢ (𝑤 = ∅ → ((𝑤 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑤 𝐷) ↔ (∅ ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 0) ⇝𝑟 0)))
13	12	imbi2d 340	. . . 4 ⊢ (𝑤 = ∅ → ((𝜑 → (𝑤 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑤 𝐷)) ↔ (𝜑 → (∅ ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 0) ⇝𝑟 0))))
14		sseq1 3969	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐵))
15		sumeq1 15573	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → Σ𝑘 ∈ 𝑤 𝐶 = Σ𝑘 ∈ 𝑦 𝐶)
16	15	mpteq2dv 5207	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) = (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶))
17		sumeq1 15573	. . . . . . 7 ⊢ (𝑤 = 𝑦 → Σ𝑘 ∈ 𝑤 𝐷 = Σ𝑘 ∈ 𝑦 𝐷)
18	16, 17	breq12d 5118	. . . . . 6 ⊢ (𝑤 = 𝑦 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑤 𝐷 ↔ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷))
19	14, 18	imbi12d 344	. . . . 5 ⊢ (𝑤 = 𝑦 → ((𝑤 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑤 𝐷) ↔ (𝑦 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷)))
20	19	imbi2d 340	. . . 4 ⊢ (𝑤 = 𝑦 → ((𝜑 → (𝑤 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑤 𝐷)) ↔ (𝜑 → (𝑦 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷))))
21		sseq1 3969	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 ⊆ 𝐵 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝐵))
22		sumeq1 15573	. . . . . . . 8 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → Σ𝑘 ∈ 𝑤 𝐶 = Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶)
23	22	mpteq2dv 5207	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) = (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶))
24		sumeq1 15573	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → Σ𝑘 ∈ 𝑤 𝐷 = Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷)
25	23, 24	breq12d 5118	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑤 𝐷 ↔ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷))
26	21, 25	imbi12d 344	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝑤 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑤 𝐷) ↔ ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷)))
27	26	imbi2d 340	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝜑 → (𝑤 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑤 𝐷)) ↔ (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷))))
28		sseq1 3969	. . . . . 6 ⊢ (𝑤 = 𝐵 → (𝑤 ⊆ 𝐵 ↔ 𝐵 ⊆ 𝐵))
29		sumeq1 15573	. . . . . . . 8 ⊢ (𝑤 = 𝐵 → Σ𝑘 ∈ 𝑤 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)
30	29	mpteq2dv 5207	. . . . . . 7 ⊢ (𝑤 = 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) = (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶))
31		sumeq1 15573	. . . . . . 7 ⊢ (𝑤 = 𝐵 → Σ𝑘 ∈ 𝑤 𝐷 = Σ𝑘 ∈ 𝐵 𝐷)
32	30, 31	breq12d 5118	. . . . . 6 ⊢ (𝑤 = 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑤 𝐷 ↔ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝐵 𝐷))
33	28, 32	imbi12d 344	. . . . 5 ⊢ (𝑤 = 𝐵 → ((𝑤 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑤 𝐷) ↔ (𝐵 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝐵 𝐷)))
34	33	imbi2d 340	. . . 4 ⊢ (𝑤 = 𝐵 → ((𝜑 → (𝑤 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑤 𝐷)) ↔ (𝜑 → (𝐵 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝐵 𝐷))))
35		fsumrlim.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
36		0cn 11147	. . . . . 6 ⊢ 0 ∈ ℂ
37		rlimconst 15426	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 0 ∈ ℂ) → (𝑥 ∈ 𝐴 ↦ 0) ⇝𝑟 0)
38	35, 36, 37	sylancl 586	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 0) ⇝𝑟 0)
39	38	a1d 25	. . . 4 ⊢ (𝜑 → (∅ ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 0) ⇝𝑟 0))
40		ssun1 4132	. . . . . . . . . 10 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
41		sstr 3952	. . . . . . . . . 10 ⊢ ((𝑦 ⊆ (𝑦 ∪ {𝑧}) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵) → 𝑦 ⊆ 𝐵)
42	40, 41	mpan 688	. . . . . . . . 9 ⊢ ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → 𝑦 ⊆ 𝐵)
43	42	imim1i 63	. . . . . . . 8 ⊢ ((𝑦 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷))
44		sumex 15572	. . . . . . . . . . . . . 14 ⊢ Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶 ∈ V
45	44	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) ∧ 𝑤 ∈ 𝐴) → Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶 ∈ V)
46		simprr 771	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∪ {𝑧}) ⊆ 𝐵)
47	46	unssbd 4148	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → {𝑧} ⊆ 𝐵)
48		vex 3449	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑧 ∈ V
49	48	snss 4746	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝐵 ↔ {𝑧} ⊆ 𝐵)
50	47, 49	sylibr 233	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝑧 ∈ 𝐵)
51	50	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → 𝑧 ∈ 𝐵)
52		fsumrlim.3	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ 𝑉)
53	52	anass1rs 653	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)
54		fsumrlim.4	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷)
55	53, 54	rlimmptrcl 15490	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)
56	55	an32s 650	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
57	56	adantllr 717	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
58	57	ralrimiva 3143	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ)
59		nfcsb1v 3880	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐶
60	59	nfel1 2923	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ
61		csbeq1a 3869	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑧 → 𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
62	61	eleq1d 2822	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑧 → (𝐶 ∈ ℂ ↔ ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ))
63	60, 62	rspc 3569	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ 𝐵 → (∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ → ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ))
64	51, 58, 63	sylc 65	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ)
65	64	ralrimiva 3143	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑥 ∈ 𝐴 ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ)
66	65	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) → ∀𝑥 ∈ 𝐴 ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ)
67		nfcsb1v 3880	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶
68	67	nfel1 2923	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ
69		csbeq1a 3869	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑤 → ⦋𝑧 / 𝑘⦌𝐶 = ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶)
70	69	eleq1d 2822	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑤 → (⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ ↔ ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ))
71	68, 70	rspc 3569	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ → ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ))
72	66, 71	mpan9 507	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) ∧ 𝑤 ∈ 𝐴) → ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ)
73	72	elexd 3465	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) ∧ 𝑤 ∈ 𝐴) → ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶 ∈ V)
74		nfcv 2907	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑤Σ𝑘 ∈ 𝑦 𝐶
75		nfcv 2907	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥𝑦
76		nfcsb1v 3880	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝐶
77	75, 76	nfsum 15575	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶
78		csbeq1a 3869	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑤 → 𝐶 = ⦋𝑤 / 𝑥⦌𝐶)
79	78	sumeq2sdv 15589	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑤 → Σ𝑘 ∈ 𝑦 𝐶 = Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶)
80	74, 77, 79	cbvmpt 5216	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) = (𝑤 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶)
81		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷)
82	80, 81	eqbrtrrid 5141	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) → (𝑤 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷)
83		nfcv 2907	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑤⦋𝑧 / 𝑘⦌𝐶
84	83, 67, 69	cbvmpt 5216	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) = (𝑤 ∈ 𝐴 ↦ ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶)
85	54	ralrimiva 3143	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑘 ∈ 𝐵 (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷)
86	85	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑘 ∈ 𝐵 (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷)
87		nfcv 2907	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘𝐴
88	87, 59	nfmpt 5212	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘(𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶)
89		nfcv 2907	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘 ⇝𝑟
90		nfcsb1v 3880	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐷
91	88, 89, 90	nfbr 5152	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) ⇝𝑟 ⦋𝑧 / 𝑘⦌𝐷
92	61	mpteq2dv 5207	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑧 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶))
93		csbeq1a 3869	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑧 → 𝐷 = ⦋𝑧 / 𝑘⦌𝐷)
94	92, 93	breq12d 5118	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷 ↔ (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) ⇝𝑟 ⦋𝑧 / 𝑘⦌𝐷))
95	91, 94	rspc 3569	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝐵 → (∀𝑘 ∈ 𝐵 (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷 → (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) ⇝𝑟 ⦋𝑧 / 𝑘⦌𝐷))
96	50, 86, 95	sylc 65	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) ⇝𝑟 ⦋𝑧 / 𝑘⦌𝐷)
97	96	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) → (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) ⇝𝑟 ⦋𝑧 / 𝑘⦌𝐷)
98	84, 97	eqbrtrrid 5141	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) → (𝑤 ∈ 𝐴 ↦ ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶) ⇝𝑟 ⦋𝑧 / 𝑘⦌𝐷)
99	45, 73, 82, 98	rlimadd 15525	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) → (𝑤 ∈ 𝐴 ↦ (Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶 + ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶)) ⇝𝑟 (Σ𝑘 ∈ 𝑦 𝐷 + ⦋𝑧 / 𝑘⦌𝐷))
100		simprl 769	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ¬ 𝑧 ∈ 𝑦)
101		disjsn 4672	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
102	100, 101	sylibr 233	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∩ {𝑧}) = ∅)
103	102	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∩ {𝑧}) = ∅)
104		eqidd 2737	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
105	2	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝐵 ∈ Fin)
106	105, 46	ssfid 9211	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∪ {𝑧}) ∈ Fin)
107	106	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∪ {𝑧}) ∈ Fin)
108	46	sselda 3944	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑘 ∈ 𝐵)
109	108	adantlr 713	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑘 ∈ 𝐵)
110	109, 57	syldan 591	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝐶 ∈ ℂ)
111	103, 104, 107, 110	fsumsplit 15626	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶 = (Σ𝑘 ∈ 𝑦 𝐶 + Σ𝑘 ∈ {𝑧}𝐶))
112		nfcv 2907	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑤𝐶
113		nfcsb1v 3880	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘⦋𝑤 / 𝑘⦌𝐶
114		csbeq1a 3869	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑤 → 𝐶 = ⦋𝑤 / 𝑘⦌𝐶)
115	112, 113, 114	cbvsumi 15582	. . . . . . . . . . . . . . . . . 18 ⊢ Σ𝑘 ∈ {𝑧}𝐶 = Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐶
116		csbeq1 3858	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = 𝑧 → ⦋𝑤 / 𝑘⦌𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
117	116	sumsn 15631	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∈ 𝐵 ∧ ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ) → Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
118	51, 64, 117	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
119	115, 118	eqtrid 2788	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ {𝑧}𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
120	119	oveq2d 7373	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (Σ𝑘 ∈ 𝑦 𝐶 + Σ𝑘 ∈ {𝑧}𝐶) = (Σ𝑘 ∈ 𝑦 𝐶 + ⦋𝑧 / 𝑘⦌𝐶))
121	111, 120	eqtrd 2776	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶 = (Σ𝑘 ∈ 𝑦 𝐶 + ⦋𝑧 / 𝑘⦌𝐶))
122	121	mpteq2dva 5205	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) = (𝑥 ∈ 𝐴 ↦ (Σ𝑘 ∈ 𝑦 𝐶 + ⦋𝑧 / 𝑘⦌𝐶)))
123	122	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) = (𝑥 ∈ 𝐴 ↦ (Σ𝑘 ∈ 𝑦 𝐶 + ⦋𝑧 / 𝑘⦌𝐶)))
124		nfcv 2907	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑤(Σ𝑘 ∈ 𝑦 𝐶 + ⦋𝑧 / 𝑘⦌𝐶)
125		nfcv 2907	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥 +
126	77, 125, 67	nfov 7387	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶 + ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶)
127	79, 69	oveq12d 7375	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (Σ𝑘 ∈ 𝑦 𝐶 + ⦋𝑧 / 𝑘⦌𝐶) = (Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶 + ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶))
128	124, 126, 127	cbvmpt 5216	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 ↦ (Σ𝑘 ∈ 𝑦 𝐶 + ⦋𝑧 / 𝑘⦌𝐶)) = (𝑤 ∈ 𝐴 ↦ (Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶 + ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶))
129	123, 128	eqtrdi 2792	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) = (𝑤 ∈ 𝐴 ↦ (Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶 + ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶)))
130		eqidd 2737	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
131		rlimcl 15385	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷 → 𝐷 ∈ ℂ)
132	54, 131	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐷 ∈ ℂ)
133	132	adantlr 713	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑘 ∈ 𝐵) → 𝐷 ∈ ℂ)
134	108, 133	syldan 591	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝐷 ∈ ℂ)
135	102, 130, 106, 134	fsumsplit 15626	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷 = (Σ𝑘 ∈ 𝑦 𝐷 + Σ𝑘 ∈ {𝑧}𝐷))
136		nfcv 2907	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑤𝐷
137		nfcsb1v 3880	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘⦋𝑤 / 𝑘⦌𝐷
138		csbeq1a 3869	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑤 → 𝐷 = ⦋𝑤 / 𝑘⦌𝐷)
139	136, 137, 138	cbvsumi 15582	. . . . . . . . . . . . . . . 16 ⊢ Σ𝑘 ∈ {𝑧}𝐷 = Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐷
140	133	ralrimiva 3143	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑘 ∈ 𝐵 𝐷 ∈ ℂ)
141	90	nfel1 2923	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐷 ∈ ℂ
142	93	eleq1d 2822	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑧 → (𝐷 ∈ ℂ ↔ ⦋𝑧 / 𝑘⦌𝐷 ∈ ℂ))
143	141, 142	rspc 3569	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ 𝐵 → (∀𝑘 ∈ 𝐵 𝐷 ∈ ℂ → ⦋𝑧 / 𝑘⦌𝐷 ∈ ℂ))
144	50, 140, 143	sylc 65	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ⦋𝑧 / 𝑘⦌𝐷 ∈ ℂ)
145		csbeq1 3858	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑧 → ⦋𝑤 / 𝑘⦌𝐷 = ⦋𝑧 / 𝑘⦌𝐷)
146	145	sumsn 15631	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝐵 ∧ ⦋𝑧 / 𝑘⦌𝐷 ∈ ℂ) → Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐷 = ⦋𝑧 / 𝑘⦌𝐷)
147	50, 144, 146	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐷 = ⦋𝑧 / 𝑘⦌𝐷)
148	139, 147	eqtrid 2788	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → Σ𝑘 ∈ {𝑧}𝐷 = ⦋𝑧 / 𝑘⦌𝐷)
149	148	oveq2d 7373	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (Σ𝑘 ∈ 𝑦 𝐷 + Σ𝑘 ∈ {𝑧}𝐷) = (Σ𝑘 ∈ 𝑦 𝐷 + ⦋𝑧 / 𝑘⦌𝐷))
150	135, 149	eqtrd 2776	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷 = (Σ𝑘 ∈ 𝑦 𝐷 + ⦋𝑧 / 𝑘⦌𝐷))
151	150	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷 = (Σ𝑘 ∈ 𝑦 𝐷 + ⦋𝑧 / 𝑘⦌𝐷))
152	99, 129, 151	3brtr4d 5137	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷)
153	152	ex 413	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷))
154	153	expr 457	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷)))
155	154	a2d 29	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) → (((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷)))
156	43, 155	syl5 34	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) → ((𝑦 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷)))
157	156	expcom 414	. . . . . 6 ⊢ (¬ 𝑧 ∈ 𝑦 → (𝜑 → ((𝑦 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷))))
158	157	a2d 29	. . . . 5 ⊢ (¬ 𝑧 ∈ 𝑦 → ((𝜑 → (𝑦 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷)) → (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷))))
159	158	adantl 482	. . . 4 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((𝜑 → (𝑦 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷)) → (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) ⇝𝑟 Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷))))
160	13, 20, 27, 34, 39, 159	findcard2s 9109	. . 3 ⊢ (𝐵 ∈ Fin → (𝜑 → (𝐵 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝐵 𝐷)))
161	2, 160	mpcom 38	. 2 ⊢ (𝜑 → (𝐵 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝐵 𝐷))
162	1, 161	mpi 20	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝐵 𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3064  Vcvv 3445  ⦋csb 3855   ∪ cun 3908   ∩ cin 3909   ⊆ wss 3910  ∅c0 4282  {csn 4586   class class class wbr 5105   ↦ cmpt 5188  (class class class)co 7357  Fincfn 8883  ℂcc 11049  ℝcr 11050  0cc0 11051   + caddc 11054   ⇝_𝑟 crli 15367  Σcsu 15570

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-rlim 15371  df-sum 15571

This theorem is referenced by:  climfsum  15705  logexprlim  26573  signsplypnf  33162