Theorem fsumrlim 15761

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 4003	. 2 ⊢ 𝐵 ⊆ 𝐵
2		fsumrlim.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ Fin)
3		sseq1 4006	. . . . . 6 ⊢ (𝑤 = ∅ → (𝑤 ⊆ 𝐵 ↔ ∅ ⊆ 𝐵))
4		sumeq1 15639	. . . . . . . . 9 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 𝐶 = Σ𝑘 ∈ ∅ 𝐶)
5		sum0 15671	. . . . . . . . 9 ⊢ Σ𝑘 ∈ ∅ 𝐶 = 0
6	4, 5	eqtrdi 2786	. . . . . . . 8 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 𝐶 = 0)
7	6	mpteq2dv 5249	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) = (𝑥 ∈ 𝐴 ↦ 0))
8		sumeq1 15639	. . . . . . . 8 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 𝐷 = Σ𝑘 ∈ ∅ 𝐷)
9		sum0 15671	. . . . . . . 8 ⊢ Σ𝑘 ∈ ∅ 𝐷 = 0
10	8, 9	eqtrdi 2786	. . . . . . 7 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 𝐷 = 0)
11	7, 10	breq12d 5160	. . . . . 6 ⊢ (𝑤 = ∅ → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑤 𝐷 ↔ (𝑥 ∈ 𝐴 ↦ 0) ⇝𝑟 0))
12	3, 11	imbi12d 343	. . . . 5 ⊢ (𝑤 = ∅ → ((𝑤 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑤 𝐷) ↔ (∅ ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 0) ⇝𝑟 0)))
13	12	imbi2d 339	. . . 4 ⊢ (𝑤 = ∅ → ((𝜑 → (𝑤 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑤 𝐷)) ↔ (𝜑 → (∅ ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 0) ⇝𝑟 0))))
14		sseq1 4006	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐵))
15		sumeq1 15639	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → Σ𝑘 ∈ 𝑤 𝐶 = Σ𝑘 ∈ 𝑦 𝐶)
16	15	mpteq2dv 5249	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) = (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶))
17		sumeq1 15639	. . . . . . 7 ⊢ (𝑤 = 𝑦 → Σ𝑘 ∈ 𝑤 𝐷 = Σ𝑘 ∈ 𝑦 𝐷)
18	16, 17	breq12d 5160	. . . . . 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 343	. . . . 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 339	. . . 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 4006	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 ⊆ 𝐵 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝐵))
22		sumeq1 15639	. . . . . . . 8 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → Σ𝑘 ∈ 𝑤 𝐶 = Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶)
23	22	mpteq2dv 5249	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) = (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶))
24		sumeq1 15639	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → Σ𝑘 ∈ 𝑤 𝐷 = Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷)
25	23, 24	breq12d 5160	. . . . . 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 343	. . . . 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 339	. . . 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 4006	. . . . . 6 ⊢ (𝑤 = 𝐵 → (𝑤 ⊆ 𝐵 ↔ 𝐵 ⊆ 𝐵))
29		sumeq1 15639	. . . . . . . 8 ⊢ (𝑤 = 𝐵 → Σ𝑘 ∈ 𝑤 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)
30	29	mpteq2dv 5249	. . . . . . 7 ⊢ (𝑤 = 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) = (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶))
31		sumeq1 15639	. . . . . . 7 ⊢ (𝑤 = 𝐵 → Σ𝑘 ∈ 𝑤 𝐷 = Σ𝑘 ∈ 𝐵 𝐷)
32	30, 31	breq12d 5160	. . . . . 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 343	. . . . 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 339	. . . 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 11210	. . . . . 6 ⊢ 0 ∈ ℂ
37		rlimconst 15492	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 0 ∈ ℂ) → (𝑥 ∈ 𝐴 ↦ 0) ⇝𝑟 0)
38	35, 36, 37	sylancl 584	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 0) ⇝𝑟 0)
39	38	a1d 25	. . . 4 ⊢ (𝜑 → (∅ ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 0) ⇝𝑟 0))
40		ssun1 4171	. . . . . . . . . 10 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
41		sstr 3989	. . . . . . . . . 10 ⊢ ((𝑦 ⊆ (𝑦 ∪ {𝑧}) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵) → 𝑦 ⊆ 𝐵)
42	40, 41	mpan 686	. . . . . . . . 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 15638	. . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∪ {𝑧}) ⊆ 𝐵)
47	46	unssbd 4187	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → {𝑧} ⊆ 𝐵)
48		vex 3476	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑧 ∈ V
49	48	snss 4788	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝐵 ↔ {𝑧} ⊆ 𝐵)
50	47, 49	sylibr 233	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝑧 ∈ 𝐵)
51	50	adantr 479	. . . . . . . . . . . . . . . . . 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 651	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)
54		fsumrlim.4	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷)
55	53, 54	rlimmptrcl 15556	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)
56	55	an32s 648	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
57	56	adantllr 715	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
58	57	ralrimiva 3144	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ)
59		nfcsb1v 3917	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐶
60	59	nfel1 2917	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ
61		csbeq1a 3906	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑧 → 𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
62	61	eleq1d 2816	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑧 → (𝐶 ∈ ℂ ↔ ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ))
63	60, 62	rspc 3599	. . . . . . . . . . . . . . . . . 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 3144	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑥 ∈ 𝐴 ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ)
66	65	adantr 479	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) → ∀𝑥 ∈ 𝐴 ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ)
67		nfcsb1v 3917	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶
68	67	nfel1 2917	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ
69		csbeq1a 3906	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑤 → ⦋𝑧 / 𝑘⦌𝐶 = ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶)
70	69	eleq1d 2816	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑤 → (⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ ↔ ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ))
71	68, 70	rspc 3599	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ → ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ))
72	66, 71	mpan9 505	. . . . . . . . . . . . . 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 3493	. . . . . . . . . . . . 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 2901	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑤Σ𝑘 ∈ 𝑦 𝐶
75		nfcv 2901	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥𝑦
76		nfcsb1v 3917	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝐶
77	75, 76	nfsum 15641	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶
78		csbeq1a 3906	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑤 → 𝐶 = ⦋𝑤 / 𝑥⦌𝐶)
79	78	sumeq2sdv 15654	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑤 → Σ𝑘 ∈ 𝑦 𝐶 = Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶)
80	74, 77, 79	cbvmpt 5258	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) = (𝑤 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶)
81		simpr 483	. . . . . . . . . . . . . 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 5183	. . . . . . . . . . . . 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 2901	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑤⦋𝑧 / 𝑘⦌𝐶
84	83, 67, 69	cbvmpt 5258	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) = (𝑤 ∈ 𝐴 ↦ ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶)
85	54	ralrimiva 3144	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑘 ∈ 𝐵 (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷)
86	85	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑘 ∈ 𝐵 (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷)
87		nfcv 2901	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘𝐴
88	87, 59	nfmpt 5254	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘(𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶)
89		nfcv 2901	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘 ⇝𝑟
90		nfcsb1v 3917	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐷
91	88, 89, 90	nfbr 5194	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) ⇝𝑟 ⦋𝑧 / 𝑘⦌𝐷
92	61	mpteq2dv 5249	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑧 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶))
93		csbeq1a 3906	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑧 → 𝐷 = ⦋𝑧 / 𝑘⦌𝐷)
94	92, 93	breq12d 5160	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷 ↔ (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) ⇝𝑟 ⦋𝑧 / 𝑘⦌𝐷))
95	91, 94	rspc 3599	. . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . 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 5183	. . . . . . . . . . . . 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 15591	. . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ¬ 𝑧 ∈ 𝑦)
101		disjsn 4714	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
102	100, 101	sylibr 233	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∩ {𝑧}) = ∅)
103	102	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∩ {𝑧}) = ∅)
104		eqidd 2731	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
105	2	adantr 479	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝐵 ∈ Fin)
106	105, 46	ssfid 9269	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∪ {𝑧}) ∈ Fin)
107	106	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∪ {𝑧}) ∈ Fin)
108	46	sselda 3981	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑘 ∈ 𝐵)
109	108	adantlr 711	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑘 ∈ 𝐵)
110	109, 57	syldan 589	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝐶 ∈ ℂ)
111	103, 104, 107, 110	fsumsplit 15691	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶 = (Σ𝑘 ∈ 𝑦 𝐶 + Σ𝑘 ∈ {𝑧}𝐶))
112		nfcv 2901	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑤𝐶
113		nfcsb1v 3917	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘⦋𝑤 / 𝑘⦌𝐶
114		csbeq1a 3906	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑤 → 𝐶 = ⦋𝑤 / 𝑘⦌𝐶)
115	112, 113, 114	cbvsumi 15647	. . . . . . . . . . . . . . . . . 18 ⊢ Σ𝑘 ∈ {𝑧}𝐶 = Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐶
116		csbeq1 3895	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = 𝑧 → ⦋𝑤 / 𝑘⦌𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
117	116	sumsn 15696	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∈ 𝐵 ∧ ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ) → Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
118	51, 64, 117	syl2anc 582	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
119	115, 118	eqtrid 2782	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ {𝑧}𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
120	119	oveq2d 7427	. . . . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶 = (Σ𝑘 ∈ 𝑦 𝐶 + ⦋𝑧 / 𝑘⦌𝐶))
122	121	mpteq2dva 5247	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) = (𝑥 ∈ 𝐴 ↦ (Σ𝑘 ∈ 𝑦 𝐶 + ⦋𝑧 / 𝑘⦌𝐶)))
123	122	adantr 479	. . . . . . . . . . . . 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 2901	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑤(Σ𝑘 ∈ 𝑦 𝐶 + ⦋𝑧 / 𝑘⦌𝐶)
125		nfcv 2901	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥 +
126	77, 125, 67	nfov 7441	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶 + ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶)
127	79, 69	oveq12d 7429	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (Σ𝑘 ∈ 𝑦 𝐶 + ⦋𝑧 / 𝑘⦌𝐶) = (Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶 + ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶))
128	124, 126, 127	cbvmpt 5258	. . . . . . . . . . . . 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 2786	. . . . . . . . . . . 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 2731	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
131		rlimcl 15451	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷 → 𝐷 ∈ ℂ)
132	54, 131	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐷 ∈ ℂ)
133	132	adantlr 711	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑘 ∈ 𝐵) → 𝐷 ∈ ℂ)
134	108, 133	syldan 589	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝐷 ∈ ℂ)
135	102, 130, 106, 134	fsumsplit 15691	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷 = (Σ𝑘 ∈ 𝑦 𝐷 + Σ𝑘 ∈ {𝑧}𝐷))
136		nfcv 2901	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑤𝐷
137		nfcsb1v 3917	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘⦋𝑤 / 𝑘⦌𝐷
138		csbeq1a 3906	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑤 → 𝐷 = ⦋𝑤 / 𝑘⦌𝐷)
139	136, 137, 138	cbvsumi 15647	. . . . . . . . . . . . . . . 16 ⊢ Σ𝑘 ∈ {𝑧}𝐷 = Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐷
140	133	ralrimiva 3144	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑘 ∈ 𝐵 𝐷 ∈ ℂ)
141	90	nfel1 2917	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐷 ∈ ℂ
142	93	eleq1d 2816	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑧 → (𝐷 ∈ ℂ ↔ ⦋𝑧 / 𝑘⦌𝐷 ∈ ℂ))
143	141, 142	rspc 3599	. . . . . . . . . . . . . . . . . 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 3895	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑧 → ⦋𝑤 / 𝑘⦌𝐷 = ⦋𝑧 / 𝑘⦌𝐷)
146	145	sumsn 15696	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝐵 ∧ ⦋𝑧 / 𝑘⦌𝐷 ∈ ℂ) → Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐷 = ⦋𝑧 / 𝑘⦌𝐷)
147	50, 144, 146	syl2anc 582	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐷 = ⦋𝑧 / 𝑘⦌𝐷)
148	139, 147	eqtrid 2782	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → Σ𝑘 ∈ {𝑧}𝐷 = ⦋𝑧 / 𝑘⦌𝐷)
149	148	oveq2d 7427	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (Σ𝑘 ∈ 𝑦 𝐷 + Σ𝑘 ∈ {𝑧}𝐷) = (Σ𝑘 ∈ 𝑦 𝐷 + ⦋𝑧 / 𝑘⦌𝐷))
150	135, 149	eqtrd 2770	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷 = (Σ𝑘 ∈ 𝑦 𝐷 + ⦋𝑧 / 𝑘⦌𝐷))
151	150	adantr 479	. . . . . . . . . . . 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 5179	. . . . . . . . . . 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 411	. . . . . . . . . 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 455	. . . . . . . . 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 412	. . . . . 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 480	. . . 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 9167	. . 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 394   = wceq 1539   ∈ wcel 2104  ∀wral 3059  Vcvv 3472  ⦋csb 3892   ∪ cun 3945   ∩ cin 3946   ⊆ wss 3947  ∅c0 4321  {csn 4627   class class class wbr 5147   ↦ cmpt 5230  (class class class)co 7411  Fincfn 8941  ℂcc 11110  ℝcr 11111  0cc0 11112   + caddc 11115   ⇝_𝑟 crli 15433  Σcsu 15636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-oi 9507  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-z 12563  df-uz 12827  df-rp 12979  df-fz 13489  df-fzo 13632  df-seq 13971  df-exp 14032  df-hash 14295  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-clim 15436  df-rlim 15437  df-sum 15637

This theorem is referenced by:  climfsum  15770  logexprlim  26964  signsplypnf  33859