Theorem fsumrlim 16022

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 3935	. 2 ⊢ 𝐵 ⊆ 𝐵
2		fsumrlim.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ Fin)
3		sseq1 3938	. . . . . 6 ⊢ (𝑤 = ∅ → (𝑤 ⊆ 𝐵 ↔ ∅ ⊆ 𝐵))
4		sumeq1 15900	. . . . . . . . 9 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 𝐶 = Σ𝑘 ∈ ∅ 𝐶)
5		sum0 15931	. . . . . . . . 9 ⊢ Σ𝑘 ∈ ∅ 𝐶 = 0
6	4, 5	eqtrdi 2814	. . . . . . . 8 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 𝐶 = 0)
7	6	mpteq2dv 5205	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) = (𝑥 ∈ 𝐴 ↦ 0))
8		sumeq1 15900	. . . . . . . 8 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 𝐷 = Σ𝑘 ∈ ∅ 𝐷)
9		sum0 15931	. . . . . . . 8 ⊢ Σ𝑘 ∈ ∅ 𝐷 = 0
10	8, 9	eqtrdi 2814	. . . . . . 7 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 𝐷 = 0)
11	7, 10	breq12d 5121	. . . . . 6 ⊢ (𝑤 = ∅ → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑤 𝐷 ↔ (𝑥 ∈ 𝐴 ↦ 0) ⇝𝑟 0))
12	3, 11	imbi12d 347	. . . . 5 ⊢ (𝑤 = ∅ → ((𝑤 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑤 𝐷) ↔ (∅ ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 0) ⇝𝑟 0)))
13	12	imbi2d 343	. . . 4 ⊢ (𝑤 = ∅ → ((𝜑 → (𝑤 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑤 𝐷)) ↔ (𝜑 → (∅ ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 0) ⇝𝑟 0))))
14		sseq1 3938	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐵))
15		sumeq1 15900	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → Σ𝑘 ∈ 𝑤 𝐶 = Σ𝑘 ∈ 𝑦 𝐶)
16	15	mpteq2dv 5205	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) = (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶))
17		sumeq1 15900	. . . . . . 7 ⊢ (𝑤 = 𝑦 → Σ𝑘 ∈ 𝑤 𝐷 = Σ𝑘 ∈ 𝑦 𝐷)
18	16, 17	breq12d 5121	. . . . . 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 347	. . . . 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 343	. . . 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 3938	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 ⊆ 𝐵 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝐵))
22		sumeq1 15900	. . . . . . . 8 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → Σ𝑘 ∈ 𝑤 𝐶 = Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶)
23	22	mpteq2dv 5205	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) = (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶))
24		sumeq1 15900	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → Σ𝑘 ∈ 𝑤 𝐷 = Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷)
25	23, 24	breq12d 5121	. . . . . 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 347	. . . . 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 343	. . . 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 3938	. . . . . 6 ⊢ (𝑤 = 𝐵 → (𝑤 ⊆ 𝐵 ↔ 𝐵 ⊆ 𝐵))
29		sumeq1 15900	. . . . . . . 8 ⊢ (𝑤 = 𝐵 → Σ𝑘 ∈ 𝑤 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)
30	29	mpteq2dv 5205	. . . . . . 7 ⊢ (𝑤 = 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑤 𝐶) = (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶))
31		sumeq1 15900	. . . . . . 7 ⊢ (𝑤 = 𝐵 → Σ𝑘 ∈ 𝑤 𝐷 = Σ𝑘 ∈ 𝐵 𝐷)
32	30, 31	breq12d 5121	. . . . . 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 347	. . . . 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 343	. . . 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 11337	. . . . . 6 ⊢ 0 ∈ ℂ
37		rlimconst 15755	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 0 ∈ ℂ) → (𝑥 ∈ 𝐴 ↦ 0) ⇝𝑟 0)
38	35, 36, 37	sylancl 598	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 0) ⇝𝑟 0)
39	38	a1d 26	. . . 4 ⊢ (𝜑 → (∅ ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 0) ⇝𝑟 0))
40		ssun1 4179	. . . . . . . . . 10 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
41		sstr 3921	. . . . . . . . . 10 ⊢ ((𝑦 ⊆ (𝑦 ∪ {𝑧}) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵) → 𝑦 ⊆ 𝐵)
42	40, 41	mpan 703	. . . . . . . . 9 ⊢ ((𝑦 ∪ {𝑧}) ⊆ 𝐵 → 𝑦 ⊆ 𝐵)
43	42	imim1i 64	. . . . . . . 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 15899	. . . . . . . . . . . . . 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 785	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∪ {𝑧}) ⊆ 𝐵)
47	46	unssbd 4195	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → {𝑧} ⊆ 𝐵)
48		vex 3458	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑧 ∈ V
49	48	snss 4747	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝐵 ↔ {𝑧} ⊆ 𝐵)
50	47, 49	sylibr 237	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝑧 ∈ 𝐵)
51	50	adantr 486	. . . . . . . . . . . . . . . . . 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 668	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)
54		fsumrlim.4	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷)
55	53, 54	rlimmptrcl 15819	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)
56	55	an32s 665	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
57	56	adantllr 732	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
58	57	ralrimiva 3158	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ)
59		nfcsb1v 3875	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐶
60	59	nfel1 2942	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ
61		csbeq1a 3865	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑧 → 𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
62	61	eleq1d 2849	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑧 → (𝐶 ∈ ℂ ↔ ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ))
63	60, 62	rspc 3568	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ 𝐵 → (∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ → ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ))
64	51, 58, 63	sylc 66	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ)
65	64	ralrimiva 3158	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑥 ∈ 𝐴 ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ)
66	65	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) → ∀𝑥 ∈ 𝐴 ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ)
67		nfcsb1v 3875	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶
68	67	nfel1 2942	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ
69		csbeq1a 3865	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑤 → ⦋𝑧 / 𝑘⦌𝐶 = ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶)
70	69	eleq1d 2849	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑤 → (⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ ↔ ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ))
71	68, 70	rspc 3568	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ → ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ))
72	66, 71	mpan9 516	. . . . . . . . . . . . . 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 3477	. . . . . . . . . . . . 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 2926	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑤Σ𝑘 ∈ 𝑦 𝐶
75		nfcv 2926	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥𝑦
76		nfcsb1v 3875	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝐶
77	75, 76	nfsum 15902	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶
78		csbeq1a 3865	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑤 → 𝐶 = ⦋𝑤 / 𝑥⦌𝐶)
79	78	sumeq2sdv 15914	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑤 → Σ𝑘 ∈ 𝑦 𝐶 = Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶)
80	74, 77, 79	cbvmpt 5213	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) = (𝑤 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶)
81		simpr 490	. . . . . . . . . . . . . 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 5147	. . . . . . . . . . . . 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 2926	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑤⦋𝑧 / 𝑘⦌𝐶
84	83, 67, 69	cbvmpt 5213	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) = (𝑤 ∈ 𝐴 ↦ ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶)
85	54	ralrimiva 3158	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑘 ∈ 𝐵 (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷)
86	85	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑘 ∈ 𝐵 (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷)
87		nfcv 2926	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘𝐴
88	87, 59	nfmpt 5209	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘(𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶)
89		nfcv 2926	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘 ⇝𝑟
90		nfcsb1v 3875	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐷
91	88, 89, 90	nfbr 5158	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) ⇝𝑟 ⦋𝑧 / 𝑘⦌𝐷
92	61	mpteq2dv 5205	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑧 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶))
93		csbeq1a 3865	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑧 → 𝐷 = ⦋𝑧 / 𝑘⦌𝐷)
94	92, 93	breq12d 5121	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷 ↔ (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) ⇝𝑟 ⦋𝑧 / 𝑘⦌𝐷))
95	91, 94	rspc 3568	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝐵 → (∀𝑘 ∈ 𝐵 (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷 → (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) ⇝𝑟 ⦋𝑧 / 𝑘⦌𝐷))
96	50, 86, 95	sylc 66	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) ⇝𝑟 ⦋𝑧 / 𝑘⦌𝐷)
97	96	adantr 486	. . . . . . . . . . . . . 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 5147	. . . . . . . . . . . . 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 15854	. . . . . . . . . . . 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 783	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ¬ 𝑧 ∈ 𝑦)
101		disjsn 4673	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
102	100, 101	sylibr 237	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∩ {𝑧}) = ∅)
103	102	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∩ {𝑧}) = ∅)
104		eqidd 2764	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
105	2	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝐵 ∈ Fin)
106	105, 46	ssfid 9298	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∪ {𝑧}) ∈ Fin)
107	106	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∪ {𝑧}) ∈ Fin)
108	46	sselda 3913	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑘 ∈ 𝐵)
109	108	adantlr 728	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑘 ∈ 𝐵)
110	109, 57	syldan 603	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝐶 ∈ ℂ)
111	103, 104, 107, 110	fsumsplit 15951	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶 = (Σ𝑘 ∈ 𝑦 𝐶 + Σ𝑘 ∈ {𝑧}𝐶))
112		csbeq1a 3865	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑤 → 𝐶 = ⦋𝑤 / 𝑘⦌𝐶)
113		nfcv 2926	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑤𝐶
114		nfcsb1v 3875	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘⦋𝑤 / 𝑘⦌𝐶
115	112, 113, 114	cbvsum 15906	. . . . . . . . . . . . . . . . . 18 ⊢ Σ𝑘 ∈ {𝑧}𝐶 = Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐶
116		csbeq1 3854	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = 𝑧 → ⦋𝑤 / 𝑘⦌𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
117	116	sumsn 15956	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∈ 𝐵 ∧ ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ) → Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
118	51, 64, 117	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
119	115, 118	eqtrid 2810	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ {𝑧}𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
120	119	oveq2d 7467	. . . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶 = (Σ𝑘 ∈ 𝑦 𝐶 + ⦋𝑧 / 𝑘⦌𝐶))
122	121	mpteq2dva 5204	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶) = (𝑥 ∈ 𝐴 ↦ (Σ𝑘 ∈ 𝑦 𝐶 + ⦋𝑧 / 𝑘⦌𝐶)))
123	122	adantr 486	. . . . . . . . . . . . 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 2926	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑤(Σ𝑘 ∈ 𝑦 𝐶 + ⦋𝑧 / 𝑘⦌𝐶)
125		nfcv 2926	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥 +
126	77, 125, 67	nfov 7481	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶 + ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶)
127	79, 69	oveq12d 7469	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (Σ𝑘 ∈ 𝑦 𝐶 + ⦋𝑧 / 𝑘⦌𝐶) = (Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶 + ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶))
128	124, 126, 127	cbvmpt 5213	. . . . . . . . . . . . 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 2814	. . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
131		rlimcl 15714	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷 → 𝐷 ∈ ℂ)
132	54, 131	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐷 ∈ ℂ)
133	132	adantlr 728	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑘 ∈ 𝐵) → 𝐷 ∈ ℂ)
134	108, 133	syldan 603	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝐷 ∈ ℂ)
135	102, 130, 106, 134	fsumsplit 15951	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷 = (Σ𝑘 ∈ 𝑦 𝐷 + Σ𝑘 ∈ {𝑧}𝐷))
136		csbeq1a 3865	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑤 → 𝐷 = ⦋𝑤 / 𝑘⦌𝐷)
137		nfcv 2926	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑤𝐷
138		nfcsb1v 3875	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘⦋𝑤 / 𝑘⦌𝐷
139	136, 137, 138	cbvsum 15906	. . . . . . . . . . . . . . . 16 ⊢ Σ𝑘 ∈ {𝑧}𝐷 = Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐷
140	133	ralrimiva 3158	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑘 ∈ 𝐵 𝐷 ∈ ℂ)
141	90	nfel1 2942	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐷 ∈ ℂ
142	93	eleq1d 2849	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑧 → (𝐷 ∈ ℂ ↔ ⦋𝑧 / 𝑘⦌𝐷 ∈ ℂ))
143	141, 142	rspc 3568	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ 𝐵 → (∀𝑘 ∈ 𝐵 𝐷 ∈ ℂ → ⦋𝑧 / 𝑘⦌𝐷 ∈ ℂ))
144	50, 140, 143	sylc 66	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ⦋𝑧 / 𝑘⦌𝐷 ∈ ℂ)
145		csbeq1 3854	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑧 → ⦋𝑤 / 𝑘⦌𝐷 = ⦋𝑧 / 𝑘⦌𝐷)
146	145	sumsn 15956	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝐵 ∧ ⦋𝑧 / 𝑘⦌𝐷 ∈ ℂ) → Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐷 = ⦋𝑧 / 𝑘⦌𝐷)
147	50, 144, 146	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐷 = ⦋𝑧 / 𝑘⦌𝐷)
148	139, 147	eqtrid 2810	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → Σ𝑘 ∈ {𝑧}𝐷 = ⦋𝑧 / 𝑘⦌𝐷)
149	148	oveq2d 7467	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (Σ𝑘 ∈ 𝑦 𝐷 + Σ𝑘 ∈ {𝑧}𝐷) = (Σ𝑘 ∈ 𝑦 𝐷 + ⦋𝑧 / 𝑘⦌𝐷))
150	135, 149	eqtrd 2798	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷 = (Σ𝑘 ∈ 𝑦 𝐷 + ⦋𝑧 / 𝑘⦌𝐷))
151	150	adantr 486	. . . . . . . . . . . 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 5143	. . . . . . . . . . 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 418	. . . . . . . . . 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 462	. . . . . . . . 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 30	. . . . . . . 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 35	. . . . . . 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 419	. . . . . 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 30	. . . . 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 487	. . . 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 9219	. . 3 ⊢ (𝐵 ∈ Fin → (𝜑 → (𝐵 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝐵 𝐷)))
161	2, 160	mpcom 39	. 2 ⊢ (𝜑 → (𝐵 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝐵 𝐷))
162	1, 161	mpi 21	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝐵 𝐷)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  Vcvv 3454  ⦋csb 3851   ⊆ wss 3900   ∩ cin 4092   ∪ cun 4152  ∅c0 4281  {csn 4588   class class class wbr 5108   ↦ cmpt 5192  (class class class)co 7451  Fincfn 9011  ℂcc 11237  ℝcr 11238  0cc0 11239   + caddc 11242   ⇝_𝑟 crli 15696  Σcsu 15897

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-inf2 9679  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316  ax-pre-sup 11317

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-er 8755  df-pm 8888  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-sup 9471  df-oi 9541  df-card 10057  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-div 12014  df-nn 12376  df-2 12445  df-3 12446  df-n0 12647  df-z 12734  df-uz 13006  df-rp 13161  df-fz 13680  df-fzo 13829  df-seq 14185  df-exp 14245  df-hash 14516  df-cj 15309  df-re 15310  df-im 15311  df-sqrt 15445  df-abs 15446  df-clim 15699  df-rlim 15700  df-sum 15898

This theorem is used by:  climfsum  16031  logexprlim  27609  signsplypnf  35290