Theorem fsumrlim 15859

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 3959	. 2 ⊢ 𝐵 ⊆ 𝐵
2		fsumrlim.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ Fin)
3		sseq1 3962	. . . . . 6 ⊢ (𝑤 = ∅ → (𝑤 ⊆ 𝐵 ↔ ∅ ⊆ 𝐵))
4		sumeq1 15736	. . . . . . . . 9 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 𝐶 = Σ𝑘 ∈ ∅ 𝐶)
5		sum0 15768	. . . . . . . . 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 15736	. . . . . . . 8 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 𝐷 = Σ𝑘 ∈ ∅ 𝐷)
9		sum0 15768	. . . . . . . 8 ⊢ Σ𝑘 ∈ ∅ 𝐷 = 0
10	8, 9	eqtrdi 2814	. . . . . . 7 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 𝐷 = 0)
11	7, 10	breq12d 5122	. . . . . 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 3962	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐵))
15		sumeq1 15736	. . . . . . . 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 15736	. . . . . . 7 ⊢ (𝑤 = 𝑦 → Σ𝑘 ∈ 𝑤 𝐷 = Σ𝑘 ∈ 𝑦 𝐷)
18	16, 17	breq12d 5122	. . . . . 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 3962	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 ⊆ 𝐵 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝐵))
22		sumeq1 15736	. . . . . . . 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 15736	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → Σ𝑘 ∈ 𝑤 𝐷 = Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷)
25	23, 24	breq12d 5122	. . . . . 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 3962	. . . . . 6 ⊢ (𝑤 = 𝐵 → (𝑤 ⊆ 𝐵 ↔ 𝐵 ⊆ 𝐵))
29		sumeq1 15736	. . . . . . . 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 15736	. . . . . . 7 ⊢ (𝑤 = 𝐵 → Σ𝑘 ∈ 𝑤 𝐷 = Σ𝑘 ∈ 𝐵 𝐷)
32	30, 31	breq12d 5122	. . . . . 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 11193	. . . . . 6 ⊢ 0 ∈ ℂ
37		rlimconst 15591	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 0 ∈ ℂ) → (𝑥 ∈ 𝐴 ↦ 0) ⇝𝑟 0)
38	35, 36, 37	sylancl 597	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 0) ⇝𝑟 0)
39	38	a1d 26	. . . 4 ⊢ (𝜑 → (∅ ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 0) ⇝𝑟 0))
40		ssun1 4131	. . . . . . . . . 10 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
41		sstr 3945	. . . . . . . . . 10 ⊢ ((𝑦 ⊆ (𝑦 ∪ {𝑧}) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵) → 𝑦 ⊆ 𝐵)
42	40, 41	mpan 702	. . . . . . . . 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 15735	. . . . . . . . . . . . . 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 784	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∪ {𝑧}) ⊆ 𝐵)
47	46	unssbd 4147	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → {𝑧} ⊆ 𝐵)
48		vex 3459	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑧 ∈ V
49	48	snss 4750	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝐵 ↔ {𝑧} ⊆ 𝐵)
50	47, 49	sylibr 237	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝑧 ∈ 𝐵)
51	50	adantr 485	. . . . . . . . . . . . . . . . . 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 667	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)
54		fsumrlim.4	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷)
55	53, 54	rlimmptrcl 15655	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)
56	55	an32s 664	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
57	56	adantllr 731	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
58	57	ralrimiva 3157	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ)
59		nfcsb1v 3877	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐶
60	59	nfel1 2941	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ
61		csbeq1a 3867	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑧 → 𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
62	61	eleq1d 2848	. . . . . . . . . . . . . . . . . . 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 66	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ)
65	64	ralrimiva 3157	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑥 ∈ 𝐴 ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ)
66	65	adantr 485	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ (𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝑦 𝐶) ⇝𝑟 Σ𝑘 ∈ 𝑦 𝐷) → ∀𝑥 ∈ 𝐴 ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ)
67		nfcsb1v 3877	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶
68	67	nfel1 2941	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ
69		csbeq1a 3867	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑤 → ⦋𝑧 / 𝑘⦌𝐶 = ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶)
70	69	eleq1d 2848	. . . . . . . . . . . . . . . 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 515	. . . . . . . . . . . . . 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 3478	. . . . . . . . . . . . 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 2925	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑤Σ𝑘 ∈ 𝑦 𝐶
75		nfcv 2925	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥𝑦
76		nfcsb1v 3877	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝐶
77	75, 76	nfsum 15738	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶
78		csbeq1a 3867	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑤 → 𝐶 = ⦋𝑤 / 𝑥⦌𝐶)
79	78	sumeq2sdv 15750	. . . . . . . . . . . . . . 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 489	. . . . . . . . . . . . . 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 2925	. . . . . . . . . . . . . . 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 3157	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑘 ∈ 𝐵 (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷)
86	85	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑘 ∈ 𝐵 (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷)
87		nfcv 2925	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘𝐴
88	87, 59	nfmpt 5209	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘(𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶)
89		nfcv 2925	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘 ⇝𝑟
90		nfcsb1v 3877	. . . . . . . . . . . . . . . . . 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 3867	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑧 → 𝐷 = ⦋𝑧 / 𝑘⦌𝐷)
94	92, 93	breq12d 5122	. . . . . . . . . . . . . . . . 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 66	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑥 ∈ 𝐴 ↦ ⦋𝑧 / 𝑘⦌𝐶) ⇝𝑟 ⦋𝑧 / 𝑘⦌𝐷)
97	96	adantr 485	. . . . . . . . . . . . . 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 15690	. . . . . . . . . . . 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 782	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ¬ 𝑧 ∈ 𝑦)
101		disjsn 4677	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
102	100, 101	sylibr 237	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∩ {𝑧}) = ∅)
103	102	adantr 485	. . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → 𝐵 ∈ Fin)
106	105, 46	ssfid 9225	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → (𝑦 ∪ {𝑧}) ∈ Fin)
107	106	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∪ {𝑧}) ∈ Fin)
108	46	sselda 3937	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑘 ∈ 𝐵)
109	108	adantlr 727	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑘 ∈ 𝐵)
110	109, 57	syldan 602	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝐶 ∈ ℂ)
111	103, 104, 107, 110	fsumsplit 15788	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑥 ∈ 𝐴) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐶 = (Σ𝑘 ∈ 𝑦 𝐶 + Σ𝑘 ∈ {𝑧}𝐶))
112		csbeq1a 3867	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑤 → 𝐶 = ⦋𝑤 / 𝑘⦌𝐶)
113		nfcv 2925	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑤𝐶
114		nfcsb1v 3877	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘⦋𝑤 / 𝑘⦌𝐶
115	112, 113, 114	cbvsum 15742	. . . . . . . . . . . . . . . . . 18 ⊢ Σ𝑘 ∈ {𝑧}𝐶 = Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐶
116		csbeq1 3856	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = 𝑧 → ⦋𝑤 / 𝑘⦌𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
117	116	sumsn 15793	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∈ 𝐵 ∧ ⦋𝑧 / 𝑘⦌𝐶 ∈ ℂ) → Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐶 = ⦋𝑧 / 𝑘⦌𝐶)
118	51, 64, 117	syl2anc 595	. . . . . . . . . . . . . . . . . 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 7426	. . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . 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 2925	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑤(Σ𝑘 ∈ 𝑦 𝐶 + ⦋𝑧 / 𝑘⦌𝐶)
125		nfcv 2925	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥 +
126	77, 125, 67	nfov 7440	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐶 + ⦋𝑤 / 𝑥⦌⦋𝑧 / 𝑘⦌𝐶)
127	79, 69	oveq12d 7428	. . . . . . . . . . . . . 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 15550	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷 → 𝐷 ∈ ℂ)
132	54, 131	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐷 ∈ ℂ)
133	132	adantlr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑘 ∈ 𝐵) → 𝐷 ∈ ℂ)
134	108, 133	syldan 602	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝐷 ∈ ℂ)
135	102, 130, 106, 134	fsumsplit 15788	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐷 = (Σ𝑘 ∈ 𝑦 𝐷 + Σ𝑘 ∈ {𝑧}𝐷))
136		csbeq1a 3867	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑤 → 𝐷 = ⦋𝑤 / 𝑘⦌𝐷)
137		nfcv 2925	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑤𝐷
138		nfcsb1v 3877	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘⦋𝑤 / 𝑘⦌𝐷
139	136, 137, 138	cbvsum 15742	. . . . . . . . . . . . . . . 16 ⊢ Σ𝑘 ∈ {𝑧}𝐷 = Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐷
140	133	ralrimiva 3157	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ∀𝑘 ∈ 𝐵 𝐷 ∈ ℂ)
141	90	nfel1 2941	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐷 ∈ ℂ
142	93	eleq1d 2848	. . . . . . . . . . . . . . . . . . 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 66	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (¬ 𝑧 ∈ 𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐵)) → ⦋𝑧 / 𝑘⦌𝐷 ∈ ℂ)
145		csbeq1 3856	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑧 → ⦋𝑤 / 𝑘⦌𝐷 = ⦋𝑧 / 𝑘⦌𝐷)
146	145	sumsn 15793	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝐵 ∧ ⦋𝑧 / 𝑘⦌𝐷 ∈ ℂ) → Σ𝑤 ∈ {𝑧}⦋𝑤 / 𝑘⦌𝐷 = ⦋𝑧 / 𝑘⦌𝐷)
147	50, 144, 146	syl2anc 595	. . . . . . . . . . . . . . . 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 7426	. . . . . . . . . . . . . 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 485	. . . . . . . . . . . 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 417	. . . . . . . . . 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 461	. . . . . . . . 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 418	. . . . . 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 486	. . . 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 9146	. . 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)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  Vcvv 3455  ⦋csb 3853   ∪ cun 3903   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286  {csn 4589   class class class wbr 5109   ↦ cmpt 5192  (class class class)co 7410  Fincfn 8939  ℂcc 11093  ℝcr 11094  0cc0 11095   + caddc 11098   ⇝_𝑟 crli 15532  Σcsu 15733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9606  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172  ax-pre-sup 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9398  df-oi 9468  df-card 9921  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-div 11867  df-nn 12229  df-2 12298  df-3 12299  df-n0 12500  df-z 12587  df-uz 12858  df-rp 13012  df-fz 13531  df-fzo 13679  df-seq 14034  df-exp 14094  df-hash 14363  df-cj 15146  df-re 15147  df-im 15148  df-sqrt 15282  df-abs 15283  df-clim 15535  df-rlim 15536  df-sum 15734

This theorem is referenced by:  climfsum  15868  logexprlim  27389  signsplypnf  34937