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Theorem fsumcn 24847

Description: A finite sum of functions to complex numbers from a common topological space is continuous. The class expression for 𝐵 normally contains free variables 𝑘 and 𝑥 to index it. (Contributed by NM, 8-Aug-2007.) (Revised by Mario Carneiro, 23-Aug-2014.)

**Hypotheses**
Ref	Expression
fsumcn.3	⊢ 𝐾 = (TopOpen‘ℂ_fld)
fsumcn.4	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
fsumcn.5	⊢ (𝜑 → 𝐴 ∈ Fin)
fsumcn.6	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾))

**Assertion**
Ref	Expression
fsumcn	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾))

Distinct variable groups: 𝑥,𝑘,𝐴 𝑘,𝐽,𝑥 𝜑,𝑘,𝑥 𝑘,𝐾,𝑥 𝑘,𝑋,𝑥 Allowed substitution hints: 𝐵(𝑥,𝑘)

Proof of Theorem fsumcn Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssid 3945	. 2 ⊢ 𝐴 ⊆ 𝐴
2		fsumcn.5	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
3		sseq1 3948	. . . . . 6 ⊢ (𝑤 = ∅ → (𝑤 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
4		sumeq1 15642	. . . . . . . 8 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
5	4	mpteq2dv 5180	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ ∅ 𝐵))
6	5	eleq1d 2822	. . . . . 6 ⊢ (𝑤 = ∅ → ((𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) ∈ (𝐽 Cn 𝐾) ↔ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ ∅ 𝐵) ∈ (𝐽 Cn 𝐾)))
7	3, 6	imbi12d 344	. . . . 5 ⊢ (𝑤 = ∅ → ((𝑤 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) ∈ (𝐽 Cn 𝐾)) ↔ (∅ ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ ∅ 𝐵) ∈ (𝐽 Cn 𝐾))))
8	7	imbi2d 340	. . . 4 ⊢ (𝑤 = ∅ → ((𝜑 → (𝑤 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) ∈ (𝐽 Cn 𝐾))) ↔ (𝜑 → (∅ ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ ∅ 𝐵) ∈ (𝐽 Cn 𝐾)))))
9		sseq1 3948	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
10		sumeq1 15642	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ 𝑦 𝐵)
11	10	mpteq2dv 5180	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵))
12	11	eleq1d 2822	. . . . . 6 ⊢ (𝑤 = 𝑦 → ((𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) ∈ (𝐽 Cn 𝐾) ↔ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾)))
13	9, 12	imbi12d 344	. . . . 5 ⊢ (𝑤 = 𝑦 → ((𝑤 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) ∈ (𝐽 Cn 𝐾)) ↔ (𝑦 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))))
14	13	imbi2d 340	. . . 4 ⊢ (𝑤 = 𝑦 → ((𝜑 → (𝑤 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) ∈ (𝐽 Cn 𝐾))) ↔ (𝜑 → (𝑦 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾)))))
15		sseq1 3948	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 ⊆ 𝐴 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝐴))
16		sumeq1 15642	. . . . . . . 8 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
17	16	mpteq2dv 5180	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
18	17	eleq1d 2822	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) ∈ (𝐽 Cn 𝐾) ↔ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ (𝐽 Cn 𝐾)))
19	15, 18	imbi12d 344	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝑤 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) ∈ (𝐽 Cn 𝐾)) ↔ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ (𝐽 Cn 𝐾))))
20	19	imbi2d 340	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝜑 → (𝑤 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) ∈ (𝐽 Cn 𝐾))) ↔ (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ (𝐽 Cn 𝐾)))))
21		sseq1 3948	. . . . . 6 ⊢ (𝑤 = 𝐴 → (𝑤 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
22		sumeq1 15642	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ 𝐴 𝐵)
23	22	mpteq2dv 5180	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵))
24	23	eleq1d 2822	. . . . . 6 ⊢ (𝑤 = 𝐴 → ((𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) ∈ (𝐽 Cn 𝐾) ↔ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾)))
25	21, 24	imbi12d 344	. . . . 5 ⊢ (𝑤 = 𝐴 → ((𝑤 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) ∈ (𝐽 Cn 𝐾)) ↔ (𝐴 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾))))
26	25	imbi2d 340	. . . 4 ⊢ (𝑤 = 𝐴 → ((𝜑 → (𝑤 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) ∈ (𝐽 Cn 𝐾))) ↔ (𝜑 → (𝐴 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾)))))
27		sum0 15674	. . . . . . 7 ⊢ Σ𝑘 ∈ ∅ 𝐵 = 0
28	27	mpteq2i 5182	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ ∅ 𝐵) = (𝑥 ∈ 𝑋 ↦ 0)
29		fsumcn.4	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
30		fsumcn.3	. . . . . . . . 9 ⊢ 𝐾 = (TopOpen‘ℂ_fld)
31	30	cnfldtopon 24757	. . . . . . . 8 ⊢ 𝐾 ∈ (TopOn‘ℂ)
32	31	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘ℂ))
33		0cnd 11128	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℂ)
34	29, 32, 33	cnmptc 23637	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 0) ∈ (𝐽 Cn 𝐾))
35	28, 34	eqeltrid 2841	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ ∅ 𝐵) ∈ (𝐽 Cn 𝐾))
36	35	a1d 25	. . . 4 ⊢ (𝜑 → (∅ ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ ∅ 𝐵) ∈ (𝐽 Cn 𝐾)))
37		ssun1 4119	. . . . . . . . . 10 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
38		sstr 3931	. . . . . . . . . 10 ⊢ ((𝑦 ⊆ (𝑦 ∪ {𝑧}) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → 𝑦 ⊆ 𝐴)
39	37, 38	mpan 691	. . . . . . . . 9 ⊢ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → 𝑦 ⊆ 𝐴)
40	39	imim1i 63	. . . . . . . 8 ⊢ ((𝑦 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾)) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾)))
41		simplr 769	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → ¬ 𝑧 ∈ 𝑦)
42		disjsn 4656	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
43	41, 42	sylibr 234	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → (𝑦 ∩ {𝑧}) = ∅)
44		eqidd 2738	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
45	2	ad2antrr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → 𝐴 ∈ Fin)
46		simprl 771	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → (𝑦 ∪ {𝑧}) ⊆ 𝐴)
47	45, 46	ssfid 9172	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → (𝑦 ∪ {𝑧}) ∈ Fin)
48		simplll 775	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝜑)
49	46	sselda 3922	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑘 ∈ 𝐴)
50		simplrr 778	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑥 ∈ 𝑋)
51	29	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐽 ∈ (TopOn‘𝑋))
52	31	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐾 ∈ (TopOn‘ℂ))
53		fsumcn.6	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾))
54		cnf2 23224	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘ℂ) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℂ)
55	51, 52, 53, 54	syl3anc 1374	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℂ)
56		eqid 2737	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐵)
57	56	fmpt 7056	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑥 ∈ 𝑋 𝐵 ∈ ℂ ↔ (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℂ)
58	55, 57	sylibr 234	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ∀𝑥 ∈ 𝑋 𝐵 ∈ ℂ)
59		rsp 3226	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑥 ∈ 𝑋 𝐵 ∈ ℂ → (𝑥 ∈ 𝑋 → 𝐵 ∈ ℂ))
60	58, 59	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 → 𝐵 ∈ ℂ))
61	60	imp 406	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ)
62	48, 49, 50, 61	syl21anc 838	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝐵 ∈ ℂ)
63	43, 44, 47, 62	fsumsplit 15694	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (Σ𝑘 ∈ 𝑦 𝐵 + Σ𝑘 ∈ {𝑧}𝐵))
64		simpr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → (𝑦 ∪ {𝑧}) ⊆ 𝐴)
65	64	unssbd 4135	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → {𝑧} ⊆ 𝐴)
66		vex 3434	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑧 ∈ V
67	66	snss 4729	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝐴 ↔ {𝑧} ⊆ 𝐴)
68	65, 67	sylibr 234	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → 𝑧 ∈ 𝐴)
69	68	adantrr 718	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → 𝑧 ∈ 𝐴)
70	60	impancom 451	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑘 ∈ 𝐴 → 𝐵 ∈ ℂ))
71	70	ralrimiv 3129	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
72	71	ad2ant2rl 750	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
73		nfcsb1v 3862	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵
74	73	nfel1 2916	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ
75		csbeq1a 3852	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
76	75	eleq1d 2822	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑧 → (𝐵 ∈ ℂ ↔ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ))
77	74, 76	rspc 3553	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ))
78	69, 72, 77	sylc 65	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)
79		sumsns 15703	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ 𝐴 ∧ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑧}𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
80	69, 78, 79	syl2anc 585	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → Σ𝑘 ∈ {𝑧}𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
81	80	oveq2d 7376	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → (Σ𝑘 ∈ 𝑦 𝐵 + Σ𝑘 ∈ {𝑧}𝐵) = (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵))
82	63, 81	eqtrd 2772	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵))
83	82	anassrs 467	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵))
84	83	mpteq2dva 5179	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (𝑥 ∈ 𝑋 ↦ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵)))
85	84	adantrr 718	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (𝑥 ∈ 𝑋 ↦ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵)))
86		nfcv 2899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑤(Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵)
87		nfcv 2899	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥𝑦
88		nfcsb1v 3862	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝐵
89	87, 88	nfsum 15644	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵
90		nfcv 2899	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥 +
91		nfcv 2899	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥𝑧
92	91, 88	nfcsbw 3864	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵
93	89, 90, 92	nfov 7390	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵 + ⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵)
94		csbeq1a 3852	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑤 → 𝐵 = ⦋𝑤 / 𝑥⦌𝐵)
95	94	sumeq2sdv 15656	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → Σ𝑘 ∈ 𝑦 𝐵 = Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵)
96	94	csbeq2dv 3845	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → ⦋𝑧 / 𝑘⦌𝐵 = ⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵)
97	95, 96	oveq12d 7378	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵) = (Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵 + ⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵))
98	86, 93, 97	cbvmpt 5188	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋 ↦ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵)) = (𝑤 ∈ 𝑋 ↦ (Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵 + ⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵))
99	85, 98	eqtrdi 2788	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (𝑤 ∈ 𝑋 ↦ (Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵 + ⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵)))
100	29	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) → 𝐽 ∈ (TopOn‘𝑋))
101		nfcv 2899	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑤Σ𝑘 ∈ 𝑦 𝐵
102	101, 89, 95	cbvmpt 5188	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) = (𝑤 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵)
103		simprr 773	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))
104	102, 103	eqeltrrid 2842	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) → (𝑤 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵) ∈ (𝐽 Cn 𝐾))
105		nfcv 2899	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑤⦋𝑧 / 𝑘⦌𝐵
106	105, 92, 96	cbvmpt 5188	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌𝐵) = (𝑤 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵)
107	68	adantrr 718	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) → 𝑧 ∈ 𝐴)
108	53	ralrimiva 3130	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾))
109	108	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) → ∀𝑘 ∈ 𝐴 (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾))
110		nfcv 2899	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘𝑋
111	110, 73	nfmpt 5184	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘(𝑥 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌𝐵)
112	111	nfel1 2916	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝑥 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾)
113	75	mpteq2dv 5180	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑧 → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌𝐵))
114	113	eleq1d 2822	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑧 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾) ↔ (𝑥 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾)))
115	112, 114	rspc 3553	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾) → (𝑥 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾)))
116	107, 109, 115	sylc 65	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) → (𝑥 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾))
117	106, 116	eqeltrrid 2842	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) → (𝑤 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵) ∈ (𝐽 Cn 𝐾))
118	30	addcn 24841	. . . . . . . . . . . . 13 ⊢ + ∈ ((𝐾 ×_t 𝐾) Cn 𝐾)
119	118	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) → + ∈ ((𝐾 ×_t 𝐾) Cn 𝐾))
120	100, 104, 117, 119	cnmpt12f 23641	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) → (𝑤 ∈ 𝑋 ↦ (Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵 + ⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵)) ∈ (𝐽 Cn 𝐾))
121	99, 120	eqeltrd 2837	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ (𝐽 Cn 𝐾))
122	121	exp32 420	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → ((𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾) → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ (𝐽 Cn 𝐾))))
123	122	a2d 29	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) → (((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾)) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ (𝐽 Cn 𝐾))))
124	40, 123	syl5 34	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) → ((𝑦 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾)) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ (𝐽 Cn 𝐾))))
125	124	expcom 413	. . . . . 6 ⊢ (¬ 𝑧 ∈ 𝑦 → (𝜑 → ((𝑦 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾)) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ (𝐽 Cn 𝐾)))))
126	125	adantl 481	. . . . 5 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (𝜑 → ((𝑦 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾)) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ (𝐽 Cn 𝐾)))))
127	126	a2d 29	. . . 4 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((𝜑 → (𝑦 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) → (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ∈ (𝐽 Cn 𝐾)))))
128	8, 14, 20, 26, 36, 127	findcard2s 9093	. . 3 ⊢ (𝐴 ∈ Fin → (𝜑 → (𝐴 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾))))
129	2, 128	mpcom 38	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾)))
130	1, 129	mpi 20	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ⦋csb 3838 ∪ cun 3888 ∩ cin 3889 ⊆ wss 3890 ∅c0 4274 {csn 4568 ↦ cmpt 5167 ⟶wf 6488 ‘cfv 6492 (class class class)co 7360 Fincfn 8886 ℂcc 11027 0cc0 11029 + caddc 11032 Σcsu 15639 TopOpenctopn 17375 ℂ_fldccnfld 21344 TopOnctopon 22885 Cn ccn 23199 ×_t ctx 23535

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-inf2 9553 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8104 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-er 8636 df-map 8768 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9268 df-fi 9317 df-sup 9348 df-inf 9349 df-oi 9418 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-icc 13296 df-fz 13453 df-fzo 13600 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15441 df-sum 15640 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-mulg 19035 df-cntz 19283 df-cmn 19748 df-psmet 21336 df-xmet 21337 df-met 21338 df-bl 21339 df-mopn 21340 df-cnfld 21345 df-top 22869 df-topon 22886 df-topsp 22908 df-bases 22921 df-cn 23202 df-cnp 23203 df-tx 23537 df-hmeo 23730 df-xms 24295 df-ms 24296 df-tms 24297

This theorem is referenced by: fsum2cn 24848 lebnumlem2 24939 plycn 26236 psercn2 26401 knoppcnlem11 36779 fsumcnf 45470

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