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Theorem fsumcn 23939

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 3939	. 2 ⊢ 𝐴 ⊆ 𝐴
2		fsumcn.5	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
3		sseq1 3942	. . . . . 6 ⊢ (𝑤 = ∅ → (𝑤 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
4		sumeq1 15328	. . . . . . . 8 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
5	4	mpteq2dv 5172	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ ∅ 𝐵))
6	5	eleq1d 2823	. . . . . 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 3942	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
10		sumeq1 15328	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ 𝑦 𝐵)
11	10	mpteq2dv 5172	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵))
12	11	eleq1d 2823	. . . . . 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 3942	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 ⊆ 𝐴 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝐴))
16		sumeq1 15328	. . . . . . . 8 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
17	16	mpteq2dv 5172	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
18	17	eleq1d 2823	. . . . . 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 3942	. . . . . 6 ⊢ (𝑤 = 𝐴 → (𝑤 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
22		sumeq1 15328	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ 𝐴 𝐵)
23	22	mpteq2dv 5172	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵))
24	23	eleq1d 2823	. . . . . 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 15361	. . . . . . 7 ⊢ Σ𝑘 ∈ ∅ 𝐵 = 0
28	27	mpteq2i 5175	. . . . . 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 23852	. . . . . . . 8 ⊢ 𝐾 ∈ (TopOn‘ℂ)
32	31	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘ℂ))
33		0cnd 10899	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℂ)
34	29, 32, 33	cnmptc 22721	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 0) ∈ (𝐽 Cn 𝐾))
35	28, 34	eqeltrid 2843	. . . . 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 4102	. . . . . . . . . 10 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
38		sstr 3925	. . . . . . . . . 10 ⊢ ((𝑦 ⊆ (𝑦 ∪ {𝑧}) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → 𝑦 ⊆ 𝐴)
39	37, 38	mpan 686	. . . . . . . . 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 765	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → ¬ 𝑧 ∈ 𝑦)
42		disjsn 4644	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
43	41, 42	sylibr 233	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → (𝑦 ∩ {𝑧}) = ∅)
44		eqidd 2739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
45	2	ad2antrr 722	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → 𝐴 ∈ Fin)
46		simprl 767	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → (𝑦 ∪ {𝑧}) ⊆ 𝐴)
47	45, 46	ssfid 8971	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → (𝑦 ∪ {𝑧}) ∈ Fin)
48		simplll 771	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝜑)
49	46	sselda 3917	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑘 ∈ 𝐴)
50		simplrr 774	. . . . . . . . . . . . . . . . . 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 22308	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘ℂ) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℂ)
55	51, 52, 53, 54	syl3anc 1369	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℂ)
56		eqid 2738	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐵)
57	56	fmpt 6966	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑥 ∈ 𝑋 𝐵 ∈ ℂ ↔ (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℂ)
58	55, 57	sylibr 233	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ∀𝑥 ∈ 𝑋 𝐵 ∈ ℂ)
59		rsp 3129	. . . . . . . . . . . . . . . . . . . 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 834	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝐵 ∈ ℂ)
63	43, 44, 47, 62	fsumsplit 15381	. . . . . . . . . . . . . . . 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 4118	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → {𝑧} ⊆ 𝐴)
66		vex 3426	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑧 ∈ V
67	66	snss 4716	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝐴 ↔ {𝑧} ⊆ 𝐴)
68	65, 67	sylibr 233	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → 𝑧 ∈ 𝐴)
69	68	adantrr 713	. . . . . . . . . . . . . . . . . 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 3106	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
72	71	ad2ant2rl 745	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
73		nfcsb1v 3853	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵
74	73	nfel1 2922	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ
75		csbeq1a 3842	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
76	75	eleq1d 2823	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑧 → (𝐵 ∈ ℂ ↔ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ))
77	74, 76	rspc 3539	. . . . . . . . . . . . . . . . . . 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 15390	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ 𝐴 ∧ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑧}𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
80	69, 78, 79	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → Σ𝑘 ∈ {𝑧}𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
81	80	oveq2d 7271	. . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . 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 5170	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (𝑥 ∈ 𝑋 ↦ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵)))
85	84	adantrr 713	. . . . . . . . . . . 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 2906	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑤(Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵)
87		nfcv 2906	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥𝑦
88		nfcsb1v 3853	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝐵
89	87, 88	nfsum 15330	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵
90		nfcv 2906	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥 +
91		nfcv 2906	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥𝑧
92	91, 88	nfcsbw 3855	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵
93	89, 90, 92	nfov 7285	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵 + ⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵)
94		csbeq1a 3842	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑤 → 𝐵 = ⦋𝑤 / 𝑥⦌𝐵)
95	94	sumeq2sdv 15344	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → Σ𝑘 ∈ 𝑦 𝐵 = Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵)
96	94	csbeq2dv 3835	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → ⦋𝑧 / 𝑘⦌𝐵 = ⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵)
97	95, 96	oveq12d 7273	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵) = (Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵 + ⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵))
98	86, 93, 97	cbvmpt 5181	. . . . . . . . . . . 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 2795	. . . . . . . . . . 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 722	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) → 𝐽 ∈ (TopOn‘𝑋))
101		nfcv 2906	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑤Σ𝑘 ∈ 𝑦 𝐵
102	101, 89, 95	cbvmpt 5181	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) = (𝑤 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵)
103		simprr 769	. . . . . . . . . . . . 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 2844	. . . . . . . . . . . 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 2906	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑤⦋𝑧 / 𝑘⦌𝐵
106	105, 92, 96	cbvmpt 5181	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌𝐵) = (𝑤 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵)
107	68	adantrr 713	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) → 𝑧 ∈ 𝐴)
108	53	ralrimiva 3107	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾))
109	108	ad2antrr 722	. . . . . . . . . . . . . 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 2906	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘𝑋
111	110, 73	nfmpt 5177	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘(𝑥 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌𝐵)
112	111	nfel1 2922	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝑥 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾)
113	75	mpteq2dv 5172	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑧 → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌𝐵))
114	113	eleq1d 2823	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑧 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾) ↔ (𝑥 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾)))
115	112, 114	rspc 3539	. . . . . . . . . . . . . 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 2844	. . . . . . . . . . . 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 23934	. . . . . . . . . . . . 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 22725	. . . . . . . . . . 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 2839	. . . . . . . . . 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 8910	. . 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 1539 ∈ wcel 2108 ∀wral 3063 ⦋csb 3828 ∪ cun 3881 ∩ cin 3882 ⊆ wss 3883 ∅c0 4253 {csn 4558 ↦ cmpt 5153 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 Fincfn 8691 ℂcc 10800 0cc0 10802 + caddc 10805 Σcsu 15325 TopOpenctopn 17049 ℂ_fldccnfld 20510 TopOnctopon 21967 Cn ccn 22283 ×_t ctx 22619

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-icc 13015 df-fz 13169 df-fzo 13312 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-sum 15326 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cn 22286 df-cnp 22287 df-tx 22621 df-hmeo 22814 df-xms 23381 df-ms 23382 df-tms 23383

This theorem is referenced by: fsum2cn 23940 lebnumlem2 24031 plycn 25327 psercn2 25487 knoppcnlem11 34610 fsumcnf 42453

W3C validator