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Theorem fsumcn 25191

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 3953	. 2 ⊢ 𝐴 ⊆ 𝐴
2		fsumcn.5	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
3		sseq1 3956	. . . . . 6 ⊢ (𝑤 = ∅ → (𝑤 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
4		sumeq1 15856	. . . . . . . 8 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
5	4	mpteq2dv 5199	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ ∅ 𝐵))
6	5	eleq1d 2846	. . . . . 6 ⊢ (𝑤 = ∅ → ((𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) ∈ (𝐽 Cn 𝐾) ↔ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ ∅ 𝐵) ∈ (𝐽 Cn 𝐾)))
7	3, 6	imbi12d 347	. . . . 5 ⊢ (𝑤 = ∅ → ((𝑤 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) ∈ (𝐽 Cn 𝐾)) ↔ (∅ ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ ∅ 𝐵) ∈ (𝐽 Cn 𝐾))))
8	7	imbi2d 343	. . . 4 ⊢ (𝑤 = ∅ → ((𝜑 → (𝑤 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) ∈ (𝐽 Cn 𝐾))) ↔ (𝜑 → (∅ ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ ∅ 𝐵) ∈ (𝐽 Cn 𝐾)))))
9		sseq1 3956	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
10		sumeq1 15856	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ 𝑦 𝐵)
11	10	mpteq2dv 5199	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵))
12	11	eleq1d 2846	. . . . . 6 ⊢ (𝑤 = 𝑦 → ((𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) ∈ (𝐽 Cn 𝐾) ↔ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾)))
13	9, 12	imbi12d 347	. . . . 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 343	. . . 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 3956	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 ⊆ 𝐴 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝐴))
16		sumeq1 15856	. . . . . . . 8 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
17	16	mpteq2dv 5199	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
18	17	eleq1d 2846	. . . . . 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 347	. . . . 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 343	. . . 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 3956	. . . . . 6 ⊢ (𝑤 = 𝐴 → (𝑤 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
22		sumeq1 15856	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ 𝐴 𝐵)
23	22	mpteq2dv 5199	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵))
24	23	eleq1d 2846	. . . . . 6 ⊢ (𝑤 = 𝐴 → ((𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) ∈ (𝐽 Cn 𝐾) ↔ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾)))
25	21, 24	imbi12d 347	. . . . 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 343	. . . 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 15887	. . . . . . 7 ⊢ Σ𝑘 ∈ ∅ 𝐵 = 0
28	27	mpteq2i 5201	. . . . . 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 25101	. . . . . . . 8 ⊢ 𝐾 ∈ (TopOn‘ℂ)
32	31	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘ℂ))
33		0cnd 11299	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℂ)
34	29, 32, 33	cnmptc 23981	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 0) ∈ (𝐽 Cn 𝐾))
35	28, 34	eqeltrid 2865	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ ∅ 𝐵) ∈ (𝐽 Cn 𝐾))
36	35	a1d 26	. . . 4 ⊢ (𝜑 → (∅ ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ ∅ 𝐵) ∈ (𝐽 Cn 𝐾)))
37		ssun1 4124	. . . . . . . . . 10 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
38		sstr 3939	. . . . . . . . . 10 ⊢ ((𝑦 ⊆ (𝑦 ∪ {𝑧}) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → 𝑦 ⊆ 𝐴)
39	37, 38	mpan 703	. . . . . . . . 9 ⊢ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → 𝑦 ⊆ 𝐴)
40	39	imim1i 64	. . . . . . . 8 ⊢ ((𝑦 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾)) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾)))
41		simplr 781	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → ¬ 𝑧 ∈ 𝑦)
42		disjsn 4672	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
43	41, 42	sylibr 237	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → (𝑦 ∩ {𝑧}) = ∅)
44		eqidd 2762	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
45	2	ad2antrr 739	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → 𝐴 ∈ Fin)
46		simprl 783	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → (𝑦 ∪ {𝑧}) ⊆ 𝐴)
47	45, 46	ssfid 9260	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → (𝑦 ∪ {𝑧}) ∈ Fin)
48		simplll 787	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝜑)
49	46	sselda 3931	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑘 ∈ 𝐴)
50		simplrr 790	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑥 ∈ 𝑋)
51	29	adantr 486	. . . . . . . . . . . . . . . . . . . . . 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 23567	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘ℂ) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℂ)
55	51, 52, 53, 54	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℂ)
56		eqid 2761	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐵)
57	56	fmpt 7110	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑥 ∈ 𝑋 𝐵 ∈ ℂ ↔ (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℂ)
58	55, 57	sylibr 237	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ∀𝑥 ∈ 𝑋 𝐵 ∈ ℂ)
59		rsp 3251	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑥 ∈ 𝑋 𝐵 ∈ ℂ → (𝑥 ∈ 𝑋 → 𝐵 ∈ ℂ))
60	58, 59	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 → 𝐵 ∈ ℂ))
61	60	imp 412	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ)
62	48, 49, 50, 61	syl21anc 851	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝐵 ∈ ℂ)
63	43, 44, 47, 62	fsumsplit 15907	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (Σ𝑘 ∈ 𝑦 𝐵 + Σ𝑘 ∈ {𝑧}𝐵))
64		simpr 490	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → (𝑦 ∪ {𝑧}) ⊆ 𝐴)
65	64	unssbd 4140	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → {𝑧} ⊆ 𝐴)
66		vex 3455	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑧 ∈ V
67	66	snss 4745	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝐴 ↔ {𝑧} ⊆ 𝐴)
68	65, 67	sylibr 237	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → 𝑧 ∈ 𝐴)
69	68	adantrr 730	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → 𝑧 ∈ 𝐴)
70	60	impancom 457	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑘 ∈ 𝐴 → 𝐵 ∈ ℂ))
71	70	ralrimiv 3154	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
72	71	ad2ant2rl 762	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
73		nfcsb1v 3871	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵
74	73	nfel1 2939	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ
75		csbeq1a 3861	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
76	75	eleq1d 2846	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑧 → (𝐵 ∈ ℂ ↔ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ))
77	74, 76	rspc 3565	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ))
78	69, 72, 77	sylc 66	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)
79		sumsns 15916	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ 𝐴 ∧ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑧}𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
80	69, 78, 79	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → Σ𝑘 ∈ {𝑧}𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
81	80	oveq2d 7436	. . . . . . . . . . . . . . . 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 2796	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵))
83	82	anassrs 473	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵))
84	83	mpteq2dva 5198	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (𝑥 ∈ 𝑋 ↦ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵)))
85	84	adantrr 730	. . . . . . . . . . . 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 2923	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑤(Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵)
87		nfcv 2923	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥𝑦
88		nfcsb1v 3871	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝐵
89	87, 88	nfsum 15858	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵
90		nfcv 2923	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥 +
91		nfcv 2923	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥𝑧
92	91, 88	nfcsbw 3873	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵
93	89, 90, 92	nfov 7450	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵 + ⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵)
94		csbeq1a 3861	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑤 → 𝐵 = ⦋𝑤 / 𝑥⦌𝐵)
95	94	sumeq2sdv 15870	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → Σ𝑘 ∈ 𝑦 𝐵 = Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵)
96	94	csbeq2dv 3854	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → ⦋𝑧 / 𝑘⦌𝐵 = ⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵)
97	95, 96	oveq12d 7438	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵) = (Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵 + ⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵))
98	86, 93, 97	cbvmpt 5207	. . . . . . . . . . . 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 2812	. . . . . . . . . . 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 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) → 𝐽 ∈ (TopOn‘𝑋))
101		nfcv 2923	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑤Σ𝑘 ∈ 𝑦 𝐵
102	101, 89, 95	cbvmpt 5207	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) = (𝑤 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵)
103		simprr 785	. . . . . . . . . . . . 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 2866	. . . . . . . . . . . 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 2923	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑤⦋𝑧 / 𝑘⦌𝐵
106	105, 92, 96	cbvmpt 5207	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌𝐵) = (𝑤 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵)
107	68	adantrr 730	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) → 𝑧 ∈ 𝐴)
108	53	ralrimiva 3155	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾))
109	108	ad2antrr 739	. . . . . . . . . . . . . 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 2923	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘𝑋
111	110, 73	nfmpt 5203	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘(𝑥 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌𝐵)
112	111	nfel1 2939	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝑥 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾)
113	75	mpteq2dv 5199	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑧 → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌𝐵))
114	113	eleq1d 2846	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑧 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾) ↔ (𝑥 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾)))
115	112, 114	rspc 3565	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾) → (𝑥 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾)))
116	107, 109, 115	sylc 66	. . . . . . . . . . . . 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 2866	. . . . . . . . . . . 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 25185	. . . . . . . . . . . . 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 23985	. . . . . . . . . . 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 2861	. . . . . . . . . 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 426	. . . . . . . . 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 30	. . . . . . . 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 35	. . . . . . 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 419	. . . . . 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 487	. . . . 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 30	. . . 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 9181	. . 3 ⊢ (𝐴 ∈ Fin → (𝜑 → (𝐴 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾))))
129	2, 128	mpcom 39	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾)))
130	1, 129	mpi 21	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∀wral 3077 ⦋csb 3847 ∪ cun 3897 ∩ cin 3898 ⊆ wss 3899 ∅c0 4279 {csn 4584 ↦ cmpt 5186 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 Fincfn 8973 ℂcc 11198 0cc0 11200 + caddc 11203 Σcsu 15853 TopOpenctopn 17592 ℂ_fldccnfld 21678 TopOnctopon 23228 Cn ccn 23542 ×_t ctx 23879

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-inf2 9642 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278 ax-addf 11279

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 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-er 8717 df-map 8849 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9354 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9504 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-z 12694 df-dec 12815 df-uz 12966 df-q 13076 df-rp 13121 df-xneg 13241 df-xadd 13242 df-xmul 13243 df-icc 13483 df-fz 13640 df-fzo 13789 df-seq 14145 df-exp 14205 df-hash 14475 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-clim 15655 df-sum 15854 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-starv 17443 df-sca 17444 df-vsca 17445 df-ip 17446 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-hom 17452 df-cco 17453 df-rest 17593 df-topn 17594 df-0g 17612 df-gsum 17613 df-topgen 17614 df-pt 17615 df-prds 17618 df-xrs 17674 df-qtop 17679 df-imas 17680 df-xps 17682 df-mre 17756 df-mrc 17757 df-acs 17759 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-submnd 18979 df-mulg 19278 df-cntz 19531 df-cmn 19996 df-psmet 21670 df-xmet 21671 df-met 21672 df-bl 21673 df-mopn 21674 df-cnfld 21679 df-top 23212 df-topon 23229 df-topsp 23251 df-bases 23264 df-cn 23545 df-cnp 23546 df-tx 23881 df-hmeo 24074 df-xms 24639 df-ms 24640 df-tms 24641

This theorem is used by: fsum2cn 25192 lebnumlem2 25283 plycn 26580 psercn2 26750 knoppcnlem11 37369 fsumcnf 46037

W3C validator