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Theorem fsumcn 23473

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 3982	. 2 ⊢ 𝐴 ⊆ 𝐴
2		fsumcn.5	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
3		sseq1 3985	. . . . . 6 ⊢ (𝑤 = ∅ → (𝑤 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
4		sumeq1 15040	. . . . . . . 8 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
5	4	mpteq2dv 5155	. . . . . . 7 ⊢ (𝑤 = ∅ → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ ∅ 𝐵))
6	5	eleq1d 2896	. . . . . 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 3985	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
10		sumeq1 15040	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ 𝑦 𝐵)
11	10	mpteq2dv 5155	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵))
12	11	eleq1d 2896	. . . . . 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 3985	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 ⊆ 𝐴 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝐴))
16		sumeq1 15040	. . . . . . . 8 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
17	16	mpteq2dv 5155	. . . . . . 7 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
18	17	eleq1d 2896	. . . . . 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 3985	. . . . . 6 ⊢ (𝑤 = 𝐴 → (𝑤 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
22		sumeq1 15040	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → Σ𝑘 ∈ 𝑤 𝐵 = Σ𝑘 ∈ 𝐴 𝐵)
23	22	mpteq2dv 5155	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑤 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵))
24	23	eleq1d 2896	. . . . . 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 15073	. . . . . . 7 ⊢ Σ𝑘 ∈ ∅ 𝐵 = 0
28	27	mpteq2i 5151	. . . . . 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 23386	. . . . . . . 8 ⊢ 𝐾 ∈ (TopOn‘ℂ)
32	31	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘ℂ))
33		0cnd 10627	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℂ)
34	29, 32, 33	cnmptc 22265	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 0) ∈ (𝐽 Cn 𝐾))
35	28, 34	eqeltrid 2916	. . . . 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 4141	. . . . . . . . . 10 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
38		sstr 3968	. . . . . . . . . 10 ⊢ ((𝑦 ⊆ (𝑦 ∪ {𝑧}) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → 𝑦 ⊆ 𝐴)
39	37, 38	mpan 688	. . . . . . . . 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 767	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → ¬ 𝑧 ∈ 𝑦)
42		disjsn 4640	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
43	41, 42	sylibr 236	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → (𝑦 ∩ {𝑧}) = ∅)
44		eqidd 2821	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
45	2	ad2antrr 724	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → 𝐴 ∈ Fin)
46		simprl 769	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → (𝑦 ∪ {𝑧}) ⊆ 𝐴)
47	45, 46	ssfid 8734	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → (𝑦 ∪ {𝑧}) ∈ Fin)
48		simplll 773	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝜑)
49	46	sselda 3960	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑘 ∈ 𝐴)
50		simplrr 776	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝑥 ∈ 𝑋)
51	29	adantr 483	. . . . . . . . . . . . . . . . . . . . . 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 21852	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘ℂ) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℂ)
55	51, 52, 53, 54	syl3anc 1366	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℂ)
56		eqid 2820	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐵)
57	56	fmpt 6867	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑥 ∈ 𝑋 𝐵 ∈ ℂ ↔ (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℂ)
58	55, 57	sylibr 236	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ∀𝑥 ∈ 𝑋 𝐵 ∈ ℂ)
59		rsp 3204	. . . . . . . . . . . . . . . . . . . 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 409	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ)
62	48, 49, 50, 61	syl21anc 835	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 𝐵 ∈ ℂ)
63	43, 44, 47, 62	fsumsplit 15092	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (Σ𝑘 ∈ 𝑦 𝐵 + Σ𝑘 ∈ {𝑧}𝐵))
64		simpr 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → (𝑦 ∪ {𝑧}) ⊆ 𝐴)
65	64	unssbd 4157	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → {𝑧} ⊆ 𝐴)
66		vex 3494	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑧 ∈ V
67	66	snss 4711	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝐴 ↔ {𝑧} ⊆ 𝐴)
68	65, 67	sylibr 236	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → 𝑧 ∈ 𝐴)
69	68	adantrr 715	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → 𝑧 ∈ 𝐴)
70	60	impancom 454	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑘 ∈ 𝐴 → 𝐵 ∈ ℂ))
71	70	ralrimiv 3180	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
72	71	ad2ant2rl 747	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
73		nfcsb1v 3900	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵
74	73	nfel1 2993	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ
75		csbeq1a 3890	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
76	75	eleq1d 2896	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑧 → (𝐵 ∈ ℂ ↔ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ))
77	74, 76	rspc 3608	. . . . . . . . . . . . . . . . . . 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 15100	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ 𝐴 ∧ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑧}𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
80	69, 78, 79	syl2anc 586	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → Σ𝑘 ∈ {𝑧}𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
81	80	oveq2d 7165	. . . . . . . . . . . . . . . 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 2855	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋)) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵))
83	82	anassrs 470	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵))
84	83	mpteq2dva 5154	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (𝑥 ∈ 𝑋 ↦ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵)))
85	84	adantrr 715	. . . . . . . . . . . 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 2976	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑤(Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵)
87		nfcv 2976	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥𝑦
88		nfcsb1v 3900	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝐵
89	87, 88	nfsumw 15042	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵
90		nfcv 2976	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥 +
91		nfcv 2976	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥𝑧
92	91, 88	nfcsbw 3902	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵
93	89, 90, 92	nfov 7179	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵 + ⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵)
94		csbeq1a 3890	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑤 → 𝐵 = ⦋𝑤 / 𝑥⦌𝐵)
95	94	sumeq2sdv 15056	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → Σ𝑘 ∈ 𝑦 𝐵 = Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵)
96	94	csbeq2dv 3883	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → ⦋𝑧 / 𝑘⦌𝐵 = ⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵)
97	95, 96	oveq12d 7167	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵) = (Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵 + ⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵))
98	86, 93, 97	cbvmpt 5160	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋 ↦ (Σ𝑘 ∈ 𝑦 𝐵 + ⦋𝑧 / 𝑘⦌𝐵)) = (𝑤 ∈ 𝑋 ↦ (Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵 + ⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵))
99	85, 98	syl6eq 2871	. . . . . . . . . . 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 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) → 𝐽 ∈ (TopOn‘𝑋))
101		nfcv 2976	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑤Σ𝑘 ∈ 𝑦 𝐵
102	101, 89, 95	cbvmpt 5160	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) = (𝑤 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 ⦋𝑤 / 𝑥⦌𝐵)
103		simprr 771	. . . . . . . . . . . . 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 2917	. . . . . . . . . . . 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 2976	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑤⦋𝑧 / 𝑘⦌𝐵
106	105, 92, 96	cbvmpt 5160	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌𝐵) = (𝑤 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌⦋𝑤 / 𝑥⦌𝐵)
107	68	adantrr 715	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑦 𝐵) ∈ (𝐽 Cn 𝐾))) → 𝑧 ∈ 𝐴)
108	53	ralrimiva 3181	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾))
109	108	ad2antrr 724	. . . . . . . . . . . . . 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 2976	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘𝑋
111	110, 73	nfmpt 5156	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘(𝑥 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌𝐵)
112	111	nfel1 2993	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝑥 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾)
113	75	mpteq2dv 5155	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑧 → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌𝐵))
114	113	eleq1d 2896	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑧 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾) ↔ (𝑥 ∈ 𝑋 ↦ ⦋𝑧 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾)))
115	112, 114	rspc 3608	. . . . . . . . . . . . . 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 2917	. . . . . . . . . . . 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 23468	. . . . . . . . . . . . 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 22269	. . . . . . . . . . 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 2912	. . . . . . . . . 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 423	. . . . . . . . 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 416	. . . . . 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 484	. . . . 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 8752	. . 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 398 = wceq 1536 ∈ wcel 2113 ∀wral 3137 ⦋csb 3876 ∪ cun 3927 ∩ cin 3928 ⊆ wss 3929 ∅c0 4284 {csn 4560 ↦ cmpt 5139 ⟶wf 6344 ‘cfv 6348 (class class class)co 7149 Fincfn 8502 ℂcc 10528 0cc0 10530 + caddc 10533 Σcsu 15037 TopOpenctopn 16690 ℂ_fldccnfld 20540 TopOnctopon 21513 Cn ccn 21827 ×_t ctx 22163

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-inf2 9097 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 ax-addf 10609

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-of 7402 df-om 7574 df-1st 7682 df-2nd 7683 df-supp 7824 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-2o 8096 df-oadd 8099 df-er 8282 df-map 8401 df-ixp 8455 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-fsupp 8827 df-fi 8868 df-sup 8899 df-inf 8900 df-oi 8967 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-icc 12739 df-fz 12890 df-fzo 13031 df-seq 13367 df-exp 13427 df-hash 13688 df-cj 14453 df-re 14454 df-im 14455 df-sqrt 14589 df-abs 14590 df-clim 14840 df-sum 15038 df-struct 16480 df-ndx 16481 df-slot 16482 df-base 16484 df-sets 16485 df-ress 16486 df-plusg 16573 df-mulr 16574 df-starv 16575 df-sca 16576 df-vsca 16577 df-ip 16578 df-tset 16579 df-ple 16580 df-ds 16582 df-unif 16583 df-hom 16584 df-cco 16585 df-rest 16691 df-topn 16692 df-0g 16710 df-gsum 16711 df-topgen 16712 df-pt 16713 df-prds 16716 df-xrs 16770 df-qtop 16775 df-imas 16776 df-xps 16778 df-mre 16852 df-mrc 16853 df-acs 16855 df-mgm 17847 df-sgrp 17896 df-mnd 17907 df-submnd 17952 df-mulg 18220 df-cntz 18442 df-cmn 18903 df-psmet 20532 df-xmet 20533 df-met 20534 df-bl 20535 df-mopn 20536 df-cnfld 20541 df-top 21497 df-topon 21514 df-topsp 21536 df-bases 21549 df-cn 21830 df-cnp 21831 df-tx 22165 df-hmeo 22358 df-xms 22925 df-ms 22926 df-tms 22927

This theorem is referenced by: fsum2cn 23474 lebnumlem2 23561 plycn 24849 psercn2 25009 knoppcnlem11 33863 fsumcnf 41352

W3C validator