Theorem fsumvma 27432

Description: Rewrite a sum over the von Mangoldt function as a sum over prime powers. (Contributed by Mario Carneiro, 15-Apr-2016.)

Hypotheses

Ref	Expression
fsumvma.1	⊢ (𝑥 = (𝑝↑𝑘) → 𝐵 = 𝐶)
fsumvma.2	⊢ (𝜑 → 𝐴 ∈ Fin)
fsumvma.3	⊢ (𝜑 → 𝐴 ⊆ ℕ)
fsumvma.4	⊢ (𝜑 → 𝑃 ∈ Fin)
fsumvma.5	⊢ (𝜑 → ((𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾) ↔ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ (𝑝↑𝑘) ∈ 𝐴)))
fsumvma.6	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
fsumvma.7	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ (Λ‘𝑥) = 0)) → 𝐵 = 0)

Assertion

Ref	Expression
fsumvma	⊢ (𝜑 → Σ𝑥 ∈ 𝐴 𝐵 = Σ𝑝 ∈ 𝑃 Σ𝑘 ∈ 𝐾 𝐶)

Distinct variable groups:   𝑘,𝑝,𝑥,𝐴   𝑥,𝐶   𝑘,𝐾,𝑥   𝜑,𝑘,𝑝,𝑥   𝐵,𝑘,𝑝   𝑃,𝑘,𝑝,𝑥

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑘, 𝑝)   𝐾(𝑝)

Proof of Theorem fsumvma

Dummy variables 𝑎 𝑧 𝑏 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvexd 6900	. . . 4 ⊢ (𝑧 = ⟨𝑝, 𝑘⟩ → (↑‘𝑧) ∈ V)
2		fveq2 6885	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑝, 𝑘⟩ → (↑‘𝑧) = (↑‘⟨𝑝, 𝑘⟩))
3		df-ov 7422	. . . . . . . 8 ⊢ (𝑝↑𝑘) = (↑‘⟨𝑝, 𝑘⟩)
4	2, 3	eqtr4di 2818	. . . . . . 7 ⊢ (𝑧 = ⟨𝑝, 𝑘⟩ → (↑‘𝑧) = (𝑝↑𝑘))
5	4	eqeq2d 2776	. . . . . 6 ⊢ (𝑧 = ⟨𝑝, 𝑘⟩ → (𝑥 = (↑‘𝑧) ↔ 𝑥 = (𝑝↑𝑘)))
6	5	biimpa 482	. . . . 5 ⊢ ((𝑧 = ⟨𝑝, 𝑘⟩ ∧ 𝑥 = (↑‘𝑧)) → 𝑥 = (𝑝↑𝑘))
7		fsumvma.1	. . . . 5 ⊢ (𝑥 = (𝑝↑𝑘) → 𝐵 = 𝐶)
8	6, 7	syl 18	. . . 4 ⊢ ((𝑧 = ⟨𝑝, 𝑘⟩ ∧ 𝑥 = (↑‘𝑧)) → 𝐵 = 𝐶)
9	1, 8	csbied 3890	. . 3 ⊢ (𝑧 = ⟨𝑝, 𝑘⟩ → ⦋(↑‘𝑧) / 𝑥⦌𝐵 = 𝐶)
10		fsumvma.4	. . 3 ⊢ (𝜑 → 𝑃 ∈ Fin)
11		fsumvma.2	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ Fin)
12	11	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → 𝐴 ∈ Fin)
13		fsumvma.5	. . . . . . . . 9 ⊢ (𝜑 → ((𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾) ↔ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ (𝑝↑𝑘) ∈ 𝐴)))
14	13	biimpd 232	. . . . . . . 8 ⊢ (𝜑 → ((𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾) → ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ (𝑝↑𝑘) ∈ 𝐴)))
15	14	impl 461	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ 𝑘 ∈ 𝐾) → ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ (𝑝↑𝑘) ∈ 𝐴))
16	15	simprd 501	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ 𝑘 ∈ 𝐾) → (𝑝↑𝑘) ∈ 𝐴)
17	16	ex 418	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → (𝑘 ∈ 𝐾 → (𝑝↑𝑘) ∈ 𝐴))
18	15	simpld 500	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ 𝑘 ∈ 𝐾) → (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ))
19	18	simpld 500	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ 𝑘 ∈ 𝐾) → 𝑝 ∈ ℙ)
20	19	adantrr 730	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → 𝑝 ∈ ℙ)
21	18	simprd 501	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ 𝑘 ∈ 𝐾) → 𝑘 ∈ ℕ)
22	21	adantrr 730	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → 𝑘 ∈ ℕ)
23	21	ex 418	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → (𝑘 ∈ 𝐾 → 𝑘 ∈ ℕ))
24	23	ssrdv 3944	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → 𝐾 ⊆ ℕ)
25	24	sselda 3938	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ 𝑧 ∈ 𝐾) → 𝑧 ∈ ℕ)
26	25	adantrl 729	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → 𝑧 ∈ ℕ)
27		eqid 2765	. . . . . . . 8 ⊢ 𝑝 = 𝑝
28		prmexpb 16804	. . . . . . . . 9 ⊢ (((𝑝 ∈ ℙ ∧ 𝑝 ∈ ℙ) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((𝑝↑𝑘) = (𝑝↑𝑧) ↔ (𝑝 = 𝑝 ∧ 𝑘 = 𝑧)))
29	28	baibd 549	. . . . . . . 8 ⊢ ((((𝑝 ∈ ℙ ∧ 𝑝 ∈ ℙ) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ ℕ)) ∧ 𝑝 = 𝑝) → ((𝑝↑𝑘) = (𝑝↑𝑧) ↔ 𝑘 = 𝑧))
30	27, 29	mpan2 704	. . . . . . 7 ⊢ (((𝑝 ∈ ℙ ∧ 𝑝 ∈ ℙ) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((𝑝↑𝑘) = (𝑝↑𝑧) ↔ 𝑘 = 𝑧))
31	20, 20, 22, 26, 30	syl22anc 852	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑝↑𝑘) = (𝑝↑𝑧) ↔ 𝑘 = 𝑧))
32	31	ex 418	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) → ((𝑝↑𝑘) = (𝑝↑𝑧) ↔ 𝑘 = 𝑧)))
33	17, 32	dom2lem 8995	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → (𝑘 ∈ 𝐾 ↦ (𝑝↑𝑘)):𝐾–1-1→𝐴)
34		f1fi 9281	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ (𝑘 ∈ 𝐾 ↦ (𝑝↑𝑘)):𝐾–1-1→𝐴) → 𝐾 ∈ Fin)
35	12, 33, 34	syl2anc 596	. . 3 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → 𝐾 ∈ Fin)
36	7	eleq1d 2850	. . . 4 ⊢ (𝑥 = (𝑝↑𝑘) → (𝐵 ∈ ℂ ↔ 𝐶 ∈ ℂ))
37		fsumvma.6	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
38	37	ralrimiva 3159	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ ℂ)
39	38	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) → ∀𝑥 ∈ 𝐴 𝐵 ∈ ℂ)
40	13	simplbda 505	. . . 4 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) → (𝑝↑𝑘) ∈ 𝐴)
41	36, 39, 40	rspcdva 3584	. . 3 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) → 𝐶 ∈ ℂ)
42	9, 10, 35, 41	fsum2d 15849	. 2 ⊢ (𝜑 → Σ𝑝 ∈ 𝑃 Σ𝑘 ∈ 𝐾 𝐶 = Σ𝑧 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)⦋(↑‘𝑧) / 𝑥⦌𝐵)
43		csbeq1a 3868	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
44		nfcv 2927	. . . 4 ⊢ Ⅎ𝑦𝐵
45		nfcsb1v 3878	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
46	43, 44, 45	cbvsum 15774	. . 3 ⊢ Σ𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))𝐵 = Σ𝑦 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))⦋𝑦 / 𝑥⦌𝐵
47		csbeq1 3857	. . . 4 ⊢ (𝑦 = (↑‘𝑧) → ⦋𝑦 / 𝑥⦌𝐵 = ⦋(↑‘𝑧) / 𝑥⦌𝐵)
48		snfi 9047	. . . . . . 7 ⊢ {𝑝} ∈ Fin
49		xpfi 9286	. . . . . . 7 ⊢ (({𝑝} ∈ Fin ∧ 𝐾 ∈ Fin) → ({𝑝} × 𝐾) ∈ Fin)
50	48, 35, 49	sylancr 599	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → ({𝑝} × 𝐾) ∈ Fin)
51	50	ralrimiva 3159	. . . . 5 ⊢ (𝜑 → ∀𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ∈ Fin)
52		iunfi 9307	. . . . 5 ⊢ ((𝑃 ∈ Fin ∧ ∀𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ∈ Fin) → ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ∈ Fin)
53	10, 51, 52	syl2anc 596	. . . 4 ⊢ (𝜑 → ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ∈ Fin)
54		fvex 6898	. . . . . . 7 ⊢ (↑‘𝑎) ∈ V
55	54	2a1i 12	. . . . . 6 ⊢ (𝜑 → (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) → (↑‘𝑎) ∈ V))
56		eliunxp 5825	. . . . . . . . 9 ⊢ (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↔ ∃𝑝∃𝑘(𝑎 = ⟨𝑝, 𝑘⟩ ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)))
57	13	simprbda 504	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) → (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ))
58		opelxp 5699	. . . . . . . . . . . . . 14 ⊢ (⟨𝑝, 𝑘⟩ ∈ (ℙ × ℕ) ↔ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ))
59	57, 58	sylibr 237	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) → ⟨𝑝, 𝑘⟩ ∈ (ℙ × ℕ))
60		eleq1 2853	. . . . . . . . . . . . 13 ⊢ (𝑎 = ⟨𝑝, 𝑘⟩ → (𝑎 ∈ (ℙ × ℕ) ↔ ⟨𝑝, 𝑘⟩ ∈ (ℙ × ℕ)))
61	59, 60	syl5ibrcom 250	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) → (𝑎 = ⟨𝑝, 𝑘⟩ → 𝑎 ∈ (ℙ × ℕ)))
62	61	impancom 457	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 = ⟨𝑝, 𝑘⟩) → ((𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾) → 𝑎 ∈ (ℙ × ℕ)))
63	62	expimpd 459	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑎 = ⟨𝑝, 𝑘⟩ ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) → 𝑎 ∈ (ℙ × ℕ)))
64	63	exlimdvv 1967	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑝∃𝑘(𝑎 = ⟨𝑝, 𝑘⟩ ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) → 𝑎 ∈ (ℙ × ℕ)))
65	56, 64	biimtrid 245	. . . . . . . 8 ⊢ (𝜑 → (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) → 𝑎 ∈ (ℙ × ℕ)))
66	65	ssrdv 3944	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ⊆ (ℙ × ℕ))
67	66	sseld 3937	. . . . . . . 8 ⊢ (𝜑 → (𝑏 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) → 𝑏 ∈ (ℙ × ℕ)))
68	65, 67	anim12d 621	. . . . . . 7 ⊢ (𝜑 → ((𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ∧ 𝑏 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)) → (𝑎 ∈ (ℙ × ℕ) ∧ 𝑏 ∈ (ℙ × ℕ))))
69		1st2nd2 8031	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (ℙ × ℕ) → 𝑎 = ⟨(1st ‘𝑎), (2nd ‘𝑎)⟩)
70	69	fveq2d 6889	. . . . . . . . . 10 ⊢ (𝑎 ∈ (ℙ × ℕ) → (↑‘𝑎) = (↑‘⟨(1st ‘𝑎), (2nd ‘𝑎)⟩))
71		df-ov 7422	. . . . . . . . . 10 ⊢ ((1st ‘𝑎)↑(2nd ‘𝑎)) = (↑‘⟨(1st ‘𝑎), (2nd ‘𝑎)⟩)
72	70, 71	eqtr4di 2818	. . . . . . . . 9 ⊢ (𝑎 ∈ (ℙ × ℕ) → (↑‘𝑎) = ((1st ‘𝑎)↑(2nd ‘𝑎)))
73		1st2nd2 8031	. . . . . . . . . . 11 ⊢ (𝑏 ∈ (ℙ × ℕ) → 𝑏 = ⟨(1st ‘𝑏), (2nd ‘𝑏)⟩)
74	73	fveq2d 6889	. . . . . . . . . 10 ⊢ (𝑏 ∈ (ℙ × ℕ) → (↑‘𝑏) = (↑‘⟨(1st ‘𝑏), (2nd ‘𝑏)⟩))
75		df-ov 7422	. . . . . . . . . 10 ⊢ ((1st ‘𝑏)↑(2nd ‘𝑏)) = (↑‘⟨(1st ‘𝑏), (2nd ‘𝑏)⟩)
76	74, 75	eqtr4di 2818	. . . . . . . . 9 ⊢ (𝑏 ∈ (ℙ × ℕ) → (↑‘𝑏) = ((1st ‘𝑏)↑(2nd ‘𝑏)))
77	72, 76	eqeqan12d 2779	. . . . . . . 8 ⊢ ((𝑎 ∈ (ℙ × ℕ) ∧ 𝑏 ∈ (ℙ × ℕ)) → ((↑‘𝑎) = (↑‘𝑏) ↔ ((1st ‘𝑎)↑(2nd ‘𝑎)) = ((1st ‘𝑏)↑(2nd ‘𝑏))))
78		xp1st 8024	. . . . . . . . . 10 ⊢ (𝑎 ∈ (ℙ × ℕ) → (1st ‘𝑎) ∈ ℙ)
79		xp2nd 8025	. . . . . . . . . 10 ⊢ (𝑎 ∈ (ℙ × ℕ) → (2nd ‘𝑎) ∈ ℕ)
80	78, 79	jca 521	. . . . . . . . 9 ⊢ (𝑎 ∈ (ℙ × ℕ) → ((1st ‘𝑎) ∈ ℙ ∧ (2nd ‘𝑎) ∈ ℕ))
81		xp1st 8024	. . . . . . . . . 10 ⊢ (𝑏 ∈ (ℙ × ℕ) → (1st ‘𝑏) ∈ ℙ)
82		xp2nd 8025	. . . . . . . . . 10 ⊢ (𝑏 ∈ (ℙ × ℕ) → (2nd ‘𝑏) ∈ ℕ)
83	81, 82	jca 521	. . . . . . . . 9 ⊢ (𝑏 ∈ (ℙ × ℕ) → ((1st ‘𝑏) ∈ ℙ ∧ (2nd ‘𝑏) ∈ ℕ))
84		prmexpb 16804	. . . . . . . . . 10 ⊢ ((((1st ‘𝑎) ∈ ℙ ∧ (1st ‘𝑏) ∈ ℙ) ∧ ((2nd ‘𝑎) ∈ ℕ ∧ (2nd ‘𝑏) ∈ ℕ)) → (((1st ‘𝑎)↑(2nd ‘𝑎)) = ((1st ‘𝑏)↑(2nd ‘𝑏)) ↔ ((1st ‘𝑎) = (1st ‘𝑏) ∧ (2nd ‘𝑎) = (2nd ‘𝑏))))
85	84	an4s 673	. . . . . . . . 9 ⊢ ((((1st ‘𝑎) ∈ ℙ ∧ (2nd ‘𝑎) ∈ ℕ) ∧ ((1st ‘𝑏) ∈ ℙ ∧ (2nd ‘𝑏) ∈ ℕ)) → (((1st ‘𝑎)↑(2nd ‘𝑎)) = ((1st ‘𝑏)↑(2nd ‘𝑏)) ↔ ((1st ‘𝑎) = (1st ‘𝑏) ∧ (2nd ‘𝑎) = (2nd ‘𝑏))))
86	80, 83, 85	syl2an 608	. . . . . . . 8 ⊢ ((𝑎 ∈ (ℙ × ℕ) ∧ 𝑏 ∈ (ℙ × ℕ)) → (((1st ‘𝑎)↑(2nd ‘𝑎)) = ((1st ‘𝑏)↑(2nd ‘𝑏)) ↔ ((1st ‘𝑎) = (1st ‘𝑏) ∧ (2nd ‘𝑎) = (2nd ‘𝑏))))
87		xpopth 8033	. . . . . . . 8 ⊢ ((𝑎 ∈ (ℙ × ℕ) ∧ 𝑏 ∈ (ℙ × ℕ)) → (((1st ‘𝑎) = (1st ‘𝑏) ∧ (2nd ‘𝑎) = (2nd ‘𝑏)) ↔ 𝑎 = 𝑏))
88	77, 86, 87	3bitrd 308	. . . . . . 7 ⊢ ((𝑎 ∈ (ℙ × ℕ) ∧ 𝑏 ∈ (ℙ × ℕ)) → ((↑‘𝑎) = (↑‘𝑏) ↔ 𝑎 = 𝑏))
89	68, 88	syl6 36	. . . . . 6 ⊢ (𝜑 → ((𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ∧ 𝑏 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)) → ((↑‘𝑎) = (↑‘𝑏) ↔ 𝑎 = 𝑏)))
90	55, 89	dom2lem 8995	. . . . 5 ⊢ (𝜑 → (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎)):∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)–1-1→V)
91		f1f1orn 6836	. . . . 5 ⊢ ((𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎)):∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)–1-1→V → (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎)):∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)–1-1-onto→ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎)))
92	90, 91	syl 18	. . . 4 ⊢ (𝜑 → (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎)):∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)–1-1-onto→ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎)))
93		fveq2 6885	. . . . . 6 ⊢ (𝑎 = 𝑧 → (↑‘𝑎) = (↑‘𝑧))
94		eqid 2765	. . . . . 6 ⊢ (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎)) = (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))
95		fvex 6898	. . . . . 6 ⊢ (↑‘𝑧) ∈ V
96	93, 94, 95	fvmpt 6993	. . . . 5 ⊢ (𝑧 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) → ((𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))‘𝑧) = (↑‘𝑧))
97	96	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)) → ((𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))‘𝑧) = (↑‘𝑧))
98		fveq2 6885	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = ⟨𝑝, 𝑘⟩ → (↑‘𝑎) = (↑‘⟨𝑝, 𝑘⟩))
99	98, 3	eqtr4di 2818	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = ⟨𝑝, 𝑘⟩ → (↑‘𝑎) = (𝑝↑𝑘))
100	99	eleq1d 2850	. . . . . . . . . . . . . 14 ⊢ (𝑎 = ⟨𝑝, 𝑘⟩ → ((↑‘𝑎) ∈ 𝐴 ↔ (𝑝↑𝑘) ∈ 𝐴))
101	40, 100	syl5ibrcom 250	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) → (𝑎 = ⟨𝑝, 𝑘⟩ → (↑‘𝑎) ∈ 𝐴))
102	101	impancom 457	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 = ⟨𝑝, 𝑘⟩) → ((𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾) → (↑‘𝑎) ∈ 𝐴))
103	102	expimpd 459	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑎 = ⟨𝑝, 𝑘⟩ ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) → (↑‘𝑎) ∈ 𝐴))
104	103	exlimdvv 1967	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑝∃𝑘(𝑎 = ⟨𝑝, 𝑘⟩ ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) → (↑‘𝑎) ∈ 𝐴))
105	56, 104	biimtrid 245	. . . . . . . . 9 ⊢ (𝜑 → (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) → (↑‘𝑎) ∈ 𝐴))
106	105	imp 412	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)) → (↑‘𝑎) ∈ 𝐴)
107	106	fmpttd 7114	. . . . . . 7 ⊢ (𝜑 → (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎)):∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)⟶𝐴)
108	107	frnd 6718	. . . . . 6 ⊢ (𝜑 → ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎)) ⊆ 𝐴)
109	108	sselda 3938	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))) → 𝑦 ∈ 𝐴)
110	45	nfel1 2943	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 ∈ ℂ
111	43	eleq1d 2850	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐵 ∈ ℂ ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ ℂ))
112	110, 111	rspc 3571	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝐵 ∈ ℂ → ⦋𝑦 / 𝑥⦌𝐵 ∈ ℂ))
113	38, 112	mpan9 516	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ∈ ℂ)
114	109, 113	syldan 603	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))) → ⦋𝑦 / 𝑥⦌𝐵 ∈ ℂ)
115	47, 53, 92, 97, 114	fsumf1o 15801	. . 3 ⊢ (𝜑 → Σ𝑦 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))⦋𝑦 / 𝑥⦌𝐵 = Σ𝑧 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)⦋(↑‘𝑧) / 𝑥⦌𝐵)
116	46, 115	eqtrid 2812	. 2 ⊢ (𝜑 → Σ𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))𝐵 = Σ𝑧 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)⦋(↑‘𝑧) / 𝑥⦌𝐵)
117	108	sselda 3938	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))) → 𝑥 ∈ 𝐴)
118	117, 37	syldan 603	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))) → 𝐵 ∈ ℂ)
119		eldif 3916	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))))
120	94, 54	elrnmpti 5954	. . . . . . . . . 10 ⊢ (𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎)) ↔ ∃𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)𝑥 = (↑‘𝑎))
121	99	eqeq2d 2776	. . . . . . . . . . 11 ⊢ (𝑎 = ⟨𝑝, 𝑘⟩ → (𝑥 = (↑‘𝑎) ↔ 𝑥 = (𝑝↑𝑘)))
122	121	rexiunxp 5828	. . . . . . . . . 10 ⊢ (∃𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)𝑥 = (↑‘𝑎) ↔ ∃𝑝 ∈ 𝑃 ∃𝑘 ∈ 𝐾 𝑥 = (𝑝↑𝑘))
123	120, 122	bitri 278	. . . . . . . . 9 ⊢ (𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎)) ↔ ∃𝑝 ∈ 𝑃 ∃𝑘 ∈ 𝐾 𝑥 = (𝑝↑𝑘))
124		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = (𝑝↑𝑘)) → 𝑥 = (𝑝↑𝑘))
125		simplr 781	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = (𝑝↑𝑘)) → 𝑥 ∈ 𝐴)
126	124, 125	eqeltrrd 2866	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = (𝑝↑𝑘)) → (𝑝↑𝑘) ∈ 𝐴)
127	13	rbaibd 550	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑝↑𝑘) ∈ 𝐴) → ((𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾) ↔ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)))
128	127	adantlr 728	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝑝↑𝑘) ∈ 𝐴) → ((𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾) ↔ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)))
129	126, 128	syldan 603	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = (𝑝↑𝑘)) → ((𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾) ↔ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)))
130	129	pm5.32da 590	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 = (𝑝↑𝑘) ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) ↔ (𝑥 = (𝑝↑𝑘) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ))))
131		ancom 466	. . . . . . . . . . . . 13 ⊢ (((𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾) ∧ 𝑥 = (𝑝↑𝑘)) ↔ (𝑥 = (𝑝↑𝑘) ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)))
132		ancom 466	. . . . . . . . . . . . 13 ⊢ (((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 𝑥 = (𝑝↑𝑘)) ↔ (𝑥 = (𝑝↑𝑘) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)))
133	130, 131, 132	3bitr4g 317	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾) ∧ 𝑥 = (𝑝↑𝑘)) ↔ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 𝑥 = (𝑝↑𝑘))))
134	133	2exbidv 1957	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑝∃𝑘((𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾) ∧ 𝑥 = (𝑝↑𝑘)) ↔ ∃𝑝∃𝑘((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 𝑥 = (𝑝↑𝑘))))
135		r2ex 3204	. . . . . . . . . . 11 ⊢ (∃𝑝 ∈ 𝑃 ∃𝑘 ∈ 𝐾 𝑥 = (𝑝↑𝑘) ↔ ∃𝑝∃𝑘((𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾) ∧ 𝑥 = (𝑝↑𝑘)))
136		r2ex 3204	. . . . . . . . . . 11 ⊢ (∃𝑝 ∈ ℙ ∃𝑘 ∈ ℕ 𝑥 = (𝑝↑𝑘) ↔ ∃𝑝∃𝑘((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 𝑥 = (𝑝↑𝑘)))
137	134, 135, 136	3bitr4g 317	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑝 ∈ 𝑃 ∃𝑘 ∈ 𝐾 𝑥 = (𝑝↑𝑘) ↔ ∃𝑝 ∈ ℙ ∃𝑘 ∈ ℕ 𝑥 = (𝑝↑𝑘)))
138		fsumvma.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ⊆ ℕ)
139	138	sselda 3938	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℕ)
140		isppw2 27334	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → ((Λ‘𝑥) ≠ 0 ↔ ∃𝑝 ∈ ℙ ∃𝑘 ∈ ℕ 𝑥 = (𝑝↑𝑘)))
141	139, 140	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((Λ‘𝑥) ≠ 0 ↔ ∃𝑝 ∈ ℙ ∃𝑘 ∈ ℕ 𝑥 = (𝑝↑𝑘)))
142	137, 141	bitr4d 285	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑝 ∈ 𝑃 ∃𝑘 ∈ 𝐾 𝑥 = (𝑝↑𝑘) ↔ (Λ‘𝑥) ≠ 0))
143	123, 142	bitrid 286	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎)) ↔ (Λ‘𝑥) ≠ 0))
144	143	necon2bbid 3003	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((Λ‘𝑥) = 0 ↔ ¬ 𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))))
145	144	pm5.32da 590	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ (Λ‘𝑥) = 0) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎)))))
146		fsumvma.7	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ (Λ‘𝑥) = 0)) → 𝐵 = 0)
147	146	ex 418	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ (Λ‘𝑥) = 0) → 𝐵 = 0))
148	145, 147	sylbird 263	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))) → 𝐵 = 0))
149	119, 148	biimtrid 245	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))) → 𝐵 = 0))
150	149	imp 412	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎)))) → 𝐵 = 0)
151	108, 118, 150, 11	fsumss 15803	. 2 ⊢ (𝜑 → Σ𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))𝐵 = Σ𝑥 ∈ 𝐴 𝐵)
152	42, 116, 151	3eqtr2rd 2807	1 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 𝐵 = Σ𝑝 ∈ 𝑃 Σ𝑘 ∈ 𝐾 𝐶)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  ∃wrex 3091  Vcvv 3457  ⦋csb 3854   ∖ cdif 3903   ⊆ wss 3906  {csn 4591  ⟨cop 4597  ∪ ciun 4958   ↦ cmpt 5194   × cxp 5661  ran crn 5664  –1-1→wf1 6537  –1-1-onto→wf1o 6539  ‘cfv 6540  (class class class)co 7419  1^st c1st 7990  2^nd c2nd 7991  Fincfn 8949  ℂcc 11117  0cc0 11119  ℕcn 12252  ↑cexp 14119  Σcsu 15765  ℙcprime 16755  Λcvma 27311

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-inf2 9617  ax-cnex 11175  ax-resscn 11176  ax-1cn 11177  ax-icn 11178  ax-addcl 11179  ax-addrcl 11180  ax-mulcl 11181  ax-mulrcl 11182  ax-mulcom 11183  ax-addass 11184  ax-mulass 11185  ax-distr 11186  ax-i2m1 11187  ax-1ne0 11188  ax-1rid 11189  ax-rnegex 11190  ax-rrecex 11191  ax-cnre 11192  ax-pre-lttri 11193  ax-pre-lttrn 11194  ax-pre-ltadd 11195  ax-pre-mulgt0 11196  ax-pre-sup 11197  ax-addf 11198

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7684  df-om 7869  df-1st 7992  df-2nd 7993  df-supp 8163  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-oadd 8463  df-er 8700  df-map 8832  df-pm 8833  df-ixp 8902  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-fsupp 9329  df-fi 9378  df-sup 9409  df-inf 9410  df-oi 9479  df-dju 9903  df-card 9941  df-pnf 11264  df-mnf 11265  df-xr 11266  df-ltxr 11267  df-le 11268  df-sub 11462  df-neg 11463  df-div 11891  df-nn 12253  df-2 12322  df-3 12323  df-4 12324  df-5 12325  df-6 12326  df-7 12327  df-8 12328  df-9 12329  df-n0 12524  df-z 12611  df-dec 12732  df-uz 12883  df-q 12993  df-rp 13037  df-xneg 13157  df-xadd 13158  df-xmul 13159  df-ioo 13396  df-ioc 13397  df-ico 13398  df-icc 13399  df-fz 13556  df-fzo 13704  df-fl 13847  df-mod 13925  df-seq 14060  df-exp 14120  df-fac 14332  df-bc 14361  df-hash 14389  df-shft 15132  df-cj 15178  df-re 15179  df-im 15180  df-sqrt 15314  df-abs 15315  df-limsup 15550  df-clim 15567  df-rlim 15568  df-sum 15766  df-ef 16147  df-sin 16149  df-cos 16150  df-pi 16152  df-dvds 16337  df-gcd 16579  df-prm 16756  df-pc 16923  df-struct 17233  df-sets 17250  df-slot 17268  df-ndx 17280  df-base 17296  df-ress 17317  df-plusg 17349  df-mulr 17350  df-starv 17351  df-sca 17352  df-vsca 17353  df-ip 17354  df-tset 17355  df-ple 17356  df-ds 17358  df-unif 17359  df-hom 17360  df-cco 17361  df-rest 17501  df-topn 17502  df-0g 17520  df-gsum 17521  df-topgen 17522  df-pt 17523  df-prds 17526  df-xrs 17582  df-qtop 17587  df-imas 17588  df-xps 17590  df-mre 17664  df-mrc 17665  df-acs 17667  df-mgm 18724  df-sgrp 18813  df-mnd 18829  df-submnd 18883  df-mulg 19182  df-cntz 19435  df-cmn 19900  df-psmet 21568  df-xmet 21569  df-met 21570  df-bl 21571  df-mopn 21572  df-fbas 21573  df-fg 21574  df-cnfld 21577  df-top 23105  df-topon 23122  df-topsp 23144  df-bases 23157  df-cld 23230  df-ntr 23231  df-cls 23232  df-nei 23309  df-lp 23347  df-perf 23348  df-cn 23438  df-cnp 23439  df-haus 23526  df-tx 23774  df-hmeo 23967  df-fil 24058  df-fm 24150  df-flim 24151  df-flf 24152  df-xms 24532  df-ms 24533  df-tms 24534  df-cncf 25092  df-limc 26080  df-dv 26081  df-log 26776  df-vma 27317

This theorem is used by:  fsumvma2  27433  vmasum  27435