Theorem fsumvma 25789

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 6685	. . . 4 ⊢ (𝑧 = ⟨𝑝, 𝑘⟩ → (↑‘𝑧) ∈ V)
2		fveq2 6670	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑝, 𝑘⟩ → (↑‘𝑧) = (↑‘⟨𝑝, 𝑘⟩))
3		df-ov 7159	. . . . . . . 8 ⊢ (𝑝↑𝑘) = (↑‘⟨𝑝, 𝑘⟩)
4	2, 3	syl6eqr 2874	. . . . . . 7 ⊢ (𝑧 = ⟨𝑝, 𝑘⟩ → (↑‘𝑧) = (𝑝↑𝑘))
5	4	eqeq2d 2832	. . . . . 6 ⊢ (𝑧 = ⟨𝑝, 𝑘⟩ → (𝑥 = (↑‘𝑧) ↔ 𝑥 = (𝑝↑𝑘)))
6	5	biimpa 479	. . . . 5 ⊢ ((𝑧 = ⟨𝑝, 𝑘⟩ ∧ 𝑥 = (↑‘𝑧)) → 𝑥 = (𝑝↑𝑘))
7		fsumvma.1	. . . . 5 ⊢ (𝑥 = (𝑝↑𝑘) → 𝐵 = 𝐶)
8	6, 7	syl 17	. . . 4 ⊢ ((𝑧 = ⟨𝑝, 𝑘⟩ ∧ 𝑥 = (↑‘𝑧)) → 𝐵 = 𝐶)
9	1, 8	csbied 3919	. . 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 483	. . . 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 231	. . . . . . . 8 ⊢ (𝜑 → ((𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾) → ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ (𝑝↑𝑘) ∈ 𝐴)))
15	14	impl 458	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ 𝑘 ∈ 𝐾) → ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ (𝑝↑𝑘) ∈ 𝐴))
16	15	simprd 498	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ 𝑘 ∈ 𝐾) → (𝑝↑𝑘) ∈ 𝐴)
17	16	ex 415	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → (𝑘 ∈ 𝐾 → (𝑝↑𝑘) ∈ 𝐴))
18	15	simpld 497	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ 𝑘 ∈ 𝐾) → (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ))
19	18	simpld 497	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ 𝑘 ∈ 𝐾) → 𝑝 ∈ ℙ)
20	19	adantrr 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → 𝑝 ∈ ℙ)
21	18	simprd 498	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ 𝑘 ∈ 𝐾) → 𝑘 ∈ ℕ)
22	21	adantrr 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → 𝑘 ∈ ℕ)
23	21	ex 415	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → (𝑘 ∈ 𝐾 → 𝑘 ∈ ℕ))
24	23	ssrdv 3973	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → 𝐾 ⊆ ℕ)
25	24	sselda 3967	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ 𝑧 ∈ 𝐾) → 𝑧 ∈ ℕ)
26	25	adantrl 714	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → 𝑧 ∈ ℕ)
27		eqid 2821	. . . . . . . 8 ⊢ 𝑝 = 𝑝
28		prmexpb 16062	. . . . . . . . 9 ⊢ (((𝑝 ∈ ℙ ∧ 𝑝 ∈ ℙ) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((𝑝↑𝑘) = (𝑝↑𝑧) ↔ (𝑝 = 𝑝 ∧ 𝑘 = 𝑧)))
29	28	baibd 542	. . . . . . . 8 ⊢ ((((𝑝 ∈ ℙ ∧ 𝑝 ∈ ℙ) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ ℕ)) ∧ 𝑝 = 𝑝) → ((𝑝↑𝑘) = (𝑝↑𝑧) ↔ 𝑘 = 𝑧))
30	27, 29	mpan2 689	. . . . . . 7 ⊢ (((𝑝 ∈ ℙ ∧ 𝑝 ∈ ℙ) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((𝑝↑𝑘) = (𝑝↑𝑧) ↔ 𝑘 = 𝑧))
31	20, 20, 22, 26, 30	syl22anc 836	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑝↑𝑘) = (𝑝↑𝑧) ↔ 𝑘 = 𝑧))
32	31	ex 415	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) → ((𝑝↑𝑘) = (𝑝↑𝑧) ↔ 𝑘 = 𝑧)))
33	17, 32	dom2lem 8549	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → (𝑘 ∈ 𝐾 ↦ (𝑝↑𝑘)):𝐾–1-1→𝐴)
34		f1fi 8811	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ (𝑘 ∈ 𝐾 ↦ (𝑝↑𝑘)):𝐾–1-1→𝐴) → 𝐾 ∈ Fin)
35	12, 33, 34	syl2anc 586	. . 3 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → 𝐾 ∈ Fin)
36	7	eleq1d 2897	. . . 4 ⊢ (𝑥 = (𝑝↑𝑘) → (𝐵 ∈ ℂ ↔ 𝐶 ∈ ℂ))
37		fsumvma.6	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
38	37	ralrimiva 3182	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ ℂ)
39	38	adantr 483	. . . 4 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) → ∀𝑥 ∈ 𝐴 𝐵 ∈ ℂ)
40	13	simplbda 502	. . . 4 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) → (𝑝↑𝑘) ∈ 𝐴)
41	36, 39, 40	rspcdva 3625	. . 3 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) → 𝐶 ∈ ℂ)
42	9, 10, 35, 41	fsum2d 15126	. 2 ⊢ (𝜑 → Σ𝑝 ∈ 𝑃 Σ𝑘 ∈ 𝐾 𝐶 = Σ𝑧 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)⦋(↑‘𝑧) / 𝑥⦌𝐵)
43		nfcv 2977	. . . 4 ⊢ Ⅎ𝑦𝐵
44		nfcsb1v 3907	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
45		csbeq1a 3897	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
46	43, 44, 45	cbvsumi 15054	. . 3 ⊢ Σ𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))𝐵 = Σ𝑦 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))⦋𝑦 / 𝑥⦌𝐵
47		csbeq1 3886	. . . 4 ⊢ (𝑦 = (↑‘𝑧) → ⦋𝑦 / 𝑥⦌𝐵 = ⦋(↑‘𝑧) / 𝑥⦌𝐵)
48		snfi 8594	. . . . . . 7 ⊢ {𝑝} ∈ Fin
49		xpfi 8789	. . . . . . 7 ⊢ (({𝑝} ∈ Fin ∧ 𝐾 ∈ Fin) → ({𝑝} × 𝐾) ∈ Fin)
50	48, 35, 49	sylancr 589	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → ({𝑝} × 𝐾) ∈ Fin)
51	50	ralrimiva 3182	. . . . 5 ⊢ (𝜑 → ∀𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ∈ Fin)
52		iunfi 8812	. . . . 5 ⊢ ((𝑃 ∈ Fin ∧ ∀𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ∈ Fin) → ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ∈ Fin)
53	10, 51, 52	syl2anc 586	. . . 4 ⊢ (𝜑 → ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ∈ Fin)
54		fvex 6683	. . . . . . 7 ⊢ (↑‘𝑎) ∈ V
55	54	2a1i 12	. . . . . 6 ⊢ (𝜑 → (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) → (↑‘𝑎) ∈ V))
56		eliunxp 5708	. . . . . . . . 9 ⊢ (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↔ ∃𝑝∃𝑘(𝑎 = ⟨𝑝, 𝑘⟩ ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)))
57	13	simprbda 501	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) → (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ))
58		opelxp 5591	. . . . . . . . . . . . . 14 ⊢ (⟨𝑝, 𝑘⟩ ∈ (ℙ × ℕ) ↔ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ))
59	57, 58	sylibr 236	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) → ⟨𝑝, 𝑘⟩ ∈ (ℙ × ℕ))
60		eleq1 2900	. . . . . . . . . . . . 13 ⊢ (𝑎 = ⟨𝑝, 𝑘⟩ → (𝑎 ∈ (ℙ × ℕ) ↔ ⟨𝑝, 𝑘⟩ ∈ (ℙ × ℕ)))
61	59, 60	syl5ibrcom 249	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) → (𝑎 = ⟨𝑝, 𝑘⟩ → 𝑎 ∈ (ℙ × ℕ)))
62	61	impancom 454	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 = ⟨𝑝, 𝑘⟩) → ((𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾) → 𝑎 ∈ (ℙ × ℕ)))
63	62	expimpd 456	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑎 = ⟨𝑝, 𝑘⟩ ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) → 𝑎 ∈ (ℙ × ℕ)))
64	63	exlimdvv 1935	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑝∃𝑘(𝑎 = ⟨𝑝, 𝑘⟩ ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) → 𝑎 ∈ (ℙ × ℕ)))
65	56, 64	syl5bi 244	. . . . . . . 8 ⊢ (𝜑 → (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) → 𝑎 ∈ (ℙ × ℕ)))
66	65	ssrdv 3973	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ⊆ (ℙ × ℕ))
67	66	sseld 3966	. . . . . . . 8 ⊢ (𝜑 → (𝑏 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) → 𝑏 ∈ (ℙ × ℕ)))
68	65, 67	anim12d 610	. . . . . . 7 ⊢ (𝜑 → ((𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ∧ 𝑏 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)) → (𝑎 ∈ (ℙ × ℕ) ∧ 𝑏 ∈ (ℙ × ℕ))))
69		1st2nd2 7728	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (ℙ × ℕ) → 𝑎 = ⟨(1st ‘𝑎), (2nd ‘𝑎)⟩)
70	69	fveq2d 6674	. . . . . . . . . 10 ⊢ (𝑎 ∈ (ℙ × ℕ) → (↑‘𝑎) = (↑‘⟨(1st ‘𝑎), (2nd ‘𝑎)⟩))
71		df-ov 7159	. . . . . . . . . 10 ⊢ ((1st ‘𝑎)↑(2nd ‘𝑎)) = (↑‘⟨(1st ‘𝑎), (2nd ‘𝑎)⟩)
72	70, 71	syl6eqr 2874	. . . . . . . . 9 ⊢ (𝑎 ∈ (ℙ × ℕ) → (↑‘𝑎) = ((1st ‘𝑎)↑(2nd ‘𝑎)))
73		1st2nd2 7728	. . . . . . . . . . 11 ⊢ (𝑏 ∈ (ℙ × ℕ) → 𝑏 = ⟨(1st ‘𝑏), (2nd ‘𝑏)⟩)
74	73	fveq2d 6674	. . . . . . . . . 10 ⊢ (𝑏 ∈ (ℙ × ℕ) → (↑‘𝑏) = (↑‘⟨(1st ‘𝑏), (2nd ‘𝑏)⟩))
75		df-ov 7159	. . . . . . . . . 10 ⊢ ((1st ‘𝑏)↑(2nd ‘𝑏)) = (↑‘⟨(1st ‘𝑏), (2nd ‘𝑏)⟩)
76	74, 75	syl6eqr 2874	. . . . . . . . 9 ⊢ (𝑏 ∈ (ℙ × ℕ) → (↑‘𝑏) = ((1st ‘𝑏)↑(2nd ‘𝑏)))
77	72, 76	eqeqan12d 2838	. . . . . . . 8 ⊢ ((𝑎 ∈ (ℙ × ℕ) ∧ 𝑏 ∈ (ℙ × ℕ)) → ((↑‘𝑎) = (↑‘𝑏) ↔ ((1st ‘𝑎)↑(2nd ‘𝑎)) = ((1st ‘𝑏)↑(2nd ‘𝑏))))
78		xp1st 7721	. . . . . . . . . 10 ⊢ (𝑎 ∈ (ℙ × ℕ) → (1st ‘𝑎) ∈ ℙ)
79		xp2nd 7722	. . . . . . . . . 10 ⊢ (𝑎 ∈ (ℙ × ℕ) → (2nd ‘𝑎) ∈ ℕ)
80	78, 79	jca 514	. . . . . . . . 9 ⊢ (𝑎 ∈ (ℙ × ℕ) → ((1st ‘𝑎) ∈ ℙ ∧ (2nd ‘𝑎) ∈ ℕ))
81		xp1st 7721	. . . . . . . . . 10 ⊢ (𝑏 ∈ (ℙ × ℕ) → (1st ‘𝑏) ∈ ℙ)
82		xp2nd 7722	. . . . . . . . . 10 ⊢ (𝑏 ∈ (ℙ × ℕ) → (2nd ‘𝑏) ∈ ℕ)
83	81, 82	jca 514	. . . . . . . . 9 ⊢ (𝑏 ∈ (ℙ × ℕ) → ((1st ‘𝑏) ∈ ℙ ∧ (2nd ‘𝑏) ∈ ℕ))
84		prmexpb 16062	. . . . . . . . . 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 658	. . . . . . . . 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 597	. . . . . . . 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 7730	. . . . . . . 8 ⊢ ((𝑎 ∈ (ℙ × ℕ) ∧ 𝑏 ∈ (ℙ × ℕ)) → (((1st ‘𝑎) = (1st ‘𝑏) ∧ (2nd ‘𝑎) = (2nd ‘𝑏)) ↔ 𝑎 = 𝑏))
88	77, 86, 87	3bitrd 307	. . . . . . 7 ⊢ ((𝑎 ∈ (ℙ × ℕ) ∧ 𝑏 ∈ (ℙ × ℕ)) → ((↑‘𝑎) = (↑‘𝑏) ↔ 𝑎 = 𝑏))
89	68, 88	syl6 35	. . . . . 6 ⊢ (𝜑 → ((𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ∧ 𝑏 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)) → ((↑‘𝑎) = (↑‘𝑏) ↔ 𝑎 = 𝑏)))
90	55, 89	dom2lem 8549	. . . . 5 ⊢ (𝜑 → (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎)):∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)–1-1→V)
91		f1f1orn 6626	. . . . 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 17	. . . 4 ⊢ (𝜑 → (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎)):∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)–1-1-onto→ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎)))
93		fveq2 6670	. . . . . 6 ⊢ (𝑎 = 𝑧 → (↑‘𝑎) = (↑‘𝑧))
94		eqid 2821	. . . . . 6 ⊢ (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎)) = (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))
95		fvex 6683	. . . . . 6 ⊢ (↑‘𝑧) ∈ V
96	93, 94, 95	fvmpt 6768	. . . . 5 ⊢ (𝑧 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) → ((𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))‘𝑧) = (↑‘𝑧))
97	96	adantl 484	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)) → ((𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))‘𝑧) = (↑‘𝑧))
98		fveq2 6670	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = ⟨𝑝, 𝑘⟩ → (↑‘𝑎) = (↑‘⟨𝑝, 𝑘⟩))
99	98, 3	syl6eqr 2874	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = ⟨𝑝, 𝑘⟩ → (↑‘𝑎) = (𝑝↑𝑘))
100	99	eleq1d 2897	. . . . . . . . . . . . . 14 ⊢ (𝑎 = ⟨𝑝, 𝑘⟩ → ((↑‘𝑎) ∈ 𝐴 ↔ (𝑝↑𝑘) ∈ 𝐴))
101	40, 100	syl5ibrcom 249	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) → (𝑎 = ⟨𝑝, 𝑘⟩ → (↑‘𝑎) ∈ 𝐴))
102	101	impancom 454	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 = ⟨𝑝, 𝑘⟩) → ((𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾) → (↑‘𝑎) ∈ 𝐴))
103	102	expimpd 456	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑎 = ⟨𝑝, 𝑘⟩ ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) → (↑‘𝑎) ∈ 𝐴))
104	103	exlimdvv 1935	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑝∃𝑘(𝑎 = ⟨𝑝, 𝑘⟩ ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) → (↑‘𝑎) ∈ 𝐴))
105	56, 104	syl5bi 244	. . . . . . . . 9 ⊢ (𝜑 → (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) → (↑‘𝑎) ∈ 𝐴))
106	105	imp 409	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)) → (↑‘𝑎) ∈ 𝐴)
107	106	fmpttd 6879	. . . . . . 7 ⊢ (𝜑 → (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎)):∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)⟶𝐴)
108	107	frnd 6521	. . . . . 6 ⊢ (𝜑 → ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎)) ⊆ 𝐴)
109	108	sselda 3967	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))) → 𝑦 ∈ 𝐴)
110	44	nfel1 2994	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 ∈ ℂ
111	45	eleq1d 2897	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐵 ∈ ℂ ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ ℂ))
112	110, 111	rspc 3611	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝐵 ∈ ℂ → ⦋𝑦 / 𝑥⦌𝐵 ∈ ℂ))
113	38, 112	mpan9 509	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ∈ ℂ)
114	109, 113	syldan 593	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))) → ⦋𝑦 / 𝑥⦌𝐵 ∈ ℂ)
115	47, 53, 92, 97, 114	fsumf1o 15080	. . 3 ⊢ (𝜑 → Σ𝑦 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))⦋𝑦 / 𝑥⦌𝐵 = Σ𝑧 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)⦋(↑‘𝑧) / 𝑥⦌𝐵)
116	46, 115	syl5eq 2868	. 2 ⊢ (𝜑 → Σ𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))𝐵 = Σ𝑧 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)⦋(↑‘𝑧) / 𝑥⦌𝐵)
117	108	sselda 3967	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))) → 𝑥 ∈ 𝐴)
118	117, 37	syldan 593	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))) → 𝐵 ∈ ℂ)
119		eldif 3946	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))))
120	94, 54	elrnmpti 5832	. . . . . . . . . 10 ⊢ (𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎)) ↔ ∃𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)𝑥 = (↑‘𝑎))
121	99	eqeq2d 2832	. . . . . . . . . . 11 ⊢ (𝑎 = ⟨𝑝, 𝑘⟩ → (𝑥 = (↑‘𝑎) ↔ 𝑥 = (𝑝↑𝑘)))
122	121	rexiunxp 5711	. . . . . . . . . 10 ⊢ (∃𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)𝑥 = (↑‘𝑎) ↔ ∃𝑝 ∈ 𝑃 ∃𝑘 ∈ 𝐾 𝑥 = (𝑝↑𝑘))
123	120, 122	bitri 277	. . . . . . . . 9 ⊢ (𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎)) ↔ ∃𝑝 ∈ 𝑃 ∃𝑘 ∈ 𝐾 𝑥 = (𝑝↑𝑘))
124		simpr 487	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = (𝑝↑𝑘)) → 𝑥 = (𝑝↑𝑘))
125		simplr 767	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = (𝑝↑𝑘)) → 𝑥 ∈ 𝐴)
126	124, 125	eqeltrrd 2914	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = (𝑝↑𝑘)) → (𝑝↑𝑘) ∈ 𝐴)
127	13	rbaibd 543	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑝↑𝑘) ∈ 𝐴) → ((𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾) ↔ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)))
128	127	adantlr 713	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝑝↑𝑘) ∈ 𝐴) → ((𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾) ↔ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)))
129	126, 128	syldan 593	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = (𝑝↑𝑘)) → ((𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾) ↔ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)))
130	129	pm5.32da 581	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 = (𝑝↑𝑘) ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) ↔ (𝑥 = (𝑝↑𝑘) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ))))
131		ancom 463	. . . . . . . . . . . . 13 ⊢ (((𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾) ∧ 𝑥 = (𝑝↑𝑘)) ↔ (𝑥 = (𝑝↑𝑘) ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)))
132		ancom 463	. . . . . . . . . . . . 13 ⊢ (((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 𝑥 = (𝑝↑𝑘)) ↔ (𝑥 = (𝑝↑𝑘) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)))
133	130, 131, 132	3bitr4g 316	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾) ∧ 𝑥 = (𝑝↑𝑘)) ↔ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 𝑥 = (𝑝↑𝑘))))
134	133	2exbidv 1925	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑝∃𝑘((𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾) ∧ 𝑥 = (𝑝↑𝑘)) ↔ ∃𝑝∃𝑘((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 𝑥 = (𝑝↑𝑘))))
135		r2ex 3303	. . . . . . . . . . 11 ⊢ (∃𝑝 ∈ 𝑃 ∃𝑘 ∈ 𝐾 𝑥 = (𝑝↑𝑘) ↔ ∃𝑝∃𝑘((𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾) ∧ 𝑥 = (𝑝↑𝑘)))
136		r2ex 3303	. . . . . . . . . . 11 ⊢ (∃𝑝 ∈ ℙ ∃𝑘 ∈ ℕ 𝑥 = (𝑝↑𝑘) ↔ ∃𝑝∃𝑘((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 𝑥 = (𝑝↑𝑘)))
137	134, 135, 136	3bitr4g 316	. . . . . . . . . 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 3967	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℕ)
140		isppw2 25692	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → ((Λ‘𝑥) ≠ 0 ↔ ∃𝑝 ∈ ℙ ∃𝑘 ∈ ℕ 𝑥 = (𝑝↑𝑘)))
141	139, 140	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((Λ‘𝑥) ≠ 0 ↔ ∃𝑝 ∈ ℙ ∃𝑘 ∈ ℕ 𝑥 = (𝑝↑𝑘)))
142	137, 141	bitr4d 284	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑝 ∈ 𝑃 ∃𝑘 ∈ 𝐾 𝑥 = (𝑝↑𝑘) ↔ (Λ‘𝑥) ≠ 0))
143	123, 142	syl5bb 285	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎)) ↔ (Λ‘𝑥) ≠ 0))
144	143	necon2bbid 3059	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((Λ‘𝑥) = 0 ↔ ¬ 𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))))
145	144	pm5.32da 581	. . . . . 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 415	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ (Λ‘𝑥) = 0) → 𝐵 = 0))
148	145, 147	sylbird 262	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))) → 𝐵 = 0))
149	119, 148	syl5bi 244	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))) → 𝐵 = 0))
150	149	imp 409	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎)))) → 𝐵 = 0)
151	108, 118, 150, 11	fsumss 15082	. 2 ⊢ (𝜑 → Σ𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))𝐵 = Σ𝑥 ∈ 𝐴 𝐵)
152	42, 116, 151	3eqtr2rd 2863	1 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 𝐵 = Σ𝑝 ∈ 𝑃 Σ𝑘 ∈ 𝐾 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  Vcvv 3494  ⦋csb 3883   ∖ cdif 3933   ⊆ wss 3936  {csn 4567  ⟨cop 4573  ∪ ciun 4919   ↦ cmpt 5146   × cxp 5553  ran crn 5556  –1-1→wf1 6352  –1-1-onto→wf1o 6354  ‘cfv 6355  (class class class)co 7156  1^st c1st 7687  2^nd c2nd 7688  Fincfn 8509  ℂcc 10535  0cc0 10537  ℕcn 11638  ↑cexp 13430  Σcsu 15042  ℙcprime 16015  Λcvma 25669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-inf2 9104  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614  ax-pre-sup 10615  ax-addf 10616  ax-mulf 10617

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-iin 4922  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-isom 6364  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-of 7409  df-om 7581  df-1st 7689  df-2nd 7690  df-supp 7831  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-2o 8103  df-oadd 8106  df-er 8289  df-map 8408  df-pm 8409  df-ixp 8462  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-fsupp 8834  df-fi 8875  df-sup 8906  df-inf 8907  df-oi 8974  df-dju 9330  df-card 9368  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-div 11298  df-nn 11639  df-2 11701  df-3 11702  df-4 11703  df-5 11704  df-6 11705  df-7 11706  df-8 11707  df-9 11708  df-n0 11899  df-z 11983  df-dec 12100  df-uz 12245  df-q 12350  df-rp 12391  df-xneg 12508  df-xadd 12509  df-xmul 12510  df-ioo 12743  df-ioc 12744  df-ico 12745  df-icc 12746  df-fz 12894  df-fzo 13035  df-fl 13163  df-mod 13239  df-seq 13371  df-exp 13431  df-fac 13635  df-bc 13664  df-hash 13692  df-shft 14426  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-limsup 14828  df-clim 14845  df-rlim 14846  df-sum 15043  df-ef 15421  df-sin 15423  df-cos 15424  df-pi 15426  df-dvds 15608  df-gcd 15844  df-prm 16016  df-pc 16174  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-mulr 16579  df-starv 16580  df-sca 16581  df-vsca 16582  df-ip 16583  df-tset 16584  df-ple 16585  df-ds 16587  df-unif 16588  df-hom 16589  df-cco 16590  df-rest 16696  df-topn 16697  df-0g 16715  df-gsum 16716  df-topgen 16717  df-pt 16718  df-prds 16721  df-xrs 16775  df-qtop 16780  df-imas 16781  df-xps 16783  df-mre 16857  df-mrc 16858  df-acs 16860  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-submnd 17957  df-mulg 18225  df-cntz 18447  df-cmn 18908  df-psmet 20537  df-xmet 20538  df-met 20539  df-bl 20540  df-mopn 20541  df-fbas 20542  df-fg 20543  df-cnfld 20546  df-top 21502  df-topon 21519  df-topsp 21541  df-bases 21554  df-cld 21627  df-ntr 21628  df-cls 21629  df-nei 21706  df-lp 21744  df-perf 21745  df-cn 21835  df-cnp 21836  df-haus 21923  df-tx 22170  df-hmeo 22363  df-fil 22454  df-fm 22546  df-flim 22547  df-flf 22548  df-xms 22930  df-ms 22931  df-tms 22932  df-cncf 23486  df-limc 24464  df-dv 24465  df-log 25140  df-vma 25675

This theorem is referenced by:  fsumvma2  25790  vmasum  25792