Theorem fsumvma 27131

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 6876	. . . 4 ⊢ (𝑧 = ⟨𝑝, 𝑘⟩ → (↑‘𝑧) ∈ V)
2		fveq2 6861	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑝, 𝑘⟩ → (↑‘𝑧) = (↑‘⟨𝑝, 𝑘⟩))
3		df-ov 7393	. . . . . . . 8 ⊢ (𝑝↑𝑘) = (↑‘⟨𝑝, 𝑘⟩)
4	2, 3	eqtr4di 2783	. . . . . . 7 ⊢ (𝑧 = ⟨𝑝, 𝑘⟩ → (↑‘𝑧) = (𝑝↑𝑘))
5	4	eqeq2d 2741	. . . . . 6 ⊢ (𝑧 = ⟨𝑝, 𝑘⟩ → (𝑥 = (↑‘𝑧) ↔ 𝑥 = (𝑝↑𝑘)))
6	5	biimpa 476	. . . . 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 3901	. . 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 480	. . . 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 229	. . . . . . . 8 ⊢ (𝜑 → ((𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾) → ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ (𝑝↑𝑘) ∈ 𝐴)))
15	14	impl 455	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ 𝑘 ∈ 𝐾) → ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ (𝑝↑𝑘) ∈ 𝐴))
16	15	simprd 495	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ 𝑘 ∈ 𝐾) → (𝑝↑𝑘) ∈ 𝐴)
17	16	ex 412	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → (𝑘 ∈ 𝐾 → (𝑝↑𝑘) ∈ 𝐴))
18	15	simpld 494	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ 𝑘 ∈ 𝐾) → (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ))
19	18	simpld 494	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ 𝑘 ∈ 𝐾) → 𝑝 ∈ ℙ)
20	19	adantrr 717	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → 𝑝 ∈ ℙ)
21	18	simprd 495	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ 𝑘 ∈ 𝐾) → 𝑘 ∈ ℕ)
22	21	adantrr 717	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → 𝑘 ∈ ℕ)
23	21	ex 412	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → (𝑘 ∈ 𝐾 → 𝑘 ∈ ℕ))
24	23	ssrdv 3955	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → 𝐾 ⊆ ℕ)
25	24	sselda 3949	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ 𝑧 ∈ 𝐾) → 𝑧 ∈ ℕ)
26	25	adantrl 716	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → 𝑧 ∈ ℕ)
27		eqid 2730	. . . . . . . 8 ⊢ 𝑝 = 𝑝
28		prmexpb 16696	. . . . . . . . 9 ⊢ (((𝑝 ∈ ℙ ∧ 𝑝 ∈ ℙ) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((𝑝↑𝑘) = (𝑝↑𝑧) ↔ (𝑝 = 𝑝 ∧ 𝑘 = 𝑧)))
29	28	baibd 539	. . . . . . . 8 ⊢ ((((𝑝 ∈ ℙ ∧ 𝑝 ∈ ℙ) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ ℕ)) ∧ 𝑝 = 𝑝) → ((𝑝↑𝑘) = (𝑝↑𝑧) ↔ 𝑘 = 𝑧))
30	27, 29	mpan2 691	. . . . . . 7 ⊢ (((𝑝 ∈ ℙ ∧ 𝑝 ∈ ℙ) ∧ (𝑘 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((𝑝↑𝑘) = (𝑝↑𝑧) ↔ 𝑘 = 𝑧))
31	20, 20, 22, 26, 30	syl22anc 838	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ (𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑝↑𝑘) = (𝑝↑𝑧) ↔ 𝑘 = 𝑧))
32	31	ex 412	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → ((𝑘 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) → ((𝑝↑𝑘) = (𝑝↑𝑧) ↔ 𝑘 = 𝑧)))
33	17, 32	dom2lem 8966	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → (𝑘 ∈ 𝐾 ↦ (𝑝↑𝑘)):𝐾–1-1→𝐴)
34		f1fi 9270	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ (𝑘 ∈ 𝐾 ↦ (𝑝↑𝑘)):𝐾–1-1→𝐴) → 𝐾 ∈ Fin)
35	12, 33, 34	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → 𝐾 ∈ Fin)
36	7	eleq1d 2814	. . . 4 ⊢ (𝑥 = (𝑝↑𝑘) → (𝐵 ∈ ℂ ↔ 𝐶 ∈ ℂ))
37		fsumvma.6	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
38	37	ralrimiva 3126	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ ℂ)
39	38	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) → ∀𝑥 ∈ 𝐴 𝐵 ∈ ℂ)
40	13	simplbda 499	. . . 4 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) → (𝑝↑𝑘) ∈ 𝐴)
41	36, 39, 40	rspcdva 3592	. . 3 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) → 𝐶 ∈ ℂ)
42	9, 10, 35, 41	fsum2d 15744	. 2 ⊢ (𝜑 → Σ𝑝 ∈ 𝑃 Σ𝑘 ∈ 𝐾 𝐶 = Σ𝑧 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)⦋(↑‘𝑧) / 𝑥⦌𝐵)
43		csbeq1a 3879	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
44		nfcv 2892	. . . 4 ⊢ Ⅎ𝑦𝐵
45		nfcsb1v 3889	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
46	43, 44, 45	cbvsum 15668	. . 3 ⊢ Σ𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))𝐵 = Σ𝑦 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))⦋𝑦 / 𝑥⦌𝐵
47		csbeq1 3868	. . . 4 ⊢ (𝑦 = (↑‘𝑧) → ⦋𝑦 / 𝑥⦌𝐵 = ⦋(↑‘𝑧) / 𝑥⦌𝐵)
48		snfi 9017	. . . . . . 7 ⊢ {𝑝} ∈ Fin
49		xpfi 9276	. . . . . . 7 ⊢ (({𝑝} ∈ Fin ∧ 𝐾 ∈ Fin) → ({𝑝} × 𝐾) ∈ Fin)
50	48, 35, 49	sylancr 587	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → ({𝑝} × 𝐾) ∈ Fin)
51	50	ralrimiva 3126	. . . . 5 ⊢ (𝜑 → ∀𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ∈ Fin)
52		iunfi 9301	. . . . 5 ⊢ ((𝑃 ∈ Fin ∧ ∀𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ∈ Fin) → ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ∈ Fin)
53	10, 51, 52	syl2anc 584	. . . 4 ⊢ (𝜑 → ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ∈ Fin)
54		fvex 6874	. . . . . . 7 ⊢ (↑‘𝑎) ∈ V
55	54	2a1i 12	. . . . . 6 ⊢ (𝜑 → (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) → (↑‘𝑎) ∈ V))
56		eliunxp 5804	. . . . . . . . 9 ⊢ (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↔ ∃𝑝∃𝑘(𝑎 = ⟨𝑝, 𝑘⟩ ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)))
57	13	simprbda 498	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) → (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ))
58		opelxp 5677	. . . . . . . . . . . . . 14 ⊢ (⟨𝑝, 𝑘⟩ ∈ (ℙ × ℕ) ↔ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ))
59	57, 58	sylibr 234	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) → ⟨𝑝, 𝑘⟩ ∈ (ℙ × ℕ))
60		eleq1 2817	. . . . . . . . . . . . 13 ⊢ (𝑎 = ⟨𝑝, 𝑘⟩ → (𝑎 ∈ (ℙ × ℕ) ↔ ⟨𝑝, 𝑘⟩ ∈ (ℙ × ℕ)))
61	59, 60	syl5ibrcom 247	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) → (𝑎 = ⟨𝑝, 𝑘⟩ → 𝑎 ∈ (ℙ × ℕ)))
62	61	impancom 451	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 = ⟨𝑝, 𝑘⟩) → ((𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾) → 𝑎 ∈ (ℙ × ℕ)))
63	62	expimpd 453	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑎 = ⟨𝑝, 𝑘⟩ ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) → 𝑎 ∈ (ℙ × ℕ)))
64	63	exlimdvv 1934	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑝∃𝑘(𝑎 = ⟨𝑝, 𝑘⟩ ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) → 𝑎 ∈ (ℙ × ℕ)))
65	56, 64	biimtrid 242	. . . . . . . 8 ⊢ (𝜑 → (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) → 𝑎 ∈ (ℙ × ℕ)))
66	65	ssrdv 3955	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ⊆ (ℙ × ℕ))
67	66	sseld 3948	. . . . . . . 8 ⊢ (𝜑 → (𝑏 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) → 𝑏 ∈ (ℙ × ℕ)))
68	65, 67	anim12d 609	. . . . . . 7 ⊢ (𝜑 → ((𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ∧ 𝑏 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)) → (𝑎 ∈ (ℙ × ℕ) ∧ 𝑏 ∈ (ℙ × ℕ))))
69		1st2nd2 8010	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (ℙ × ℕ) → 𝑎 = ⟨(1st ‘𝑎), (2nd ‘𝑎)⟩)
70	69	fveq2d 6865	. . . . . . . . . 10 ⊢ (𝑎 ∈ (ℙ × ℕ) → (↑‘𝑎) = (↑‘⟨(1st ‘𝑎), (2nd ‘𝑎)⟩))
71		df-ov 7393	. . . . . . . . . 10 ⊢ ((1st ‘𝑎)↑(2nd ‘𝑎)) = (↑‘⟨(1st ‘𝑎), (2nd ‘𝑎)⟩)
72	70, 71	eqtr4di 2783	. . . . . . . . 9 ⊢ (𝑎 ∈ (ℙ × ℕ) → (↑‘𝑎) = ((1st ‘𝑎)↑(2nd ‘𝑎)))
73		1st2nd2 8010	. . . . . . . . . . 11 ⊢ (𝑏 ∈ (ℙ × ℕ) → 𝑏 = ⟨(1st ‘𝑏), (2nd ‘𝑏)⟩)
74	73	fveq2d 6865	. . . . . . . . . 10 ⊢ (𝑏 ∈ (ℙ × ℕ) → (↑‘𝑏) = (↑‘⟨(1st ‘𝑏), (2nd ‘𝑏)⟩))
75		df-ov 7393	. . . . . . . . . 10 ⊢ ((1st ‘𝑏)↑(2nd ‘𝑏)) = (↑‘⟨(1st ‘𝑏), (2nd ‘𝑏)⟩)
76	74, 75	eqtr4di 2783	. . . . . . . . 9 ⊢ (𝑏 ∈ (ℙ × ℕ) → (↑‘𝑏) = ((1st ‘𝑏)↑(2nd ‘𝑏)))
77	72, 76	eqeqan12d 2744	. . . . . . . 8 ⊢ ((𝑎 ∈ (ℙ × ℕ) ∧ 𝑏 ∈ (ℙ × ℕ)) → ((↑‘𝑎) = (↑‘𝑏) ↔ ((1st ‘𝑎)↑(2nd ‘𝑎)) = ((1st ‘𝑏)↑(2nd ‘𝑏))))
78		xp1st 8003	. . . . . . . . . 10 ⊢ (𝑎 ∈ (ℙ × ℕ) → (1st ‘𝑎) ∈ ℙ)
79		xp2nd 8004	. . . . . . . . . 10 ⊢ (𝑎 ∈ (ℙ × ℕ) → (2nd ‘𝑎) ∈ ℕ)
80	78, 79	jca 511	. . . . . . . . 9 ⊢ (𝑎 ∈ (ℙ × ℕ) → ((1st ‘𝑎) ∈ ℙ ∧ (2nd ‘𝑎) ∈ ℕ))
81		xp1st 8003	. . . . . . . . . 10 ⊢ (𝑏 ∈ (ℙ × ℕ) → (1st ‘𝑏) ∈ ℙ)
82		xp2nd 8004	. . . . . . . . . 10 ⊢ (𝑏 ∈ (ℙ × ℕ) → (2nd ‘𝑏) ∈ ℕ)
83	81, 82	jca 511	. . . . . . . . 9 ⊢ (𝑏 ∈ (ℙ × ℕ) → ((1st ‘𝑏) ∈ ℙ ∧ (2nd ‘𝑏) ∈ ℕ))
84		prmexpb 16696	. . . . . . . . . 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 660	. . . . . . . . 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 596	. . . . . . . 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 8012	. . . . . . . 8 ⊢ ((𝑎 ∈ (ℙ × ℕ) ∧ 𝑏 ∈ (ℙ × ℕ)) → (((1st ‘𝑎) = (1st ‘𝑏) ∧ (2nd ‘𝑎) = (2nd ‘𝑏)) ↔ 𝑎 = 𝑏))
88	77, 86, 87	3bitrd 305	. . . . . . 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 8966	. . . . 5 ⊢ (𝜑 → (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎)):∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)–1-1→V)
91		f1f1orn 6814	. . . . 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 6861	. . . . . 6 ⊢ (𝑎 = 𝑧 → (↑‘𝑎) = (↑‘𝑧))
94		eqid 2730	. . . . . 6 ⊢ (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎)) = (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))
95		fvex 6874	. . . . . 6 ⊢ (↑‘𝑧) ∈ V
96	93, 94, 95	fvmpt 6971	. . . . 5 ⊢ (𝑧 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) → ((𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))‘𝑧) = (↑‘𝑧))
97	96	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)) → ((𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))‘𝑧) = (↑‘𝑧))
98		fveq2 6861	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = ⟨𝑝, 𝑘⟩ → (↑‘𝑎) = (↑‘⟨𝑝, 𝑘⟩))
99	98, 3	eqtr4di 2783	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = ⟨𝑝, 𝑘⟩ → (↑‘𝑎) = (𝑝↑𝑘))
100	99	eleq1d 2814	. . . . . . . . . . . . . 14 ⊢ (𝑎 = ⟨𝑝, 𝑘⟩ → ((↑‘𝑎) ∈ 𝐴 ↔ (𝑝↑𝑘) ∈ 𝐴))
101	40, 100	syl5ibrcom 247	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) → (𝑎 = ⟨𝑝, 𝑘⟩ → (↑‘𝑎) ∈ 𝐴))
102	101	impancom 451	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 = ⟨𝑝, 𝑘⟩) → ((𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾) → (↑‘𝑎) ∈ 𝐴))
103	102	expimpd 453	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑎 = ⟨𝑝, 𝑘⟩ ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) → (↑‘𝑎) ∈ 𝐴))
104	103	exlimdvv 1934	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑝∃𝑘(𝑎 = ⟨𝑝, 𝑘⟩ ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) → (↑‘𝑎) ∈ 𝐴))
105	56, 104	biimtrid 242	. . . . . . . . 9 ⊢ (𝜑 → (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) → (↑‘𝑎) ∈ 𝐴))
106	105	imp 406	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)) → (↑‘𝑎) ∈ 𝐴)
107	106	fmpttd 7090	. . . . . . 7 ⊢ (𝜑 → (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎)):∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)⟶𝐴)
108	107	frnd 6699	. . . . . 6 ⊢ (𝜑 → ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎)) ⊆ 𝐴)
109	108	sselda 3949	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))) → 𝑦 ∈ 𝐴)
110	45	nfel1 2909	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 ∈ ℂ
111	43	eleq1d 2814	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐵 ∈ ℂ ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ ℂ))
112	110, 111	rspc 3579	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝐵 ∈ ℂ → ⦋𝑦 / 𝑥⦌𝐵 ∈ ℂ))
113	38, 112	mpan9 506	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ∈ ℂ)
114	109, 113	syldan 591	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))) → ⦋𝑦 / 𝑥⦌𝐵 ∈ ℂ)
115	47, 53, 92, 97, 114	fsumf1o 15696	. . 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 2777	. 2 ⊢ (𝜑 → Σ𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))𝐵 = Σ𝑧 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)⦋(↑‘𝑧) / 𝑥⦌𝐵)
117	108	sselda 3949	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))) → 𝑥 ∈ 𝐴)
118	117, 37	syldan 591	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))) → 𝐵 ∈ ℂ)
119		eldif 3927	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))))
120	94, 54	elrnmpti 5929	. . . . . . . . . 10 ⊢ (𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎)) ↔ ∃𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)𝑥 = (↑‘𝑎))
121	99	eqeq2d 2741	. . . . . . . . . . 11 ⊢ (𝑎 = ⟨𝑝, 𝑘⟩ → (𝑥 = (↑‘𝑎) ↔ 𝑥 = (𝑝↑𝑘)))
122	121	rexiunxp 5807	. . . . . . . . . 10 ⊢ (∃𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾)𝑥 = (↑‘𝑎) ↔ ∃𝑝 ∈ 𝑃 ∃𝑘 ∈ 𝐾 𝑥 = (𝑝↑𝑘))
123	120, 122	bitri 275	. . . . . . . . 9 ⊢ (𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎)) ↔ ∃𝑝 ∈ 𝑃 ∃𝑘 ∈ 𝐾 𝑥 = (𝑝↑𝑘))
124		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = (𝑝↑𝑘)) → 𝑥 = (𝑝↑𝑘))
125		simplr 768	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = (𝑝↑𝑘)) → 𝑥 ∈ 𝐴)
126	124, 125	eqeltrrd 2830	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = (𝑝↑𝑘)) → (𝑝↑𝑘) ∈ 𝐴)
127	13	rbaibd 540	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑝↑𝑘) ∈ 𝐴) → ((𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾) ↔ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)))
128	127	adantlr 715	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝑝↑𝑘) ∈ 𝐴) → ((𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾) ↔ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)))
129	126, 128	syldan 591	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = (𝑝↑𝑘)) → ((𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾) ↔ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)))
130	129	pm5.32da 579	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 = (𝑝↑𝑘) ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)) ↔ (𝑥 = (𝑝↑𝑘) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ))))
131		ancom 460	. . . . . . . . . . . . 13 ⊢ (((𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾) ∧ 𝑥 = (𝑝↑𝑘)) ↔ (𝑥 = (𝑝↑𝑘) ∧ (𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾)))
132		ancom 460	. . . . . . . . . . . . 13 ⊢ (((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 𝑥 = (𝑝↑𝑘)) ↔ (𝑥 = (𝑝↑𝑘) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)))
133	130, 131, 132	3bitr4g 314	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾) ∧ 𝑥 = (𝑝↑𝑘)) ↔ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 𝑥 = (𝑝↑𝑘))))
134	133	2exbidv 1924	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑝∃𝑘((𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾) ∧ 𝑥 = (𝑝↑𝑘)) ↔ ∃𝑝∃𝑘((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 𝑥 = (𝑝↑𝑘))))
135		r2ex 3175	. . . . . . . . . . 11 ⊢ (∃𝑝 ∈ 𝑃 ∃𝑘 ∈ 𝐾 𝑥 = (𝑝↑𝑘) ↔ ∃𝑝∃𝑘((𝑝 ∈ 𝑃 ∧ 𝑘 ∈ 𝐾) ∧ 𝑥 = (𝑝↑𝑘)))
136		r2ex 3175	. . . . . . . . . . 11 ⊢ (∃𝑝 ∈ ℙ ∃𝑘 ∈ ℕ 𝑥 = (𝑝↑𝑘) ↔ ∃𝑝∃𝑘((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 𝑥 = (𝑝↑𝑘)))
137	134, 135, 136	3bitr4g 314	. . . . . . . . . 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 3949	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℕ)
140		isppw2 27032	. . . . . . . . . . 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 282	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑝 ∈ 𝑃 ∃𝑘 ∈ 𝐾 𝑥 = (𝑝↑𝑘) ↔ (Λ‘𝑥) ≠ 0))
143	123, 142	bitrid 283	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎)) ↔ (Λ‘𝑥) ≠ 0))
144	143	necon2bbid 2969	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((Λ‘𝑥) = 0 ↔ ¬ 𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))))
145	144	pm5.32da 579	. . . . . 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 412	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ (Λ‘𝑥) = 0) → 𝐵 = 0))
148	145, 147	sylbird 260	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))) → 𝐵 = 0))
149	119, 148	biimtrid 242	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))) → 𝐵 = 0))
150	149	imp 406	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎)))) → 𝐵 = 0)
151	108, 118, 150, 11	fsumss 15698	. 2 ⊢ (𝜑 → Σ𝑥 ∈ ran (𝑎 ∈ ∪ 𝑝 ∈ 𝑃 ({𝑝} × 𝐾) ↦ (↑‘𝑎))𝐵 = Σ𝑥 ∈ 𝐴 𝐵)
152	42, 116, 151	3eqtr2rd 2772	1 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 𝐵 = Σ𝑝 ∈ 𝑃 Σ𝑘 ∈ 𝐾 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  ∃wrex 3054  Vcvv 3450  ⦋csb 3865   ∖ cdif 3914   ⊆ wss 3917  {csn 4592  ⟨cop 4598  ∪ ciun 4958   ↦ cmpt 5191   × cxp 5639  ran crn 5642  –1-1→wf1 6511  –1-1-onto→wf1o 6513  ‘cfv 6514  (class class class)co 7390  1^st c1st 7969  2^nd c2nd 7970  Fincfn 8921  ℂcc 11073  0cc0 11075  ℕcn 12193  ↑cexp 14033  Σcsu 15659  ℙcprime 16648  Λcvma 27009

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-inf2 9601  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153  ax-addf 11154

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7656  df-om 7846  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-oadd 8441  df-er 8674  df-map 8804  df-pm 8805  df-ixp 8874  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-fsupp 9320  df-fi 9369  df-sup 9400  df-inf 9401  df-oi 9470  df-dju 9861  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-z 12537  df-dec 12657  df-uz 12801  df-q 12915  df-rp 12959  df-xneg 13079  df-xadd 13080  df-xmul 13081  df-ioo 13317  df-ioc 13318  df-ico 13319  df-icc 13320  df-fz 13476  df-fzo 13623  df-fl 13761  df-mod 13839  df-seq 13974  df-exp 14034  df-fac 14246  df-bc 14275  df-hash 14303  df-shft 15040  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-limsup 15444  df-clim 15461  df-rlim 15462  df-sum 15660  df-ef 16040  df-sin 16042  df-cos 16043  df-pi 16045  df-dvds 16230  df-gcd 16472  df-prm 16649  df-pc 16815  df-struct 17124  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-plusg 17240  df-mulr 17241  df-starv 17242  df-sca 17243  df-vsca 17244  df-ip 17245  df-tset 17246  df-ple 17247  df-ds 17249  df-unif 17250  df-hom 17251  df-cco 17252  df-rest 17392  df-topn 17393  df-0g 17411  df-gsum 17412  df-topgen 17413  df-pt 17414  df-prds 17417  df-xrs 17472  df-qtop 17477  df-imas 17478  df-xps 17480  df-mre 17554  df-mrc 17555  df-acs 17557  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-submnd 18718  df-mulg 19007  df-cntz 19256  df-cmn 19719  df-psmet 21263  df-xmet 21264  df-met 21265  df-bl 21266  df-mopn 21267  df-fbas 21268  df-fg 21269  df-cnfld 21272  df-top 22788  df-topon 22805  df-topsp 22827  df-bases 22840  df-cld 22913  df-ntr 22914  df-cls 22915  df-nei 22992  df-lp 23030  df-perf 23031  df-cn 23121  df-cnp 23122  df-haus 23209  df-tx 23456  df-hmeo 23649  df-fil 23740  df-fm 23832  df-flim 23833  df-flf 23834  df-xms 24215  df-ms 24216  df-tms 24217  df-cncf 24778  df-limc 25774  df-dv 25775  df-log 26472  df-vma 27015

This theorem is referenced by:  fsumvma2  27132  vmasum  27134