Theorem gsummpt2co 33031

Description: Split a finite sum into a sum of a collection of sums over disjoint subsets. (Contributed by Thierry Arnoux, 27-Mar-2018.)

Hypotheses

Ref	Expression
gsummpt2co.b	⊢ 𝐵 = (Base‘𝑊)
gsummpt2co.z	⊢ 0 = (0g‘𝑊)
gsummpt2co.w	⊢ (𝜑 → 𝑊 ∈ CMnd)
gsummpt2co.a	⊢ (𝜑 → 𝐴 ∈ Fin)
gsummpt2co.e	⊢ (𝜑 → 𝐸 ∈ 𝑉)
gsummpt2co.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
gsummpt2co.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝐸)
gsummpt2co.3	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐷)

Assertion

Ref	Expression
gsummpt2co	⊢ (𝜑 → (𝑊 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑊 Σg (𝑦 ∈ 𝐸 ↦ (𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑦}) ↦ 𝐶)))))

Distinct variable groups:   𝑥, 0 ,𝑦   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑦,𝐶   𝑥,𝐸,𝑦   𝑥,𝐹,𝑦   𝑦,𝑉   𝑥,𝑊,𝑦   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑦)   𝐶(𝑥)   𝐷(𝑥,𝑦)   𝑉(𝑥)

Proof of Theorem gsummpt2co

Dummy variables 𝑧 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcsb1v 3946	. . . 4 ⊢ Ⅎ𝑥⦋(2nd ‘𝑝) / 𝑥⦌𝐶
2		gsummpt2co.b	. . . 4 ⊢ 𝐵 = (Base‘𝑊)
3		gsummpt2co.z	. . . 4 ⊢ 0 = (0g‘𝑊)
4		csbeq1a 3935	. . . 4 ⊢ (𝑥 = (2nd ‘𝑝) → 𝐶 = ⦋(2nd ‘𝑝) / 𝑥⦌𝐶)
5		gsummpt2co.w	. . . 4 ⊢ (𝜑 → 𝑊 ∈ CMnd)
6		gsummpt2co.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin)
7		ssidd 4032	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝐵)
8		gsummpt2co.1	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
9		elcnv 5901	. . . . . 6 ⊢ (𝑝 ∈ ◡𝐹 ↔ ∃𝑧∃𝑥(𝑝 = ⟨𝑧, 𝑥⟩ ∧ 𝑥𝐹𝑧))
10		vex 3492	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
11		vex 3492	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
12	10, 11	op2ndd 8041	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑧, 𝑥⟩ → (2nd ‘𝑝) = 𝑥)
13	12	adantr 480	. . . . . . . 8 ⊢ ((𝑝 = ⟨𝑧, 𝑥⟩ ∧ 𝑥𝐹𝑧) → (2nd ‘𝑝) = 𝑥)
14		gsummpt2co.3	. . . . . . . . . . 11 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐷)
15	14	dmmptss 6272	. . . . . . . . . 10 ⊢ dom 𝐹 ⊆ 𝐴
16	11, 10	breldm 5933	. . . . . . . . . 10 ⊢ (𝑥𝐹𝑧 → 𝑥 ∈ dom 𝐹)
17	15, 16	sselid 4006	. . . . . . . . 9 ⊢ (𝑥𝐹𝑧 → 𝑥 ∈ 𝐴)
18	17	adantl 481	. . . . . . . 8 ⊢ ((𝑝 = ⟨𝑧, 𝑥⟩ ∧ 𝑥𝐹𝑧) → 𝑥 ∈ 𝐴)
19	13, 18	eqeltrd 2844	. . . . . . 7 ⊢ ((𝑝 = ⟨𝑧, 𝑥⟩ ∧ 𝑥𝐹𝑧) → (2nd ‘𝑝) ∈ 𝐴)
20	19	exlimivv 1931	. . . . . 6 ⊢ (∃𝑧∃𝑥(𝑝 = ⟨𝑧, 𝑥⟩ ∧ 𝑥𝐹𝑧) → (2nd ‘𝑝) ∈ 𝐴)
21	9, 20	sylbi 217	. . . . 5 ⊢ (𝑝 ∈ ◡𝐹 → (2nd ‘𝑝) ∈ 𝐴)
22	21	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ ◡𝐹) → (2nd ‘𝑝) ∈ 𝐴)
23	14	funmpt2 6617	. . . . . . 7 ⊢ Fun 𝐹
24		funcnvcnv 6645	. . . . . . 7 ⊢ (Fun 𝐹 → Fun ◡◡𝐹)
25	23, 24	ax-mp 5	. . . . . 6 ⊢ Fun ◡◡𝐹
26	25	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Fun ◡◡𝐹)
27		dfdm4 5920	. . . . . . . 8 ⊢ dom 𝐹 = ran ◡𝐹
28	14	dmeqi 5929	. . . . . . . . 9 ⊢ dom 𝐹 = dom (𝑥 ∈ 𝐴 ↦ 𝐷)
29		gsummpt2co.2	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝐸)
30	29	ralrimiva 3152	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐷 ∈ 𝐸)
31		dmmptg 6273	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐴 𝐷 ∈ 𝐸 → dom (𝑥 ∈ 𝐴 ↦ 𝐷) = 𝐴)
32	30, 31	syl 17	. . . . . . . . 9 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐷) = 𝐴)
33	28, 32	eqtrid 2792	. . . . . . . 8 ⊢ (𝜑 → dom 𝐹 = 𝐴)
34	27, 33	eqtr3id 2794	. . . . . . 7 ⊢ (𝜑 → ran ◡𝐹 = 𝐴)
35	34	eleq2d 2830	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ran ◡𝐹 ↔ 𝑥 ∈ 𝐴))
36	35	biimpar 477	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ran ◡𝐹)
37		relcnv 6134	. . . . . 6 ⊢ Rel ◡𝐹
38		fcnvgreu 32691	. . . . . 6 ⊢ (((Rel ◡𝐹 ∧ Fun ◡◡𝐹) ∧ 𝑥 ∈ ran ◡𝐹) → ∃!𝑝 ∈ ◡ 𝐹𝑥 = (2nd ‘𝑝))
39	37, 38	mpanl1 699	. . . . 5 ⊢ ((Fun ◡◡𝐹 ∧ 𝑥 ∈ ran ◡𝐹) → ∃!𝑝 ∈ ◡ 𝐹𝑥 = (2nd ‘𝑝))
40	26, 36, 39	syl2anc 583	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃!𝑝 ∈ ◡ 𝐹𝑥 = (2nd ‘𝑝))
41	1, 2, 3, 4, 5, 6, 7, 8, 22, 40	gsummptf1o 20005	. . 3 ⊢ (𝜑 → (𝑊 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑊 Σg (𝑝 ∈ ◡𝐹 ↦ ⦋(2nd ‘𝑝) / 𝑥⦌𝐶)))
42	14	rnmptss 7157	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 𝐷 ∈ 𝐸 → ran 𝐹 ⊆ 𝐸)
43	30, 42	syl 17	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐸)
44		dfcnv2 32694	. . . . . . 7 ⊢ (ran 𝐹 ⊆ 𝐸 → ◡𝐹 = ∪ 𝑧 ∈ 𝐸 ({𝑧} × (◡𝐹 “ {𝑧})))
45	43, 44	syl 17	. . . . . 6 ⊢ (𝜑 → ◡𝐹 = ∪ 𝑧 ∈ 𝐸 ({𝑧} × (◡𝐹 “ {𝑧})))
46	45	mpteq1d 5261	. . . . 5 ⊢ (𝜑 → (𝑝 ∈ ◡𝐹 ↦ ⦋(2nd ‘𝑝) / 𝑥⦌𝐶) = (𝑝 ∈ ∪ 𝑧 ∈ 𝐸 ({𝑧} × (◡𝐹 “ {𝑧})) ↦ ⦋(2nd ‘𝑝) / 𝑥⦌𝐶))
47		nfcv 2908	. . . . . 6 ⊢ Ⅎ𝑧⦋(2nd ‘𝑝) / 𝑥⦌𝐶
48		csbeq1 3924	. . . . . . . 8 ⊢ ((2nd ‘𝑝) = 𝑥 → ⦋(2nd ‘𝑝) / 𝑥⦌𝐶 = ⦋𝑥 / 𝑥⦌𝐶)
49	12, 48	syl 17	. . . . . . 7 ⊢ (𝑝 = ⟨𝑧, 𝑥⟩ → ⦋(2nd ‘𝑝) / 𝑥⦌𝐶 = ⦋𝑥 / 𝑥⦌𝐶)
50		csbid 3934	. . . . . . 7 ⊢ ⦋𝑥 / 𝑥⦌𝐶 = 𝐶
51	49, 50	eqtrdi 2796	. . . . . 6 ⊢ (𝑝 = ⟨𝑧, 𝑥⟩ → ⦋(2nd ‘𝑝) / 𝑥⦌𝐶 = 𝐶)
52	47, 1, 51	mpomptxf 32695	. . . . 5 ⊢ (𝑝 ∈ ∪ 𝑧 ∈ 𝐸 ({𝑧} × (◡𝐹 “ {𝑧})) ↦ ⦋(2nd ‘𝑝) / 𝑥⦌𝐶) = (𝑧 ∈ 𝐸, 𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶)
53	46, 52	eqtrdi 2796	. . . 4 ⊢ (𝜑 → (𝑝 ∈ ◡𝐹 ↦ ⦋(2nd ‘𝑝) / 𝑥⦌𝐶) = (𝑧 ∈ 𝐸, 𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶))
54	53	oveq2d 7464	. . 3 ⊢ (𝜑 → (𝑊 Σg (𝑝 ∈ ◡𝐹 ↦ ⦋(2nd ‘𝑝) / 𝑥⦌𝐶)) = (𝑊 Σg (𝑧 ∈ 𝐸, 𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶)))
55		gsummpt2co.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝑉)
56		mptfi 9421	. . . . . . . 8 ⊢ (𝐴 ∈ Fin → (𝑥 ∈ 𝐴 ↦ 𝐷) ∈ Fin)
57	14, 56	eqeltrid 2848	. . . . . . 7 ⊢ (𝐴 ∈ Fin → 𝐹 ∈ Fin)
58		cnvfi 9243	. . . . . . 7 ⊢ (𝐹 ∈ Fin → ◡𝐹 ∈ Fin)
59	6, 57, 58	3syl 18	. . . . . 6 ⊢ (𝜑 → ◡𝐹 ∈ Fin)
60		imaexg 7953	. . . . . 6 ⊢ (◡𝐹 ∈ Fin → (◡𝐹 “ {𝑧}) ∈ V)
61	59, 60	syl 17	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ {𝑧}) ∈ V)
62	61	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐸) → (◡𝐹 “ {𝑧}) ∈ V)
63		simpll 766	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐸) ∧ 𝑥 ∈ (◡𝐹 “ {𝑧})) → 𝜑)
64		imassrn 6100	. . . . . . . . 9 ⊢ (◡𝐹 “ {𝑧}) ⊆ ran ◡𝐹
65	64, 27	sseqtrri 4046	. . . . . . . 8 ⊢ (◡𝐹 “ {𝑧}) ⊆ dom 𝐹
66	65, 15	sstri 4018	. . . . . . 7 ⊢ (◡𝐹 “ {𝑧}) ⊆ 𝐴
67	10, 11	elimasn 6119	. . . . . . . . . 10 ⊢ (𝑥 ∈ (◡𝐹 “ {𝑧}) ↔ ⟨𝑧, 𝑥⟩ ∈ ◡𝐹)
68	67	biimpi 216	. . . . . . . . 9 ⊢ (𝑥 ∈ (◡𝐹 “ {𝑧}) → ⟨𝑧, 𝑥⟩ ∈ ◡𝐹)
69	68	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐸) ∧ 𝑥 ∈ (◡𝐹 “ {𝑧})) → ⟨𝑧, 𝑥⟩ ∈ ◡𝐹)
70	69, 67	sylibr 234	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐸) ∧ 𝑥 ∈ (◡𝐹 “ {𝑧})) → 𝑥 ∈ (◡𝐹 “ {𝑧}))
71	66, 70	sselid 4006	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐸) ∧ 𝑥 ∈ (◡𝐹 “ {𝑧})) → 𝑥 ∈ 𝐴)
72	63, 71, 8	syl2anc 583	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐸) ∧ 𝑥 ∈ (◡𝐹 “ {𝑧})) → 𝐶 ∈ 𝐵)
73	72	anasss 466	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐸 ∧ 𝑥 ∈ (◡𝐹 “ {𝑧}))) → 𝐶 ∈ 𝐵)
74		df-br 5167	. . . . . . . . 9 ⊢ (𝑧◡𝐹𝑥 ↔ ⟨𝑧, 𝑥⟩ ∈ ◡𝐹)
75	69, 74	sylibr 234	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐸) ∧ 𝑥 ∈ (◡𝐹 “ {𝑧})) → 𝑧◡𝐹𝑥)
76	75	anasss 466	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐸 ∧ 𝑥 ∈ (◡𝐹 “ {𝑧}))) → 𝑧◡𝐹𝑥)
77	76	pm2.24d 151	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐸 ∧ 𝑥 ∈ (◡𝐹 “ {𝑧}))) → (¬ 𝑧◡𝐹𝑥 → 𝐶 = 0 ))
78	77	imp 406	. . . . 5 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐸 ∧ 𝑥 ∈ (◡𝐹 “ {𝑧}))) ∧ ¬ 𝑧◡𝐹𝑥) → 𝐶 = 0 )
79	78	anasss 466	. . . 4 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝐸 ∧ 𝑥 ∈ (◡𝐹 “ {𝑧})) ∧ ¬ 𝑧◡𝐹𝑥)) → 𝐶 = 0 )
80	2, 3, 5, 55, 62, 73, 59, 79	gsum2d2 20016	. . 3 ⊢ (𝜑 → (𝑊 Σg (𝑧 ∈ 𝐸, 𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶)) = (𝑊 Σg (𝑧 ∈ 𝐸 ↦ (𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶)))))
81	41, 54, 80	3eqtrd 2784	. 2 ⊢ (𝜑 → (𝑊 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑊 Σg (𝑧 ∈ 𝐸 ↦ (𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶)))))
82		nfcv 2908	. . . 4 ⊢ Ⅎ𝑧(𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑦}) ↦ 𝐶))
83		nfcv 2908	. . . 4 ⊢ Ⅎ𝑦(𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶))
84		sneq 4658	. . . . . . 7 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
85	84	imaeq2d 6089	. . . . . 6 ⊢ (𝑦 = 𝑧 → (◡𝐹 “ {𝑦}) = (◡𝐹 “ {𝑧}))
86	85	mpteq1d 5261	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑥 ∈ (◡𝐹 “ {𝑦}) ↦ 𝐶) = (𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶))
87	86	oveq2d 7464	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑦}) ↦ 𝐶)) = (𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶)))
88	82, 83, 87	cbvmpt 5277	. . 3 ⊢ (𝑦 ∈ 𝐸 ↦ (𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑦}) ↦ 𝐶))) = (𝑧 ∈ 𝐸 ↦ (𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶)))
89	88	oveq2i 7459	. 2 ⊢ (𝑊 Σg (𝑦 ∈ 𝐸 ↦ (𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑦}) ↦ 𝐶)))) = (𝑊 Σg (𝑧 ∈ 𝐸 ↦ (𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶))))
90	81, 89	eqtr4di 2798	1 ⊢ (𝜑 → (𝑊 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑊 Σg (𝑦 ∈ 𝐸 ↦ (𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑦}) ↦ 𝐶)))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  ∀wral 3067  ∃!wreu 3386  Vcvv 3488  ⦋csb 3921   ⊆ wss 3976  {csn 4648  ⟨cop 4654  ∪ ciun 5015   class class class wbr 5166   ↦ cmpt 5249   × cxp 5698  ^◡ccnv 5699  dom cdm 5700  ran crn 5701   “ cima 5703  Rel wrel 5705  Fun wfun 6567  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  2^nd c2nd 8029  Fincfn 9003  Basecbs 17258  0_gc0g 17499   Σ_g cgsu 17500  CMndccmn 19822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-om 7904  df-1st 8030  df-2nd 8031  df-supp 8202  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fsupp 9432  df-oi 9579  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-n0 12554  df-z 12640  df-uz 12904  df-fz 13568  df-fzo 13712  df-seq 14053  df-hash 14380  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-0g 17501  df-gsum 17502  df-mre 17644  df-mrc 17645  df-acs 17647  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-submnd 18819  df-mulg 19108  df-cntz 19357  df-cmn 19824

This theorem is referenced by:  gsummpt2d  33032