Theorem gsummpt2co 33035

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 3870	. . . 4 ⊢ Ⅎ𝑥⦋(2nd ‘𝑝) / 𝑥⦌𝐶
2		gsummpt2co.b	. . . 4 ⊢ 𝐵 = (Base‘𝑊)
3		gsummpt2co.z	. . . 4 ⊢ 0 = (0g‘𝑊)
4		csbeq1a 3860	. . . 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 3954	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝐵)
8		gsummpt2co.1	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
9		elcnv 5820	. . . . . 6 ⊢ (𝑝 ∈ ◡𝐹 ↔ ∃𝑧∃𝑥(𝑝 = ⟨𝑧, 𝑥⟩ ∧ 𝑥𝐹𝑧))
10		vex 3441	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
11		vex 3441	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
12	10, 11	op2ndd 7938	. . . . . . . . 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 6193	. . . . . . . . . 10 ⊢ dom 𝐹 ⊆ 𝐴
16	11, 10	breldm 5852	. . . . . . . . . 10 ⊢ (𝑥𝐹𝑧 → 𝑥 ∈ dom 𝐹)
17	15, 16	sselid 3928	. . . . . . . . 9 ⊢ (𝑥𝐹𝑧 → 𝑥 ∈ 𝐴)
18	17	adantl 481	. . . . . . . 8 ⊢ ((𝑝 = ⟨𝑧, 𝑥⟩ ∧ 𝑥𝐹𝑧) → 𝑥 ∈ 𝐴)
19	13, 18	eqeltrd 2833	. . . . . . 7 ⊢ ((𝑝 = ⟨𝑧, 𝑥⟩ ∧ 𝑥𝐹𝑧) → (2nd ‘𝑝) ∈ 𝐴)
20	19	exlimivv 1933	. . . . . 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 6525	. . . . . . 7 ⊢ Fun 𝐹
24		funcnvcnv 6553	. . . . . . 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 5839	. . . . . . . 8 ⊢ dom 𝐹 = ran ◡𝐹
28	14	dmeqi 5848	. . . . . . . . 9 ⊢ dom 𝐹 = dom (𝑥 ∈ 𝐴 ↦ 𝐷)
29		gsummpt2co.2	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝐸)
30	29	ralrimiva 3125	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐷 ∈ 𝐸)
31		dmmptg 6194	. . . . . . . . . 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 2780	. . . . . . . 8 ⊢ (𝜑 → dom 𝐹 = 𝐴)
34	27, 33	eqtr3id 2782	. . . . . . 7 ⊢ (𝜑 → ran ◡𝐹 = 𝐴)
35	34	eleq2d 2819	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ran ◡𝐹 ↔ 𝑥 ∈ 𝐴))
36	35	biimpar 477	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ran ◡𝐹)
37		relcnv 6057	. . . . . 6 ⊢ Rel ◡𝐹
38		fcnvgreu 32657	. . . . . 6 ⊢ (((Rel ◡𝐹 ∧ Fun ◡◡𝐹) ∧ 𝑥 ∈ ran ◡𝐹) → ∃!𝑝 ∈ ◡ 𝐹𝑥 = (2nd ‘𝑝))
39	37, 38	mpanl1 700	. . . . 5 ⊢ ((Fun ◡◡𝐹 ∧ 𝑥 ∈ ran ◡𝐹) → ∃!𝑝 ∈ ◡ 𝐹𝑥 = (2nd ‘𝑝))
40	26, 36, 39	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃!𝑝 ∈ ◡ 𝐹𝑥 = (2nd ‘𝑝))
41	1, 2, 3, 4, 5, 6, 7, 8, 22, 40	gsummptf1o 19877	. . 3 ⊢ (𝜑 → (𝑊 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑊 Σg (𝑝 ∈ ◡𝐹 ↦ ⦋(2nd ‘𝑝) / 𝑥⦌𝐶)))
42	14	rnmptss 7062	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 𝐷 ∈ 𝐸 → ran 𝐹 ⊆ 𝐸)
43	30, 42	syl 17	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐸)
44		dfcnv2 32660	. . . . . . 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 5183	. . . . 5 ⊢ (𝜑 → (𝑝 ∈ ◡𝐹 ↦ ⦋(2nd ‘𝑝) / 𝑥⦌𝐶) = (𝑝 ∈ ∪ 𝑧 ∈ 𝐸 ({𝑧} × (◡𝐹 “ {𝑧})) ↦ ⦋(2nd ‘𝑝) / 𝑥⦌𝐶))
47		nfcv 2895	. . . . . 6 ⊢ Ⅎ𝑧⦋(2nd ‘𝑝) / 𝑥⦌𝐶
48		csbeq1 3849	. . . . . . . 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 3859	. . . . . . 7 ⊢ ⦋𝑥 / 𝑥⦌𝐶 = 𝐶
51	49, 50	eqtrdi 2784	. . . . . 6 ⊢ (𝑝 = ⟨𝑧, 𝑥⟩ → ⦋(2nd ‘𝑝) / 𝑥⦌𝐶 = 𝐶)
52	47, 1, 51	mpomptxf 32663	. . . . 5 ⊢ (𝑝 ∈ ∪ 𝑧 ∈ 𝐸 ({𝑧} × (◡𝐹 “ {𝑧})) ↦ ⦋(2nd ‘𝑝) / 𝑥⦌𝐶) = (𝑧 ∈ 𝐸, 𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶)
53	46, 52	eqtrdi 2784	. . . 4 ⊢ (𝜑 → (𝑝 ∈ ◡𝐹 ↦ ⦋(2nd ‘𝑝) / 𝑥⦌𝐶) = (𝑧 ∈ 𝐸, 𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶))
54	53	oveq2d 7368	. . 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 9242	. . . . . . . 8 ⊢ (𝐴 ∈ Fin → (𝑥 ∈ 𝐴 ↦ 𝐷) ∈ Fin)
57	14, 56	eqeltrid 2837	. . . . . . 7 ⊢ (𝐴 ∈ Fin → 𝐹 ∈ Fin)
58		cnvfi 9092	. . . . . . 7 ⊢ (𝐹 ∈ Fin → ◡𝐹 ∈ Fin)
59	6, 57, 58	3syl 18	. . . . . 6 ⊢ (𝜑 → ◡𝐹 ∈ Fin)
60		imaexg 7849	. . . . . 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 6024	. . . . . . . . 9 ⊢ (◡𝐹 “ {𝑧}) ⊆ ran ◡𝐹
65	64, 27	sseqtrri 3980	. . . . . . . 8 ⊢ (◡𝐹 “ {𝑧}) ⊆ dom 𝐹
66	65, 15	sstri 3940	. . . . . . 7 ⊢ (◡𝐹 “ {𝑧}) ⊆ 𝐴
67	10, 11	elimasn 6043	. . . . . . . . . 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 3928	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐸) ∧ 𝑥 ∈ (◡𝐹 “ {𝑧})) → 𝑥 ∈ 𝐴)
72	63, 71, 8	syl2anc 584	. . . . 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 5094	. . . . . . . . 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 19888	. . 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 2772	. 2 ⊢ (𝜑 → (𝑊 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑊 Σg (𝑧 ∈ 𝐸 ↦ (𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶)))))
82		nfcv 2895	. . . 4 ⊢ Ⅎ𝑧(𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑦}) ↦ 𝐶))
83		nfcv 2895	. . . 4 ⊢ Ⅎ𝑦(𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶))
84		sneq 4585	. . . . . . 7 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
85	84	imaeq2d 6013	. . . . . 6 ⊢ (𝑦 = 𝑧 → (◡𝐹 “ {𝑦}) = (◡𝐹 “ {𝑧}))
86	85	mpteq1d 5183	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑥 ∈ (◡𝐹 “ {𝑦}) ↦ 𝐶) = (𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶))
87	86	oveq2d 7368	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑦}) ↦ 𝐶)) = (𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶)))
88	82, 83, 87	cbvmpt 5195	. . 3 ⊢ (𝑦 ∈ 𝐸 ↦ (𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑦}) ↦ 𝐶))) = (𝑧 ∈ 𝐸 ↦ (𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶)))
89	88	oveq2i 7363	. 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 2786	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 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3048  ∃!wreu 3345  Vcvv 3437  ⦋csb 3846   ⊆ wss 3898  {csn 4575  ⟨cop 4581  ∪ ciun 4941   class class class wbr 5093   ↦ cmpt 5174   × cxp 5617  ^◡ccnv 5618  dom cdm 5619  ran crn 5620   “ cima 5622  Rel wrel 5624  Fun wfun 6480  ‘cfv 6486  (class class class)co 7352   ∈ cmpo 7354  2^nd c2nd 7926  Fincfn 8875  Basecbs 17122  0_gc0g 17345   Σ_g cgsu 17346  CMndccmn 19694

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-iin 4944  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-of 7616  df-om 7803  df-1st 7927  df-2nd 7928  df-supp 8097  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-2o 8392  df-er 8628  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-fsupp 9253  df-oi 9403  df-card 9839  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-nn 12133  df-2 12195  df-n0 12389  df-z 12476  df-uz 12739  df-fz 13410  df-fzo 13557  df-seq 13911  df-hash 14240  df-sets 17077  df-slot 17095  df-ndx 17107  df-base 17123  df-ress 17144  df-plusg 17176  df-0g 17347  df-gsum 17348  df-mre 17490  df-mrc 17491  df-acs 17493  df-mgm 18550  df-sgrp 18629  df-mnd 18645  df-submnd 18694  df-mulg 18983  df-cntz 19231  df-cmn 19696

This theorem is referenced by:  gsummpt2d  33036  elrgspnsubrunlem2  33222