Theorem gsummpt2co 33051

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 3923	. . . 4 ⊢ Ⅎ𝑥⦋(2nd ‘𝑝) / 𝑥⦌𝐶
2		gsummpt2co.b	. . . 4 ⊢ 𝐵 = (Base‘𝑊)
3		gsummpt2co.z	. . . 4 ⊢ 0 = (0g‘𝑊)
4		csbeq1a 3913	. . . 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 4007	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝐵)
8		gsummpt2co.1	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
9		elcnv 5887	. . . . . 6 ⊢ (𝑝 ∈ ◡𝐹 ↔ ∃𝑧∃𝑥(𝑝 = ⟨𝑧, 𝑥⟩ ∧ 𝑥𝐹𝑧))
10		vex 3484	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
11		vex 3484	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
12	10, 11	op2ndd 8025	. . . . . . . . 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 6261	. . . . . . . . . 10 ⊢ dom 𝐹 ⊆ 𝐴
16	11, 10	breldm 5919	. . . . . . . . . 10 ⊢ (𝑥𝐹𝑧 → 𝑥 ∈ dom 𝐹)
17	15, 16	sselid 3981	. . . . . . . . 9 ⊢ (𝑥𝐹𝑧 → 𝑥 ∈ 𝐴)
18	17	adantl 481	. . . . . . . 8 ⊢ ((𝑝 = ⟨𝑧, 𝑥⟩ ∧ 𝑥𝐹𝑧) → 𝑥 ∈ 𝐴)
19	13, 18	eqeltrd 2841	. . . . . . 7 ⊢ ((𝑝 = ⟨𝑧, 𝑥⟩ ∧ 𝑥𝐹𝑧) → (2nd ‘𝑝) ∈ 𝐴)
20	19	exlimivv 1932	. . . . . 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 6605	. . . . . . 7 ⊢ Fun 𝐹
24		funcnvcnv 6633	. . . . . . 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 5906	. . . . . . . 8 ⊢ dom 𝐹 = ran ◡𝐹
28	14	dmeqi 5915	. . . . . . . . 9 ⊢ dom 𝐹 = dom (𝑥 ∈ 𝐴 ↦ 𝐷)
29		gsummpt2co.2	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝐸)
30	29	ralrimiva 3146	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐷 ∈ 𝐸)
31		dmmptg 6262	. . . . . . . . . 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 2789	. . . . . . . 8 ⊢ (𝜑 → dom 𝐹 = 𝐴)
34	27, 33	eqtr3id 2791	. . . . . . 7 ⊢ (𝜑 → ran ◡𝐹 = 𝐴)
35	34	eleq2d 2827	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ran ◡𝐹 ↔ 𝑥 ∈ 𝐴))
36	35	biimpar 477	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ran ◡𝐹)
37		relcnv 6122	. . . . . 6 ⊢ Rel ◡𝐹
38		fcnvgreu 32683	. . . . . 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 19981	. . 3 ⊢ (𝜑 → (𝑊 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑊 Σg (𝑝 ∈ ◡𝐹 ↦ ⦋(2nd ‘𝑝) / 𝑥⦌𝐶)))
42	14	rnmptss 7143	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 𝐷 ∈ 𝐸 → ran 𝐹 ⊆ 𝐸)
43	30, 42	syl 17	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐸)
44		dfcnv2 32686	. . . . . . 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 5237	. . . . 5 ⊢ (𝜑 → (𝑝 ∈ ◡𝐹 ↦ ⦋(2nd ‘𝑝) / 𝑥⦌𝐶) = (𝑝 ∈ ∪ 𝑧 ∈ 𝐸 ({𝑧} × (◡𝐹 “ {𝑧})) ↦ ⦋(2nd ‘𝑝) / 𝑥⦌𝐶))
47		nfcv 2905	. . . . . 6 ⊢ Ⅎ𝑧⦋(2nd ‘𝑝) / 𝑥⦌𝐶
48		csbeq1 3902	. . . . . . . 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 3912	. . . . . . 7 ⊢ ⦋𝑥 / 𝑥⦌𝐶 = 𝐶
51	49, 50	eqtrdi 2793	. . . . . 6 ⊢ (𝑝 = ⟨𝑧, 𝑥⟩ → ⦋(2nd ‘𝑝) / 𝑥⦌𝐶 = 𝐶)
52	47, 1, 51	mpomptxf 32687	. . . . 5 ⊢ (𝑝 ∈ ∪ 𝑧 ∈ 𝐸 ({𝑧} × (◡𝐹 “ {𝑧})) ↦ ⦋(2nd ‘𝑝) / 𝑥⦌𝐶) = (𝑧 ∈ 𝐸, 𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶)
53	46, 52	eqtrdi 2793	. . . 4 ⊢ (𝜑 → (𝑝 ∈ ◡𝐹 ↦ ⦋(2nd ‘𝑝) / 𝑥⦌𝐶) = (𝑧 ∈ 𝐸, 𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶))
54	53	oveq2d 7447	. . 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 9391	. . . . . . . 8 ⊢ (𝐴 ∈ Fin → (𝑥 ∈ 𝐴 ↦ 𝐷) ∈ Fin)
57	14, 56	eqeltrid 2845	. . . . . . 7 ⊢ (𝐴 ∈ Fin → 𝐹 ∈ Fin)
58		cnvfi 9216	. . . . . . 7 ⊢ (𝐹 ∈ Fin → ◡𝐹 ∈ Fin)
59	6, 57, 58	3syl 18	. . . . . 6 ⊢ (𝜑 → ◡𝐹 ∈ Fin)
60		imaexg 7935	. . . . . 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 767	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐸) ∧ 𝑥 ∈ (◡𝐹 “ {𝑧})) → 𝜑)
64		imassrn 6089	. . . . . . . . 9 ⊢ (◡𝐹 “ {𝑧}) ⊆ ran ◡𝐹
65	64, 27	sseqtrri 4033	. . . . . . . 8 ⊢ (◡𝐹 “ {𝑧}) ⊆ dom 𝐹
66	65, 15	sstri 3993	. . . . . . 7 ⊢ (◡𝐹 “ {𝑧}) ⊆ 𝐴
67	10, 11	elimasn 6108	. . . . . . . . . 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 3981	. . . . . 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 5144	. . . . . . . . 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 19992	. . 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 2781	. 2 ⊢ (𝜑 → (𝑊 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑊 Σg (𝑧 ∈ 𝐸 ↦ (𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶)))))
82		nfcv 2905	. . . 4 ⊢ Ⅎ𝑧(𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑦}) ↦ 𝐶))
83		nfcv 2905	. . . 4 ⊢ Ⅎ𝑦(𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶))
84		sneq 4636	. . . . . . 7 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
85	84	imaeq2d 6078	. . . . . 6 ⊢ (𝑦 = 𝑧 → (◡𝐹 “ {𝑦}) = (◡𝐹 “ {𝑧}))
86	85	mpteq1d 5237	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑥 ∈ (◡𝐹 “ {𝑦}) ↦ 𝐶) = (𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶))
87	86	oveq2d 7447	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑦}) ↦ 𝐶)) = (𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶)))
88	82, 83, 87	cbvmpt 5253	. . 3 ⊢ (𝑦 ∈ 𝐸 ↦ (𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑦}) ↦ 𝐶))) = (𝑧 ∈ 𝐸 ↦ (𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶)))
89	88	oveq2i 7442	. 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 2795	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 1540  ∃wex 1779   ∈ wcel 2108  ∀wral 3061  ∃!wreu 3378  Vcvv 3480  ⦋csb 3899   ⊆ wss 3951  {csn 4626  ⟨cop 4632  ∪ ciun 4991   class class class wbr 5143   ↦ cmpt 5225   × cxp 5683  ^◡ccnv 5684  dom cdm 5685  ran crn 5686   “ cima 5688  Rel wrel 5690  Fun wfun 6555  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433  2^nd c2nd 8013  Fincfn 8985  Basecbs 17247  0_gc0g 17484   Σ_g cgsu 17485  CMndccmn 19798

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-0g 17486  df-gsum 17487  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-mulg 19086  df-cntz 19335  df-cmn 19800

This theorem is referenced by:  gsummpt2d  33052  elrgspnsubrunlem2  33252