Theorem gsummpt2co 33189

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 3876	. . . 4 ⊢ Ⅎ𝑥⦋(2nd ‘𝑝) / 𝑥⦌𝐶
2		gsummpt2co.b	. . . 4 ⊢ 𝐵 = (Base‘𝑊)
3		gsummpt2co.z	. . . 4 ⊢ 0 = (0g‘𝑊)
4		csbeq1a 3866	. . . 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 3959	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝐵)
8		gsummpt2co.1	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
9		elcnv 5846	. . . . . 6 ⊢ (𝑝 ∈ ◡𝐹 ↔ ∃𝑧∃𝑥(𝑝 = ⟨𝑧, 𝑥⟩ ∧ 𝑥𝐹𝑧))
10		vex 3457	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
11		vex 3457	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
12	10, 11	op2ndd 7977	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑧, 𝑥⟩ → (2nd ‘𝑝) = 𝑥)
13	12	adantr 484	. . . . . . . 8 ⊢ ((𝑝 = ⟨𝑧, 𝑥⟩ ∧ 𝑥𝐹𝑧) → (2nd ‘𝑝) = 𝑥)
14		gsummpt2co.3	. . . . . . . . . . 11 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐷)
15	14	dmmptss 6224	. . . . . . . . . 10 ⊢ dom 𝐹 ⊆ 𝐴
16	11, 10	breldm 5882	. . . . . . . . . 10 ⊢ (𝑥𝐹𝑧 → 𝑥 ∈ dom 𝐹)
17	15, 16	sselid 3934	. . . . . . . . 9 ⊢ (𝑥𝐹𝑧 → 𝑥 ∈ 𝐴)
18	17	adantl 485	. . . . . . . 8 ⊢ ((𝑝 = ⟨𝑧, 𝑥⟩ ∧ 𝑥𝐹𝑧) → 𝑥 ∈ 𝐴)
19	13, 18	eqeltrd 2861	. . . . . . 7 ⊢ ((𝑝 = ⟨𝑧, 𝑥⟩ ∧ 𝑥𝐹𝑧) → (2nd ‘𝑝) ∈ 𝐴)
20	19	exlimivv 1951	. . . . . 6 ⊢ (∃𝑧∃𝑥(𝑝 = ⟨𝑧, 𝑥⟩ ∧ 𝑥𝐹𝑧) → (2nd ‘𝑝) ∈ 𝐴)
21	9, 20	sylbi 219	. . . . 5 ⊢ (𝑝 ∈ ◡𝐹 → (2nd ‘𝑝) ∈ 𝐴)
22	21	adantl 485	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ ◡𝐹) → (2nd ‘𝑝) ∈ 𝐴)
23	14	funmpt2 6556	. . . . . . 7 ⊢ Fun 𝐹
24		funcnvcnv 6584	. . . . . . 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 5869	. . . . . . . 8 ⊢ dom 𝐹 = ran ◡𝐹
28	14	dmeqi 5878	. . . . . . . . 9 ⊢ dom 𝐹 = dom (𝑥 ∈ 𝐴 ↦ 𝐷)
29		gsummpt2co.2	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝐸)
30	29	ralrimiva 3153	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐷 ∈ 𝐸)
31		dmmptg 6225	. . . . . . . . . 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 2808	. . . . . . . 8 ⊢ (𝜑 → dom 𝐹 = 𝐴)
34	27, 33	eqtr3id 2810	. . . . . . 7 ⊢ (𝜑 → ran ◡𝐹 = 𝐴)
35	34	eleq2d 2847	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ran ◡𝐹 ↔ 𝑥 ∈ 𝐴))
36	35	biimpar 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ran ◡𝐹)
37		relcnv 6090	. . . . . 6 ⊢ Rel ◡𝐹
38		fcnvgreu 32824	. . . . . 6 ⊢ (((Rel ◡𝐹 ∧ Fun ◡◡𝐹) ∧ 𝑥 ∈ ran ◡𝐹) → ∃!𝑝 ∈ ◡ 𝐹𝑥 = (2nd ‘𝑝))
39	37, 38	mpanl1 710	. . . . 5 ⊢ ((Fun ◡◡𝐹 ∧ 𝑥 ∈ ran ◡𝐹) → ∃!𝑝 ∈ ◡ 𝐹𝑥 = (2nd ‘𝑝))
40	26, 36, 39	syl2anc 593	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃!𝑝 ∈ ◡ 𝐹𝑥 = (2nd ‘𝑝))
41	1, 2, 3, 4, 5, 6, 7, 8, 22, 40	gsummptf1o 19986	. . 3 ⊢ (𝜑 → (𝑊 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑊 Σg (𝑝 ∈ ◡𝐹 ↦ ⦋(2nd ‘𝑝) / 𝑥⦌𝐶)))
42	14	rnmptss 7100	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 𝐷 ∈ 𝐸 → ran 𝐹 ⊆ 𝐸)
43	30, 42	syl 17	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐸)
44		dfcnv2 32827	. . . . . . 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 5189	. . . . 5 ⊢ (𝜑 → (𝑝 ∈ ◡𝐹 ↦ ⦋(2nd ‘𝑝) / 𝑥⦌𝐶) = (𝑝 ∈ ∪ 𝑧 ∈ 𝐸 ({𝑧} × (◡𝐹 “ {𝑧})) ↦ ⦋(2nd ‘𝑝) / 𝑥⦌𝐶))
47		nfcv 2923	. . . . . 6 ⊢ Ⅎ𝑧⦋(2nd ‘𝑝) / 𝑥⦌𝐶
48		csbeq1 3855	. . . . . . . 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 3865	. . . . . . 7 ⊢ ⦋𝑥 / 𝑥⦌𝐶 = 𝐶
51	49, 50	eqtrdi 2812	. . . . . 6 ⊢ (𝑝 = ⟨𝑧, 𝑥⟩ → ⦋(2nd ‘𝑝) / 𝑥⦌𝐶 = 𝐶)
52	47, 1, 51	mpomptxf 32830	. . . . 5 ⊢ (𝑝 ∈ ∪ 𝑧 ∈ 𝐸 ({𝑧} × (◡𝐹 “ {𝑧})) ↦ ⦋(2nd ‘𝑝) / 𝑥⦌𝐶) = (𝑧 ∈ 𝐸, 𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶)
53	46, 52	eqtrdi 2812	. . . 4 ⊢ (𝜑 → (𝑝 ∈ ◡𝐹 ↦ ⦋(2nd ‘𝑝) / 𝑥⦌𝐶) = (𝑧 ∈ 𝐸, 𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶))
54	53	oveq2d 7408	. . 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 9291	. . . . . . . 8 ⊢ (𝐴 ∈ Fin → (𝑥 ∈ 𝐴 ↦ 𝐷) ∈ Fin)
57	14, 56	eqeltrid 2865	. . . . . . 7 ⊢ (𝐴 ∈ Fin → 𝐹 ∈ Fin)
58		cnvfi 9140	. . . . . . 7 ⊢ (𝐹 ∈ Fin → ◡𝐹 ∈ Fin)
59	6, 57, 58	3syl 18	. . . . . 6 ⊢ (𝜑 → ◡𝐹 ∈ Fin)
60		imaexg 7890	. . . . . 6 ⊢ (◡𝐹 ∈ Fin → (◡𝐹 “ {𝑧}) ∈ V)
61	59, 60	syl 17	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ {𝑧}) ∈ V)
62	61	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐸) → (◡𝐹 “ {𝑧}) ∈ V)
63		simpll 776	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐸) ∧ 𝑥 ∈ (◡𝐹 “ {𝑧})) → 𝜑)
64		imassrn 6057	. . . . . . . . 9 ⊢ (◡𝐹 “ {𝑧}) ⊆ ran ◡𝐹
65	64, 27	sseqtrri 3985	. . . . . . . 8 ⊢ (◡𝐹 “ {𝑧}) ⊆ dom 𝐹
66	65, 15	sstri 3945	. . . . . . 7 ⊢ (◡𝐹 “ {𝑧}) ⊆ 𝐴
67	10, 11	elimasn 6076	. . . . . . . . 9 ⊢ (𝑥 ∈ (◡𝐹 “ {𝑧}) ↔ ⟨𝑧, 𝑥⟩ ∈ ◡𝐹)
68	67	bilani 508	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐸) ∧ 𝑥 ∈ (◡𝐹 “ {𝑧})) → ⟨𝑧, 𝑥⟩ ∈ ◡𝐹)
69	68, 67	sylibr 236	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐸) ∧ 𝑥 ∈ (◡𝐹 “ {𝑧})) → 𝑥 ∈ (◡𝐹 “ {𝑧}))
70	66, 69	sselid 3934	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐸) ∧ 𝑥 ∈ (◡𝐹 “ {𝑧})) → 𝑥 ∈ 𝐴)
71	63, 70, 8	syl2anc 593	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐸) ∧ 𝑥 ∈ (◡𝐹 “ {𝑧})) → 𝐶 ∈ 𝐵)
72	71	anasss 470	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐸 ∧ 𝑥 ∈ (◡𝐹 “ {𝑧}))) → 𝐶 ∈ 𝐵)
73		df-br 5100	. . . . . . . . 9 ⊢ (𝑧◡𝐹𝑥 ↔ ⟨𝑧, 𝑥⟩ ∈ ◡𝐹)
74	68, 73	sylibr 236	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐸) ∧ 𝑥 ∈ (◡𝐹 “ {𝑧})) → 𝑧◡𝐹𝑥)
75	74	anasss 470	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐸 ∧ 𝑥 ∈ (◡𝐹 “ {𝑧}))) → 𝑧◡𝐹𝑥)
76	75	pm2.24d 151	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐸 ∧ 𝑥 ∈ (◡𝐹 “ {𝑧}))) → (¬ 𝑧◡𝐹𝑥 → 𝐶 = 0 ))
77	76	imp 410	. . . . 5 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐸 ∧ 𝑥 ∈ (◡𝐹 “ {𝑧}))) ∧ ¬ 𝑧◡𝐹𝑥) → 𝐶 = 0 )
78	77	anasss 470	. . . 4 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝐸 ∧ 𝑥 ∈ (◡𝐹 “ {𝑧})) ∧ ¬ 𝑧◡𝐹𝑥)) → 𝐶 = 0 )
79	2, 3, 5, 55, 62, 72, 59, 78	gsum2d2 19997	. . 3 ⊢ (𝜑 → (𝑊 Σg (𝑧 ∈ 𝐸, 𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶)) = (𝑊 Σg (𝑧 ∈ 𝐸 ↦ (𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶)))))
80	41, 54, 79	3eqtrd 2800	. 2 ⊢ (𝜑 → (𝑊 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑊 Σg (𝑧 ∈ 𝐸 ↦ (𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶)))))
81		nfcv 2923	. . . 4 ⊢ Ⅎ𝑧(𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑦}) ↦ 𝐶))
82		nfcv 2923	. . . 4 ⊢ Ⅎ𝑦(𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶))
83		sneq 4591	. . . . . . 7 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
84	83	imaeq2d 6046	. . . . . 6 ⊢ (𝑦 = 𝑧 → (◡𝐹 “ {𝑦}) = (◡𝐹 “ {𝑧}))
85	84	mpteq1d 5189	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑥 ∈ (◡𝐹 “ {𝑦}) ↦ 𝐶) = (𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶))
86	85	oveq2d 7408	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑦}) ↦ 𝐶)) = (𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶)))
87	81, 82, 86	cbvmpt 5201	. . 3 ⊢ (𝑦 ∈ 𝐸 ↦ (𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑦}) ↦ 𝐶))) = (𝑧 ∈ 𝐸 ↦ (𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶)))
88	87	oveq2i 7403	. 2 ⊢ (𝑊 Σg (𝑦 ∈ 𝐸 ↦ (𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑦}) ↦ 𝐶)))) = (𝑊 Σg (𝑧 ∈ 𝐸 ↦ (𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑧}) ↦ 𝐶))))
89	80, 88	eqtr4di 2814	1 ⊢ (𝜑 → (𝑊 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑊 Σg (𝑦 ∈ 𝐸 ↦ (𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑦}) ↦ 𝐶)))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1559  ∃wex 1798   ∈ wcel 2141  ∀wral 3075  ∃!wreu 3364  Vcvv 3453  ⦋csb 3852   ⊆ wss 3904  {csn 4581  ⟨cop 4587  ∪ ciun 4948   class class class wbr 5099   ↦ cmpt 5180   × cxp 5643  ^◡ccnv 5644  dom cdm 5645  ran crn 5646   “ cima 5648  Rel wrel 5650  Fun wfun 6511  ‘cfv 6517  (class class class)co 7392   ∈ cmpo 7394  2^nd c2nd 7965  Fincfn 8923  Basecbs 17228  0_gc0g 17451   Σ_g cgsu 17452  CMndccmn 19803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-iin 4951  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-se 5599  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-isom 6526  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-of 7656  df-om 7843  df-1st 7966  df-2nd 7967  df-supp 8136  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-2o 8433  df-er 8673  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-fsupp 9305  df-oi 9455  df-card 9894  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-nn 12208  df-2 12277  df-n0 12479  df-z 12566  df-uz 12837  df-fz 13510  df-fzo 13657  df-seq 14012  df-hash 14341  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17250  df-plusg 17282  df-0g 17453  df-gsum 17454  df-mre 17597  df-mrc 17598  df-acs 17600  df-mgm 18657  df-sgrp 18736  df-mnd 18752  df-submnd 18801  df-mulg 19093  df-cntz 19340  df-cmn 19805

This theorem is referenced by:  gsummpt2d  33190  elrgspnsubrunlem2  33390