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Theorem dprdf1o 19967

Description: Rearrange the index set of a direct product family. (Contributed by Mario Carneiro, 25-Apr-2016.)

**Hypotheses**
Ref	Expression
dprdf1o.1	⊢ (𝜑 → 𝐺dom DProd 𝑆)
dprdf1o.2	⊢ (𝜑 → dom 𝑆 = 𝐼)
dprdf1o.3	⊢ (𝜑 → 𝐹:𝐽–1-1-onto→𝐼)

**Assertion**
Ref	Expression
dprdf1o	⊢ (𝜑 → (𝐺dom DProd (𝑆 ∘ 𝐹) ∧ (𝐺 DProd (𝑆 ∘ 𝐹)) = (𝐺 DProd 𝑆)))

Proof of Theorem dprdf1o Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2737	. . 3 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
2		eqid 2737	. . 3 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
3		eqid 2737	. . 3 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
4		dprdf1o.1	. . . 4 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
5		dprdgrp 19940	. . . 4 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → 𝐺 ∈ Grp)
7		dprdf1o.3	. . . . 5 ⊢ (𝜑 → 𝐹:𝐽–1-1-onto→𝐼)
8		f1of1 6774	. . . . 5 ⊢ (𝐹:𝐽–1-1-onto→𝐼 → 𝐹:𝐽–1-1→𝐼)
9	7, 8	syl 17	. . . 4 ⊢ (𝜑 → 𝐹:𝐽–1-1→𝐼)
10		dprdf1o.2	. . . . 5 ⊢ (𝜑 → dom 𝑆 = 𝐼)
11	4, 10	dprddomcld 19936	. . . 4 ⊢ (𝜑 → 𝐼 ∈ V)
12		f1dmex 7903	. . . 4 ⊢ ((𝐹:𝐽–1-1→𝐼 ∧ 𝐼 ∈ V) → 𝐽 ∈ V)
13	9, 11, 12	syl2anc 585	. . 3 ⊢ (𝜑 → 𝐽 ∈ V)
14	4, 10	dprdf2 19942	. . . 4 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
15		f1of 6775	. . . . 5 ⊢ (𝐹:𝐽–1-1-onto→𝐼 → 𝐹:𝐽⟶𝐼)
16	7, 15	syl 17	. . . 4 ⊢ (𝜑 → 𝐹:𝐽⟶𝐼)
17		fco 6687	. . . 4 ⊢ ((𝑆:𝐼⟶(SubGrp‘𝐺) ∧ 𝐹:𝐽⟶𝐼) → (𝑆 ∘ 𝐹):𝐽⟶(SubGrp‘𝐺))
18	14, 16, 17	syl2anc 585	. . 3 ⊢ (𝜑 → (𝑆 ∘ 𝐹):𝐽⟶(SubGrp‘𝐺))
19	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → 𝐺dom DProd 𝑆)
20	10	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → dom 𝑆 = 𝐼)
21	16	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → 𝐹:𝐽⟶𝐼)
22		simpr1 1196	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ 𝐽)
23	21, 22	ffvelcdmd 7032	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → (𝐹‘𝑥) ∈ 𝐼)
24		simpr2 1197	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ 𝐽)
25	21, 24	ffvelcdmd 7032	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → (𝐹‘𝑦) ∈ 𝐼)
26		simpr3 1198	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦)
27	9	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → 𝐹:𝐽–1-1→𝐼)
28		f1fveq 7210	. . . . . . . 8 ⊢ ((𝐹:𝐽–1-1→𝐼 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ 𝑥 = 𝑦))
29	27, 22, 24, 28	syl12anc 837	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ 𝑥 = 𝑦))
30	29	necon3bid 2977	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ((𝐹‘𝑥) ≠ (𝐹‘𝑦) ↔ 𝑥 ≠ 𝑦))
31	26, 30	mpbird 257	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → (𝐹‘𝑥) ≠ (𝐹‘𝑦))
32	19, 20, 23, 25, 31, 1	dprdcntz 19943	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → (𝑆‘(𝐹‘𝑥)) ⊆ ((Cntz‘𝐺)‘(𝑆‘(𝐹‘𝑦))))
33		fvco3 6934	. . . . 5 ⊢ ((𝐹:𝐽⟶𝐼 ∧ 𝑥 ∈ 𝐽) → ((𝑆 ∘ 𝐹)‘𝑥) = (𝑆‘(𝐹‘𝑥)))
34	21, 22, 33	syl2anc 585	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ((𝑆 ∘ 𝐹)‘𝑥) = (𝑆‘(𝐹‘𝑥)))
35		fvco3 6934	. . . . . 6 ⊢ ((𝐹:𝐽⟶𝐼 ∧ 𝑦 ∈ 𝐽) → ((𝑆 ∘ 𝐹)‘𝑦) = (𝑆‘(𝐹‘𝑦)))
36	21, 24, 35	syl2anc 585	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ((𝑆 ∘ 𝐹)‘𝑦) = (𝑆‘(𝐹‘𝑦)))
37	36	fveq2d 6839	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ((Cntz‘𝐺)‘((𝑆 ∘ 𝐹)‘𝑦)) = ((Cntz‘𝐺)‘(𝑆‘(𝐹‘𝑦))))
38	32, 34, 37	3sstr4d 3990	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ((𝑆 ∘ 𝐹)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆 ∘ 𝐹)‘𝑦)))
39	16, 33	sylan 581	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ((𝑆 ∘ 𝐹)‘𝑥) = (𝑆‘(𝐹‘𝑥)))
40		imaco 6210	. . . . . . . . 9 ⊢ ((𝑆 ∘ 𝐹) “ (𝐽 ∖ {𝑥})) = (𝑆 “ (𝐹 “ (𝐽 ∖ {𝑥})))
41	7	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝐹:𝐽–1-1-onto→𝐼)
42		dff1o3 6781	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐽–1-1-onto→𝐼 ↔ (𝐹:𝐽–onto→𝐼 ∧ Fun ^◡𝐹))
43	42	simprbi 496	. . . . . . . . . . . 12 ⊢ (𝐹:𝐽–1-1-onto→𝐼 → Fun ^◡𝐹)
44		imadif 6577	. . . . . . . . . . . 12 ⊢ (Fun ^◡𝐹 → (𝐹 “ (𝐽 ∖ {𝑥})) = ((𝐹 “ 𝐽) ∖ (𝐹 “ {𝑥})))
45	41, 43, 44	3syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ (𝐽 ∖ {𝑥})) = ((𝐹 “ 𝐽) ∖ (𝐹 “ {𝑥})))
46		f1ofo 6782	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐽–1-1-onto→𝐼 → 𝐹:𝐽–onto→𝐼)
47		foima 6752	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐽–onto→𝐼 → (𝐹 “ 𝐽) = 𝐼)
48	41, 46, 47	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝐽) = 𝐼)
49		f1ofn 6776	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝐽–1-1-onto→𝐼 → 𝐹 Fn 𝐽)
50	7, 49	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 Fn 𝐽)
51		fnsnfv 6914	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn 𝐽 ∧ 𝑥 ∈ 𝐽) → {(𝐹‘𝑥)} = (𝐹 “ {𝑥}))
52	50, 51	sylan 581	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → {(𝐹‘𝑥)} = (𝐹 “ {𝑥}))
53	52	eqcomd 2743	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ {𝑥}) = {(𝐹‘𝑥)})
54	48, 53	difeq12d 4080	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ((𝐹 “ 𝐽) ∖ (𝐹 “ {𝑥})) = (𝐼 ∖ {(𝐹‘𝑥)}))
55	45, 54	eqtrd 2772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ (𝐽 ∖ {𝑥})) = (𝐼 ∖ {(𝐹‘𝑥)}))
56	55	imaeq2d 6020	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝑆 “ (𝐹 “ (𝐽 ∖ {𝑥}))) = (𝑆 “ (𝐼 ∖ {(𝐹‘𝑥)})))
57	40, 56	eqtrid 2784	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ((𝑆 ∘ 𝐹) “ (𝐽 ∖ {𝑥})) = (𝑆 “ (𝐼 ∖ {(𝐹‘𝑥)})))
58	57	unieqd 4877	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ∪ ((𝑆 ∘ 𝐹) “ (𝐽 ∖ {𝑥})) = ∪ (𝑆 “ (𝐼 ∖ {(𝐹‘𝑥)})))
59	58	fveq2d 6839	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ∘ 𝐹) “ (𝐽 ∖ {𝑥}))) = ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {(𝐹‘𝑥)}))))
60	39, 59	ineq12d 4174	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (((𝑆 ∘ 𝐹)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ∘ 𝐹) “ (𝐽 ∖ {𝑥})))) = ((𝑆‘(𝐹‘𝑥)) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {(𝐹‘𝑥)})))))
61	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝐺dom DProd 𝑆)
62	10	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → dom 𝑆 = 𝐼)
63	16	ffvelcdmda 7031	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹‘𝑥) ∈ 𝐼)
64	61, 62, 63, 2, 3	dprddisj 19944	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ((𝑆‘(𝐹‘𝑥)) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {(𝐹‘𝑥)})))) = {(0_g‘𝐺)})
65	60, 64	eqtrd 2772	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (((𝑆 ∘ 𝐹)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ∘ 𝐹) “ (𝐽 ∖ {𝑥})))) = {(0_g‘𝐺)})
66		eqimss 3993	. . . 4 ⊢ ((((𝑆 ∘ 𝐹)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ∘ 𝐹) “ (𝐽 ∖ {𝑥})))) = {(0_g‘𝐺)} → (((𝑆 ∘ 𝐹)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ∘ 𝐹) “ (𝐽 ∖ {𝑥})))) ⊆ {(0_g‘𝐺)})
67	65, 66	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (((𝑆 ∘ 𝐹)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ∘ 𝐹) “ (𝐽 ∖ {𝑥})))) ⊆ {(0_g‘𝐺)})
68	1, 2, 3, 6, 13, 18, 38, 67	dmdprdd 19934	. 2 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ∘ 𝐹))
69		rnco2 6213	. . . . . 6 ⊢ ran (𝑆 ∘ 𝐹) = (𝑆 “ ran 𝐹)
70		forn 6750	. . . . . . . . 9 ⊢ (𝐹:𝐽–onto→𝐼 → ran 𝐹 = 𝐼)
71	7, 46, 70	3syl 18	. . . . . . . 8 ⊢ (𝜑 → ran 𝐹 = 𝐼)
72	71	imaeq2d 6020	. . . . . . 7 ⊢ (𝜑 → (𝑆 “ ran 𝐹) = (𝑆 “ 𝐼))
73		ffn 6663	. . . . . . . 8 ⊢ (𝑆:𝐼⟶(SubGrp‘𝐺) → 𝑆 Fn 𝐼)
74		fnima 6623	. . . . . . . 8 ⊢ (𝑆 Fn 𝐼 → (𝑆 “ 𝐼) = ran 𝑆)
75	14, 73, 74	3syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑆 “ 𝐼) = ran 𝑆)
76	72, 75	eqtrd 2772	. . . . . 6 ⊢ (𝜑 → (𝑆 “ ran 𝐹) = ran 𝑆)
77	69, 76	eqtrid 2784	. . . . 5 ⊢ (𝜑 → ran (𝑆 ∘ 𝐹) = ran 𝑆)
78	77	unieqd 4877	. . . 4 ⊢ (𝜑 → ∪ ran (𝑆 ∘ 𝐹) = ∪ ran 𝑆)
79	78	fveq2d 6839	. . 3 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘∪ ran (𝑆 ∘ 𝐹)) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆))
80	3	dprdspan 19962	. . . 4 ⊢ (𝐺dom DProd (𝑆 ∘ 𝐹) → (𝐺 DProd (𝑆 ∘ 𝐹)) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran (𝑆 ∘ 𝐹)))
81	68, 80	syl 17	. . 3 ⊢ (𝜑 → (𝐺 DProd (𝑆 ∘ 𝐹)) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran (𝑆 ∘ 𝐹)))
82	3	dprdspan 19962	. . . 4 ⊢ (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆))
83	4, 82	syl 17	. . 3 ⊢ (𝜑 → (𝐺 DProd 𝑆) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆))
84	79, 81, 83	3eqtr4d 2782	. 2 ⊢ (𝜑 → (𝐺 DProd (𝑆 ∘ 𝐹)) = (𝐺 DProd 𝑆))
85	68, 84	jca 511	1 ⊢ (𝜑 → (𝐺dom DProd (𝑆 ∘ 𝐹) ∧ (𝐺 DProd (𝑆 ∘ 𝐹)) = (𝐺 DProd 𝑆)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 Vcvv 3441 ∖ cdif 3899 ∩ cin 3901 ⊆ wss 3902 {csn 4581 ∪ cuni 4864 class class class wbr 5099 ^◡ccnv 5624 dom cdm 5625 ran crn 5626 “ cima 5628 ∘ ccom 5629 Fun wfun 6487 Fn wfn 6488 ⟶wf 6489 –1-1→wf1 6490 –onto→wfo 6491 –1-1-onto→wf1o 6492 ‘cfv 6493 (class class class)co 7360 0_gc0g 17363 mrClscmrc 17506 Grpcgrp 18867 SubGrpcsubg 19054 Cntzccntz 19248 DProd cdprd 19928

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-iin 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-tpos 8170 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-oi 9419 df-card 9855 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12150 df-2 12212 df-n0 12406 df-z 12493 df-uz 12756 df-fz 13428 df-fzo 13575 df-seq 13929 df-hash 14258 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17141 df-ress 17162 df-plusg 17194 df-0g 17365 df-gsum 17366 df-mre 17509 df-mrc 17510 df-acs 17512 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-mhm 18712 df-submnd 18713 df-grp 18870 df-minusg 18871 df-sbg 18872 df-mulg 19002 df-subg 19057 df-ghm 19146 df-gim 19192 df-cntz 19250 df-oppg 19279 df-cmn 19715 df-dprd 19930

This theorem is referenced by: dprdf1 19968 ablfaclem2 20021

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