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Theorem dprdf1o 20020

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 2736	. . 3 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
2		eqid 2736	. . 3 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
3		eqid 2736	. . 3 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
4		dprdf1o.1	. . . 4 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
5		dprdgrp 19993	. . . 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 6822	. . . . 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 19989	. . . 4 ⊢ (𝜑 → 𝐼 ∈ V)
12		f1dmex 7960	. . . 4 ⊢ ((𝐹:𝐽–1-1→𝐼 ∧ 𝐼 ∈ V) → 𝐽 ∈ V)
13	9, 11, 12	syl2anc 584	. . 3 ⊢ (𝜑 → 𝐽 ∈ V)
14	4, 10	dprdf2 19995	. . . 4 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
15		f1of 6823	. . . . 5 ⊢ (𝐹:𝐽–1-1-onto→𝐼 → 𝐹:𝐽⟶𝐼)
16	7, 15	syl 17	. . . 4 ⊢ (𝜑 → 𝐹:𝐽⟶𝐼)
17		fco 6735	. . . 4 ⊢ ((𝑆:𝐼⟶(SubGrp‘𝐺) ∧ 𝐹:𝐽⟶𝐼) → (𝑆 ∘ 𝐹):𝐽⟶(SubGrp‘𝐺))
18	14, 16, 17	syl2anc 584	. . 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 1195	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ 𝐽)
23	21, 22	ffvelcdmd 7080	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → (𝐹‘𝑥) ∈ 𝐼)
24		simpr2 1196	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ 𝐽)
25	21, 24	ffvelcdmd 7080	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → (𝐹‘𝑦) ∈ 𝐼)
26		simpr3 1197	. . . . . 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 7260	. . . . . . . 8 ⊢ ((𝐹:𝐽–1-1→𝐼 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ 𝑥 = 𝑦))
29	27, 22, 24, 28	syl12anc 836	. . . . . . 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 19996	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → (𝑆‘(𝐹‘𝑥)) ⊆ ((Cntz‘𝐺)‘(𝑆‘(𝐹‘𝑦))))
33		fvco3 6983	. . . . 5 ⊢ ((𝐹:𝐽⟶𝐼 ∧ 𝑥 ∈ 𝐽) → ((𝑆 ∘ 𝐹)‘𝑥) = (𝑆‘(𝐹‘𝑥)))
34	21, 22, 33	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ((𝑆 ∘ 𝐹)‘𝑥) = (𝑆‘(𝐹‘𝑥)))
35		fvco3 6983	. . . . . 6 ⊢ ((𝐹:𝐽⟶𝐼 ∧ 𝑦 ∈ 𝐽) → ((𝑆 ∘ 𝐹)‘𝑦) = (𝑆‘(𝐹‘𝑦)))
36	21, 24, 35	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ((𝑆 ∘ 𝐹)‘𝑦) = (𝑆‘(𝐹‘𝑦)))
37	36	fveq2d 6885	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ((Cntz‘𝐺)‘((𝑆 ∘ 𝐹)‘𝑦)) = ((Cntz‘𝐺)‘(𝑆‘(𝐹‘𝑦))))
38	32, 34, 37	3sstr4d 4019	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ((𝑆 ∘ 𝐹)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆 ∘ 𝐹)‘𝑦)))
39	16, 33	sylan 580	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ((𝑆 ∘ 𝐹)‘𝑥) = (𝑆‘(𝐹‘𝑥)))
40		imaco 6245	. . . . . . . . 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 6829	. . . . . . . . . . . . 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 6625	. . . . . . . . . . . 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 6830	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐽–1-1-onto→𝐼 → 𝐹:𝐽–onto→𝐼)
47		foima 6800	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐽–onto→𝐼 → (𝐹 “ 𝐽) = 𝐼)
48	41, 46, 47	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝐽) = 𝐼)
49		f1ofn 6824	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝐽–1-1-onto→𝐼 → 𝐹 Fn 𝐽)
50	7, 49	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 Fn 𝐽)
51		fnsnfv 6963	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn 𝐽 ∧ 𝑥 ∈ 𝐽) → {(𝐹‘𝑥)} = (𝐹 “ {𝑥}))
52	50, 51	sylan 580	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → {(𝐹‘𝑥)} = (𝐹 “ {𝑥}))
53	52	eqcomd 2742	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ {𝑥}) = {(𝐹‘𝑥)})
54	48, 53	difeq12d 4107	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ((𝐹 “ 𝐽) ∖ (𝐹 “ {𝑥})) = (𝐼 ∖ {(𝐹‘𝑥)}))
55	45, 54	eqtrd 2771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ (𝐽 ∖ {𝑥})) = (𝐼 ∖ {(𝐹‘𝑥)}))
56	55	imaeq2d 6052	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝑆 “ (𝐹 “ (𝐽 ∖ {𝑥}))) = (𝑆 “ (𝐼 ∖ {(𝐹‘𝑥)})))
57	40, 56	eqtrid 2783	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ((𝑆 ∘ 𝐹) “ (𝐽 ∖ {𝑥})) = (𝑆 “ (𝐼 ∖ {(𝐹‘𝑥)})))
58	57	unieqd 4901	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ∪ ((𝑆 ∘ 𝐹) “ (𝐽 ∖ {𝑥})) = ∪ (𝑆 “ (𝐼 ∖ {(𝐹‘𝑥)})))
59	58	fveq2d 6885	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ∘ 𝐹) “ (𝐽 ∖ {𝑥}))) = ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {(𝐹‘𝑥)}))))
60	39, 59	ineq12d 4201	. . . . 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 7079	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹‘𝑥) ∈ 𝐼)
64	61, 62, 63, 2, 3	dprddisj 19997	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ((𝑆‘(𝐹‘𝑥)) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {(𝐹‘𝑥)})))) = {(0_g‘𝐺)})
65	60, 64	eqtrd 2771	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (((𝑆 ∘ 𝐹)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ∘ 𝐹) “ (𝐽 ∖ {𝑥})))) = {(0_g‘𝐺)})
66		eqimss 4022	. . . 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 19987	. 2 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ∘ 𝐹))
69		rnco2 6247	. . . . . 6 ⊢ ran (𝑆 ∘ 𝐹) = (𝑆 “ ran 𝐹)
70		forn 6798	. . . . . . . . 9 ⊢ (𝐹:𝐽–onto→𝐼 → ran 𝐹 = 𝐼)
71	7, 46, 70	3syl 18	. . . . . . . 8 ⊢ (𝜑 → ran 𝐹 = 𝐼)
72	71	imaeq2d 6052	. . . . . . 7 ⊢ (𝜑 → (𝑆 “ ran 𝐹) = (𝑆 “ 𝐼))
73		ffn 6711	. . . . . . . 8 ⊢ (𝑆:𝐼⟶(SubGrp‘𝐺) → 𝑆 Fn 𝐼)
74		fnima 6673	. . . . . . . 8 ⊢ (𝑆 Fn 𝐼 → (𝑆 “ 𝐼) = ran 𝑆)
75	14, 73, 74	3syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑆 “ 𝐼) = ran 𝑆)
76	72, 75	eqtrd 2771	. . . . . 6 ⊢ (𝜑 → (𝑆 “ ran 𝐹) = ran 𝑆)
77	69, 76	eqtrid 2783	. . . . 5 ⊢ (𝜑 → ran (𝑆 ∘ 𝐹) = ran 𝑆)
78	77	unieqd 4901	. . . 4 ⊢ (𝜑 → ∪ ran (𝑆 ∘ 𝐹) = ∪ ran 𝑆)
79	78	fveq2d 6885	. . 3 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘∪ ran (𝑆 ∘ 𝐹)) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆))
80	3	dprdspan 20015	. . . 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 20015	. . . 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 2781	. 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 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2933 Vcvv 3464 ∖ cdif 3928 ∩ cin 3930 ⊆ wss 3931 {csn 4606 ∪ cuni 4888 class class class wbr 5124 ^◡ccnv 5658 dom cdm 5659 ran crn 5660 “ cima 5662 ∘ ccom 5663 Fun wfun 6530 Fn wfn 6531 ⟶wf 6532 –1-1→wf1 6533 –onto→wfo 6534 –1-1-onto→wf1o 6535 ‘cfv 6536 (class class class)co 7410 0_gc0g 17458 mrClscmrc 17600 Grpcgrp 18921 SubGrpcsubg 19108 Cntzccntz 19303 DProd cdprd 19981

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7676 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8724 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9379 df-oi 9529 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-nn 12246 df-2 12308 df-n0 12507 df-z 12594 df-uz 12858 df-fz 13530 df-fzo 13677 df-seq 14025 df-hash 14354 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-0g 17460 df-gsum 17461 df-mre 17603 df-mrc 17604 df-acs 17606 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-mhm 18766 df-submnd 18767 df-grp 18924 df-minusg 18925 df-sbg 18926 df-mulg 19056 df-subg 19111 df-ghm 19201 df-gim 19247 df-cntz 19305 df-oppg 19334 df-cmn 19768 df-dprd 19983

This theorem is referenced by: dprdf1 20021 ablfaclem2 20074

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