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Theorem dprdf1o 20228

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 2761	. . 3 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
2		eqid 2761	. . 3 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
3		eqid 2761	. . 3 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
4		dprdf1o.1	. . . 4 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
5		dprdgrp 20201	. . . 4 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp)
6	4, 5	syl 18	. . 3 ⊢ (𝜑 → 𝐺 ∈ Grp)
7		dprdf1o.3	. . . . 5 ⊢ (𝜑 → 𝐹:𝐽–1-1-onto→𝐼)
8		f1of1 6815	. . . . 5 ⊢ (𝐹:𝐽–1-1-onto→𝐼 → 𝐹:𝐽–1-1→𝐼)
9	7, 8	syl 18	. . . 4 ⊢ (𝜑 → 𝐹:𝐽–1-1→𝐼)
10		dprdf1o.2	. . . . 5 ⊢ (𝜑 → dom 𝑆 = 𝐼)
11	4, 10	dprddomcld 20197	. . . 4 ⊢ (𝜑 → 𝐼 ∈ V)
12		f1dmex 7958	. . . 4 ⊢ ((𝐹:𝐽–1-1→𝐼 ∧ 𝐼 ∈ V) → 𝐽 ∈ V)
13	9, 11, 12	syl2anc 596	. . 3 ⊢ (𝜑 → 𝐽 ∈ V)
14	4, 10	dprdf2 20203	. . . 4 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
15		f1of 6816	. . . . 5 ⊢ (𝐹:𝐽–1-1-onto→𝐼 → 𝐹:𝐽⟶𝐼)
16	7, 15	syl 18	. . . 4 ⊢ (𝜑 → 𝐹:𝐽⟶𝐼)
17		fco 6726	. . . 4 ⊢ ((𝑆:𝐼⟶(SubGrp‘𝐺) ∧ 𝐹:𝐽⟶𝐼) → (𝑆 ∘ 𝐹):𝐽⟶(SubGrp‘𝐺))
18	14, 16, 17	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑆 ∘ 𝐹):𝐽⟶(SubGrp‘𝐺))
19	4	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → 𝐺dom DProd 𝑆)
20	10	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → dom 𝑆 = 𝐼)
21	16	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → 𝐹:𝐽⟶𝐼)
22		simpr1 1213	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ 𝐽)
23	21, 22	ffvelcdmd 7077	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → (𝐹‘𝑥) ∈ 𝐼)
24		simpr2 1214	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ 𝐽)
25	21, 24	ffvelcdmd 7077	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → (𝐹‘𝑦) ∈ 𝐼)
26		simpr3 1215	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦)
27	9	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → 𝐹:𝐽–1-1→𝐼)
28		f1fveq 7258	. . . . . . . 8 ⊢ ((𝐹:𝐽–1-1→𝐼 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ 𝑥 = 𝑦))
29	27, 22, 24, 28	syl12anc 850	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ 𝑥 = 𝑦))
30	29	necon3bid 3000	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ((𝐹‘𝑥) ≠ (𝐹‘𝑦) ↔ 𝑥 ≠ 𝑦))
31	26, 30	mpbird 260	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → (𝐹‘𝑥) ≠ (𝐹‘𝑦))
32	19, 20, 23, 25, 31, 1	dprdcntz 20204	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → (𝑆‘(𝐹‘𝑥)) ⊆ ((Cntz‘𝐺)‘(𝑆‘(𝐹‘𝑦))))
33		fvco3 6977	. . . . 5 ⊢ ((𝐹:𝐽⟶𝐼 ∧ 𝑥 ∈ 𝐽) → ((𝑆 ∘ 𝐹)‘𝑥) = (𝑆‘(𝐹‘𝑥)))
34	21, 22, 33	syl2anc 596	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ((𝑆 ∘ 𝐹)‘𝑥) = (𝑆‘(𝐹‘𝑥)))
35		fvco3 6977	. . . . . 6 ⊢ ((𝐹:𝐽⟶𝐼 ∧ 𝑦 ∈ 𝐽) → ((𝑆 ∘ 𝐹)‘𝑦) = (𝑆‘(𝐹‘𝑦)))
36	21, 24, 35	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ((𝑆 ∘ 𝐹)‘𝑦) = (𝑆‘(𝐹‘𝑦)))
37	36	fveq2d 6881	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ((Cntz‘𝐺)‘((𝑆 ∘ 𝐹)‘𝑦)) = ((Cntz‘𝐺)‘(𝑆‘(𝐹‘𝑦))))
38	32, 34, 37	3sstr4d 3986	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ((𝑆 ∘ 𝐹)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆 ∘ 𝐹)‘𝑦)))
39	16, 33	sylan 592	. . . . . 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 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝐹:𝐽–1-1-onto→𝐼)
42		dff1o3 6823	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐽–1-1-onto→𝐼 ↔ (𝐹:𝐽–onto→𝐼 ∧ Fun ^◡𝐹))
43	42	simprbi 503	. . . . . . . . . . . 12 ⊢ (𝐹:𝐽–1-1-onto→𝐼 → Fun ^◡𝐹)
44		imadif 6616	. . . . . . . . . . . 12 ⊢ (Fun ^◡𝐹 → (𝐹 “ (𝐽 ∖ {𝑥})) = ((𝐹 “ 𝐽) ∖ (𝐹 “ {𝑥})))
45	41, 43, 44	3syl 19	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ (𝐽 ∖ {𝑥})) = ((𝐹 “ 𝐽) ∖ (𝐹 “ {𝑥})))
46		f1ofo 6824	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐽–1-1-onto→𝐼 → 𝐹:𝐽–onto→𝐼)
47		foima 6793	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐽–onto→𝐼 → (𝐹 “ 𝐽) = 𝐼)
48	41, 46, 47	3syl 19	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝐽) = 𝐼)
49		f1ofn 6817	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝐽–1-1-onto→𝐼 → 𝐹 Fn 𝐽)
50	7, 49	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 Fn 𝐽)
51		fnsnfv 6956	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn 𝐽 ∧ 𝑥 ∈ 𝐽) → {(𝐹‘𝑥)} = (𝐹 “ {𝑥}))
52	50, 51	sylan 592	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → {(𝐹‘𝑥)} = (𝐹 “ {𝑥}))
53	52	eqcomd 2767	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ {𝑥}) = {(𝐹‘𝑥)})
54	48, 53	difeq12d 4075	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ((𝐹 “ 𝐽) ∖ (𝐹 “ {𝑥})) = (𝐼 ∖ {(𝐹‘𝑥)}))
55	45, 54	eqtrd 2796	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ (𝐽 ∖ {𝑥})) = (𝐼 ∖ {(𝐹‘𝑥)}))
56	55	imaeq2d 6054	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝑆 “ (𝐹 “ (𝐽 ∖ {𝑥}))) = (𝑆 “ (𝐼 ∖ {(𝐹‘𝑥)})))
57	40, 56	eqtrid 2808	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ((𝑆 ∘ 𝐹) “ (𝐽 ∖ {𝑥})) = (𝑆 “ (𝐼 ∖ {(𝐹‘𝑥)})))
58	57	unieqd 4880	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ∪ ((𝑆 ∘ 𝐹) “ (𝐽 ∖ {𝑥})) = ∪ (𝑆 “ (𝐼 ∖ {(𝐹‘𝑥)})))
59	58	fveq2d 6881	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ∘ 𝐹) “ (𝐽 ∖ {𝑥}))) = ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {(𝐹‘𝑥)}))))
60	39, 59	ineq12d 4167	. . . . 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 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝐺dom DProd 𝑆)
62	10	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → dom 𝑆 = 𝐼)
63	16	ffvelcdmda 7076	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹‘𝑥) ∈ 𝐼)
64	61, 62, 63, 2, 3	dprddisj 20205	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ((𝑆‘(𝐹‘𝑥)) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {(𝐹‘𝑥)})))) = {(0_g‘𝐺)})
65	60, 64	eqtrd 2796	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (((𝑆 ∘ 𝐹)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ∘ 𝐹) “ (𝐽 ∖ {𝑥})))) = {(0_g‘𝐺)})
66		eqimss 3989	. . . 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 18	. . 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 20195	. 2 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ∘ 𝐹))
69		rnco2 6248	. . . . . 6 ⊢ ran (𝑆 ∘ 𝐹) = (𝑆 “ ran 𝐹)
70		forn 6791	. . . . . . . . 9 ⊢ (𝐹:𝐽–onto→𝐼 → ran 𝐹 = 𝐼)
71	7, 46, 70	3syl 19	. . . . . . . 8 ⊢ (𝜑 → ran 𝐹 = 𝐼)
72	71	imaeq2d 6054	. . . . . . 7 ⊢ (𝜑 → (𝑆 “ ran 𝐹) = (𝑆 “ 𝐼))
73		ffn 6701	. . . . . . . 8 ⊢ (𝑆:𝐼⟶(SubGrp‘𝐺) → 𝑆 Fn 𝐼)
74		fnima 6661	. . . . . . . 8 ⊢ (𝑆 Fn 𝐼 → (𝑆 “ 𝐼) = ran 𝑆)
75	14, 73, 74	3syl 19	. . . . . . 7 ⊢ (𝜑 → (𝑆 “ 𝐼) = ran 𝑆)
76	72, 75	eqtrd 2796	. . . . . 6 ⊢ (𝜑 → (𝑆 “ ran 𝐹) = ran 𝑆)
77	69, 76	eqtrid 2808	. . . . 5 ⊢ (𝜑 → ran (𝑆 ∘ 𝐹) = ran 𝑆)
78	77	unieqd 4880	. . . 4 ⊢ (𝜑 → ∪ ran (𝑆 ∘ 𝐹) = ∪ ran 𝑆)
79	78	fveq2d 6881	. . 3 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘∪ ran (𝑆 ∘ 𝐹)) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆))
80	3	dprdspan 20223	. . . 4 ⊢ (𝐺dom DProd (𝑆 ∘ 𝐹) → (𝐺 DProd (𝑆 ∘ 𝐹)) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran (𝑆 ∘ 𝐹)))
81	68, 80	syl 18	. . 3 ⊢ (𝜑 → (𝐺 DProd (𝑆 ∘ 𝐹)) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran (𝑆 ∘ 𝐹)))
82	3	dprdspan 20223	. . . 4 ⊢ (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆))
83	4, 82	syl 18	. . 3 ⊢ (𝜑 → (𝐺 DProd 𝑆) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆))
84	79, 81, 83	3eqtr4d 2806	. 2 ⊢ (𝜑 → (𝐺 DProd (𝑆 ∘ 𝐹)) = (𝐺 DProd 𝑆))
85	68, 84	jca 521	1 ⊢ (𝜑 → (𝐺dom DProd (𝑆 ∘ 𝐹) ∧ (𝐺 DProd (𝑆 ∘ 𝐹)) = (𝐺 DProd 𝑆)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 Vcvv 3451 ∖ cdif 3896 ∩ cin 3898 ⊆ wss 3899 {csn 4584 ∪ cuni 4867 class class class wbr 5103 ^◡ccnv 5650 dom cdm 5651 ran crn 5652 “ cima 5654 ∘ ccom 5655 Fun wfun 6525 Fn wfn 6526 ⟶wf 6527 –1-1→wf1 6528 –onto→wfo 6529 –1-1-onto→wf1o 6530 ‘cfv 6531 (class class class)co 7412 0_gc0g 17590 mrClscmrc 17733 Grpcgrp 19124 SubGrpcsubg 19310 Cntzccntz 19509 DProd cdprd 20189

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7682 df-om 7867 df-1st 7990 df-2nd 7991 df-supp 8162 df-tpos 8227 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-2o 8461 df-er 8701 df-map 8833 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9338 df-oi 9488 df-card 10001 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-nn 12317 df-2 12386 df-n0 12588 df-z 12675 df-uz 12947 df-fz 13621 df-fzo 13769 df-seq 14125 df-hash 14455 df-sets 17322 df-slot 17340 df-ndx 17352 df-base 17368 df-ress 17389 df-plusg 17421 df-0g 17592 df-gsum 17593 df-mre 17736 df-mrc 17737 df-acs 17739 df-mgm 18796 df-sgrp 18888 df-mnd 18904 df-mhm 18958 df-submnd 18959 df-grp 19127 df-minusg 19128 df-sbg 19129 df-mulg 19258 df-subg 19313 df-ghm 19408 df-gim 19453 df-cntz 19511 df-oppg 19540 df-cmn 19976 df-dprd 20191

This theorem is used by: dprdf1 20229 ablfaclem2 20282

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