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Theorem dprdf1o 19156

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 2823	. . 3 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
2		eqid 2823	. . 3 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
3		eqid 2823	. . 3 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
4		dprdf1o.1	. . . 4 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
5		dprdgrp 19129	. . . 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 6616	. . . . 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 19125	. . . 4 ⊢ (𝜑 → 𝐼 ∈ V)
12		f1dmex 7660	. . . 4 ⊢ ((𝐹:𝐽–1-1→𝐼 ∧ 𝐼 ∈ V) → 𝐽 ∈ V)
13	9, 11, 12	syl2anc 586	. . 3 ⊢ (𝜑 → 𝐽 ∈ V)
14	4, 10	dprdf2 19131	. . . 4 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
15		f1of 6617	. . . . 5 ⊢ (𝐹:𝐽–1-1-onto→𝐼 → 𝐹:𝐽⟶𝐼)
16	7, 15	syl 17	. . . 4 ⊢ (𝜑 → 𝐹:𝐽⟶𝐼)
17		fco 6533	. . . 4 ⊢ ((𝑆:𝐼⟶(SubGrp‘𝐺) ∧ 𝐹:𝐽⟶𝐼) → (𝑆 ∘ 𝐹):𝐽⟶(SubGrp‘𝐺))
18	14, 16, 17	syl2anc 586	. . 3 ⊢ (𝜑 → (𝑆 ∘ 𝐹):𝐽⟶(SubGrp‘𝐺))
19	4	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → 𝐺dom DProd 𝑆)
20	10	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → dom 𝑆 = 𝐼)
21	16	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → 𝐹:𝐽⟶𝐼)
22		simpr1 1190	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ 𝐽)
23	21, 22	ffvelrnd 6854	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → (𝐹‘𝑥) ∈ 𝐼)
24		simpr2 1191	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ 𝐽)
25	21, 24	ffvelrnd 6854	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → (𝐹‘𝑦) ∈ 𝐼)
26		simpr3 1192	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦)
27	9	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → 𝐹:𝐽–1-1→𝐼)
28		f1fveq 7022	. . . . . . . 8 ⊢ ((𝐹:𝐽–1-1→𝐼 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ 𝑥 = 𝑦))
29	27, 22, 24, 28	syl12anc 834	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ 𝑥 = 𝑦))
30	29	necon3bid 3062	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ((𝐹‘𝑥) ≠ (𝐹‘𝑦) ↔ 𝑥 ≠ 𝑦))
31	26, 30	mpbird 259	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → (𝐹‘𝑥) ≠ (𝐹‘𝑦))
32	19, 20, 23, 25, 31, 1	dprdcntz 19132	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → (𝑆‘(𝐹‘𝑥)) ⊆ ((Cntz‘𝐺)‘(𝑆‘(𝐹‘𝑦))))
33		fvco3 6762	. . . . 5 ⊢ ((𝐹:𝐽⟶𝐼 ∧ 𝑥 ∈ 𝐽) → ((𝑆 ∘ 𝐹)‘𝑥) = (𝑆‘(𝐹‘𝑥)))
34	21, 22, 33	syl2anc 586	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ((𝑆 ∘ 𝐹)‘𝑥) = (𝑆‘(𝐹‘𝑥)))
35		fvco3 6762	. . . . . 6 ⊢ ((𝐹:𝐽⟶𝐼 ∧ 𝑦 ∈ 𝐽) → ((𝑆 ∘ 𝐹)‘𝑦) = (𝑆‘(𝐹‘𝑦)))
36	21, 24, 35	syl2anc 586	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ((𝑆 ∘ 𝐹)‘𝑦) = (𝑆‘(𝐹‘𝑦)))
37	36	fveq2d 6676	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ((Cntz‘𝐺)‘((𝑆 ∘ 𝐹)‘𝑦)) = ((Cntz‘𝐺)‘(𝑆‘(𝐹‘𝑦))))
38	32, 34, 37	3sstr4d 4016	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ((𝑆 ∘ 𝐹)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆 ∘ 𝐹)‘𝑦)))
39	16, 33	sylan 582	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ((𝑆 ∘ 𝐹)‘𝑥) = (𝑆‘(𝐹‘𝑥)))
40		imaco 6106	. . . . . . . . 9 ⊢ ((𝑆 ∘ 𝐹) “ (𝐽 ∖ {𝑥})) = (𝑆 “ (𝐹 “ (𝐽 ∖ {𝑥})))
41	7	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝐹:𝐽–1-1-onto→𝐼)
42		dff1o3 6623	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐽–1-1-onto→𝐼 ↔ (𝐹:𝐽–onto→𝐼 ∧ Fun ^◡𝐹))
43	42	simprbi 499	. . . . . . . . . . . 12 ⊢ (𝐹:𝐽–1-1-onto→𝐼 → Fun ^◡𝐹)
44		imadif 6440	. . . . . . . . . . . 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 6624	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐽–1-1-onto→𝐼 → 𝐹:𝐽–onto→𝐼)
47		foima 6597	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐽–onto→𝐼 → (𝐹 “ 𝐽) = 𝐼)
48	41, 46, 47	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝐽) = 𝐼)
49		f1ofn 6618	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝐽–1-1-onto→𝐼 → 𝐹 Fn 𝐽)
50	7, 49	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 Fn 𝐽)
51		fnsnfv 6745	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn 𝐽 ∧ 𝑥 ∈ 𝐽) → {(𝐹‘𝑥)} = (𝐹 “ {𝑥}))
52	50, 51	sylan 582	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → {(𝐹‘𝑥)} = (𝐹 “ {𝑥}))
53	52	eqcomd 2829	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ {𝑥}) = {(𝐹‘𝑥)})
54	48, 53	difeq12d 4102	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ((𝐹 “ 𝐽) ∖ (𝐹 “ {𝑥})) = (𝐼 ∖ {(𝐹‘𝑥)}))
55	45, 54	eqtrd 2858	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ (𝐽 ∖ {𝑥})) = (𝐼 ∖ {(𝐹‘𝑥)}))
56	55	imaeq2d 5931	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝑆 “ (𝐹 “ (𝐽 ∖ {𝑥}))) = (𝑆 “ (𝐼 ∖ {(𝐹‘𝑥)})))
57	40, 56	syl5eq 2870	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ((𝑆 ∘ 𝐹) “ (𝐽 ∖ {𝑥})) = (𝑆 “ (𝐼 ∖ {(𝐹‘𝑥)})))
58	57	unieqd 4854	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ∪ ((𝑆 ∘ 𝐹) “ (𝐽 ∖ {𝑥})) = ∪ (𝑆 “ (𝐼 ∖ {(𝐹‘𝑥)})))
59	58	fveq2d 6676	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ∘ 𝐹) “ (𝐽 ∖ {𝑥}))) = ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {(𝐹‘𝑥)}))))
60	39, 59	ineq12d 4192	. . . . 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 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝐺dom DProd 𝑆)
62	10	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → dom 𝑆 = 𝐼)
63	16	ffvelrnda 6853	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹‘𝑥) ∈ 𝐼)
64	61, 62, 63, 2, 3	dprddisj 19133	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ((𝑆‘(𝐹‘𝑥)) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {(𝐹‘𝑥)})))) = {(0_g‘𝐺)})
65	60, 64	eqtrd 2858	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (((𝑆 ∘ 𝐹)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆 ∘ 𝐹) “ (𝐽 ∖ {𝑥})))) = {(0_g‘𝐺)})
66		eqimss 4025	. . . 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 19123	. 2 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ∘ 𝐹))
69		rnco2 6108	. . . . . 6 ⊢ ran (𝑆 ∘ 𝐹) = (𝑆 “ ran 𝐹)
70		forn 6595	. . . . . . . . 9 ⊢ (𝐹:𝐽–onto→𝐼 → ran 𝐹 = 𝐼)
71	7, 46, 70	3syl 18	. . . . . . . 8 ⊢ (𝜑 → ran 𝐹 = 𝐼)
72	71	imaeq2d 5931	. . . . . . 7 ⊢ (𝜑 → (𝑆 “ ran 𝐹) = (𝑆 “ 𝐼))
73		ffn 6516	. . . . . . . 8 ⊢ (𝑆:𝐼⟶(SubGrp‘𝐺) → 𝑆 Fn 𝐼)
74		fnima 6480	. . . . . . . 8 ⊢ (𝑆 Fn 𝐼 → (𝑆 “ 𝐼) = ran 𝑆)
75	14, 73, 74	3syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑆 “ 𝐼) = ran 𝑆)
76	72, 75	eqtrd 2858	. . . . . 6 ⊢ (𝜑 → (𝑆 “ ran 𝐹) = ran 𝑆)
77	69, 76	syl5eq 2870	. . . . 5 ⊢ (𝜑 → ran (𝑆 ∘ 𝐹) = ran 𝑆)
78	77	unieqd 4854	. . . 4 ⊢ (𝜑 → ∪ ran (𝑆 ∘ 𝐹) = ∪ ran 𝑆)
79	78	fveq2d 6676	. . 3 ⊢ (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘∪ ran (𝑆 ∘ 𝐹)) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆))
80	3	dprdspan 19151	. . . 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 19151	. . . 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 2868	. 2 ⊢ (𝜑 → (𝐺 DProd (𝑆 ∘ 𝐹)) = (𝐺 DProd 𝑆))
85	68, 84	jca 514	1 ⊢ (𝜑 → (𝐺dom DProd (𝑆 ∘ 𝐹) ∧ (𝐺 DProd (𝑆 ∘ 𝐹)) = (𝐺 DProd 𝑆)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 Vcvv 3496 ∖ cdif 3935 ∩ cin 3937 ⊆ wss 3938 {csn 4569 ∪ cuni 4840 class class class wbr 5068 ^◡ccnv 5556 dom cdm 5557 ran crn 5558 “ cima 5560 ∘ ccom 5561 Fun wfun 6351 Fn wfn 6352 ⟶wf 6353 –1-1→wf1 6354 –onto→wfo 6355 –1-1-onto→wf1o 6356 ‘cfv 6357 (class class class)co 7158 0_gc0g 16715 mrClscmrc 16856 Grpcgrp 18105 SubGrpcsubg 18275 Cntzccntz 18447 DProd cdprd 19117

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-0g 16717 df-gsum 16718 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-subg 18278 df-ghm 18358 df-gim 18401 df-cntz 18449 df-oppg 18476 df-cmn 18910 df-dprd 19119

This theorem is referenced by: dprdf1 19157 ablfaclem2 19210

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