Theorem lflnegcl 34362

Description: Closure of the negative of a functional. (This is specialized for the purpose of proving ldualgrp 34433, and we do not define a general operation here.) (Contributed by NM, 22-Oct-2014.)

Hypotheses

Ref	Expression
lflnegcl.v	⊢ 𝑉 = (Base‘𝑊)
lflnegcl.r	⊢ 𝑅 = (Scalar‘𝑊)
lflnegcl.i	⊢ 𝐼 = (invg‘𝑅)
lflnegcl.n	⊢ 𝑁 = (𝑥 ∈ 𝑉 ↦ (𝐼‘(𝐺‘𝑥)))
lflnegcl.f	⊢ 𝐹 = (LFnl‘𝑊)
lflnegcl.w	⊢ (𝜑 → 𝑊 ∈ LMod)
lflnegcl.g	⊢ (𝜑 → 𝐺 ∈ 𝐹)

Assertion

Ref	Expression
lflnegcl	⊢ (𝜑 → 𝑁 ∈ 𝐹)

Distinct variable groups:   𝑥,𝐺   𝑥,𝐼   𝑥,𝑅   𝑥,𝑉   𝑥,𝑊   𝜑,𝑥

Allowed substitution hints:   𝐹(𝑥)   𝑁(𝑥)

Proof of Theorem lflnegcl

Dummy variables 𝑦 𝑘 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lflnegcl.w	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LMod)
2		lflnegcl.r	. . . . . . . 8 ⊢ 𝑅 = (Scalar‘𝑊)
3	2	lmodring 18871	. . . . . . 7 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring)
4	1, 3	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Ring)
5		ringgrp 18552	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
6	4, 5	syl 17	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Grp)
7	6	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑅 ∈ Grp)
8	1	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑊 ∈ LMod)
9		lflnegcl.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
10	9	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝐺 ∈ 𝐹)
11		simpr 477	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ 𝑉)
12		eqid 2622	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
13		lflnegcl.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑊)
14		lflnegcl.f	. . . . . 6 ⊢ 𝐹 = (LFnl‘𝑊)
15	2, 12, 13, 14	lflcl 34351	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑥 ∈ 𝑉) → (𝐺‘𝑥) ∈ (Base‘𝑅))
16	8, 10, 11, 15	syl3anc 1326	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐺‘𝑥) ∈ (Base‘𝑅))
17		lflnegcl.i	. . . . 5 ⊢ 𝐼 = (invg‘𝑅)
18	12, 17	grpinvcl 17467	. . . 4 ⊢ ((𝑅 ∈ Grp ∧ (𝐺‘𝑥) ∈ (Base‘𝑅)) → (𝐼‘(𝐺‘𝑥)) ∈ (Base‘𝑅))
19	7, 16, 18	syl2anc 693	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐼‘(𝐺‘𝑥)) ∈ (Base‘𝑅))
20		lflnegcl.n	. . 3 ⊢ 𝑁 = (𝑥 ∈ 𝑉 ↦ (𝐼‘(𝐺‘𝑥)))
21	19, 20	fmptd 6385	. 2 ⊢ (𝜑 → 𝑁:𝑉⟶(Base‘𝑅))
22		ringabl 18580	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Abel)
23	4, 22	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Abel)
24	23	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑅 ∈ Abel)
25	4	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑅 ∈ Ring)
26		simpr1 1067	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑘 ∈ (Base‘𝑅))
27	1	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑊 ∈ LMod)
28	9	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝐺 ∈ 𝐹)
29		simpr2 1068	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑦 ∈ 𝑉)
30	2, 12, 13, 14	lflcl 34351	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑦 ∈ 𝑉) → (𝐺‘𝑦) ∈ (Base‘𝑅))
31	27, 28, 29, 30	syl3anc 1326	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐺‘𝑦) ∈ (Base‘𝑅))
32		eqid 2622	. . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅)
33	12, 32	ringcl 18561	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑘 ∈ (Base‘𝑅) ∧ (𝐺‘𝑦) ∈ (Base‘𝑅)) → (𝑘(.r‘𝑅)(𝐺‘𝑦)) ∈ (Base‘𝑅))
34	25, 26, 31, 33	syl3anc 1326	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑘(.r‘𝑅)(𝐺‘𝑦)) ∈ (Base‘𝑅))
35		simpr3 1069	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑧 ∈ 𝑉)
36	2, 12, 13, 14	lflcl 34351	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑧 ∈ 𝑉) → (𝐺‘𝑧) ∈ (Base‘𝑅))
37	27, 28, 35, 36	syl3anc 1326	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐺‘𝑧) ∈ (Base‘𝑅))
38		eqid 2622	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
39	12, 38, 17	ablinvadd 18215	. . . . . 6 ⊢ ((𝑅 ∈ Abel ∧ (𝑘(.r‘𝑅)(𝐺‘𝑦)) ∈ (Base‘𝑅) ∧ (𝐺‘𝑧) ∈ (Base‘𝑅)) → (𝐼‘((𝑘(.r‘𝑅)(𝐺‘𝑦))(+g‘𝑅)(𝐺‘𝑧))) = ((𝐼‘(𝑘(.r‘𝑅)(𝐺‘𝑦)))(+g‘𝑅)(𝐼‘(𝐺‘𝑧))))
40	24, 34, 37, 39	syl3anc 1326	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐼‘((𝑘(.r‘𝑅)(𝐺‘𝑦))(+g‘𝑅)(𝐺‘𝑧))) = ((𝐼‘(𝑘(.r‘𝑅)(𝐺‘𝑦)))(+g‘𝑅)(𝐼‘(𝐺‘𝑧))))
41		eqid 2622	. . . . . . . 8 ⊢ (+g‘𝑊) = (+g‘𝑊)
42		eqid 2622	. . . . . . . 8 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
43	13, 41, 2, 42, 12, 38, 32, 14	lfli 34348	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑘 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐺‘((𝑘( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧)) = ((𝑘(.r‘𝑅)(𝐺‘𝑦))(+g‘𝑅)(𝐺‘𝑧)))
44	27, 28, 26, 29, 35, 43	syl113anc 1338	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐺‘((𝑘( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧)) = ((𝑘(.r‘𝑅)(𝐺‘𝑦))(+g‘𝑅)(𝐺‘𝑧)))
45	44	fveq2d 6195	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐼‘(𝐺‘((𝑘( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧))) = (𝐼‘((𝑘(.r‘𝑅)(𝐺‘𝑦))(+g‘𝑅)(𝐺‘𝑧))))
46	12, 32, 17, 25, 26, 31	ringmneg2 18597	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑘(.r‘𝑅)(𝐼‘(𝐺‘𝑦))) = (𝐼‘(𝑘(.r‘𝑅)(𝐺‘𝑦))))
47	46	oveq1d 6665	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑘(.r‘𝑅)(𝐼‘(𝐺‘𝑦)))(+g‘𝑅)(𝐼‘(𝐺‘𝑧))) = ((𝐼‘(𝑘(.r‘𝑅)(𝐺‘𝑦)))(+g‘𝑅)(𝐼‘(𝐺‘𝑧))))
48	40, 45, 47	3eqtr4d 2666	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐼‘(𝐺‘((𝑘( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧))) = ((𝑘(.r‘𝑅)(𝐼‘(𝐺‘𝑦)))(+g‘𝑅)(𝐼‘(𝐺‘𝑧))))
49	13, 2, 42, 12	lmodvscl 18880	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑘 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝑉) → (𝑘( ·𝑠 ‘𝑊)𝑦) ∈ 𝑉)
50	27, 26, 29, 49	syl3anc 1326	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑘( ·𝑠 ‘𝑊)𝑦) ∈ 𝑉)
51	13, 41	lmodvacl 18877	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑘( ·𝑠 ‘𝑊)𝑦) ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝑘( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧) ∈ 𝑉)
52	27, 50, 35, 51	syl3anc 1326	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑘( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧) ∈ 𝑉)
53		fveq2 6191	. . . . . . 7 ⊢ (𝑥 = ((𝑘( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧) → (𝐺‘𝑥) = (𝐺‘((𝑘( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧)))
54	53	fveq2d 6195	. . . . . 6 ⊢ (𝑥 = ((𝑘( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧) → (𝐼‘(𝐺‘𝑥)) = (𝐼‘(𝐺‘((𝑘( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧))))
55		fvex 6201	. . . . . 6 ⊢ (𝐼‘(𝐺‘((𝑘( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧))) ∈ V
56	54, 20, 55	fvmpt 6282	. . . . 5 ⊢ (((𝑘( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧) ∈ 𝑉 → (𝑁‘((𝑘( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧)) = (𝐼‘(𝐺‘((𝑘( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧))))
57	52, 56	syl 17	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑁‘((𝑘( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧)) = (𝐼‘(𝐺‘((𝑘( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧))))
58		fveq2 6191	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐺‘𝑥) = (𝐺‘𝑦))
59	58	fveq2d 6195	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐼‘(𝐺‘𝑥)) = (𝐼‘(𝐺‘𝑦)))
60		fvex 6201	. . . . . . . 8 ⊢ (𝐼‘(𝐺‘𝑦)) ∈ V
61	59, 20, 60	fvmpt 6282	. . . . . . 7 ⊢ (𝑦 ∈ 𝑉 → (𝑁‘𝑦) = (𝐼‘(𝐺‘𝑦)))
62	29, 61	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑁‘𝑦) = (𝐼‘(𝐺‘𝑦)))
63	62	oveq2d 6666	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑘(.r‘𝑅)(𝑁‘𝑦)) = (𝑘(.r‘𝑅)(𝐼‘(𝐺‘𝑦))))
64		fveq2 6191	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐺‘𝑥) = (𝐺‘𝑧))
65	64	fveq2d 6195	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝐼‘(𝐺‘𝑥)) = (𝐼‘(𝐺‘𝑧)))
66		fvex 6201	. . . . . . 7 ⊢ (𝐼‘(𝐺‘𝑧)) ∈ V
67	65, 20, 66	fvmpt 6282	. . . . . 6 ⊢ (𝑧 ∈ 𝑉 → (𝑁‘𝑧) = (𝐼‘(𝐺‘𝑧)))
68	35, 67	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑁‘𝑧) = (𝐼‘(𝐺‘𝑧)))
69	63, 68	oveq12d 6668	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑘(.r‘𝑅)(𝑁‘𝑦))(+g‘𝑅)(𝑁‘𝑧)) = ((𝑘(.r‘𝑅)(𝐼‘(𝐺‘𝑦)))(+g‘𝑅)(𝐼‘(𝐺‘𝑧))))
70	48, 57, 69	3eqtr4d 2666	. . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑅) ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑁‘((𝑘( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧)) = ((𝑘(.r‘𝑅)(𝑁‘𝑦))(+g‘𝑅)(𝑁‘𝑧)))
71	70	ralrimivvva 2972	. 2 ⊢ (𝜑 → ∀𝑘 ∈ (Base‘𝑅)∀𝑦 ∈ 𝑉 ∀𝑧 ∈ 𝑉 (𝑁‘((𝑘( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧)) = ((𝑘(.r‘𝑅)(𝑁‘𝑦))(+g‘𝑅)(𝑁‘𝑧)))
72	13, 41, 2, 42, 12, 38, 32, 14	islfl 34347	. . 3 ⊢ (𝑊 ∈ LMod → (𝑁 ∈ 𝐹 ↔ (𝑁:𝑉⟶(Base‘𝑅) ∧ ∀𝑘 ∈ (Base‘𝑅)∀𝑦 ∈ 𝑉 ∀𝑧 ∈ 𝑉 (𝑁‘((𝑘( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧)) = ((𝑘(.r‘𝑅)(𝑁‘𝑦))(+g‘𝑅)(𝑁‘𝑧)))))
73	1, 72	syl 17	. 2 ⊢ (𝜑 → (𝑁 ∈ 𝐹 ↔ (𝑁:𝑉⟶(Base‘𝑅) ∧ ∀𝑘 ∈ (Base‘𝑅)∀𝑦 ∈ 𝑉 ∀𝑧 ∈ 𝑉 (𝑁‘((𝑘( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧)) = ((𝑘(.r‘𝑅)(𝑁‘𝑦))(+g‘𝑅)(𝑁‘𝑧)))))
74	21, 71, 73	mpbir2and 957	1 ⊢ (𝜑 → 𝑁 ∈ 𝐹)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ↦ cmpt 4729  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  +_gcplusg 15941  ._rcmulr 15942  Scalarcsca 15944   ·_𝑠 cvsca 15945  Grpcgrp 17422  inv_gcminusg 17423  Abelcabl 18194  Ringcrg 18547  LModclmod 18863  LFnlclfn 34344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-lmod 18865  df-lfl 34345

This theorem is referenced by:  ldualgrplem  34432