Theorem lmhmeql 19829

Description: The equalizer of two module homomorphisms is a subspace. (Contributed by Stefan O'Rear, 7-Mar-2015.)

Hypothesis

Ref	Expression
lmhmeql.u	⊢ 𝑈 = (LSubSp‘𝑆)

Assertion

Ref	Expression
lmhmeql	⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ 𝑈)

Proof of Theorem lmhmeql

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

Step	Hyp	Ref	Expression
1		lmghm 19805	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2		lmghm 19805	. . 3 ⊢ (𝐺 ∈ (𝑆 LMHom 𝑇) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
3		ghmeql 18383	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubGrp‘𝑆))
4	1, 2, 3	syl2an 597	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubGrp‘𝑆))
5		fveq2 6672	. . . . . . . 8 ⊢ (𝑧 = (𝑥( ·𝑠 ‘𝑆)𝑦) → (𝐹‘𝑧) = (𝐹‘(𝑥( ·𝑠 ‘𝑆)𝑦)))
6		fveq2 6672	. . . . . . . 8 ⊢ (𝑧 = (𝑥( ·𝑠 ‘𝑆)𝑦) → (𝐺‘𝑧) = (𝐺‘(𝑥( ·𝑠 ‘𝑆)𝑦)))
7	5, 6	eqeq12d 2839	. . . . . . 7 ⊢ (𝑧 = (𝑥( ·𝑠 ‘𝑆)𝑦) → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘(𝑥( ·𝑠 ‘𝑆)𝑦)) = (𝐺‘(𝑥( ·𝑠 ‘𝑆)𝑦))))
8		lmhmlmod1 19807	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
9	8	adantr 483	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) → 𝑆 ∈ LMod)
10	9	ad2antrr 724	. . . . . . . 8 ⊢ ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → 𝑆 ∈ LMod)
11		simplr 767	. . . . . . . 8 ⊢ ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → 𝑥 ∈ (Base‘(Scalar‘𝑆)))
12		simprl 769	. . . . . . . 8 ⊢ ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → 𝑦 ∈ (Base‘𝑆))
13		eqid 2823	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
14		eqid 2823	. . . . . . . . 9 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆)
15		eqid 2823	. . . . . . . . 9 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
16		eqid 2823	. . . . . . . . 9 ⊢ (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
17	13, 14, 15, 16	lmodvscl 19653	. . . . . . . 8 ⊢ ((𝑆 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥( ·𝑠 ‘𝑆)𝑦) ∈ (Base‘𝑆))
18	10, 11, 12, 17	syl3anc 1367	. . . . . . 7 ⊢ ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → (𝑥( ·𝑠 ‘𝑆)𝑦) ∈ (Base‘𝑆))
19		oveq2 7166	. . . . . . . . 9 ⊢ ((𝐹‘𝑦) = (𝐺‘𝑦) → (𝑥( ·𝑠 ‘𝑇)(𝐹‘𝑦)) = (𝑥( ·𝑠 ‘𝑇)(𝐺‘𝑦)))
20	19	ad2antll 727	. . . . . . . 8 ⊢ ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → (𝑥( ·𝑠 ‘𝑇)(𝐹‘𝑦)) = (𝑥( ·𝑠 ‘𝑇)(𝐺‘𝑦)))
21		simplll 773	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → 𝐹 ∈ (𝑆 LMHom 𝑇))
22		eqid 2823	. . . . . . . . . 10 ⊢ ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑇)
23	14, 16, 13, 15, 22	lmhmlin 19809	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥( ·𝑠 ‘𝑆)𝑦)) = (𝑥( ·𝑠 ‘𝑇)(𝐹‘𝑦)))
24	21, 11, 12, 23	syl3anc 1367	. . . . . . . 8 ⊢ ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → (𝐹‘(𝑥( ·𝑠 ‘𝑆)𝑦)) = (𝑥( ·𝑠 ‘𝑇)(𝐹‘𝑦)))
25		simpllr 774	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → 𝐺 ∈ (𝑆 LMHom 𝑇))
26	14, 16, 13, 15, 22	lmhmlin 19809	. . . . . . . . 9 ⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐺‘(𝑥( ·𝑠 ‘𝑆)𝑦)) = (𝑥( ·𝑠 ‘𝑇)(𝐺‘𝑦)))
27	25, 11, 12, 26	syl3anc 1367	. . . . . . . 8 ⊢ ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → (𝐺‘(𝑥( ·𝑠 ‘𝑆)𝑦)) = (𝑥( ·𝑠 ‘𝑇)(𝐺‘𝑦)))
28	20, 24, 27	3eqtr4d 2868	. . . . . . 7 ⊢ ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → (𝐹‘(𝑥( ·𝑠 ‘𝑆)𝑦)) = (𝐺‘(𝑥( ·𝑠 ‘𝑆)𝑦)))
29	7, 18, 28	elrabd 3684	. . . . . 6 ⊢ ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → (𝑥( ·𝑠 ‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)})
30	29	expr 459	. . . . 5 ⊢ ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹‘𝑦) = (𝐺‘𝑦) → (𝑥( ·𝑠 ‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}))
31	30	ralrimiva 3184	. . . 4 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) → ∀𝑦 ∈ (Base‘𝑆)((𝐹‘𝑦) = (𝐺‘𝑦) → (𝑥( ·𝑠 ‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}))
32		eqid 2823	. . . . . . . . 9 ⊢ (Base‘𝑇) = (Base‘𝑇)
33	13, 32	lmhmf 19808	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
34	33	ffnd 6517	. . . . . . 7 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 Fn (Base‘𝑆))
35	13, 32	lmhmf 19808	. . . . . . . 8 ⊢ (𝐺 ∈ (𝑆 LMHom 𝑇) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
36	35	ffnd 6517	. . . . . . 7 ⊢ (𝐺 ∈ (𝑆 LMHom 𝑇) → 𝐺 Fn (Base‘𝑆))
37		fndmin 6817	. . . . . . 7 ⊢ ((𝐹 Fn (Base‘𝑆) ∧ 𝐺 Fn (Base‘𝑆)) → dom (𝐹 ∩ 𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)})
38	34, 36, 37	syl2an 597	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) → dom (𝐹 ∩ 𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)})
39	38	adantr 483	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) → dom (𝐹 ∩ 𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)})
40		eleq2 2903	. . . . . . 7 ⊢ (dom (𝐹 ∩ 𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} → ((𝑥( ·𝑠 ‘𝑆)𝑦) ∈ dom (𝐹 ∩ 𝐺) ↔ (𝑥( ·𝑠 ‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}))
41	40	raleqbi1dv 3405	. . . . . 6 ⊢ (dom (𝐹 ∩ 𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} → (∀𝑦 ∈ dom (𝐹 ∩ 𝐺)(𝑥( ·𝑠 ‘𝑆)𝑦) ∈ dom (𝐹 ∩ 𝐺) ↔ ∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} (𝑥( ·𝑠 ‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}))
42		fveq2 6672	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝐹‘𝑧) = (𝐹‘𝑦))
43		fveq2 6672	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝐺‘𝑧) = (𝐺‘𝑦))
44	42, 43	eqeq12d 2839	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘𝑦) = (𝐺‘𝑦)))
45	44	ralrab 3687	. . . . . 6 ⊢ (∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} (𝑥( ·𝑠 ‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ↔ ∀𝑦 ∈ (Base‘𝑆)((𝐹‘𝑦) = (𝐺‘𝑦) → (𝑥( ·𝑠 ‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}))
46	41, 45	syl6bb 289	. . . . 5 ⊢ (dom (𝐹 ∩ 𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} → (∀𝑦 ∈ dom (𝐹 ∩ 𝐺)(𝑥( ·𝑠 ‘𝑆)𝑦) ∈ dom (𝐹 ∩ 𝐺) ↔ ∀𝑦 ∈ (Base‘𝑆)((𝐹‘𝑦) = (𝐺‘𝑦) → (𝑥( ·𝑠 ‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)})))
47	39, 46	syl 17	. . . 4 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) → (∀𝑦 ∈ dom (𝐹 ∩ 𝐺)(𝑥( ·𝑠 ‘𝑆)𝑦) ∈ dom (𝐹 ∩ 𝐺) ↔ ∀𝑦 ∈ (Base‘𝑆)((𝐹‘𝑦) = (𝐺‘𝑦) → (𝑥( ·𝑠 ‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)})))
48	31, 47	mpbird 259	. . 3 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) → ∀𝑦 ∈ dom (𝐹 ∩ 𝐺)(𝑥( ·𝑠 ‘𝑆)𝑦) ∈ dom (𝐹 ∩ 𝐺))
49	48	ralrimiva 3184	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) → ∀𝑥 ∈ (Base‘(Scalar‘𝑆))∀𝑦 ∈ dom (𝐹 ∩ 𝐺)(𝑥( ·𝑠 ‘𝑆)𝑦) ∈ dom (𝐹 ∩ 𝐺))
50		lmhmeql.u	. . . 4 ⊢ 𝑈 = (LSubSp‘𝑆)
51	14, 16, 13, 15, 50	islss4 19736	. . 3 ⊢ (𝑆 ∈ LMod → (dom (𝐹 ∩ 𝐺) ∈ 𝑈 ↔ (dom (𝐹 ∩ 𝐺) ∈ (SubGrp‘𝑆) ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑆))∀𝑦 ∈ dom (𝐹 ∩ 𝐺)(𝑥( ·𝑠 ‘𝑆)𝑦) ∈ dom (𝐹 ∩ 𝐺))))
52	9, 51	syl 17	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) → (dom (𝐹 ∩ 𝐺) ∈ 𝑈 ↔ (dom (𝐹 ∩ 𝐺) ∈ (SubGrp‘𝑆) ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑆))∀𝑦 ∈ dom (𝐹 ∩ 𝐺)(𝑥( ·𝑠 ‘𝑆)𝑦) ∈ dom (𝐹 ∩ 𝐺))))
53	4, 49, 52	mpbir2and 711	1 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ 𝑈)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  {crab 3144   ∩ cin 3937  dom cdm 5557   Fn wfn 6352  ‘cfv 6357  (class class class)co 7158  Basecbs 16485  Scalarcsca 16570   ·_𝑠 cvsca 16571  SubGrpcsubg 18275   GrpHom cghm 18357  LModclmod 19636  LSubSpclss 19705   LMHom clmhm 19793

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-iun 4923  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-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-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-er 8291  df-map 8410  df-en 8512  df-dom 8513  df-sdom 8514  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-ndx 16488  df-slot 16489  df-base 16491  df-sets 16492  df-ress 16493  df-plusg 16580  df-0g 16717  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-subg 18278  df-ghm 18358  df-mgp 19242  df-ur 19254  df-ring 19301  df-lmod 19638  df-lss 19706  df-lmhm 19796

This theorem is referenced by:  lspextmo  19830