Theorem lmhmeql 21331

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 21307	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2		lmghm 21307	. . 3 ⊢ (𝐺 ∈ (𝑆 LMHom 𝑇) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
3		ghmeql 19454	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubGrp‘𝑆))
4	1, 2, 3	syl2an 608	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubGrp‘𝑆))
5		fveq2 6885	. . . . . . . 8 ⊢ (𝑧 = (𝑥( ·𝑠 ‘𝑆)𝑦) → (𝐹‘𝑧) = (𝐹‘(𝑥( ·𝑠 ‘𝑆)𝑦)))
6		fveq2 6885	. . . . . . . 8 ⊢ (𝑧 = (𝑥( ·𝑠 ‘𝑆)𝑦) → (𝐺‘𝑧) = (𝐺‘(𝑥( ·𝑠 ‘𝑆)𝑦)))
7	5, 6	eqeq12d 2776	. . . . . . 7 ⊢ (𝑧 = (𝑥( ·𝑠 ‘𝑆)𝑦) → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘(𝑥( ·𝑠 ‘𝑆)𝑦)) = (𝐺‘(𝑥( ·𝑠 ‘𝑆)𝑦))))
8		lmhmlmod1 21309	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
9	8	adantr 486	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) → 𝑆 ∈ LMod)
10	9	ad2antrr 739	. . . . . . . 8 ⊢ ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → 𝑆 ∈ LMod)
11		simplr 781	. . . . . . . 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 783	. . . . . . . 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 2760	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
14		eqid 2760	. . . . . . . . 9 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆)
15		eqid 2760	. . . . . . . . 9 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
16		eqid 2760	. . . . . . . . 9 ⊢ (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
17	13, 14, 15, 16	lmodvscl 21154	. . . . . . . 8 ⊢ ((𝑆 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥( ·𝑠 ‘𝑆)𝑦) ∈ (Base‘𝑆))
18	10, 11, 12, 17	syl3anc 1398	. . . . . . 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 7428	. . . . . . . . 9 ⊢ ((𝐹‘𝑦) = (𝐺‘𝑦) → (𝑥( ·𝑠 ‘𝑇)(𝐹‘𝑦)) = (𝑥( ·𝑠 ‘𝑇)(𝐺‘𝑦)))
20	19	ad2antll 742	. . . . . . . 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 787	. . . . . . . . 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 2760	. . . . . . . . . 10 ⊢ ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑇)
23	14, 16, 13, 15, 22	lmhmlin 21311	. . . . . . . . 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 1398	. . . . . . . 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 788	. . . . . . . . 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 21311	. . . . . . . . 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 1398	. . . . . . . 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 2805	. . . . . . 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 3646	. . . . . 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 462	. . . . 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 3154	. . . 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 2760	. . . . . . . . 9 ⊢ (Base‘𝑇) = (Base‘𝑇)
33	13, 32	lmhmf 21310	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
34	33	ffnd 6710	. . . . . . 7 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 Fn (Base‘𝑆))
35	13, 32	lmhmf 21310	. . . . . . . 8 ⊢ (𝐺 ∈ (𝑆 LMHom 𝑇) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
36	35	ffnd 6710	. . . . . . 7 ⊢ (𝐺 ∈ (𝑆 LMHom 𝑇) → 𝐺 Fn (Base‘𝑆))
37		fndmin 7044	. . . . . . 7 ⊢ ((𝐹 Fn (Base‘𝑆) ∧ 𝐺 Fn (Base‘𝑆)) → dom (𝐹 ∩ 𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)})
38	34, 36, 37	syl2an 608	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) → dom (𝐹 ∩ 𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)})
39	38	adantr 486	. . . . 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 2849	. . . . . . 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 3329	. . . . . 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 6885	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝐹‘𝑧) = (𝐹‘𝑦))
43		fveq2 6885	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝐺‘𝑧) = (𝐺‘𝑦))
44	42, 43	eqeq12d 2776	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘𝑦) = (𝐺‘𝑦)))
45	44	ralrab 3651	. . . . . 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	bitrdi 290	. . . . 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 18	. . . 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 260	. . 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 3154	. 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 21238	. . 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 18	. 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 726	1 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ 𝑈)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412   ∩ cin 4089  dom cdm 5650   Fn wfn 6533  ‘cfv 6538  (class class class)co 7420  Basecbs 17388  Scalarcsca 17432   ·_𝑠 cvsca 17433  SubGrpcsubg 19331   GrpHom cghm 19428  LModclmod 21136  LSubSpclss 21207   LMHom clmhm 21295

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11257  ax-resscn 11258  ax-1cn 11259  ax-icn 11260  ax-addcl 11261  ax-addrcl 11262  ax-mulcl 11263  ax-mulrcl 11264  ax-mulcom 11265  ax-addass 11266  ax-mulass 11267  ax-distr 11268  ax-i2m1 11269  ax-1ne0 11270  ax-1rid 11271  ax-rnegex 11272  ax-rrecex 11273  ax-cnre 11274  ax-pre-lttri 11275  ax-pre-lttrn 11276  ax-pre-ltadd 11277  ax-pre-mulgt0 11278

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-rdg 8419  df-er 8718  df-map 8850  df-en 8975  df-dom 8976  df-sdom 8977  df-pnf 11346  df-mnf 11347  df-xr 11348  df-ltxr 11349  df-le 11350  df-sub 11544  df-neg 11545  df-nn 12337  df-2 12406  df-sets 17343  df-slot 17361  df-ndx 17373  df-base 17389  df-ress 17410  df-plusg 17442  df-0g 17613  df-mgm 18817  df-sgrp 18909  df-mnd 18925  df-mhm 18979  df-submnd 18980  df-grp 19148  df-minusg 19149  df-sbg 19150  df-subg 19334  df-ghm 19429  df-mgp 20362  df-ur 20409  df-ring 20462  df-lmod 21138  df-lss 21208  df-lmhm 21298

This theorem is used by:  lspextmo  21332