Theorem lmhmeql 21077

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 21053	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2		lmghm 21053	. . 3 ⊢ (𝐺 ∈ (𝑆 LMHom 𝑇) → 𝐺 ∈ (𝑆 GrpHom 𝑇))
3		ghmeql 19279	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubGrp‘𝑆))
4	1, 2, 3	syl2an 595	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubGrp‘𝑆))
5		fveq2 6920	. . . . . . . 8 ⊢ (𝑧 = (𝑥( ·𝑠 ‘𝑆)𝑦) → (𝐹‘𝑧) = (𝐹‘(𝑥( ·𝑠 ‘𝑆)𝑦)))
6		fveq2 6920	. . . . . . . 8 ⊢ (𝑧 = (𝑥( ·𝑠 ‘𝑆)𝑦) → (𝐺‘𝑧) = (𝐺‘(𝑥( ·𝑠 ‘𝑆)𝑦)))
7	5, 6	eqeq12d 2756	. . . . . . 7 ⊢ (𝑧 = (𝑥( ·𝑠 ‘𝑆)𝑦) → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘(𝑥( ·𝑠 ‘𝑆)𝑦)) = (𝐺‘(𝑥( ·𝑠 ‘𝑆)𝑦))))
8		lmhmlmod1 21055	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
9	8	adantr 480	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) → 𝑆 ∈ LMod)
10	9	ad2antrr 725	. . . . . . . 8 ⊢ ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → 𝑆 ∈ LMod)
11		simplr 768	. . . . . . . 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 770	. . . . . . . 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 2740	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
14		eqid 2740	. . . . . . . . 9 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆)
15		eqid 2740	. . . . . . . . 9 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
16		eqid 2740	. . . . . . . . 9 ⊢ (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
17	13, 14, 15, 16	lmodvscl 20898	. . . . . . . 8 ⊢ ((𝑆 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥( ·𝑠 ‘𝑆)𝑦) ∈ (Base‘𝑆))
18	10, 11, 12, 17	syl3anc 1371	. . . . . . 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 7456	. . . . . . . . 9 ⊢ ((𝐹‘𝑦) = (𝐺‘𝑦) → (𝑥( ·𝑠 ‘𝑇)(𝐹‘𝑦)) = (𝑥( ·𝑠 ‘𝑇)(𝐺‘𝑦)))
20	19	ad2antll 728	. . . . . . . 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 774	. . . . . . . . 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 2740	. . . . . . . . . 10 ⊢ ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑇)
23	14, 16, 13, 15, 22	lmhmlin 21057	. . . . . . . . 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 1371	. . . . . . . 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 775	. . . . . . . . 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 21057	. . . . . . . . 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 1371	. . . . . . . 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 2790	. . . . . . 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 3710	. . . . . 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 456	. . . . 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 3152	. . . 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 2740	. . . . . . . . 9 ⊢ (Base‘𝑇) = (Base‘𝑇)
33	13, 32	lmhmf 21056	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
34	33	ffnd 6748	. . . . . . 7 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 Fn (Base‘𝑆))
35	13, 32	lmhmf 21056	. . . . . . . 8 ⊢ (𝐺 ∈ (𝑆 LMHom 𝑇) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
36	35	ffnd 6748	. . . . . . 7 ⊢ (𝐺 ∈ (𝑆 LMHom 𝑇) → 𝐺 Fn (Base‘𝑆))
37		fndmin 7078	. . . . . . 7 ⊢ ((𝐹 Fn (Base‘𝑆) ∧ 𝐺 Fn (Base‘𝑆)) → dom (𝐹 ∩ 𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)})
38	34, 36, 37	syl2an 595	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) → dom (𝐹 ∩ 𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)})
39	38	adantr 480	. . . . 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 2833	. . . . . . 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 3346	. . . . . 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 6920	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝐹‘𝑧) = (𝐹‘𝑦))
43		fveq2 6920	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝐺‘𝑧) = (𝐺‘𝑦))
44	42, 43	eqeq12d 2756	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘𝑦) = (𝐺‘𝑦)))
45	44	ralrab 3715	. . . . . 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 287	. . . . 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 257	. . 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 3152	. 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 20983	. . 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 712	1 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ 𝑈)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  {crab 3443   ∩ cin 3975  dom cdm 5700   Fn wfn 6568  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  Scalarcsca 17314   ·_𝑠 cvsca 17315  SubGrpcsubg 19160   GrpHom cghm 19252  LModclmod 20880  LSubSpclss 20952   LMHom clmhm 21041

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-0g 17501  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-mhm 18818  df-submnd 18819  df-grp 18976  df-minusg 18977  df-sbg 18978  df-subg 19163  df-ghm 19253  df-mgp 20162  df-ur 20209  df-ring 20262  df-lmod 20882  df-lss 20953  df-lmhm 21044

This theorem is referenced by:  lspextmo  21078