Theorem ghmplusg 19362

Description: The pointwise sum of two linear functions is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Hypothesis

Ref	Expression
ghmplusg.p	⊢ + = (+g‘𝑁)

Assertion

Ref	Expression
ghmplusg	⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (𝐹 ∘f + 𝐺) ∈ (𝑀 GrpHom 𝑁))

Proof of Theorem ghmplusg

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

Step	Hyp	Ref	Expression
1		eqid 2738	. 2 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2738	. 2 ⊢ (Base‘𝑁) = (Base‘𝑁)
3		eqid 2738	. 2 ⊢ (+g‘𝑀) = (+g‘𝑀)
4		ghmplusg.p	. 2 ⊢ + = (+g‘𝑁)
5		ghmgrp1 18751	. . 3 ⊢ (𝐺 ∈ (𝑀 GrpHom 𝑁) → 𝑀 ∈ Grp)
6	5	3ad2ant3 1133	. 2 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝑀 ∈ Grp)
7		ghmgrp2 18752	. . 3 ⊢ (𝐺 ∈ (𝑀 GrpHom 𝑁) → 𝑁 ∈ Grp)
8	7	3ad2ant3 1133	. 2 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝑁 ∈ Grp)
9	2, 4	grpcl 18500	. . . . 5 ⊢ ((𝑁 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁)) → (𝑥 + 𝑦) ∈ (Base‘𝑁))
10	9	3expb 1118	. . . 4 ⊢ ((𝑁 ∈ Grp ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑥 + 𝑦) ∈ (Base‘𝑁))
11	8, 10	sylan 579	. . 3 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑥 + 𝑦) ∈ (Base‘𝑁))
12	1, 2	ghmf 18753	. . . 4 ⊢ (𝐹 ∈ (𝑀 GrpHom 𝑁) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
13	12	3ad2ant2 1132	. . 3 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
14	1, 2	ghmf 18753	. . . 4 ⊢ (𝐺 ∈ (𝑀 GrpHom 𝑁) → 𝐺:(Base‘𝑀)⟶(Base‘𝑁))
15	14	3ad2ant3 1133	. . 3 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝐺:(Base‘𝑀)⟶(Base‘𝑁))
16		fvexd 6771	. . 3 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (Base‘𝑀) ∈ V)
17		inidm 4149	. . 3 ⊢ ((Base‘𝑀) ∩ (Base‘𝑀)) = (Base‘𝑀)
18	11, 13, 15, 16, 16, 17	off 7529	. 2 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (𝐹 ∘f + 𝐺):(Base‘𝑀)⟶(Base‘𝑁))
19	1, 3, 4	ghmlin 18754	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐹‘(𝑥(+g‘𝑀)𝑦)) = ((𝐹‘𝑥) + (𝐹‘𝑦)))
20	19	3expb 1118	. . . . . 6 ⊢ ((𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹‘(𝑥(+g‘𝑀)𝑦)) = ((𝐹‘𝑥) + (𝐹‘𝑦)))
21	20	3ad2antl2 1184	. . . . 5 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹‘(𝑥(+g‘𝑀)𝑦)) = ((𝐹‘𝑥) + (𝐹‘𝑦)))
22	1, 3, 4	ghmlin 18754	. . . . . . 7 ⊢ ((𝐺 ∈ (𝑀 GrpHom 𝑁) ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐺‘(𝑥(+g‘𝑀)𝑦)) = ((𝐺‘𝑥) + (𝐺‘𝑦)))
23	22	3expb 1118	. . . . . 6 ⊢ ((𝐺 ∈ (𝑀 GrpHom 𝑁) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺‘(𝑥(+g‘𝑀)𝑦)) = ((𝐺‘𝑥) + (𝐺‘𝑦)))
24	23	3ad2antl3 1185	. . . . 5 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺‘(𝑥(+g‘𝑀)𝑦)) = ((𝐺‘𝑥) + (𝐺‘𝑦)))
25	21, 24	oveq12d 7273	. . . 4 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹‘(𝑥(+g‘𝑀)𝑦)) + (𝐺‘(𝑥(+g‘𝑀)𝑦))) = (((𝐹‘𝑥) + (𝐹‘𝑦)) + ((𝐺‘𝑥) + (𝐺‘𝑦))))
26		simpl1 1189	. . . . . 6 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑁 ∈ Abel)
27		ablcmn 19308	. . . . . 6 ⊢ (𝑁 ∈ Abel → 𝑁 ∈ CMnd)
28	26, 27	syl 17	. . . . 5 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑁 ∈ CMnd)
29	13	ffvelrnda 6943	. . . . . 6 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝐹‘𝑥) ∈ (Base‘𝑁))
30	29	adantrr 713	. . . . 5 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹‘𝑥) ∈ (Base‘𝑁))
31	13	ffvelrnda 6943	. . . . . 6 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐹‘𝑦) ∈ (Base‘𝑁))
32	31	adantrl 712	. . . . 5 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹‘𝑦) ∈ (Base‘𝑁))
33	15	ffvelrnda 6943	. . . . . 6 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝐺‘𝑥) ∈ (Base‘𝑁))
34	33	adantrr 713	. . . . 5 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺‘𝑥) ∈ (Base‘𝑁))
35	15	ffvelrnda 6943	. . . . . 6 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐺‘𝑦) ∈ (Base‘𝑁))
36	35	adantrl 712	. . . . 5 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺‘𝑦) ∈ (Base‘𝑁))
37	2, 4	cmn4 19321	. . . . 5 ⊢ ((𝑁 ∈ CMnd ∧ ((𝐹‘𝑥) ∈ (Base‘𝑁) ∧ (𝐹‘𝑦) ∈ (Base‘𝑁)) ∧ ((𝐺‘𝑥) ∈ (Base‘𝑁) ∧ (𝐺‘𝑦) ∈ (Base‘𝑁))) → (((𝐹‘𝑥) + (𝐹‘𝑦)) + ((𝐺‘𝑥) + (𝐺‘𝑦))) = (((𝐹‘𝑥) + (𝐺‘𝑥)) + ((𝐹‘𝑦) + (𝐺‘𝑦))))
38	28, 30, 32, 34, 36, 37	syl122anc 1377	. . . 4 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (((𝐹‘𝑥) + (𝐹‘𝑦)) + ((𝐺‘𝑥) + (𝐺‘𝑦))) = (((𝐹‘𝑥) + (𝐺‘𝑥)) + ((𝐹‘𝑦) + (𝐺‘𝑦))))
39	25, 38	eqtrd 2778	. . 3 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹‘(𝑥(+g‘𝑀)𝑦)) + (𝐺‘(𝑥(+g‘𝑀)𝑦))) = (((𝐹‘𝑥) + (𝐺‘𝑥)) + ((𝐹‘𝑦) + (𝐺‘𝑦))))
40	13	ffnd 6585	. . . . 5 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝐹 Fn (Base‘𝑀))
41	40	adantr 480	. . . 4 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐹 Fn (Base‘𝑀))
42	15	ffnd 6585	. . . . 5 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝐺 Fn (Base‘𝑀))
43	42	adantr 480	. . . 4 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐺 Fn (Base‘𝑀))
44		fvexd 6771	. . . 4 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (Base‘𝑀) ∈ V)
45	1, 3	grpcl 18500	. . . . . 6 ⊢ ((𝑀 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀))
46	45	3expb 1118	. . . . 5 ⊢ ((𝑀 ∈ Grp ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀))
47	6, 46	sylan 579	. . . 4 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀))
48		fnfvof 7528	. . . 4 ⊢ (((𝐹 Fn (Base‘𝑀) ∧ 𝐺 Fn (Base‘𝑀)) ∧ ((Base‘𝑀) ∈ V ∧ (𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀))) → ((𝐹 ∘f + 𝐺)‘(𝑥(+g‘𝑀)𝑦)) = ((𝐹‘(𝑥(+g‘𝑀)𝑦)) + (𝐺‘(𝑥(+g‘𝑀)𝑦))))
49	41, 43, 44, 47, 48	syl22anc 835	. . 3 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹 ∘f + 𝐺)‘(𝑥(+g‘𝑀)𝑦)) = ((𝐹‘(𝑥(+g‘𝑀)𝑦)) + (𝐺‘(𝑥(+g‘𝑀)𝑦))))
50		simprl 767	. . . . 5 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑥 ∈ (Base‘𝑀))
51		fnfvof 7528	. . . . 5 ⊢ (((𝐹 Fn (Base‘𝑀) ∧ 𝐺 Fn (Base‘𝑀)) ∧ ((Base‘𝑀) ∈ V ∧ 𝑥 ∈ (Base‘𝑀))) → ((𝐹 ∘f + 𝐺)‘𝑥) = ((𝐹‘𝑥) + (𝐺‘𝑥)))
52	41, 43, 44, 50, 51	syl22anc 835	. . . 4 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹 ∘f + 𝐺)‘𝑥) = ((𝐹‘𝑥) + (𝐺‘𝑥)))
53		simprr 769	. . . . 5 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑦 ∈ (Base‘𝑀))
54		fnfvof 7528	. . . . 5 ⊢ (((𝐹 Fn (Base‘𝑀) ∧ 𝐺 Fn (Base‘𝑀)) ∧ ((Base‘𝑀) ∈ V ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹 ∘f + 𝐺)‘𝑦) = ((𝐹‘𝑦) + (𝐺‘𝑦)))
55	41, 43, 44, 53, 54	syl22anc 835	. . . 4 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹 ∘f + 𝐺)‘𝑦) = ((𝐹‘𝑦) + (𝐺‘𝑦)))
56	52, 55	oveq12d 7273	. . 3 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (((𝐹 ∘f + 𝐺)‘𝑥) + ((𝐹 ∘f + 𝐺)‘𝑦)) = (((𝐹‘𝑥) + (𝐺‘𝑥)) + ((𝐹‘𝑦) + (𝐺‘𝑦))))
57	39, 49, 56	3eqtr4d 2788	. 2 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹 ∘f + 𝐺)‘(𝑥(+g‘𝑀)𝑦)) = (((𝐹 ∘f + 𝐺)‘𝑥) + ((𝐹 ∘f + 𝐺)‘𝑦)))
58	1, 2, 3, 4, 6, 8, 18, 57	isghmd 18758	1 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (𝐹 ∘f + 𝐺) ∈ (𝑀 GrpHom 𝑁))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  Vcvv 3422   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ∘_f cof 7509  Basecbs 16840  +_gcplusg 16888  Grpcgrp 18492   GrpHom cghm 18746  CMndccmn 19301  Abelcabl 19302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-grp 18495  df-ghm 18747  df-cmn 19303  df-abl 19304

This theorem is referenced by:  lmhmplusg  20221  nmotri  23809  nghmplusg  23810