Theorem ghmplusg 19819

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 2740	. 2 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2740	. 2 ⊢ (Base‘𝑁) = (Base‘𝑁)
3		eqid 2740	. 2 ⊢ (+g‘𝑀) = (+g‘𝑀)
4		ghmplusg.p	. 2 ⊢ + = (+g‘𝑁)
5		ghmgrp1 19191	. . 3 ⊢ (𝐺 ∈ (𝑀 GrpHom 𝑁) → 𝑀 ∈ Grp)
6	5	3ad2ant3 1141	. 2 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝑀 ∈ Grp)
7		ghmgrp2 19192	. . 3 ⊢ (𝐺 ∈ (𝑀 GrpHom 𝑁) → 𝑁 ∈ Grp)
8	7	3ad2ant3 1141	. 2 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝑁 ∈ Grp)
9	2, 4	grpcl 18915	. . . . 5 ⊢ ((𝑁 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁)) → (𝑥 + 𝑦) ∈ (Base‘𝑁))
10	9	3expb 1126	. . . 4 ⊢ ((𝑁 ∈ Grp ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑥 + 𝑦) ∈ (Base‘𝑁))
11	8, 10	sylan 586	. . 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 19193	. . . 4 ⊢ (𝐹 ∈ (𝑀 GrpHom 𝑁) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
13	12	3ad2ant2 1140	. . 3 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
14	1, 2	ghmf 19193	. . . 4 ⊢ (𝐺 ∈ (𝑀 GrpHom 𝑁) → 𝐺:(Base‘𝑀)⟶(Base‘𝑁))
15	14	3ad2ant3 1141	. . 3 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝐺:(Base‘𝑀)⟶(Base‘𝑁))
16		fvexd 6849	. . 3 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (Base‘𝑀) ∈ V)
17		inidm 4162	. . 3 ⊢ ((Base‘𝑀) ∩ (Base‘𝑀)) = (Base‘𝑀)
18	11, 13, 15, 16, 16, 17	off 7645	. 2 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (𝐹 ∘f + 𝐺):(Base‘𝑀)⟶(Base‘𝑁))
19	1, 3, 4	ghmlin 19194	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐹‘(𝑥(+g‘𝑀)𝑦)) = ((𝐹‘𝑥) + (𝐹‘𝑦)))
20	19	3expb 1126	. . . . . 6 ⊢ ((𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹‘(𝑥(+g‘𝑀)𝑦)) = ((𝐹‘𝑥) + (𝐹‘𝑦)))
21	20	3ad2antl2 1193	. . . . 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 19194	. . . . . . 7 ⊢ ((𝐺 ∈ (𝑀 GrpHom 𝑁) ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐺‘(𝑥(+g‘𝑀)𝑦)) = ((𝐺‘𝑥) + (𝐺‘𝑦)))
23	22	3expb 1126	. . . . . 6 ⊢ ((𝐺 ∈ (𝑀 GrpHom 𝑁) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺‘(𝑥(+g‘𝑀)𝑦)) = ((𝐺‘𝑥) + (𝐺‘𝑦)))
24	23	3ad2antl3 1194	. . . . 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 7381	. . . 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 1198	. . . . . 6 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑁 ∈ Abel)
27		ablcmn 19760	. . . . . 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	ffvelcdmda 7032	. . . . . 6 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝐹‘𝑥) ∈ (Base‘𝑁))
30	29	adantrr 723	. . . . 5 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹‘𝑥) ∈ (Base‘𝑁))
31	13	ffvelcdmda 7032	. . . . . 6 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐹‘𝑦) ∈ (Base‘𝑁))
32	31	adantrl 722	. . . . 5 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹‘𝑦) ∈ (Base‘𝑁))
33	15	ffvelcdmda 7032	. . . . . 6 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝐺‘𝑥) ∈ (Base‘𝑁))
34	33	adantrr 723	. . . . 5 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺‘𝑥) ∈ (Base‘𝑁))
35	15	ffvelcdmda 7032	. . . . . 6 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐺‘𝑦) ∈ (Base‘𝑁))
36	35	adantrl 722	. . . . 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 19774	. . . . 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 1387	. . . 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 2775	. . 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 6663	. . . . 5 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝐹 Fn (Base‘𝑀))
41	40	adantr 481	. . . 4 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐹 Fn (Base‘𝑀))
42	15	ffnd 6663	. . . . 5 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝐺 Fn (Base‘𝑀))
43	42	adantr 481	. . . 4 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐺 Fn (Base‘𝑀))
44		fvexd 6849	. . . 4 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (Base‘𝑀) ∈ V)
45	1, 3	grpcl 18915	. . . . . 6 ⊢ ((𝑀 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀))
46	45	3expb 1126	. . . . 5 ⊢ ((𝑀 ∈ Grp ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀))
47	6, 46	sylan 586	. . . 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 7644	. . . 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 844	. . 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 776	. . . . 5 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑥 ∈ (Base‘𝑀))
51		fnfvof 7644	. . . . 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 844	. . . 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 778	. . . . 5 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑦 ∈ (Base‘𝑀))
54		fnfvof 7644	. . . . 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 844	. . . 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 7381	. . 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 2785	. 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 19198	1 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (𝐹 ∘f + 𝐺) ∈ (𝑀 GrpHom 𝑁))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  Vcvv 3432   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7363   ∘_f cof 7625  Basecbs 17177  +_gcplusg 17218  Grpcgrp 18907   GrpHom cghm 19185  CMndccmn 19753  Abelcabl 19754

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-of 7627  df-1st 7938  df-2nd 7939  df-map 8772  df-mgm 18606  df-sgrp 18685  df-mnd 18701  df-grp 18910  df-ghm 19186  df-cmn 19755  df-abl 19756

This theorem is referenced by:  lmhmplusg  21041  nmotri  24729  nghmplusg  24730