Theorem ghmplusg 19869

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 2761	. 2 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2761	. 2 ⊢ (Base‘𝑁) = (Base‘𝑁)
3		eqid 2761	. 2 ⊢ (+g‘𝑀) = (+g‘𝑀)
4		ghmplusg.p	. 2 ⊢ + = (+g‘𝑁)
5		ghmgrp1 19241	. . 3 ⊢ (𝐺 ∈ (𝑀 GrpHom 𝑁) → 𝑀 ∈ Grp)
6	5	3ad2ant3 1147	. 2 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝑀 ∈ Grp)
7		ghmgrp2 19242	. . 3 ⊢ (𝐺 ∈ (𝑀 GrpHom 𝑁) → 𝑁 ∈ Grp)
8	7	3ad2ant3 1147	. 2 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝑁 ∈ Grp)
9	2, 4	grpcl 18966	. . . . 5 ⊢ ((𝑁 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁)) → (𝑥 + 𝑦) ∈ (Base‘𝑁))
10	9	3expb 1132	. . . 4 ⊢ ((𝑁 ∈ Grp ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑥 + 𝑦) ∈ (Base‘𝑁))
11	8, 10	sylan 589	. . 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 19243	. . . 4 ⊢ (𝐹 ∈ (𝑀 GrpHom 𝑁) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
13	12	3ad2ant2 1146	. . 3 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
14	1, 2	ghmf 19243	. . . 4 ⊢ (𝐺 ∈ (𝑀 GrpHom 𝑁) → 𝐺:(Base‘𝑀)⟶(Base‘𝑁))
15	14	3ad2ant3 1147	. . 3 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝐺:(Base‘𝑀)⟶(Base‘𝑁))
16		fvexd 6878	. . 3 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (Base‘𝑀) ∈ V)
17		inidm 4178	. . 3 ⊢ ((Base‘𝑀) ∩ (Base‘𝑀)) = (Base‘𝑀)
18	11, 13, 15, 16, 16, 17	off 7674	. 2 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (𝐹 ∘f + 𝐺):(Base‘𝑀)⟶(Base‘𝑁))
19	1, 3, 4	ghmlin 19244	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐹‘(𝑥(+g‘𝑀)𝑦)) = ((𝐹‘𝑥) + (𝐹‘𝑦)))
20	19	3expb 1132	. . . . . 6 ⊢ ((𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹‘(𝑥(+g‘𝑀)𝑦)) = ((𝐹‘𝑥) + (𝐹‘𝑦)))
21	20	3ad2antl2 1199	. . . . 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 19244	. . . . . . 7 ⊢ ((𝐺 ∈ (𝑀 GrpHom 𝑁) ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐺‘(𝑥(+g‘𝑀)𝑦)) = ((𝐺‘𝑥) + (𝐺‘𝑦)))
23	22	3expb 1132	. . . . . 6 ⊢ ((𝐺 ∈ (𝑀 GrpHom 𝑁) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺‘(𝑥(+g‘𝑀)𝑦)) = ((𝐺‘𝑥) + (𝐺‘𝑦)))
24	23	3ad2antl3 1200	. . . . 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 7410	. . . 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 1204	. . . . . 6 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑁 ∈ Abel)
27		ablcmn 19810	. . . . . 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 7061	. . . . . 6 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝐹‘𝑥) ∈ (Base‘𝑁))
30	29	adantrr 727	. . . . 5 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹‘𝑥) ∈ (Base‘𝑁))
31	13	ffvelcdmda 7061	. . . . . 6 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐹‘𝑦) ∈ (Base‘𝑁))
32	31	adantrl 726	. . . . 5 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹‘𝑦) ∈ (Base‘𝑁))
33	15	ffvelcdmda 7061	. . . . . 6 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝐺‘𝑥) ∈ (Base‘𝑁))
34	33	adantrr 727	. . . . 5 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺‘𝑥) ∈ (Base‘𝑁))
35	15	ffvelcdmda 7061	. . . . . 6 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐺‘𝑦) ∈ (Base‘𝑁))
36	35	adantrl 726	. . . . 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 19824	. . . . 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 1397	. . . 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 2796	. . 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 6688	. . . . 5 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝐹 Fn (Base‘𝑀))
41	40	adantr 484	. . . 4 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐹 Fn (Base‘𝑀))
42	15	ffnd 6688	. . . . 5 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝐺 Fn (Base‘𝑀))
43	42	adantr 484	. . . 4 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐺 Fn (Base‘𝑀))
44		fvexd 6878	. . . 4 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (Base‘𝑀) ∈ V)
45	1, 3	grpcl 18966	. . . . . 6 ⊢ ((𝑀 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀))
46	45	3expb 1132	. . . . 5 ⊢ ((𝑀 ∈ Grp ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀))
47	6, 46	sylan 589	. . . 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 7673	. . . 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 849	. . 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 780	. . . . 5 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑥 ∈ (Base‘𝑀))
51		fnfvof 7673	. . . . 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 849	. . . 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 782	. . . . 5 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑦 ∈ (Base‘𝑀))
54		fnfvof 7673	. . . . 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 849	. . . 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 7410	. . 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 2806	. 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 19248	1 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (𝐹 ∘f + 𝐺) ∈ (𝑀 GrpHom 𝑁))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  Vcvv 3453   Fn wfn 6512  ⟶wf 6513  ‘cfv 6517  (class class class)co 7392   ∘_f cof 7654  Basecbs 17228  +_gcplusg 17269  Grpcgrp 18958   GrpHom cghm 19236  CMndccmn 19803  Abelcabl 19804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-of 7656  df-1st 7966  df-2nd 7967  df-map 8805  df-mgm 18657  df-sgrp 18736  df-mnd 18752  df-grp 18961  df-ghm 19237  df-cmn 19805  df-abl 19806

This theorem is referenced by:  lmhmplusg  21091  nmotri  24779  nghmplusg  24780