Theorem ghmplusg 20090

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 2763	. 2 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2763	. 2 ⊢ (Base‘𝑁) = (Base‘𝑁)
3		eqid 2763	. 2 ⊢ (+g‘𝑀) = (+g‘𝑀)
4		ghmplusg.p	. 2 ⊢ + = (+g‘𝑁)
5		ghmgrp1 19461	. . 3 ⊢ (𝐺 ∈ (𝑀 GrpHom 𝑁) → 𝑀 ∈ Grp)
6	5	3ad2ant3 1153	. 2 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝑀 ∈ Grp)
7		ghmgrp2 19462	. . 3 ⊢ (𝐺 ∈ (𝑀 GrpHom 𝑁) → 𝑁 ∈ Grp)
8	7	3ad2ant3 1153	. 2 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝑁 ∈ Grp)
9	2, 4	grpcl 19181	. . . . 5 ⊢ ((𝑁 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁)) → (𝑥 + 𝑦) ∈ (Base‘𝑁))
10	9	3expb 1138	. . . 4 ⊢ ((𝑁 ∈ Grp ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑥 + 𝑦) ∈ (Base‘𝑁))
11	8, 10	sylan 592	. . 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 19463	. . . 4 ⊢ (𝐹 ∈ (𝑀 GrpHom 𝑁) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
13	12	3ad2ant2 1152	. . 3 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
14	1, 2	ghmf 19463	. . . 4 ⊢ (𝐺 ∈ (𝑀 GrpHom 𝑁) → 𝐺:(Base‘𝑀)⟶(Base‘𝑁))
15	14	3ad2ant3 1153	. . 3 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝐺:(Base‘𝑀)⟶(Base‘𝑁))
16		fvexd 6923	. . 3 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (Base‘𝑀) ∈ V)
17		inidm 4126	. . 3 ⊢ ((Base‘𝑀) ∩ (Base‘𝑀)) = (Base‘𝑀)
18	11, 13, 15, 16, 16, 17	off 7735	. 2 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (𝐹 ∘f + 𝐺):(Base‘𝑀)⟶(Base‘𝑁))
19	1, 3, 4	ghmlin 19464	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐹‘(𝑥(+g‘𝑀)𝑦)) = ((𝐹‘𝑥) + (𝐹‘𝑦)))
20	19	3expb 1138	. . . . . 6 ⊢ ((𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹‘(𝑥(+g‘𝑀)𝑦)) = ((𝐹‘𝑥) + (𝐹‘𝑦)))
21	20	3ad2antl2 1205	. . . . 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 19464	. . . . . . 7 ⊢ ((𝐺 ∈ (𝑀 GrpHom 𝑁) ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐺‘(𝑥(+g‘𝑀)𝑦)) = ((𝐺‘𝑥) + (𝐺‘𝑦)))
23	22	3expb 1138	. . . . . 6 ⊢ ((𝐺 ∈ (𝑀 GrpHom 𝑁) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺‘(𝑥(+g‘𝑀)𝑦)) = ((𝐺‘𝑥) + (𝐺‘𝑦)))
24	23	3ad2antl3 1206	. . . . 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 7461	. . . 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 1210	. . . . . 6 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑁 ∈ Abel)
27		ablcmn 20031	. . . . . 6 ⊢ (𝑁 ∈ Abel → 𝑁 ∈ CMnd)
28	26, 27	syl 18	. . . . 5 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑁 ∈ CMnd)
29	13	ffvelcdmda 7108	. . . . . 6 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝐹‘𝑥) ∈ (Base‘𝑁))
30	29	adantrr 730	. . . . 5 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹‘𝑥) ∈ (Base‘𝑁))
31	13	ffvelcdmda 7108	. . . . . 6 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐹‘𝑦) ∈ (Base‘𝑁))
32	31	adantrl 729	. . . . 5 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹‘𝑦) ∈ (Base‘𝑁))
33	15	ffvelcdmda 7108	. . . . . 6 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝐺‘𝑥) ∈ (Base‘𝑁))
34	33	adantrr 730	. . . . 5 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺‘𝑥) ∈ (Base‘𝑁))
35	15	ffvelcdmda 7108	. . . . . 6 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐺‘𝑦) ∈ (Base‘𝑁))
36	35	adantrl 729	. . . . 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 20045	. . . . 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 1407	. . . 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 2798	. . 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 6732	. . . . 5 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝐹 Fn (Base‘𝑀))
41	40	adantr 486	. . . 4 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐹 Fn (Base‘𝑀))
42	15	ffnd 6732	. . . . 5 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝐺 Fn (Base‘𝑀))
43	42	adantr 486	. . . 4 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐺 Fn (Base‘𝑀))
44		fvexd 6923	. . . 4 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (Base‘𝑀) ∈ V)
45	1, 3	grpcl 19181	. . . . . 6 ⊢ ((𝑀 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀))
46	45	3expb 1138	. . . . 5 ⊢ ((𝑀 ∈ Grp ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀))
47	6, 46	sylan 592	. . . 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 7734	. . . 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 852	. . 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 783	. . . . 5 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑥 ∈ (Base‘𝑀))
51		fnfvof 7734	. . . . 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 852	. . . 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 785	. . . . 5 ⊢ (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑦 ∈ (Base‘𝑀))
54		fnfvof 7734	. . . . 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 852	. . . 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 7461	. . 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 2808	. 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 19468	1 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (𝐹 ∘f + 𝐺) ∈ (𝑀 GrpHom 𝑁))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  Vcvv 3454   Fn wfn 6553  ⟶wf 6554  ‘cfv 6558  (class class class)co 7443   ∘_f cof 7715  Basecbs 17416  +_gcplusg 17457  Grpcgrp 19173   GrpHom cghm 19456  CMndccmn 20024  Abelcabl 20025

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-of 7717  df-1st 8025  df-2nd 8026  df-map 8877  df-mgm 18845  df-sgrp 18937  df-mnd 18953  df-grp 19176  df-ghm 19457  df-cmn 20026  df-abl 20027

This theorem is used by:  lmhmplusg  21349  nmotri  25088  nghmplusg  25089