Theorem isrnghm 45338

Description: A function is a non-unital ring homomorphism iff it is a group homomorphism and preserves multiplication. (Contributed by AV, 22-Feb-2020.)

Hypotheses

Ref	Expression
isrnghm.b	⊢ 𝐵 = (Base‘𝑅)
isrnghm.t	⊢ · = (.r‘𝑅)
isrnghm.m	⊢ ∗ = (.r‘𝑆)

Assertion

Ref	Expression
isrnghm	⊢ (𝐹 ∈ (𝑅 RngHomo 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)))))

Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦   𝑥,𝐹,𝑦

Allowed substitution hints:   · (𝑥,𝑦)   ∗ (𝑥,𝑦)

Proof of Theorem isrnghm

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rnghmrcl 45335	. 2 ⊢ (𝐹 ∈ (𝑅 RngHomo 𝑆) → (𝑅 ∈ Rng ∧ 𝑆 ∈ Rng))
2		isrnghm.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
3		isrnghm.t	. . . . 5 ⊢ · = (.r‘𝑅)
4		isrnghm.m	. . . . 5 ⊢ ∗ = (.r‘𝑆)
5		eqid 2738	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
6		eqid 2738	. . . . 5 ⊢ (+g‘𝑅) = (+g‘𝑅)
7		eqid 2738	. . . . 5 ⊢ (+g‘𝑆) = (+g‘𝑆)
8	2, 3, 4, 5, 6, 7	rnghmval 45337	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝑅 RngHomo 𝑆) = {𝑓 ∈ ((Base‘𝑆) ↑m 𝐵) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥(+g‘𝑅)𝑦)) = ((𝑓‘𝑥)(+g‘𝑆)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) ∗ (𝑓‘𝑦)))})
9	8	eleq2d 2824	. . 3 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝐹 ∈ (𝑅 RngHomo 𝑆) ↔ 𝐹 ∈ {𝑓 ∈ ((Base‘𝑆) ↑m 𝐵) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥(+g‘𝑅)𝑦)) = ((𝑓‘𝑥)(+g‘𝑆)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) ∗ (𝑓‘𝑦)))}))
10		fveq1 6755	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥(+g‘𝑅)𝑦)) = (𝐹‘(𝑥(+g‘𝑅)𝑦)))
11		fveq1 6755	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
12		fveq1 6755	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦))
13	11, 12	oveq12d 7273	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥)(+g‘𝑆)(𝑓‘𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)))
14	10, 13	eqeq12d 2754	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑓‘(𝑥(+g‘𝑅)𝑦)) = ((𝑓‘𝑥)(+g‘𝑆)(𝑓‘𝑦)) ↔ (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦))))
15		fveq1 6755	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥 · 𝑦)) = (𝐹‘(𝑥 · 𝑦)))
16	11, 12	oveq12d 7273	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥) ∗ (𝑓‘𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)))
17	15, 16	eqeq12d 2754	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) ∗ (𝑓‘𝑦)) ↔ (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦))))
18	14, 17	anbi12d 630	. . . . . 6 ⊢ (𝑓 = 𝐹 → (((𝑓‘(𝑥(+g‘𝑅)𝑦)) = ((𝑓‘𝑥)(+g‘𝑆)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) ∗ (𝑓‘𝑦))) ↔ ((𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)))))
19	18	2ralbidv 3122	. . . . 5 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥(+g‘𝑅)𝑦)) = ((𝑓‘𝑥)(+g‘𝑆)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) ∗ (𝑓‘𝑦))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)))))
20	19	elrab 3617	. . . 4 ⊢ (𝐹 ∈ {𝑓 ∈ ((Base‘𝑆) ↑m 𝐵) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥(+g‘𝑅)𝑦)) = ((𝑓‘𝑥)(+g‘𝑆)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) ∗ (𝑓‘𝑦)))} ↔ (𝐹 ∈ ((Base‘𝑆) ↑m 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)))))
21		r19.26-2 3095	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦))) ↔ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦))))
22	21	anbi2i 622	. . . . . 6 ⊢ ((𝐹 ∈ ((Base‘𝑆) ↑m 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)))) ↔ (𝐹 ∈ ((Base‘𝑆) ↑m 𝐵) ∧ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)))))
23		anass 468	. . . . . 6 ⊢ (((𝐹 ∈ ((Base‘𝑆) ↑m 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦))) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦))) ↔ (𝐹 ∈ ((Base‘𝑆) ↑m 𝐵) ∧ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)))))
24	22, 23	bitr4i 277	. . . . 5 ⊢ ((𝐹 ∈ ((Base‘𝑆) ↑m 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)))) ↔ ((𝐹 ∈ ((Base‘𝑆) ↑m 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦))) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦))))
25	2, 5, 6, 7	isghm 18749	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) ↔ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ (𝐹:𝐵⟶(Base‘𝑆) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)))))
26		fvex 6769	. . . . . . . . . . 11 ⊢ (Base‘𝑆) ∈ V
27	2	fvexi 6770	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
28	26, 27	pm3.2i 470	. . . . . . . . . 10 ⊢ ((Base‘𝑆) ∈ V ∧ 𝐵 ∈ V)
29		elmapg 8586	. . . . . . . . . 10 ⊢ (((Base‘𝑆) ∈ V ∧ 𝐵 ∈ V) → (𝐹 ∈ ((Base‘𝑆) ↑m 𝐵) ↔ 𝐹:𝐵⟶(Base‘𝑆)))
30	28, 29	mp1i 13	. . . . . . . . 9 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝐹 ∈ ((Base‘𝑆) ↑m 𝐵) ↔ 𝐹:𝐵⟶(Base‘𝑆)))
31	30	anbi1d 629	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → ((𝐹 ∈ ((Base‘𝑆) ↑m 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦))) ↔ (𝐹:𝐵⟶(Base‘𝑆) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)))))
32		rngabl 45323	. . . . . . . . . 10 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
33		ablgrp 19306	. . . . . . . . . 10 ⊢ (𝑅 ∈ Abel → 𝑅 ∈ Grp)
34	32, 33	syl 17	. . . . . . . . 9 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
35		rngabl 45323	. . . . . . . . . 10 ⊢ (𝑆 ∈ Rng → 𝑆 ∈ Abel)
36		ablgrp 19306	. . . . . . . . . 10 ⊢ (𝑆 ∈ Abel → 𝑆 ∈ Grp)
37	35, 36	syl 17	. . . . . . . . 9 ⊢ (𝑆 ∈ Rng → 𝑆 ∈ Grp)
38		ibar 528	. . . . . . . . 9 ⊢ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) → ((𝐹:𝐵⟶(Base‘𝑆) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦))) ↔ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ (𝐹:𝐵⟶(Base‘𝑆) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦))))))
39	34, 37, 38	syl2an 595	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → ((𝐹:𝐵⟶(Base‘𝑆) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦))) ↔ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ (𝐹:𝐵⟶(Base‘𝑆) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦))))))
40	31, 39	bitr2d 279	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ (𝐹:𝐵⟶(Base‘𝑆) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)))) ↔ (𝐹 ∈ ((Base‘𝑆) ↑m 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)))))
41	25, 40	bitr2id 283	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → ((𝐹 ∈ ((Base‘𝑆) ↑m 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦))) ↔ 𝐹 ∈ (𝑅 GrpHom 𝑆)))
42	41	anbi1d 629	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (((𝐹 ∈ ((Base‘𝑆) ↑m 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦))) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦))) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)))))
43	24, 42	syl5bb 282	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → ((𝐹 ∈ ((Base‘𝑆) ↑m 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)))) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)))))
44	20, 43	syl5bb 282	. . 3 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝐹 ∈ {𝑓 ∈ ((Base‘𝑆) ↑m 𝐵) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥(+g‘𝑅)𝑦)) = ((𝑓‘𝑥)(+g‘𝑆)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) ∗ (𝑓‘𝑦)))} ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)))))
45	9, 44	bitrd 278	. 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝐹 ∈ (𝑅 RngHomo 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)))))
46	1, 45	biadanii 818	1 ⊢ (𝐹 ∈ (𝑅 RngHomo 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)))))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  {crab 3067  Vcvv 3422  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ↑_m cmap 8573  Basecbs 16840  +_gcplusg 16888  ._rcmulr 16889  Grpcgrp 18492   GrpHom cghm 18746  Abelcabl 19302  Rngcrng 45320   RngHomo crngh 45331

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-map 8575  df-ghm 18747  df-abl 19304  df-rng0 45321  df-rnghomo 45333

This theorem is referenced by:  isrnghmmul  45339  rnghmghm  45344  rnghmmul  45346  isrnghm2d  45347  zrrnghm  45363