Theorem isrnghm 20406

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 RngHom 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 20403	. 2 ⊢ (𝐹 ∈ (𝑅 RngHom 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 2736	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
6		eqid 2736	. . . . 5 ⊢ (+g‘𝑅) = (+g‘𝑅)
7		eqid 2736	. . . . 5 ⊢ (+g‘𝑆) = (+g‘𝑆)
8	2, 3, 4, 5, 6, 7	rnghmval 20405	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝑅 RngHom 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 2821	. . 3 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝐹 ∈ (𝑅 RngHom 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 6880	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥(+g‘𝑅)𝑦)) = (𝐹‘(𝑥(+g‘𝑅)𝑦)))
11		fveq1 6880	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
12		fveq1 6880	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦))
13	11, 12	oveq12d 7428	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥)(+g‘𝑆)(𝑓‘𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)))
14	10, 13	eqeq12d 2752	. . . . . . 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 6880	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥 · 𝑦)) = (𝐹‘(𝑥 · 𝑦)))
16	11, 12	oveq12d 7428	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥) ∗ (𝑓‘𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)))
17	15, 16	eqeq12d 2752	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) ∗ (𝑓‘𝑦)) ↔ (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦))))
18	14, 17	anbi12d 632	. . . . . 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 3209	. . . . 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 3676	. . . 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 3126	. . . . . . 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 623	. . . . . 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 278	. . . . 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 19203	. . . . . . 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 6894	. . . . . . . . . . 11 ⊢ (Base‘𝑆) ∈ V
27	2	fvexi 6895	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
28	26, 27	pm3.2i 470	. . . . . . . . . 10 ⊢ ((Base‘𝑆) ∈ V ∧ 𝐵 ∈ V)
29		elmapg 8858	. . . . . . . . . 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 631	. . . . . . . 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 20120	. . . . . . . . . 10 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
33		ablgrp 19771	. . . . . . . . . 10 ⊢ (𝑅 ∈ Abel → 𝑅 ∈ Grp)
34	32, 33	syl 17	. . . . . . . . 9 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
35		rngabl 20120	. . . . . . . . . 10 ⊢ (𝑆 ∈ Rng → 𝑆 ∈ Abel)
36		ablgrp 19771	. . . . . . . . . 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 596	. . . . . . . 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 280	. . . . . . 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 284	. . . . . 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 631	. . . . 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	bitrid 283	. . . 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	bitrid 283	. . 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 279	. 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝐹 ∈ (𝑅 RngHom 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)))))
46	1, 45	biadanii 821	1 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)))))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3052  {crab 3420  Vcvv 3464  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ↑_m cmap 8845  Basecbs 17233  +_gcplusg 17276  ._rcmulr 17277  Grpcgrp 18921   GrpHom cghm 19200  Abelcabl 19767  Rngcrng 20117   RngHom crnghm 20399

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7993  df-2nd 7994  df-map 8847  df-ghm 19201  df-abl 19769  df-rng 20118  df-rnghm 20401

This theorem is referenced by:  isrnghmmul  20407  rnghmghm  20412  rnghmmul  20414  isrnghm2d  20415  zrrnghm  20501  rngqiprngho  21269