Theorem isrnghm

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

	Ref	Expression
Hypotheses	isrnghm.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	isrnghm.t	⊢ · = ( .r ‘ 𝑅 )
	isrnghm.m	⊢ ∗ = ( .r ‘ 𝑆 )
Assertion	isrnghm	⊢ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ↔ ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) ∧ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ∗ ( 𝐹 ‘ 𝑦 ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		isrnghm.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		isrnghm.t	⊢ · = ( .r ‘ 𝑅 )
3		isrnghm.m	⊢ ∗ = ( .r ‘ 𝑆 )
4		rnghmrcl	⊢ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) → ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) )
5		eqid	⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
6		eqid	⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 )
7		eqid	⊢ ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 )
8	1 2 3 5 6 7	rnghmval	⊢ ( ( 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	⊢ ( ( 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	⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) ) = ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) ) )
11		fveq1	⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) )
12		fveq1	⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) )
13	11 12	oveq12d	⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑆 ) ( 𝑓 ‘ 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) )
14	10 13	eqeq12d	⊢ ( 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	⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ ( 𝑥 · 𝑦 ) ) = ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) )
16	11 12	oveq12d	⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 ‘ 𝑥 ) ∗ ( 𝑓 ‘ 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ∗ ( 𝐹 ‘ 𝑦 ) ) )
17	15 16	eqeq12d	⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ∗ ( 𝑓 ‘ 𝑦 ) ) ↔ ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ∗ ( 𝐹 ‘ 𝑦 ) ) ) )
18	14 17	anbi12d	⊢ ( 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	⊢ ( 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	⊢ ( 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	⊢ ( ∀ 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	⊢ ( ( 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	⊢ ( ( ( 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	⊢ ( ( 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	1 5 6 7	isghm	⊢ ( 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	⊢ ( Base ‘ 𝑆 ) ∈ V
27	1	fvexi	⊢ 𝐵 ∈ V
28	26 27	pm3.2i	⊢ ( ( Base ‘ 𝑆 ) ∈ V ∧ 𝐵 ∈ V )
29		elmapg	⊢ ( ( ( Base ‘ 𝑆 ) ∈ V ∧ 𝐵 ∈ V ) → ( 𝐹 ∈ ( ( Base ‘ 𝑆 ) ↑m 𝐵 ) ↔ 𝐹 : 𝐵 ⟶ ( Base ‘ 𝑆 ) ) )
30	28 29	mp1i	⊢ ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) → ( 𝐹 ∈ ( ( Base ‘ 𝑆 ) ↑m 𝐵 ) ↔ 𝐹 : 𝐵 ⟶ ( Base ‘ 𝑆 ) ) )
31	30	anbi1d	⊢ ( ( 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	⊢ ( 𝑅 ∈ Rng → 𝑅 ∈ Abel )
33		ablgrp	⊢ ( 𝑅 ∈ Abel → 𝑅 ∈ Grp )
34	32 33	syl	⊢ ( 𝑅 ∈ Rng → 𝑅 ∈ Grp )
35		rngabl	⊢ ( 𝑆 ∈ Rng → 𝑆 ∈ Abel )
36		ablgrp	⊢ ( 𝑆 ∈ Abel → 𝑆 ∈ Grp )
37	35 36	syl	⊢ ( 𝑆 ∈ Rng → 𝑆 ∈ Grp )
38		ibar	⊢ ( ( 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	⊢ ( ( 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	⊢ ( ( 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	⊢ ( ( 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	⊢ ( ( 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	⊢ ( ( 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	⊢ ( ( 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	⊢ ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) → ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ↔ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ∗ ( 𝐹 ‘ 𝑦 ) ) ) ) )
46	4 45	biadanii	⊢ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ↔ ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) ∧ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ∗ ( 𝐹 ‘ 𝑦 ) ) ) ) )