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Metamath Proof Explorer

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 ) ) ) ) )