Theorem elrhmunit

Description: Ring homomorphisms preserve unit elements. (Contributed by Thierry Arnoux, 23-Oct-2017)

	Ref	Expression
Assertion	elrhmunit	⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → ( 𝐹 ‘ 𝐴 ) ∈ ( Unit ‘ 𝑆 ) )

Proof

Step	Hyp	Ref	Expression
1		simpl	⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) )
2		eqid	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
3		eqid	⊢ ( Unit ‘ 𝑅 ) = ( Unit ‘ 𝑅 )
4	2 3	unitss	⊢ ( Unit ‘ 𝑅 ) ⊆ ( Base ‘ 𝑅 )
5		simpr	⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → 𝐴 ∈ ( Unit ‘ 𝑅 ) )
6	4 5	sselid	⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → 𝐴 ∈ ( Base ‘ 𝑅 ) )
7		rhmrcl1	⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → 𝑅 ∈ Ring )
8		eqid	⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 )
9	2 8	ringidcl	⊢ ( 𝑅 ∈ Ring → ( 1r ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) )
10	1 7 9	3syl	⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → ( 1r ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) )
11		eqid	⊢ ( ∥r ‘ 𝑅 ) = ( ∥r ‘ 𝑅 )
12		eqid	⊢ ( oppr ‘ 𝑅 ) = ( oppr ‘ 𝑅 )
13		eqid	⊢ ( ∥r ‘ ( oppr ‘ 𝑅 ) ) = ( ∥r ‘ ( oppr ‘ 𝑅 ) )
14	3 8 11 12 13	isunit	⊢ ( 𝐴 ∈ ( Unit ‘ 𝑅 ) ↔ ( 𝐴 ( ∥r ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ∧ 𝐴 ( ∥r ‘ ( oppr ‘ 𝑅 ) ) ( 1r ‘ 𝑅 ) ) )
15	14	bilani	⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → ( 𝐴 ( ∥r ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ∧ 𝐴 ( ∥r ‘ ( oppr ‘ 𝑅 ) ) ( 1r ‘ 𝑅 ) ) )
16	15	simpld	⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → 𝐴 ( ∥r ‘ 𝑅 ) ( 1r ‘ 𝑅 ) )
17		eqid	⊢ ( ∥r ‘ 𝑆 ) = ( ∥r ‘ 𝑆 )
18	2 11 17	rhmdvdsr	⊢ ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Base ‘ 𝑅 ) ∧ ( 1r ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) ) ∧ 𝐴 ( ∥r ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ) → ( 𝐹 ‘ 𝐴 ) ( ∥r ‘ 𝑆 ) ( 𝐹 ‘ ( 1r ‘ 𝑅 ) ) )
19	1 6 10 16 18	syl31anc	⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → ( 𝐹 ‘ 𝐴 ) ( ∥r ‘ 𝑆 ) ( 𝐹 ‘ ( 1r ‘ 𝑅 ) ) )
20		eqid	⊢ ( 1r ‘ 𝑆 ) = ( 1r ‘ 𝑆 )
21	8 20	rhm1	⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → ( 𝐹 ‘ ( 1r ‘ 𝑅 ) ) = ( 1r ‘ 𝑆 ) )
22	21	breq2d	⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → ( ( 𝐹 ‘ 𝐴 ) ( ∥r ‘ 𝑆 ) ( 𝐹 ‘ ( 1r ‘ 𝑅 ) ) ↔ ( 𝐹 ‘ 𝐴 ) ( ∥r ‘ 𝑆 ) ( 1r ‘ 𝑆 ) ) )
23	22	adantr	⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → ( ( 𝐹 ‘ 𝐴 ) ( ∥r ‘ 𝑆 ) ( 𝐹 ‘ ( 1r ‘ 𝑅 ) ) ↔ ( 𝐹 ‘ 𝐴 ) ( ∥r ‘ 𝑆 ) ( 1r ‘ 𝑆 ) ) )
24	19 23	mpbid	⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → ( 𝐹 ‘ 𝐴 ) ( ∥r ‘ 𝑆 ) ( 1r ‘ 𝑆 ) )
25		rhmopp	⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → 𝐹 ∈ ( ( oppr ‘ 𝑅 ) RingHom ( oppr ‘ 𝑆 ) ) )
26	25	adantr	⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → 𝐹 ∈ ( ( oppr ‘ 𝑅 ) RingHom ( oppr ‘ 𝑆 ) ) )
27	15	simprd	⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → 𝐴 ( ∥r ‘ ( oppr ‘ 𝑅 ) ) ( 1r ‘ 𝑅 ) )
28	12 2	opprbas	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ ( oppr ‘ 𝑅 ) )
29		eqid	⊢ ( ∥r ‘ ( oppr ‘ 𝑆 ) ) = ( ∥r ‘ ( oppr ‘ 𝑆 ) )
30	28 13 29	rhmdvdsr	⊢ ( ( ( 𝐹 ∈ ( ( oppr ‘ 𝑅 ) RingHom ( oppr ‘ 𝑆 ) ) ∧ 𝐴 ∈ ( Base ‘ 𝑅 ) ∧ ( 1r ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) ) ∧ 𝐴 ( ∥r ‘ ( oppr ‘ 𝑅 ) ) ( 1r ‘ 𝑅 ) ) → ( 𝐹 ‘ 𝐴 ) ( ∥r ‘ ( oppr ‘ 𝑆 ) ) ( 𝐹 ‘ ( 1r ‘ 𝑅 ) ) )
31	26 6 10 27 30	syl31anc	⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → ( 𝐹 ‘ 𝐴 ) ( ∥r ‘ ( oppr ‘ 𝑆 ) ) ( 𝐹 ‘ ( 1r ‘ 𝑅 ) ) )
32	21	breq2d	⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → ( ( 𝐹 ‘ 𝐴 ) ( ∥r ‘ ( oppr ‘ 𝑆 ) ) ( 𝐹 ‘ ( 1r ‘ 𝑅 ) ) ↔ ( 𝐹 ‘ 𝐴 ) ( ∥r ‘ ( oppr ‘ 𝑆 ) ) ( 1r ‘ 𝑆 ) ) )
33	32	adantr	⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → ( ( 𝐹 ‘ 𝐴 ) ( ∥r ‘ ( oppr ‘ 𝑆 ) ) ( 𝐹 ‘ ( 1r ‘ 𝑅 ) ) ↔ ( 𝐹 ‘ 𝐴 ) ( ∥r ‘ ( oppr ‘ 𝑆 ) ) ( 1r ‘ 𝑆 ) ) )
34	31 33	mpbid	⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → ( 𝐹 ‘ 𝐴 ) ( ∥r ‘ ( oppr ‘ 𝑆 ) ) ( 1r ‘ 𝑆 ) )
35		eqid	⊢ ( Unit ‘ 𝑆 ) = ( Unit ‘ 𝑆 )
36		eqid	⊢ ( oppr ‘ 𝑆 ) = ( oppr ‘ 𝑆 )
37	35 20 17 36 29	isunit	⊢ ( ( 𝐹 ‘ 𝐴 ) ∈ ( Unit ‘ 𝑆 ) ↔ ( ( 𝐹 ‘ 𝐴 ) ( ∥r ‘ 𝑆 ) ( 1r ‘ 𝑆 ) ∧ ( 𝐹 ‘ 𝐴 ) ( ∥r ‘ ( oppr ‘ 𝑆 ) ) ( 1r ‘ 𝑆 ) ) )
38	24 34 37	sylanbrc	⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → ( 𝐹 ‘ 𝐴 ) ∈ ( Unit ‘ 𝑆 ) )