Metamath Proof Explorer

Theorem rngisom1

Description: If there is a non-unital ring isomorphism between a unital ring and a non-unital ring, then the function value of the ring unity of the unital ring is a ring unity of the non-unital ring. (Contributed by AV, 27-Feb-2025)

Ref Expression
Hypotheses rngisom1.1 ⊢ 1 = ( 1r ‘ 𝑅 )
rngisom1.b ⊢ 𝐵 = ( Base ‘ 𝑆 )
rngisom1.t ⊢ · = ( .r ‘ 𝑆 )
Assertion rngisom1 ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) → ∀ 𝑥 ∈ 𝐵 ( ( ( 𝐹 ‘ 1 ) · 𝑥 ) = 𝑥 ∧ ( 𝑥 · ( 𝐹 ‘ 1 ) ) = 𝑥 ) )

Proof

Step Hyp Ref Expression
1 rngisom1.1 ⊢ 1 = ( 1r ‘ 𝑅 )
2 rngisom1.b ⊢ 𝐵 = ( Base ‘ 𝑆 )
3 rngisom1.t ⊢ · = ( .r ‘ 𝑆 )
4 rngimcnv ⊢ ( 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) → ◡ 𝐹 ∈ ( 𝑆 RngIso 𝑅 ) )
5 eqid ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
6 2 5 rngimrnghm ⊢ ( ◡ 𝐹 ∈ ( 𝑆 RngIso 𝑅 ) → ◡ 𝐹 ∈ ( 𝑆 RngHom 𝑅 ) )
7 4 6 syl ⊢ ( 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) → ◡ 𝐹 ∈ ( 𝑆 RngHom 𝑅 ) )
8 7 3ad2ant3 ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) → ◡ 𝐹 ∈ ( 𝑆 RngHom 𝑅 ) )
9 8 adantr ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) ∧ 𝑥 ∈ 𝐵 ) → ◡ 𝐹 ∈ ( 𝑆 RngHom 𝑅 ) )
10 1 2 rngisomfv1 ⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) → ( 𝐹 ‘ 1 ) ∈ 𝐵 )
11 10 3adant2 ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) → ( 𝐹 ‘ 1 ) ∈ 𝐵 )
12 11 adantr ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) ∧ 𝑥 ∈ 𝐵 ) → ( 𝐹 ‘ 1 ) ∈ 𝐵 )
13 simpr ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ 𝐵 )
14 eqid ⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 )
15 2 3 14 rnghmmul ⊢ ( ( ◡ 𝐹 ∈ ( 𝑆 RngHom 𝑅 ) ∧ ( 𝐹 ‘ 1 ) ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 1 ) · 𝑥 ) ) = ( ( ◡ 𝐹 ‘ ( 𝐹 ‘ 1 ) ) ( .r ‘ 𝑅 ) ( ◡ 𝐹 ‘ 𝑥 ) ) )
16 9 12 13 15 syl3anc ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) ∧ 𝑥 ∈ 𝐵 ) → ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 1 ) · 𝑥 ) ) = ( ( ◡ 𝐹 ‘ ( 𝐹 ‘ 1 ) ) ( .r ‘ 𝑅 ) ( ◡ 𝐹 ‘ 𝑥 ) ) )
17 16 fveq2d ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) ∧ 𝑥 ∈ 𝐵 ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 1 ) · 𝑥 ) ) ) = ( 𝐹 ‘ ( ( ◡ 𝐹 ‘ ( 𝐹 ‘ 1 ) ) ( .r ‘ 𝑅 ) ( ◡ 𝐹 ‘ 𝑥 ) ) ) )
18 5 2 rngimf1o ⊢ ( 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) → 𝐹 : ( Base ‘ 𝑅 ) –1-1-onto→ 𝐵 )
19 18 3ad2ant3 ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) → 𝐹 : ( Base ‘ 𝑅 ) –1-1-onto→ 𝐵 )
20 simpl2 ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) ∧ 𝑥 ∈ 𝐵 ) → 𝑆 ∈ Rng )
21 2 3 rngcl ⊢ ( ( 𝑆 ∈ Rng ∧ ( 𝐹 ‘ 1 ) ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( ( 𝐹 ‘ 1 ) · 𝑥 ) ∈ 𝐵 )
22 20 12 13 21 syl3anc ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) ∧ 𝑥 ∈ 𝐵 ) → ( ( 𝐹 ‘ 1 ) · 𝑥 ) ∈ 𝐵 )
23 f1ocnvfv2 ⊢ ( ( 𝐹 : ( Base ‘ 𝑅 ) –1-1-onto→ 𝐵 ∧ ( ( 𝐹 ‘ 1 ) · 𝑥 ) ∈ 𝐵 ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 1 ) · 𝑥 ) ) ) = ( ( 𝐹 ‘ 1 ) · 𝑥 ) )
24 19 22 23 syl2an2r ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) ∧ 𝑥 ∈ 𝐵 ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ ( ( 𝐹 ‘ 1 ) · 𝑥 ) ) ) = ( ( 𝐹 ‘ 1 ) · 𝑥 ) )
25 5 1 ringidcl ⊢ ( 𝑅 ∈ Ring → 1 ∈ ( Base ‘ 𝑅 ) )
26 25 3ad2ant1 ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) → 1 ∈ ( Base ‘ 𝑅 ) )
27 19 26 jca ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) → ( 𝐹 : ( Base ‘ 𝑅 ) –1-1-onto→ 𝐵 ∧ 1 ∈ ( Base ‘ 𝑅 ) ) )
28 27 adantr ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) ∧ 𝑥 ∈ 𝐵 ) → ( 𝐹 : ( Base ‘ 𝑅 ) –1-1-onto→ 𝐵 ∧ 1 ∈ ( Base ‘ 𝑅 ) ) )
29 f1ocnvfv1 ⊢ ( ( 𝐹 : ( Base ‘ 𝑅 ) –1-1-onto→ 𝐵 ∧ 1 ∈ ( Base ‘ 𝑅 ) ) → ( ◡ 𝐹 ‘ ( 𝐹 ‘ 1 ) ) = 1 )
30 28 29 syl ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) ∧ 𝑥 ∈ 𝐵 ) → ( ◡ 𝐹 ‘ ( 𝐹 ‘ 1 ) ) = 1 )
31 30 oveq1d ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) ∧ 𝑥 ∈ 𝐵 ) → ( ( ◡ 𝐹 ‘ ( 𝐹 ‘ 1 ) ) ( .r ‘ 𝑅 ) ( ◡ 𝐹 ‘ 𝑥 ) ) = ( 1 ( .r ‘ 𝑅 ) ( ◡ 𝐹 ‘ 𝑥 ) ) )
32 simpl1 ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) ∧ 𝑥 ∈ 𝐵 ) → 𝑅 ∈ Ring )
33 2 5 rngimf1o ⊢ ( ◡ 𝐹 ∈ ( 𝑆 RngIso 𝑅 ) → ◡ 𝐹 : 𝐵 –1-1-onto→ ( Base ‘ 𝑅 ) )
34 f1of ⊢ ( ◡ 𝐹 : 𝐵 –1-1-onto→ ( Base ‘ 𝑅 ) → ◡ 𝐹 : 𝐵 ⟶ ( Base ‘ 𝑅 ) )
35 33 34 syl ⊢ ( ◡ 𝐹 ∈ ( 𝑆 RngIso 𝑅 ) → ◡ 𝐹 : 𝐵 ⟶ ( Base ‘ 𝑅 ) )
36 4 35 syl ⊢ ( 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) → ◡ 𝐹 : 𝐵 ⟶ ( Base ‘ 𝑅 ) )
37 36 3ad2ant3 ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) → ◡ 𝐹 : 𝐵 ⟶ ( Base ‘ 𝑅 ) )
38 37 ffvelcdmda ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) ∧ 𝑥 ∈ 𝐵 ) → ( ◡ 𝐹 ‘ 𝑥 ) ∈ ( Base ‘ 𝑅 ) )
39 5 14 1 32 38 ringlidmd ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) ∧ 𝑥 ∈ 𝐵 ) → ( 1 ( .r ‘ 𝑅 ) ( ◡ 𝐹 ‘ 𝑥 ) ) = ( ◡ 𝐹 ‘ 𝑥 ) )
40 31 39 eqtrd ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) ∧ 𝑥 ∈ 𝐵 ) → ( ( ◡ 𝐹 ‘ ( 𝐹 ‘ 1 ) ) ( .r ‘ 𝑅 ) ( ◡ 𝐹 ‘ 𝑥 ) ) = ( ◡ 𝐹 ‘ 𝑥 ) )
41 40 fveq2d ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) ∧ 𝑥 ∈ 𝐵 ) → ( 𝐹 ‘ ( ( ◡ 𝐹 ‘ ( 𝐹 ‘ 1 ) ) ( .r ‘ 𝑅 ) ( ◡ 𝐹 ‘ 𝑥 ) ) ) = ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑥 ) ) )
42 f1ocnvfv2 ⊢ ( ( 𝐹 : ( Base ‘ 𝑅 ) –1-1-onto→ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑥 ) ) = 𝑥 )
43 19 42 sylan ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) ∧ 𝑥 ∈ 𝐵 ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑥 ) ) = 𝑥 )
44 41 43 eqtrd ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) ∧ 𝑥 ∈ 𝐵 ) → ( 𝐹 ‘ ( ( ◡ 𝐹 ‘ ( 𝐹 ‘ 1 ) ) ( .r ‘ 𝑅 ) ( ◡ 𝐹 ‘ 𝑥 ) ) ) = 𝑥 )
45 17 24 44 3eqtr3d ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) ∧ 𝑥 ∈ 𝐵 ) → ( ( 𝐹 ‘ 1 ) · 𝑥 ) = 𝑥 )
46 4 3ad2ant3 ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) → ◡ 𝐹 ∈ ( 𝑆 RngIso 𝑅 ) )
47 46 6 syl ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) → ◡ 𝐹 ∈ ( 𝑆 RngHom 𝑅 ) )
48 47 adantr ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) ∧ 𝑥 ∈ 𝐵 ) → ◡ 𝐹 ∈ ( 𝑆 RngHom 𝑅 ) )
49 2 3 14 rnghmmul ⊢ ( ( ◡ 𝐹 ∈ ( 𝑆 RngHom 𝑅 ) ∧ 𝑥 ∈ 𝐵 ∧ ( 𝐹 ‘ 1 ) ∈ 𝐵 ) → ( ◡ 𝐹 ‘ ( 𝑥 · ( 𝐹 ‘ 1 ) ) ) = ( ( ◡ 𝐹 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( ◡ 𝐹 ‘ ( 𝐹 ‘ 1 ) ) ) )
50 48 13 12 49 syl3anc ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) ∧ 𝑥 ∈ 𝐵 ) → ( ◡ 𝐹 ‘ ( 𝑥 · ( 𝐹 ‘ 1 ) ) ) = ( ( ◡ 𝐹 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( ◡ 𝐹 ‘ ( 𝐹 ‘ 1 ) ) ) )
51 30 oveq2d ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) ∧ 𝑥 ∈ 𝐵 ) → ( ( ◡ 𝐹 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( ◡ 𝐹 ‘ ( 𝐹 ‘ 1 ) ) ) = ( ( ◡ 𝐹 ‘ 𝑥 ) ( .r ‘ 𝑅 ) 1 ) )
52 5 14 1 32 38 ringridmd ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) ∧ 𝑥 ∈ 𝐵 ) → ( ( ◡ 𝐹 ‘ 𝑥 ) ( .r ‘ 𝑅 ) 1 ) = ( ◡ 𝐹 ‘ 𝑥 ) )
53 50 51 52 3eqtrd ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) ∧ 𝑥 ∈ 𝐵 ) → ( ◡ 𝐹 ‘ ( 𝑥 · ( 𝐹 ‘ 1 ) ) ) = ( ◡ 𝐹 ‘ 𝑥 ) )
54 53 fveq2d ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) ∧ 𝑥 ∈ 𝐵 ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ ( 𝑥 · ( 𝐹 ‘ 1 ) ) ) ) = ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑥 ) ) )
55 2 3 rngcl ⊢ ( ( 𝑆 ∈ Rng ∧ 𝑥 ∈ 𝐵 ∧ ( 𝐹 ‘ 1 ) ∈ 𝐵 ) → ( 𝑥 · ( 𝐹 ‘ 1 ) ) ∈ 𝐵 )
56 20 13 12 55 syl3anc ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) ∧ 𝑥 ∈ 𝐵 ) → ( 𝑥 · ( 𝐹 ‘ 1 ) ) ∈ 𝐵 )
57 f1ocnvfv2 ⊢ ( ( 𝐹 : ( Base ‘ 𝑅 ) –1-1-onto→ 𝐵 ∧ ( 𝑥 · ( 𝐹 ‘ 1 ) ) ∈ 𝐵 ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ ( 𝑥 · ( 𝐹 ‘ 1 ) ) ) ) = ( 𝑥 · ( 𝐹 ‘ 1 ) ) )
58 19 56 57 syl2an2r ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) ∧ 𝑥 ∈ 𝐵 ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ ( 𝑥 · ( 𝐹 ‘ 1 ) ) ) ) = ( 𝑥 · ( 𝐹 ‘ 1 ) ) )
59 54 58 43 3eqtr3d ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) ∧ 𝑥 ∈ 𝐵 ) → ( 𝑥 · ( 𝐹 ‘ 1 ) ) = 𝑥 )
60 45 59 jca ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) ∧ 𝑥 ∈ 𝐵 ) → ( ( ( 𝐹 ‘ 1 ) · 𝑥 ) = 𝑥 ∧ ( 𝑥 · ( 𝐹 ‘ 1 ) ) = 𝑥 ) )
61 60 ralrimiva ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) → ∀ 𝑥 ∈ 𝐵 ( ( ( 𝐹 ‘ 1 ) · 𝑥 ) = 𝑥 ∧ ( 𝑥 · ( 𝐹 ‘ 1 ) ) = 𝑥 ) )