Metamath Proof Explorer


Theorem frobrhm

Description: In a commutative ring with prime characteristic, the Frobenius function F is a ring endomorphism, thus named the Frobenius endomorphism. (Contributed by Thierry Arnoux, 31-May-2024)

Ref Expression
Hypotheses frobrhm.1 𝐵 = ( Base ‘ 𝑅 )
frobrhm.2 𝑃 = ( chr ‘ 𝑅 )
frobrhm.3 = ( .g ‘ ( mulGrp ‘ 𝑅 ) )
frobrhm.4 𝐹 = ( 𝑥𝐵 ↦ ( 𝑃 𝑥 ) )
frobrhm.5 ( 𝜑𝑅 ∈ CRing )
frobrhm.6 ( 𝜑𝑃 ∈ ℙ )
Assertion frobrhm ( 𝜑𝐹 ∈ ( 𝑅 RingHom 𝑅 ) )

Proof

Step Hyp Ref Expression
1 frobrhm.1 𝐵 = ( Base ‘ 𝑅 )
2 frobrhm.2 𝑃 = ( chr ‘ 𝑅 )
3 frobrhm.3 = ( .g ‘ ( mulGrp ‘ 𝑅 ) )
4 frobrhm.4 𝐹 = ( 𝑥𝐵 ↦ ( 𝑃 𝑥 ) )
5 frobrhm.5 ( 𝜑𝑅 ∈ CRing )
6 frobrhm.6 ( 𝜑𝑃 ∈ ℙ )
7 eqid ( 1r𝑅 ) = ( 1r𝑅 )
8 eqid ( .r𝑅 ) = ( .r𝑅 )
9 5 crngringd ( 𝜑𝑅 ∈ Ring )
10 simpr ( ( 𝜑𝑥 = ( 1r𝑅 ) ) → 𝑥 = ( 1r𝑅 ) )
11 10 oveq2d ( ( 𝜑𝑥 = ( 1r𝑅 ) ) → ( 𝑃 𝑥 ) = ( 𝑃 ( 1r𝑅 ) ) )
12 eqid ( mulGrp ‘ 𝑅 ) = ( mulGrp ‘ 𝑅 )
13 12 ringmgp ( 𝑅 ∈ Ring → ( mulGrp ‘ 𝑅 ) ∈ Mnd )
14 9 13 syl ( 𝜑 → ( mulGrp ‘ 𝑅 ) ∈ Mnd )
15 prmnn ( 𝑃 ∈ ℙ → 𝑃 ∈ ℕ )
16 nnnn0 ( 𝑃 ∈ ℕ → 𝑃 ∈ ℕ0 )
17 6 15 16 3syl ( 𝜑𝑃 ∈ ℕ0 )
18 12 1 mgpbas 𝐵 = ( Base ‘ ( mulGrp ‘ 𝑅 ) )
19 12 7 ringidval ( 1r𝑅 ) = ( 0g ‘ ( mulGrp ‘ 𝑅 ) )
20 18 3 19 mulgnn0z ( ( ( mulGrp ‘ 𝑅 ) ∈ Mnd ∧ 𝑃 ∈ ℕ0 ) → ( 𝑃 ( 1r𝑅 ) ) = ( 1r𝑅 ) )
21 14 17 20 syl2anc ( 𝜑 → ( 𝑃 ( 1r𝑅 ) ) = ( 1r𝑅 ) )
22 21 adantr ( ( 𝜑𝑥 = ( 1r𝑅 ) ) → ( 𝑃 ( 1r𝑅 ) ) = ( 1r𝑅 ) )
23 11 22 eqtrd ( ( 𝜑𝑥 = ( 1r𝑅 ) ) → ( 𝑃 𝑥 ) = ( 1r𝑅 ) )
24 1 7 ringidcl ( 𝑅 ∈ Ring → ( 1r𝑅 ) ∈ 𝐵 )
25 9 24 syl ( 𝜑 → ( 1r𝑅 ) ∈ 𝐵 )
26 4 23 25 25 fvmptd2 ( 𝜑 → ( 𝐹 ‘ ( 1r𝑅 ) ) = ( 1r𝑅 ) )
27 12 crngmgp ( 𝑅 ∈ CRing → ( mulGrp ‘ 𝑅 ) ∈ CMnd )
28 5 27 syl ( 𝜑 → ( mulGrp ‘ 𝑅 ) ∈ CMnd )
29 28 adantr ( ( 𝜑 ∧ ( 𝑖𝐵𝑗𝐵 ) ) → ( mulGrp ‘ 𝑅 ) ∈ CMnd )
30 17 adantr ( ( 𝜑 ∧ ( 𝑖𝐵𝑗𝐵 ) ) → 𝑃 ∈ ℕ0 )
31 simprl ( ( 𝜑 ∧ ( 𝑖𝐵𝑗𝐵 ) ) → 𝑖𝐵 )
32 simprr ( ( 𝜑 ∧ ( 𝑖𝐵𝑗𝐵 ) ) → 𝑗𝐵 )
33 12 8 mgpplusg ( .r𝑅 ) = ( +g ‘ ( mulGrp ‘ 𝑅 ) )
34 18 3 33 mulgnn0di ( ( ( mulGrp ‘ 𝑅 ) ∈ CMnd ∧ ( 𝑃 ∈ ℕ0𝑖𝐵𝑗𝐵 ) ) → ( 𝑃 ( 𝑖 ( .r𝑅 ) 𝑗 ) ) = ( ( 𝑃 𝑖 ) ( .r𝑅 ) ( 𝑃 𝑗 ) ) )
35 29 30 31 32 34 syl13anc ( ( 𝜑 ∧ ( 𝑖𝐵𝑗𝐵 ) ) → ( 𝑃 ( 𝑖 ( .r𝑅 ) 𝑗 ) ) = ( ( 𝑃 𝑖 ) ( .r𝑅 ) ( 𝑃 𝑗 ) ) )
36 simpr ( ( ( 𝜑 ∧ ( 𝑖𝐵𝑗𝐵 ) ) ∧ 𝑥 = ( 𝑖 ( .r𝑅 ) 𝑗 ) ) → 𝑥 = ( 𝑖 ( .r𝑅 ) 𝑗 ) )
37 36 oveq2d ( ( ( 𝜑 ∧ ( 𝑖𝐵𝑗𝐵 ) ) ∧ 𝑥 = ( 𝑖 ( .r𝑅 ) 𝑗 ) ) → ( 𝑃 𝑥 ) = ( 𝑃 ( 𝑖 ( .r𝑅 ) 𝑗 ) ) )
38 9 adantr ( ( 𝜑 ∧ ( 𝑖𝐵𝑗𝐵 ) ) → 𝑅 ∈ Ring )
39 1 8 ringcl ( ( 𝑅 ∈ Ring ∧ 𝑖𝐵𝑗𝐵 ) → ( 𝑖 ( .r𝑅 ) 𝑗 ) ∈ 𝐵 )
40 38 31 32 39 syl3anc ( ( 𝜑 ∧ ( 𝑖𝐵𝑗𝐵 ) ) → ( 𝑖 ( .r𝑅 ) 𝑗 ) ∈ 𝐵 )
41 ovexd ( ( 𝜑 ∧ ( 𝑖𝐵𝑗𝐵 ) ) → ( 𝑃 ( 𝑖 ( .r𝑅 ) 𝑗 ) ) ∈ V )
42 4 37 40 41 fvmptd2 ( ( 𝜑 ∧ ( 𝑖𝐵𝑗𝐵 ) ) → ( 𝐹 ‘ ( 𝑖 ( .r𝑅 ) 𝑗 ) ) = ( 𝑃 ( 𝑖 ( .r𝑅 ) 𝑗 ) ) )
43 simpr ( ( ( 𝜑 ∧ ( 𝑖𝐵𝑗𝐵 ) ) ∧ 𝑥 = 𝑖 ) → 𝑥 = 𝑖 )
44 43 oveq2d ( ( ( 𝜑 ∧ ( 𝑖𝐵𝑗𝐵 ) ) ∧ 𝑥 = 𝑖 ) → ( 𝑃 𝑥 ) = ( 𝑃 𝑖 ) )
45 ovexd ( ( 𝜑 ∧ ( 𝑖𝐵𝑗𝐵 ) ) → ( 𝑃 𝑖 ) ∈ V )
46 4 44 31 45 fvmptd2 ( ( 𝜑 ∧ ( 𝑖𝐵𝑗𝐵 ) ) → ( 𝐹𝑖 ) = ( 𝑃 𝑖 ) )
47 simpr ( ( ( 𝜑 ∧ ( 𝑖𝐵𝑗𝐵 ) ) ∧ 𝑥 = 𝑗 ) → 𝑥 = 𝑗 )
48 47 oveq2d ( ( ( 𝜑 ∧ ( 𝑖𝐵𝑗𝐵 ) ) ∧ 𝑥 = 𝑗 ) → ( 𝑃 𝑥 ) = ( 𝑃 𝑗 ) )
49 ovexd ( ( 𝜑 ∧ ( 𝑖𝐵𝑗𝐵 ) ) → ( 𝑃 𝑗 ) ∈ V )
50 4 48 32 49 fvmptd2 ( ( 𝜑 ∧ ( 𝑖𝐵𝑗𝐵 ) ) → ( 𝐹𝑗 ) = ( 𝑃 𝑗 ) )
51 46 50 oveq12d ( ( 𝜑 ∧ ( 𝑖𝐵𝑗𝐵 ) ) → ( ( 𝐹𝑖 ) ( .r𝑅 ) ( 𝐹𝑗 ) ) = ( ( 𝑃 𝑖 ) ( .r𝑅 ) ( 𝑃 𝑗 ) ) )
52 35 42 51 3eqtr4d ( ( 𝜑 ∧ ( 𝑖𝐵𝑗𝐵 ) ) → ( 𝐹 ‘ ( 𝑖 ( .r𝑅 ) 𝑗 ) ) = ( ( 𝐹𝑖 ) ( .r𝑅 ) ( 𝐹𝑗 ) ) )
53 eqid ( +g𝑅 ) = ( +g𝑅 )
54 14 adantr ( ( 𝜑𝑥𝐵 ) → ( mulGrp ‘ 𝑅 ) ∈ Mnd )
55 17 adantr ( ( 𝜑𝑥𝐵 ) → 𝑃 ∈ ℕ0 )
56 simpr ( ( 𝜑𝑥𝐵 ) → 𝑥𝐵 )
57 18 3 mulgnn0cl ( ( ( mulGrp ‘ 𝑅 ) ∈ Mnd ∧ 𝑃 ∈ ℕ0𝑥𝐵 ) → ( 𝑃 𝑥 ) ∈ 𝐵 )
58 54 55 56 57 syl3anc ( ( 𝜑𝑥𝐵 ) → ( 𝑃 𝑥 ) ∈ 𝐵 )
59 58 4 fmptd ( 𝜑𝐹 : 𝐵𝐵 )
60 5 adantr ( ( 𝜑 ∧ ( 𝑖𝐵𝑗𝐵 ) ) → 𝑅 ∈ CRing )
61 6 adantr ( ( 𝜑 ∧ ( 𝑖𝐵𝑗𝐵 ) ) → 𝑃 ∈ ℙ )
62 1 53 3 2 60 61 31 32 freshmansdream ( ( 𝜑 ∧ ( 𝑖𝐵𝑗𝐵 ) ) → ( 𝑃 ( 𝑖 ( +g𝑅 ) 𝑗 ) ) = ( ( 𝑃 𝑖 ) ( +g𝑅 ) ( 𝑃 𝑗 ) ) )
63 simpr ( ( ( 𝜑 ∧ ( 𝑖𝐵𝑗𝐵 ) ) ∧ 𝑥 = ( 𝑖 ( +g𝑅 ) 𝑗 ) ) → 𝑥 = ( 𝑖 ( +g𝑅 ) 𝑗 ) )
64 63 oveq2d ( ( ( 𝜑 ∧ ( 𝑖𝐵𝑗𝐵 ) ) ∧ 𝑥 = ( 𝑖 ( +g𝑅 ) 𝑗 ) ) → ( 𝑃 𝑥 ) = ( 𝑃 ( 𝑖 ( +g𝑅 ) 𝑗 ) ) )
65 1 53 ringacl ( ( 𝑅 ∈ Ring ∧ 𝑖𝐵𝑗𝐵 ) → ( 𝑖 ( +g𝑅 ) 𝑗 ) ∈ 𝐵 )
66 38 31 32 65 syl3anc ( ( 𝜑 ∧ ( 𝑖𝐵𝑗𝐵 ) ) → ( 𝑖 ( +g𝑅 ) 𝑗 ) ∈ 𝐵 )
67 ovexd ( ( 𝜑 ∧ ( 𝑖𝐵𝑗𝐵 ) ) → ( 𝑃 ( 𝑖 ( +g𝑅 ) 𝑗 ) ) ∈ V )
68 4 64 66 67 fvmptd2 ( ( 𝜑 ∧ ( 𝑖𝐵𝑗𝐵 ) ) → ( 𝐹 ‘ ( 𝑖 ( +g𝑅 ) 𝑗 ) ) = ( 𝑃 ( 𝑖 ( +g𝑅 ) 𝑗 ) ) )
69 46 50 oveq12d ( ( 𝜑 ∧ ( 𝑖𝐵𝑗𝐵 ) ) → ( ( 𝐹𝑖 ) ( +g𝑅 ) ( 𝐹𝑗 ) ) = ( ( 𝑃 𝑖 ) ( +g𝑅 ) ( 𝑃 𝑗 ) ) )
70 62 68 69 3eqtr4d ( ( 𝜑 ∧ ( 𝑖𝐵𝑗𝐵 ) ) → ( 𝐹 ‘ ( 𝑖 ( +g𝑅 ) 𝑗 ) ) = ( ( 𝐹𝑖 ) ( +g𝑅 ) ( 𝐹𝑗 ) ) )
71 1 7 7 8 8 9 9 26 52 1 53 53 59 70 isrhmd ( 𝜑𝐹 ∈ ( 𝑅 RingHom 𝑅 ) )