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 = ( 𝑥 ∈ 𝐵 ↦ ( 𝑃 ↑ 𝑥 ) ) |
5 |
|
frobrhm.5 |
⊢ ( 𝜑 → 𝑅 ∈ CRing ) |
6 |
|
frobrhm.6 |
⊢ ( 𝜑 → 𝑃 ∈ ℙ ) |
7 |
|
eqid |
⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 ) |
8 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
9 |
5
|
crngringd |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
10 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 = ( 1r ‘ 𝑅 ) ) → 𝑥 = ( 1r ‘ 𝑅 ) ) |
11 |
10
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑥 = ( 1r ‘ 𝑅 ) ) → ( 𝑃 ↑ 𝑥 ) = ( 𝑃 ↑ ( 1r ‘ 𝑅 ) ) ) |
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 |
⊢ ( 𝜑 → 𝑃 ∈ ℕ0 ) |
18 |
12 1
|
mgpbas |
⊢ 𝐵 = ( Base ‘ ( mulGrp ‘ 𝑅 ) ) |
19 |
12 7
|
ringidval |
⊢ ( 1r ‘ 𝑅 ) = ( 0g ‘ ( mulGrp ‘ 𝑅 ) ) |
20 |
18 3 19
|
mulgnn0z |
⊢ ( ( ( mulGrp ‘ 𝑅 ) ∈ Mnd ∧ 𝑃 ∈ ℕ0 ) → ( 𝑃 ↑ ( 1r ‘ 𝑅 ) ) = ( 1r ‘ 𝑅 ) ) |
21 |
14 17 20
|
syl2anc |
⊢ ( 𝜑 → ( 𝑃 ↑ ( 1r ‘ 𝑅 ) ) = ( 1r ‘ 𝑅 ) ) |
22 |
21
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 = ( 1r ‘ 𝑅 ) ) → ( 𝑃 ↑ ( 1r ‘ 𝑅 ) ) = ( 1r ‘ 𝑅 ) ) |
23 |
11 22
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 = ( 1r ‘ 𝑅 ) ) → ( 𝑃 ↑ 𝑥 ) = ( 1r ‘ 𝑅 ) ) |
24 |
1 7
|
ringidcl |
⊢ ( 𝑅 ∈ Ring → ( 1r ‘ 𝑅 ) ∈ 𝐵 ) |
25 |
9 24
|
syl |
⊢ ( 𝜑 → ( 1r ‘ 𝑅 ) ∈ 𝐵 ) |
26 |
4 23 25 25
|
fvmptd2 |
⊢ ( 𝜑 → ( 𝐹 ‘ ( 1r ‘ 𝑅 ) ) = ( 1r ‘ 𝑅 ) ) |
27 |
12
|
crngmgp |
⊢ ( 𝑅 ∈ CRing → ( mulGrp ‘ 𝑅 ) ∈ CMnd ) |
28 |
5 27
|
syl |
⊢ ( 𝜑 → ( mulGrp ‘ 𝑅 ) ∈ CMnd ) |
29 |
28
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) ) → ( mulGrp ‘ 𝑅 ) ∈ CMnd ) |
30 |
17
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) ) → 𝑃 ∈ ℕ0 ) |
31 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) ) → 𝑖 ∈ 𝐵 ) |
32 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) ) → 𝑗 ∈ 𝐵 ) |
33 |
12 8
|
mgpplusg |
⊢ ( .r ‘ 𝑅 ) = ( +g ‘ ( mulGrp ‘ 𝑅 ) ) |
34 |
18 3 33
|
mulgnn0di |
⊢ ( ( ( mulGrp ‘ 𝑅 ) ∈ CMnd ∧ ( 𝑃 ∈ ℕ0 ∧ 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) ) → ( 𝑃 ↑ ( 𝑖 ( .r ‘ 𝑅 ) 𝑗 ) ) = ( ( 𝑃 ↑ 𝑖 ) ( .r ‘ 𝑅 ) ( 𝑃 ↑ 𝑗 ) ) ) |
35 |
29 30 31 32 34
|
syl13anc |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) ) → ( 𝑃 ↑ ( 𝑖 ( .r ‘ 𝑅 ) 𝑗 ) ) = ( ( 𝑃 ↑ 𝑖 ) ( .r ‘ 𝑅 ) ( 𝑃 ↑ 𝑗 ) ) ) |
36 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) ) ∧ 𝑥 = ( 𝑖 ( .r ‘ 𝑅 ) 𝑗 ) ) → 𝑥 = ( 𝑖 ( .r ‘ 𝑅 ) 𝑗 ) ) |
37 |
36
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) ) ∧ 𝑥 = ( 𝑖 ( .r ‘ 𝑅 ) 𝑗 ) ) → ( 𝑃 ↑ 𝑥 ) = ( 𝑃 ↑ ( 𝑖 ( .r ‘ 𝑅 ) 𝑗 ) ) ) |
38 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) ) → 𝑅 ∈ Ring ) |
39 |
1 8
|
ringcl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) → ( 𝑖 ( .r ‘ 𝑅 ) 𝑗 ) ∈ 𝐵 ) |
40 |
38 31 32 39
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) ) → ( 𝑖 ( .r ‘ 𝑅 ) 𝑗 ) ∈ 𝐵 ) |
41 |
|
ovexd |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) ) → ( 𝑃 ↑ ( 𝑖 ( .r ‘ 𝑅 ) 𝑗 ) ) ∈ V ) |
42 |
4 37 40 41
|
fvmptd2 |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) ) → ( 𝐹 ‘ ( 𝑖 ( .r ‘ 𝑅 ) 𝑗 ) ) = ( 𝑃 ↑ ( 𝑖 ( .r ‘ 𝑅 ) 𝑗 ) ) ) |
43 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) ) ∧ 𝑥 = 𝑖 ) → 𝑥 = 𝑖 ) |
44 |
43
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) ) ∧ 𝑥 = 𝑖 ) → ( 𝑃 ↑ 𝑥 ) = ( 𝑃 ↑ 𝑖 ) ) |
45 |
|
ovexd |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) ) → ( 𝑃 ↑ 𝑖 ) ∈ V ) |
46 |
4 44 31 45
|
fvmptd2 |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) ) → ( 𝐹 ‘ 𝑖 ) = ( 𝑃 ↑ 𝑖 ) ) |
47 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) ) ∧ 𝑥 = 𝑗 ) → 𝑥 = 𝑗 ) |
48 |
47
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) ) ∧ 𝑥 = 𝑗 ) → ( 𝑃 ↑ 𝑥 ) = ( 𝑃 ↑ 𝑗 ) ) |
49 |
|
ovexd |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) ) → ( 𝑃 ↑ 𝑗 ) ∈ V ) |
50 |
4 48 32 49
|
fvmptd2 |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) ) → ( 𝐹 ‘ 𝑗 ) = ( 𝑃 ↑ 𝑗 ) ) |
51 |
46 50
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) ) → ( ( 𝐹 ‘ 𝑖 ) ( .r ‘ 𝑅 ) ( 𝐹 ‘ 𝑗 ) ) = ( ( 𝑃 ↑ 𝑖 ) ( .r ‘ 𝑅 ) ( 𝑃 ↑ 𝑗 ) ) ) |
52 |
35 42 51
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) ) → ( 𝐹 ‘ ( 𝑖 ( .r ‘ 𝑅 ) 𝑗 ) ) = ( ( 𝐹 ‘ 𝑖 ) ( .r ‘ 𝑅 ) ( 𝐹 ‘ 𝑗 ) ) ) |
53 |
|
eqid |
⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 ) |
54 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( mulGrp ‘ 𝑅 ) ∈ Mnd ) |
55 |
17
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝑃 ∈ ℕ0 ) |
56 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ 𝐵 ) |
57 |
18 3
|
mulgnn0cl |
⊢ ( ( ( mulGrp ‘ 𝑅 ) ∈ Mnd ∧ 𝑃 ∈ ℕ0 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑃 ↑ 𝑥 ) ∈ 𝐵 ) |
58 |
54 55 56 57
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑃 ↑ 𝑥 ) ∈ 𝐵 ) |
59 |
58 4
|
fmptd |
⊢ ( 𝜑 → 𝐹 : 𝐵 ⟶ 𝐵 ) |
60 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) ) → 𝑅 ∈ CRing ) |
61 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) ) → 𝑃 ∈ ℙ ) |
62 |
1 53 3 2 60 61 31 32
|
freshmansdream |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) ) → ( 𝑃 ↑ ( 𝑖 ( +g ‘ 𝑅 ) 𝑗 ) ) = ( ( 𝑃 ↑ 𝑖 ) ( +g ‘ 𝑅 ) ( 𝑃 ↑ 𝑗 ) ) ) |
63 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) ) ∧ 𝑥 = ( 𝑖 ( +g ‘ 𝑅 ) 𝑗 ) ) → 𝑥 = ( 𝑖 ( +g ‘ 𝑅 ) 𝑗 ) ) |
64 |
63
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) ) ∧ 𝑥 = ( 𝑖 ( +g ‘ 𝑅 ) 𝑗 ) ) → ( 𝑃 ↑ 𝑥 ) = ( 𝑃 ↑ ( 𝑖 ( +g ‘ 𝑅 ) 𝑗 ) ) ) |
65 |
1 53
|
ringacl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) → ( 𝑖 ( +g ‘ 𝑅 ) 𝑗 ) ∈ 𝐵 ) |
66 |
38 31 32 65
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) ) → ( 𝑖 ( +g ‘ 𝑅 ) 𝑗 ) ∈ 𝐵 ) |
67 |
|
ovexd |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) ) → ( 𝑃 ↑ ( 𝑖 ( +g ‘ 𝑅 ) 𝑗 ) ) ∈ V ) |
68 |
4 64 66 67
|
fvmptd2 |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) ) → ( 𝐹 ‘ ( 𝑖 ( +g ‘ 𝑅 ) 𝑗 ) ) = ( 𝑃 ↑ ( 𝑖 ( +g ‘ 𝑅 ) 𝑗 ) ) ) |
69 |
46 50
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) ) → ( ( 𝐹 ‘ 𝑖 ) ( +g ‘ 𝑅 ) ( 𝐹 ‘ 𝑗 ) ) = ( ( 𝑃 ↑ 𝑖 ) ( +g ‘ 𝑅 ) ( 𝑃 ↑ 𝑗 ) ) ) |
70 |
62 68 69
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) ) → ( 𝐹 ‘ ( 𝑖 ( +g ‘ 𝑅 ) 𝑗 ) ) = ( ( 𝐹 ‘ 𝑖 ) ( +g ‘ 𝑅 ) ( 𝐹 ‘ 𝑗 ) ) ) |
71 |
1 7 7 8 8 9 9 26 52 1 53 53 59 70
|
isrhmd |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝑅 RingHom 𝑅 ) ) |