Step |
Hyp |
Ref |
Expression |
1 |
|
proot1hash.g |
⊢ 𝐺 = ( ( mulGrp ‘ 𝑅 ) ↾s ( Unit ‘ 𝑅 ) ) |
2 |
|
proot1hash.o |
⊢ 𝑂 = ( od ‘ 𝐺 ) |
3 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
4 |
3 2
|
odf |
⊢ 𝑂 : ( Base ‘ 𝐺 ) ⟶ ℕ0 |
5 |
|
ffn |
⊢ ( 𝑂 : ( Base ‘ 𝐺 ) ⟶ ℕ0 → 𝑂 Fn ( Base ‘ 𝐺 ) ) |
6 |
|
fniniseg2 |
⊢ ( 𝑂 Fn ( Base ‘ 𝐺 ) → ( ◡ 𝑂 “ { 𝑁 } ) = { 𝑥 ∈ ( Base ‘ 𝐺 ) ∣ ( 𝑂 ‘ 𝑥 ) = 𝑁 } ) |
7 |
4 5 6
|
mp2b |
⊢ ( ◡ 𝑂 “ { 𝑁 } ) = { 𝑥 ∈ ( Base ‘ 𝐺 ) ∣ ( 𝑂 ‘ 𝑥 ) = 𝑁 } |
8 |
|
simp3 |
⊢ ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) → 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) |
9 |
|
fniniseg |
⊢ ( 𝑂 Fn ( Base ‘ 𝐺 ) → ( 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ↔ ( 𝑋 ∈ ( Base ‘ 𝐺 ) ∧ ( 𝑂 ‘ 𝑋 ) = 𝑁 ) ) ) |
10 |
4 5 9
|
mp2b |
⊢ ( 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ↔ ( 𝑋 ∈ ( Base ‘ 𝐺 ) ∧ ( 𝑂 ‘ 𝑋 ) = 𝑁 ) ) |
11 |
8 10
|
sylib |
⊢ ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) → ( 𝑋 ∈ ( Base ‘ 𝐺 ) ∧ ( 𝑂 ‘ 𝑋 ) = 𝑁 ) ) |
12 |
11
|
simprd |
⊢ ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) → ( 𝑂 ‘ 𝑋 ) = 𝑁 ) |
13 |
12
|
eqeq2d |
⊢ ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) → ( ( 𝑂 ‘ 𝑥 ) = ( 𝑂 ‘ 𝑋 ) ↔ ( 𝑂 ‘ 𝑥 ) = 𝑁 ) ) |
14 |
13
|
rabbidv |
⊢ ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) → { 𝑥 ∈ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ { 𝑋 } ) ∣ ( 𝑂 ‘ 𝑥 ) = ( 𝑂 ‘ 𝑋 ) } = { 𝑥 ∈ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ { 𝑋 } ) ∣ ( 𝑂 ‘ 𝑥 ) = 𝑁 } ) |
15 |
|
isidom |
⊢ ( 𝑅 ∈ IDomn ↔ ( 𝑅 ∈ CRing ∧ 𝑅 ∈ Domn ) ) |
16 |
15
|
simprbi |
⊢ ( 𝑅 ∈ IDomn → 𝑅 ∈ Domn ) |
17 |
16
|
3ad2ant1 |
⊢ ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) → 𝑅 ∈ Domn ) |
18 |
|
domnring |
⊢ ( 𝑅 ∈ Domn → 𝑅 ∈ Ring ) |
19 |
|
eqid |
⊢ ( Unit ‘ 𝑅 ) = ( Unit ‘ 𝑅 ) |
20 |
19 1
|
unitgrp |
⊢ ( 𝑅 ∈ Ring → 𝐺 ∈ Grp ) |
21 |
17 18 20
|
3syl |
⊢ ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) → 𝐺 ∈ Grp ) |
22 |
3
|
subgacs |
⊢ ( 𝐺 ∈ Grp → ( SubGrp ‘ 𝐺 ) ∈ ( ACS ‘ ( Base ‘ 𝐺 ) ) ) |
23 |
|
acsmre |
⊢ ( ( SubGrp ‘ 𝐺 ) ∈ ( ACS ‘ ( Base ‘ 𝐺 ) ) → ( SubGrp ‘ 𝐺 ) ∈ ( Moore ‘ ( Base ‘ 𝐺 ) ) ) |
24 |
21 22 23
|
3syl |
⊢ ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) → ( SubGrp ‘ 𝐺 ) ∈ ( Moore ‘ ( Base ‘ 𝐺 ) ) ) |
25 |
|
eqid |
⊢ ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) = ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) |
26 |
25
|
mrcssv |
⊢ ( ( SubGrp ‘ 𝐺 ) ∈ ( Moore ‘ ( Base ‘ 𝐺 ) ) → ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ { 𝑋 } ) ⊆ ( Base ‘ 𝐺 ) ) |
27 |
|
dfrab3ss |
⊢ ( ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ { 𝑋 } ) ⊆ ( Base ‘ 𝐺 ) → { 𝑥 ∈ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ { 𝑋 } ) ∣ ( 𝑂 ‘ 𝑥 ) = 𝑁 } = ( ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ { 𝑋 } ) ∩ { 𝑥 ∈ ( Base ‘ 𝐺 ) ∣ ( 𝑂 ‘ 𝑥 ) = 𝑁 } ) ) |
28 |
24 26 27
|
3syl |
⊢ ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) → { 𝑥 ∈ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ { 𝑋 } ) ∣ ( 𝑂 ‘ 𝑥 ) = 𝑁 } = ( ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ { 𝑋 } ) ∩ { 𝑥 ∈ ( Base ‘ 𝐺 ) ∣ ( 𝑂 ‘ 𝑥 ) = 𝑁 } ) ) |
29 |
|
incom |
⊢ ( ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ { 𝑋 } ) ∩ { 𝑥 ∈ ( Base ‘ 𝐺 ) ∣ ( 𝑂 ‘ 𝑥 ) = 𝑁 } ) = ( { 𝑥 ∈ ( Base ‘ 𝐺 ) ∣ ( 𝑂 ‘ 𝑥 ) = 𝑁 } ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ { 𝑋 } ) ) |
30 |
|
simpl1 |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ∧ 𝑥 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) → 𝑅 ∈ IDomn ) |
31 |
|
simpl2 |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ∧ 𝑥 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) → 𝑁 ∈ ℕ ) |
32 |
|
simpr |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ∧ 𝑥 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) → 𝑥 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) |
33 |
|
simpl3 |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ∧ 𝑥 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) → 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) |
34 |
1 2 25
|
proot1mul |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑥 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ∧ 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ) → 𝑥 ∈ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ { 𝑋 } ) ) |
35 |
30 31 32 33 34
|
syl22anc |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ∧ 𝑥 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) → 𝑥 ∈ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ { 𝑋 } ) ) |
36 |
35
|
ex |
⊢ ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) → ( 𝑥 ∈ ( ◡ 𝑂 “ { 𝑁 } ) → 𝑥 ∈ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ { 𝑋 } ) ) ) |
37 |
36
|
ssrdv |
⊢ ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) → ( ◡ 𝑂 “ { 𝑁 } ) ⊆ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ { 𝑋 } ) ) |
38 |
7 37
|
eqsstrrid |
⊢ ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) → { 𝑥 ∈ ( Base ‘ 𝐺 ) ∣ ( 𝑂 ‘ 𝑥 ) = 𝑁 } ⊆ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ { 𝑋 } ) ) |
39 |
|
df-ss |
⊢ ( { 𝑥 ∈ ( Base ‘ 𝐺 ) ∣ ( 𝑂 ‘ 𝑥 ) = 𝑁 } ⊆ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ { 𝑋 } ) ↔ ( { 𝑥 ∈ ( Base ‘ 𝐺 ) ∣ ( 𝑂 ‘ 𝑥 ) = 𝑁 } ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ { 𝑋 } ) ) = { 𝑥 ∈ ( Base ‘ 𝐺 ) ∣ ( 𝑂 ‘ 𝑥 ) = 𝑁 } ) |
40 |
38 39
|
sylib |
⊢ ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) → ( { 𝑥 ∈ ( Base ‘ 𝐺 ) ∣ ( 𝑂 ‘ 𝑥 ) = 𝑁 } ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ { 𝑋 } ) ) = { 𝑥 ∈ ( Base ‘ 𝐺 ) ∣ ( 𝑂 ‘ 𝑥 ) = 𝑁 } ) |
41 |
29 40
|
syl5eq |
⊢ ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) → ( ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ { 𝑋 } ) ∩ { 𝑥 ∈ ( Base ‘ 𝐺 ) ∣ ( 𝑂 ‘ 𝑥 ) = 𝑁 } ) = { 𝑥 ∈ ( Base ‘ 𝐺 ) ∣ ( 𝑂 ‘ 𝑥 ) = 𝑁 } ) |
42 |
14 28 41
|
3eqtrrd |
⊢ ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) → { 𝑥 ∈ ( Base ‘ 𝐺 ) ∣ ( 𝑂 ‘ 𝑥 ) = 𝑁 } = { 𝑥 ∈ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ { 𝑋 } ) ∣ ( 𝑂 ‘ 𝑥 ) = ( 𝑂 ‘ 𝑋 ) } ) |
43 |
7 42
|
syl5eq |
⊢ ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) → ( ◡ 𝑂 “ { 𝑁 } ) = { 𝑥 ∈ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ { 𝑋 } ) ∣ ( 𝑂 ‘ 𝑥 ) = ( 𝑂 ‘ 𝑋 ) } ) |
44 |
43
|
fveq2d |
⊢ ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) → ( ♯ ‘ ( ◡ 𝑂 “ { 𝑁 } ) ) = ( ♯ ‘ { 𝑥 ∈ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ { 𝑋 } ) ∣ ( 𝑂 ‘ 𝑥 ) = ( 𝑂 ‘ 𝑋 ) } ) ) |
45 |
11
|
simpld |
⊢ ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) → 𝑋 ∈ ( Base ‘ 𝐺 ) ) |
46 |
|
simp2 |
⊢ ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) → 𝑁 ∈ ℕ ) |
47 |
12 46
|
eqeltrd |
⊢ ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) → ( 𝑂 ‘ 𝑋 ) ∈ ℕ ) |
48 |
3 2 25
|
odngen |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( Base ‘ 𝐺 ) ∧ ( 𝑂 ‘ 𝑋 ) ∈ ℕ ) → ( ♯ ‘ { 𝑥 ∈ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ { 𝑋 } ) ∣ ( 𝑂 ‘ 𝑥 ) = ( 𝑂 ‘ 𝑋 ) } ) = ( ϕ ‘ ( 𝑂 ‘ 𝑋 ) ) ) |
49 |
21 45 47 48
|
syl3anc |
⊢ ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) → ( ♯ ‘ { 𝑥 ∈ ( ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ‘ { 𝑋 } ) ∣ ( 𝑂 ‘ 𝑥 ) = ( 𝑂 ‘ 𝑋 ) } ) = ( ϕ ‘ ( 𝑂 ‘ 𝑋 ) ) ) |
50 |
12
|
fveq2d |
⊢ ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) → ( ϕ ‘ ( 𝑂 ‘ 𝑋 ) ) = ( ϕ ‘ 𝑁 ) ) |
51 |
44 49 50
|
3eqtrd |
⊢ ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) → ( ♯ ‘ ( ◡ 𝑂 “ { 𝑁 } ) ) = ( ϕ ‘ 𝑁 ) ) |