Step |
Hyp |
Ref |
Expression |
1 |
|
idomsubgmo.g |
⊢ 𝐺 = ( ( mulGrp ‘ 𝑅 ) ↾s ( Unit ‘ 𝑅 ) ) |
2 |
|
proot1mul.o |
⊢ 𝑂 = ( od ‘ 𝐺 ) |
3 |
|
proot1mul.k |
⊢ 𝐾 = ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) |
4 |
|
simpll |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ∧ 𝑌 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ) → 𝑅 ∈ IDomn ) |
5 |
|
isidom |
⊢ ( 𝑅 ∈ IDomn ↔ ( 𝑅 ∈ CRing ∧ 𝑅 ∈ Domn ) ) |
6 |
5
|
simprbi |
⊢ ( 𝑅 ∈ IDomn → 𝑅 ∈ Domn ) |
7 |
|
domnring |
⊢ ( 𝑅 ∈ Domn → 𝑅 ∈ Ring ) |
8 |
|
eqid |
⊢ ( Unit ‘ 𝑅 ) = ( Unit ‘ 𝑅 ) |
9 |
8 1
|
unitgrp |
⊢ ( 𝑅 ∈ Ring → 𝐺 ∈ Grp ) |
10 |
4 6 7 9
|
4syl |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ∧ 𝑌 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ) → 𝐺 ∈ Grp ) |
11 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
12 |
11
|
subgacs |
⊢ ( 𝐺 ∈ Grp → ( SubGrp ‘ 𝐺 ) ∈ ( ACS ‘ ( Base ‘ 𝐺 ) ) ) |
13 |
|
acsmre |
⊢ ( ( SubGrp ‘ 𝐺 ) ∈ ( ACS ‘ ( Base ‘ 𝐺 ) ) → ( SubGrp ‘ 𝐺 ) ∈ ( Moore ‘ ( Base ‘ 𝐺 ) ) ) |
14 |
10 12 13
|
3syl |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ∧ 𝑌 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ) → ( SubGrp ‘ 𝐺 ) ∈ ( Moore ‘ ( Base ‘ 𝐺 ) ) ) |
15 |
|
simprl |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ∧ 𝑌 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ) → 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) |
16 |
11 2
|
odf |
⊢ 𝑂 : ( Base ‘ 𝐺 ) ⟶ ℕ0 |
17 |
|
ffn |
⊢ ( 𝑂 : ( Base ‘ 𝐺 ) ⟶ ℕ0 → 𝑂 Fn ( Base ‘ 𝐺 ) ) |
18 |
|
fniniseg |
⊢ ( 𝑂 Fn ( Base ‘ 𝐺 ) → ( 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ↔ ( 𝑋 ∈ ( Base ‘ 𝐺 ) ∧ ( 𝑂 ‘ 𝑋 ) = 𝑁 ) ) ) |
19 |
16 17 18
|
mp2b |
⊢ ( 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ↔ ( 𝑋 ∈ ( Base ‘ 𝐺 ) ∧ ( 𝑂 ‘ 𝑋 ) = 𝑁 ) ) |
20 |
15 19
|
sylib |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ∧ 𝑌 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ) → ( 𝑋 ∈ ( Base ‘ 𝐺 ) ∧ ( 𝑂 ‘ 𝑋 ) = 𝑁 ) ) |
21 |
20
|
simpld |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ∧ 𝑌 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ) → 𝑋 ∈ ( Base ‘ 𝐺 ) ) |
22 |
21
|
snssd |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ∧ 𝑌 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ) → { 𝑋 } ⊆ ( Base ‘ 𝐺 ) ) |
23 |
14 3 22
|
mrcssidd |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ∧ 𝑌 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ) → { 𝑋 } ⊆ ( 𝐾 ‘ { 𝑋 } ) ) |
24 |
|
snssg |
⊢ ( 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) → ( 𝑋 ∈ ( 𝐾 ‘ { 𝑋 } ) ↔ { 𝑋 } ⊆ ( 𝐾 ‘ { 𝑋 } ) ) ) |
25 |
15 24
|
syl |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ∧ 𝑌 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ) → ( 𝑋 ∈ ( 𝐾 ‘ { 𝑋 } ) ↔ { 𝑋 } ⊆ ( 𝐾 ‘ { 𝑋 } ) ) ) |
26 |
23 25
|
mpbird |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ∧ 𝑌 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ) → 𝑋 ∈ ( 𝐾 ‘ { 𝑋 } ) ) |
27 |
1
|
idomsubgmo |
⊢ ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ) → ∃* 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ( ♯ ‘ 𝑥 ) = 𝑁 ) |
28 |
27
|
adantr |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ∧ 𝑌 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ) → ∃* 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ( ♯ ‘ 𝑥 ) = 𝑁 ) |
29 |
3
|
mrccl |
⊢ ( ( ( SubGrp ‘ 𝐺 ) ∈ ( Moore ‘ ( Base ‘ 𝐺 ) ) ∧ { 𝑋 } ⊆ ( Base ‘ 𝐺 ) ) → ( 𝐾 ‘ { 𝑋 } ) ∈ ( SubGrp ‘ 𝐺 ) ) |
30 |
14 22 29
|
syl2anc |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ∧ 𝑌 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ) → ( 𝐾 ‘ { 𝑋 } ) ∈ ( SubGrp ‘ 𝐺 ) ) |
31 |
20
|
simprd |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ∧ 𝑌 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ) → ( 𝑂 ‘ 𝑋 ) = 𝑁 ) |
32 |
|
simplr |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ∧ 𝑌 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ) → 𝑁 ∈ ℕ ) |
33 |
31 32
|
eqeltrd |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ∧ 𝑌 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ) → ( 𝑂 ‘ 𝑋 ) ∈ ℕ ) |
34 |
11 2 3
|
odhash2 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ ( Base ‘ 𝐺 ) ∧ ( 𝑂 ‘ 𝑋 ) ∈ ℕ ) → ( ♯ ‘ ( 𝐾 ‘ { 𝑋 } ) ) = ( 𝑂 ‘ 𝑋 ) ) |
35 |
10 21 33 34
|
syl3anc |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ∧ 𝑌 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ) → ( ♯ ‘ ( 𝐾 ‘ { 𝑋 } ) ) = ( 𝑂 ‘ 𝑋 ) ) |
36 |
35 31
|
eqtrd |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ∧ 𝑌 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ) → ( ♯ ‘ ( 𝐾 ‘ { 𝑋 } ) ) = 𝑁 ) |
37 |
|
simprr |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ∧ 𝑌 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ) → 𝑌 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) |
38 |
|
fniniseg |
⊢ ( 𝑂 Fn ( Base ‘ 𝐺 ) → ( 𝑌 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ↔ ( 𝑌 ∈ ( Base ‘ 𝐺 ) ∧ ( 𝑂 ‘ 𝑌 ) = 𝑁 ) ) ) |
39 |
16 17 38
|
mp2b |
⊢ ( 𝑌 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ↔ ( 𝑌 ∈ ( Base ‘ 𝐺 ) ∧ ( 𝑂 ‘ 𝑌 ) = 𝑁 ) ) |
40 |
37 39
|
sylib |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ∧ 𝑌 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ) → ( 𝑌 ∈ ( Base ‘ 𝐺 ) ∧ ( 𝑂 ‘ 𝑌 ) = 𝑁 ) ) |
41 |
40
|
simpld |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ∧ 𝑌 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ) → 𝑌 ∈ ( Base ‘ 𝐺 ) ) |
42 |
41
|
snssd |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ∧ 𝑌 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ) → { 𝑌 } ⊆ ( Base ‘ 𝐺 ) ) |
43 |
3
|
mrccl |
⊢ ( ( ( SubGrp ‘ 𝐺 ) ∈ ( Moore ‘ ( Base ‘ 𝐺 ) ) ∧ { 𝑌 } ⊆ ( Base ‘ 𝐺 ) ) → ( 𝐾 ‘ { 𝑌 } ) ∈ ( SubGrp ‘ 𝐺 ) ) |
44 |
14 42 43
|
syl2anc |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ∧ 𝑌 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ) → ( 𝐾 ‘ { 𝑌 } ) ∈ ( SubGrp ‘ 𝐺 ) ) |
45 |
40
|
simprd |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ∧ 𝑌 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ) → ( 𝑂 ‘ 𝑌 ) = 𝑁 ) |
46 |
45 32
|
eqeltrd |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ∧ 𝑌 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ) → ( 𝑂 ‘ 𝑌 ) ∈ ℕ ) |
47 |
11 2 3
|
odhash2 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ ( Base ‘ 𝐺 ) ∧ ( 𝑂 ‘ 𝑌 ) ∈ ℕ ) → ( ♯ ‘ ( 𝐾 ‘ { 𝑌 } ) ) = ( 𝑂 ‘ 𝑌 ) ) |
48 |
10 41 46 47
|
syl3anc |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ∧ 𝑌 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ) → ( ♯ ‘ ( 𝐾 ‘ { 𝑌 } ) ) = ( 𝑂 ‘ 𝑌 ) ) |
49 |
48 45
|
eqtrd |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ∧ 𝑌 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ) → ( ♯ ‘ ( 𝐾 ‘ { 𝑌 } ) ) = 𝑁 ) |
50 |
|
fveqeq2 |
⊢ ( 𝑥 = ( 𝐾 ‘ { 𝑋 } ) → ( ( ♯ ‘ 𝑥 ) = 𝑁 ↔ ( ♯ ‘ ( 𝐾 ‘ { 𝑋 } ) ) = 𝑁 ) ) |
51 |
|
fveqeq2 |
⊢ ( 𝑥 = ( 𝐾 ‘ { 𝑌 } ) → ( ( ♯ ‘ 𝑥 ) = 𝑁 ↔ ( ♯ ‘ ( 𝐾 ‘ { 𝑌 } ) ) = 𝑁 ) ) |
52 |
50 51
|
rmoi |
⊢ ( ( ∃* 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ( ♯ ‘ 𝑥 ) = 𝑁 ∧ ( ( 𝐾 ‘ { 𝑋 } ) ∈ ( SubGrp ‘ 𝐺 ) ∧ ( ♯ ‘ ( 𝐾 ‘ { 𝑋 } ) ) = 𝑁 ) ∧ ( ( 𝐾 ‘ { 𝑌 } ) ∈ ( SubGrp ‘ 𝐺 ) ∧ ( ♯ ‘ ( 𝐾 ‘ { 𝑌 } ) ) = 𝑁 ) ) → ( 𝐾 ‘ { 𝑋 } ) = ( 𝐾 ‘ { 𝑌 } ) ) |
53 |
28 30 36 44 49 52
|
syl122anc |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ∧ 𝑌 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ) → ( 𝐾 ‘ { 𝑋 } ) = ( 𝐾 ‘ { 𝑌 } ) ) |
54 |
26 53
|
eleqtrd |
⊢ ( ( ( 𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ) ∧ ( 𝑋 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ∧ 𝑌 ∈ ( ◡ 𝑂 “ { 𝑁 } ) ) ) → 𝑋 ∈ ( 𝐾 ‘ { 𝑌 } ) ) |