Step |
Hyp |
Ref |
Expression |
1 |
|
odf1o1.x |
⊢ 𝑋 = ( Base ‘ 𝐺 ) |
2 |
|
odf1o1.t |
⊢ · = ( .g ‘ 𝐺 ) |
3 |
|
odf1o1.o |
⊢ 𝑂 = ( od ‘ 𝐺 ) |
4 |
|
odf1o1.k |
⊢ 𝐾 = ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) |
5 |
|
simpl1 |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) ∧ 𝑥 ∈ ℤ ) → 𝐺 ∈ Grp ) |
6 |
1
|
subgacs |
⊢ ( 𝐺 ∈ Grp → ( SubGrp ‘ 𝐺 ) ∈ ( ACS ‘ 𝑋 ) ) |
7 |
|
acsmre |
⊢ ( ( SubGrp ‘ 𝐺 ) ∈ ( ACS ‘ 𝑋 ) → ( SubGrp ‘ 𝐺 ) ∈ ( Moore ‘ 𝑋 ) ) |
8 |
5 6 7
|
3syl |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) ∧ 𝑥 ∈ ℤ ) → ( SubGrp ‘ 𝐺 ) ∈ ( Moore ‘ 𝑋 ) ) |
9 |
|
simpl2 |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) ∧ 𝑥 ∈ ℤ ) → 𝐴 ∈ 𝑋 ) |
10 |
9
|
snssd |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) ∧ 𝑥 ∈ ℤ ) → { 𝐴 } ⊆ 𝑋 ) |
11 |
4
|
mrccl |
⊢ ( ( ( SubGrp ‘ 𝐺 ) ∈ ( Moore ‘ 𝑋 ) ∧ { 𝐴 } ⊆ 𝑋 ) → ( 𝐾 ‘ { 𝐴 } ) ∈ ( SubGrp ‘ 𝐺 ) ) |
12 |
8 10 11
|
syl2anc |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) ∧ 𝑥 ∈ ℤ ) → ( 𝐾 ‘ { 𝐴 } ) ∈ ( SubGrp ‘ 𝐺 ) ) |
13 |
|
simpr |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) ∧ 𝑥 ∈ ℤ ) → 𝑥 ∈ ℤ ) |
14 |
8 4 10
|
mrcssidd |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) ∧ 𝑥 ∈ ℤ ) → { 𝐴 } ⊆ ( 𝐾 ‘ { 𝐴 } ) ) |
15 |
|
snidg |
⊢ ( 𝐴 ∈ 𝑋 → 𝐴 ∈ { 𝐴 } ) |
16 |
9 15
|
syl |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) ∧ 𝑥 ∈ ℤ ) → 𝐴 ∈ { 𝐴 } ) |
17 |
14 16
|
sseldd |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) ∧ 𝑥 ∈ ℤ ) → 𝐴 ∈ ( 𝐾 ‘ { 𝐴 } ) ) |
18 |
2
|
subgmulgcl |
⊢ ( ( ( 𝐾 ‘ { 𝐴 } ) ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑥 ∈ ℤ ∧ 𝐴 ∈ ( 𝐾 ‘ { 𝐴 } ) ) → ( 𝑥 · 𝐴 ) ∈ ( 𝐾 ‘ { 𝐴 } ) ) |
19 |
12 13 17 18
|
syl3anc |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) ∧ 𝑥 ∈ ℤ ) → ( 𝑥 · 𝐴 ) ∈ ( 𝐾 ‘ { 𝐴 } ) ) |
20 |
19
|
ex |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) → ( 𝑥 ∈ ℤ → ( 𝑥 · 𝐴 ) ∈ ( 𝐾 ‘ { 𝐴 } ) ) ) |
21 |
|
simpl3 |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) ∧ ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ) ) → ( 𝑂 ‘ 𝐴 ) = 0 ) |
22 |
21
|
breq1d |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) ∧ ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ) ) → ( ( 𝑂 ‘ 𝐴 ) ∥ ( 𝑥 − 𝑦 ) ↔ 0 ∥ ( 𝑥 − 𝑦 ) ) ) |
23 |
|
zsubcl |
⊢ ( ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ) → ( 𝑥 − 𝑦 ) ∈ ℤ ) |
24 |
23
|
adantl |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) ∧ ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ) ) → ( 𝑥 − 𝑦 ) ∈ ℤ ) |
25 |
|
0dvds |
⊢ ( ( 𝑥 − 𝑦 ) ∈ ℤ → ( 0 ∥ ( 𝑥 − 𝑦 ) ↔ ( 𝑥 − 𝑦 ) = 0 ) ) |
26 |
24 25
|
syl |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) ∧ ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ) ) → ( 0 ∥ ( 𝑥 − 𝑦 ) ↔ ( 𝑥 − 𝑦 ) = 0 ) ) |
27 |
22 26
|
bitrd |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) ∧ ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ) ) → ( ( 𝑂 ‘ 𝐴 ) ∥ ( 𝑥 − 𝑦 ) ↔ ( 𝑥 − 𝑦 ) = 0 ) ) |
28 |
|
simpl1 |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) ∧ ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ) ) → 𝐺 ∈ Grp ) |
29 |
|
simpl2 |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) ∧ ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ) ) → 𝐴 ∈ 𝑋 ) |
30 |
|
simprl |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) ∧ ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ) ) → 𝑥 ∈ ℤ ) |
31 |
|
simprr |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) ∧ ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ) ) → 𝑦 ∈ ℤ ) |
32 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
33 |
1 3 2 32
|
odcong |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ) ) → ( ( 𝑂 ‘ 𝐴 ) ∥ ( 𝑥 − 𝑦 ) ↔ ( 𝑥 · 𝐴 ) = ( 𝑦 · 𝐴 ) ) ) |
34 |
28 29 30 31 33
|
syl112anc |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) ∧ ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ) ) → ( ( 𝑂 ‘ 𝐴 ) ∥ ( 𝑥 − 𝑦 ) ↔ ( 𝑥 · 𝐴 ) = ( 𝑦 · 𝐴 ) ) ) |
35 |
|
zcn |
⊢ ( 𝑥 ∈ ℤ → 𝑥 ∈ ℂ ) |
36 |
|
zcn |
⊢ ( 𝑦 ∈ ℤ → 𝑦 ∈ ℂ ) |
37 |
|
subeq0 |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( ( 𝑥 − 𝑦 ) = 0 ↔ 𝑥 = 𝑦 ) ) |
38 |
35 36 37
|
syl2an |
⊢ ( ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ) → ( ( 𝑥 − 𝑦 ) = 0 ↔ 𝑥 = 𝑦 ) ) |
39 |
38
|
adantl |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) ∧ ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ) ) → ( ( 𝑥 − 𝑦 ) = 0 ↔ 𝑥 = 𝑦 ) ) |
40 |
27 34 39
|
3bitr3d |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) ∧ ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ) ) → ( ( 𝑥 · 𝐴 ) = ( 𝑦 · 𝐴 ) ↔ 𝑥 = 𝑦 ) ) |
41 |
40
|
ex |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) → ( ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ) → ( ( 𝑥 · 𝐴 ) = ( 𝑦 · 𝐴 ) ↔ 𝑥 = 𝑦 ) ) ) |
42 |
20 41
|
dom2lem |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) → ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝐴 ) ) : ℤ –1-1→ ( 𝐾 ‘ { 𝐴 } ) ) |
43 |
19
|
fmpttd |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) → ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝐴 ) ) : ℤ ⟶ ( 𝐾 ‘ { 𝐴 } ) ) |
44 |
|
eqid |
⊢ ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝐴 ) ) = ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝐴 ) ) |
45 |
1 2 44 4
|
cycsubg2 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → ( 𝐾 ‘ { 𝐴 } ) = ran ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝐴 ) ) ) |
46 |
45
|
3adant3 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) → ( 𝐾 ‘ { 𝐴 } ) = ran ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝐴 ) ) ) |
47 |
46
|
eqcomd |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) → ran ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝐴 ) ) = ( 𝐾 ‘ { 𝐴 } ) ) |
48 |
|
dffo2 |
⊢ ( ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝐴 ) ) : ℤ –onto→ ( 𝐾 ‘ { 𝐴 } ) ↔ ( ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝐴 ) ) : ℤ ⟶ ( 𝐾 ‘ { 𝐴 } ) ∧ ran ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝐴 ) ) = ( 𝐾 ‘ { 𝐴 } ) ) ) |
49 |
43 47 48
|
sylanbrc |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) → ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝐴 ) ) : ℤ –onto→ ( 𝐾 ‘ { 𝐴 } ) ) |
50 |
|
df-f1o |
⊢ ( ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝐴 ) ) : ℤ –1-1-onto→ ( 𝐾 ‘ { 𝐴 } ) ↔ ( ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝐴 ) ) : ℤ –1-1→ ( 𝐾 ‘ { 𝐴 } ) ∧ ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝐴 ) ) : ℤ –onto→ ( 𝐾 ‘ { 𝐴 } ) ) ) |
51 |
42 49 50
|
sylanbrc |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑂 ‘ 𝐴 ) = 0 ) → ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝐴 ) ) : ℤ –1-1-onto→ ( 𝐾 ‘ { 𝐴 } ) ) |