Step |
Hyp |
Ref |
Expression |
1 |
|
cycsubgcyg.x |
⊢ 𝑋 = ( Base ‘ 𝐺 ) |
2 |
|
cycsubgcyg.t |
⊢ · = ( .g ‘ 𝐺 ) |
3 |
|
cycsubgcyg.s |
⊢ 𝑆 = ran ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝐴 ) ) |
4 |
|
eqid |
⊢ ( Base ‘ ( 𝐺 ↾s 𝑆 ) ) = ( Base ‘ ( 𝐺 ↾s 𝑆 ) ) |
5 |
|
eqid |
⊢ ( .g ‘ ( 𝐺 ↾s 𝑆 ) ) = ( .g ‘ ( 𝐺 ↾s 𝑆 ) ) |
6 |
|
eqid |
⊢ ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝐴 ) ) = ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝐴 ) ) |
7 |
1 2 6
|
cycsubgcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → ( ran ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝐴 ) ) ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝐴 ∈ ran ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝐴 ) ) ) ) |
8 |
7
|
simpld |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → ran ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝐴 ) ) ∈ ( SubGrp ‘ 𝐺 ) ) |
9 |
3 8
|
eqeltrid |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ) |
10 |
|
eqid |
⊢ ( 𝐺 ↾s 𝑆 ) = ( 𝐺 ↾s 𝑆 ) |
11 |
10
|
subggrp |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝐺 ↾s 𝑆 ) ∈ Grp ) |
12 |
9 11
|
syl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → ( 𝐺 ↾s 𝑆 ) ∈ Grp ) |
13 |
7
|
simprd |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → 𝐴 ∈ ran ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝐴 ) ) ) |
14 |
13 3
|
eleqtrrdi |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → 𝐴 ∈ 𝑆 ) |
15 |
10
|
subgbas |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → 𝑆 = ( Base ‘ ( 𝐺 ↾s 𝑆 ) ) ) |
16 |
9 15
|
syl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → 𝑆 = ( Base ‘ ( 𝐺 ↾s 𝑆 ) ) ) |
17 |
14 16
|
eleqtrd |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → 𝐴 ∈ ( Base ‘ ( 𝐺 ↾s 𝑆 ) ) ) |
18 |
16
|
eleq2d |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → ( 𝑦 ∈ 𝑆 ↔ 𝑦 ∈ ( Base ‘ ( 𝐺 ↾s 𝑆 ) ) ) ) |
19 |
18
|
biimpar |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑦 ∈ ( Base ‘ ( 𝐺 ↾s 𝑆 ) ) ) → 𝑦 ∈ 𝑆 ) |
20 |
|
simpr |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑦 ∈ 𝑆 ) → 𝑦 ∈ 𝑆 ) |
21 |
20 3
|
eleqtrdi |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑦 ∈ 𝑆 ) → 𝑦 ∈ ran ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝐴 ) ) ) |
22 |
|
oveq1 |
⊢ ( 𝑥 = 𝑛 → ( 𝑥 · 𝐴 ) = ( 𝑛 · 𝐴 ) ) |
23 |
22
|
cbvmptv |
⊢ ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝐴 ) ) = ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝐴 ) ) |
24 |
|
ovex |
⊢ ( 𝑛 · 𝐴 ) ∈ V |
25 |
23 24
|
elrnmpti |
⊢ ( 𝑦 ∈ ran ( 𝑥 ∈ ℤ ↦ ( 𝑥 · 𝐴 ) ) ↔ ∃ 𝑛 ∈ ℤ 𝑦 = ( 𝑛 · 𝐴 ) ) |
26 |
21 25
|
sylib |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑦 ∈ 𝑆 ) → ∃ 𝑛 ∈ ℤ 𝑦 = ( 𝑛 · 𝐴 ) ) |
27 |
9
|
ad2antrr |
⊢ ( ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑛 ∈ ℤ ) → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ) |
28 |
|
simpr |
⊢ ( ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑛 ∈ ℤ ) → 𝑛 ∈ ℤ ) |
29 |
14
|
ad2antrr |
⊢ ( ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑛 ∈ ℤ ) → 𝐴 ∈ 𝑆 ) |
30 |
2 10 5
|
subgmulg |
⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑛 ∈ ℤ ∧ 𝐴 ∈ 𝑆 ) → ( 𝑛 · 𝐴 ) = ( 𝑛 ( .g ‘ ( 𝐺 ↾s 𝑆 ) ) 𝐴 ) ) |
31 |
27 28 29 30
|
syl3anc |
⊢ ( ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑛 ∈ ℤ ) → ( 𝑛 · 𝐴 ) = ( 𝑛 ( .g ‘ ( 𝐺 ↾s 𝑆 ) ) 𝐴 ) ) |
32 |
31
|
eqeq2d |
⊢ ( ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑛 ∈ ℤ ) → ( 𝑦 = ( 𝑛 · 𝐴 ) ↔ 𝑦 = ( 𝑛 ( .g ‘ ( 𝐺 ↾s 𝑆 ) ) 𝐴 ) ) ) |
33 |
32
|
rexbidva |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑦 ∈ 𝑆 ) → ( ∃ 𝑛 ∈ ℤ 𝑦 = ( 𝑛 · 𝐴 ) ↔ ∃ 𝑛 ∈ ℤ 𝑦 = ( 𝑛 ( .g ‘ ( 𝐺 ↾s 𝑆 ) ) 𝐴 ) ) ) |
34 |
26 33
|
mpbid |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑦 ∈ 𝑆 ) → ∃ 𝑛 ∈ ℤ 𝑦 = ( 𝑛 ( .g ‘ ( 𝐺 ↾s 𝑆 ) ) 𝐴 ) ) |
35 |
19 34
|
syldan |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑦 ∈ ( Base ‘ ( 𝐺 ↾s 𝑆 ) ) ) → ∃ 𝑛 ∈ ℤ 𝑦 = ( 𝑛 ( .g ‘ ( 𝐺 ↾s 𝑆 ) ) 𝐴 ) ) |
36 |
4 5 12 17 35
|
iscygd |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → ( 𝐺 ↾s 𝑆 ) ∈ CycGrp ) |