Step |
Hyp |
Ref |
Expression |
1 |
|
cygctb.1 |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
eqid |
⊢ ( .g ‘ 𝐺 ) = ( .g ‘ 𝐺 ) |
3 |
|
simpl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐵 ≈ 1o ) → 𝐺 ∈ Grp ) |
4 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
5 |
1 4
|
grpidcl |
⊢ ( 𝐺 ∈ Grp → ( 0g ‘ 𝐺 ) ∈ 𝐵 ) |
6 |
5
|
adantr |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐵 ≈ 1o ) → ( 0g ‘ 𝐺 ) ∈ 𝐵 ) |
7 |
|
0z |
⊢ 0 ∈ ℤ |
8 |
|
en1eqsn |
⊢ ( ( ( 0g ‘ 𝐺 ) ∈ 𝐵 ∧ 𝐵 ≈ 1o ) → 𝐵 = { ( 0g ‘ 𝐺 ) } ) |
9 |
5 8
|
sylan |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐵 ≈ 1o ) → 𝐵 = { ( 0g ‘ 𝐺 ) } ) |
10 |
9
|
eleq2d |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐵 ≈ 1o ) → ( 𝑥 ∈ 𝐵 ↔ 𝑥 ∈ { ( 0g ‘ 𝐺 ) } ) ) |
11 |
10
|
biimpa |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐵 ≈ 1o ) ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ { ( 0g ‘ 𝐺 ) } ) |
12 |
|
velsn |
⊢ ( 𝑥 ∈ { ( 0g ‘ 𝐺 ) } ↔ 𝑥 = ( 0g ‘ 𝐺 ) ) |
13 |
11 12
|
sylib |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐵 ≈ 1o ) ∧ 𝑥 ∈ 𝐵 ) → 𝑥 = ( 0g ‘ 𝐺 ) ) |
14 |
1 4 2
|
mulg0 |
⊢ ( ( 0g ‘ 𝐺 ) ∈ 𝐵 → ( 0 ( .g ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) = ( 0g ‘ 𝐺 ) ) |
15 |
6 14
|
syl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐵 ≈ 1o ) → ( 0 ( .g ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) = ( 0g ‘ 𝐺 ) ) |
16 |
15
|
adantr |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐵 ≈ 1o ) ∧ 𝑥 ∈ 𝐵 ) → ( 0 ( .g ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) = ( 0g ‘ 𝐺 ) ) |
17 |
13 16
|
eqtr4d |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐵 ≈ 1o ) ∧ 𝑥 ∈ 𝐵 ) → 𝑥 = ( 0 ( .g ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) ) |
18 |
|
oveq1 |
⊢ ( 𝑛 = 0 → ( 𝑛 ( .g ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) = ( 0 ( .g ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) ) |
19 |
18
|
rspceeqv |
⊢ ( ( 0 ∈ ℤ ∧ 𝑥 = ( 0 ( .g ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) ) → ∃ 𝑛 ∈ ℤ 𝑥 = ( 𝑛 ( .g ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) ) |
20 |
7 17 19
|
sylancr |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝐵 ≈ 1o ) ∧ 𝑥 ∈ 𝐵 ) → ∃ 𝑛 ∈ ℤ 𝑥 = ( 𝑛 ( .g ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) ) |
21 |
1 2 3 6 20
|
iscygd |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐵 ≈ 1o ) → 𝐺 ∈ CycGrp ) |