Step |
Hyp |
Ref |
Expression |
1 |
|
cycsubm.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
cycsubm.t |
⊢ · = ( .g ‘ 𝐺 ) |
3 |
|
cycsubm.f |
⊢ 𝐹 = ( 𝑥 ∈ ℕ0 ↦ ( 𝑥 · 𝐴 ) ) |
4 |
|
cycsubm.c |
⊢ 𝐶 = ran 𝐹 |
5 |
1 2
|
mulgnn0cl |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑥 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵 ) → ( 𝑥 · 𝐴 ) ∈ 𝐵 ) |
6 |
5
|
3expa |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝑥 ∈ ℕ0 ) ∧ 𝐴 ∈ 𝐵 ) → ( 𝑥 · 𝐴 ) ∈ 𝐵 ) |
7 |
6
|
an32s |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ) ∧ 𝑥 ∈ ℕ0 ) → ( 𝑥 · 𝐴 ) ∈ 𝐵 ) |
8 |
7 3
|
fmptd |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ) → 𝐹 : ℕ0 ⟶ 𝐵 ) |
9 |
8
|
frnd |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ) → ran 𝐹 ⊆ 𝐵 ) |
10 |
4 9
|
eqsstrid |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ) → 𝐶 ⊆ 𝐵 ) |
11 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
12 |
11
|
a1i |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ) → 0 ∈ ℕ0 ) |
13 |
|
oveq1 |
⊢ ( 𝑖 = 0 → ( 𝑖 · 𝐴 ) = ( 0 · 𝐴 ) ) |
14 |
13
|
eqeq2d |
⊢ ( 𝑖 = 0 → ( ( 0g ‘ 𝐺 ) = ( 𝑖 · 𝐴 ) ↔ ( 0g ‘ 𝐺 ) = ( 0 · 𝐴 ) ) ) |
15 |
14
|
adantl |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ) ∧ 𝑖 = 0 ) → ( ( 0g ‘ 𝐺 ) = ( 𝑖 · 𝐴 ) ↔ ( 0g ‘ 𝐺 ) = ( 0 · 𝐴 ) ) ) |
16 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
17 |
1 16 2
|
mulg0 |
⊢ ( 𝐴 ∈ 𝐵 → ( 0 · 𝐴 ) = ( 0g ‘ 𝐺 ) ) |
18 |
17
|
adantl |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ) → ( 0 · 𝐴 ) = ( 0g ‘ 𝐺 ) ) |
19 |
18
|
eqcomd |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ) → ( 0g ‘ 𝐺 ) = ( 0 · 𝐴 ) ) |
20 |
12 15 19
|
rspcedvd |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ) → ∃ 𝑖 ∈ ℕ0 ( 0g ‘ 𝐺 ) = ( 𝑖 · 𝐴 ) ) |
21 |
1 2 3 4
|
cycsubmel |
⊢ ( ( 0g ‘ 𝐺 ) ∈ 𝐶 ↔ ∃ 𝑖 ∈ ℕ0 ( 0g ‘ 𝐺 ) = ( 𝑖 · 𝐴 ) ) |
22 |
20 21
|
sylibr |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ) → ( 0g ‘ 𝐺 ) ∈ 𝐶 ) |
23 |
|
simplr |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ) ∧ 𝑖 ∈ ℕ0 ) ∧ 𝑗 ∈ ℕ0 ) → 𝑖 ∈ ℕ0 ) |
24 |
|
simpr |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ) ∧ 𝑖 ∈ ℕ0 ) ∧ 𝑗 ∈ ℕ0 ) → 𝑗 ∈ ℕ0 ) |
25 |
23 24
|
nn0addcld |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ) ∧ 𝑖 ∈ ℕ0 ) ∧ 𝑗 ∈ ℕ0 ) → ( 𝑖 + 𝑗 ) ∈ ℕ0 ) |
26 |
25
|
adantr |
⊢ ( ( ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ) ∧ 𝑖 ∈ ℕ0 ) ∧ 𝑗 ∈ ℕ0 ) ∧ ( 𝑏 = ( 𝑗 · 𝐴 ) ∧ 𝑎 = ( 𝑖 · 𝐴 ) ) ) → ( 𝑖 + 𝑗 ) ∈ ℕ0 ) |
27 |
|
oveq1 |
⊢ ( 𝑘 = ( 𝑖 + 𝑗 ) → ( 𝑘 · 𝐴 ) = ( ( 𝑖 + 𝑗 ) · 𝐴 ) ) |
28 |
27
|
eqeq2d |
⊢ ( 𝑘 = ( 𝑖 + 𝑗 ) → ( ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) = ( 𝑘 · 𝐴 ) ↔ ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) = ( ( 𝑖 + 𝑗 ) · 𝐴 ) ) ) |
29 |
28
|
adantl |
⊢ ( ( ( ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ) ∧ 𝑖 ∈ ℕ0 ) ∧ 𝑗 ∈ ℕ0 ) ∧ ( 𝑏 = ( 𝑗 · 𝐴 ) ∧ 𝑎 = ( 𝑖 · 𝐴 ) ) ) ∧ 𝑘 = ( 𝑖 + 𝑗 ) ) → ( ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) = ( 𝑘 · 𝐴 ) ↔ ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) = ( ( 𝑖 + 𝑗 ) · 𝐴 ) ) ) |
30 |
|
oveq12 |
⊢ ( ( 𝑎 = ( 𝑖 · 𝐴 ) ∧ 𝑏 = ( 𝑗 · 𝐴 ) ) → ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) = ( ( 𝑖 · 𝐴 ) ( +g ‘ 𝐺 ) ( 𝑗 · 𝐴 ) ) ) |
31 |
30
|
ancoms |
⊢ ( ( 𝑏 = ( 𝑗 · 𝐴 ) ∧ 𝑎 = ( 𝑖 · 𝐴 ) ) → ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) = ( ( 𝑖 · 𝐴 ) ( +g ‘ 𝐺 ) ( 𝑗 · 𝐴 ) ) ) |
32 |
|
simplll |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ) ∧ 𝑖 ∈ ℕ0 ) ∧ 𝑗 ∈ ℕ0 ) → 𝐺 ∈ Mnd ) |
33 |
|
simpllr |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ) ∧ 𝑖 ∈ ℕ0 ) ∧ 𝑗 ∈ ℕ0 ) → 𝐴 ∈ 𝐵 ) |
34 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
35 |
1 2 34
|
mulgnn0dir |
⊢ ( ( 𝐺 ∈ Mnd ∧ ( 𝑖 ∈ ℕ0 ∧ 𝑗 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵 ) ) → ( ( 𝑖 + 𝑗 ) · 𝐴 ) = ( ( 𝑖 · 𝐴 ) ( +g ‘ 𝐺 ) ( 𝑗 · 𝐴 ) ) ) |
36 |
32 23 24 33 35
|
syl13anc |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ) ∧ 𝑖 ∈ ℕ0 ) ∧ 𝑗 ∈ ℕ0 ) → ( ( 𝑖 + 𝑗 ) · 𝐴 ) = ( ( 𝑖 · 𝐴 ) ( +g ‘ 𝐺 ) ( 𝑗 · 𝐴 ) ) ) |
37 |
36
|
eqcomd |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ) ∧ 𝑖 ∈ ℕ0 ) ∧ 𝑗 ∈ ℕ0 ) → ( ( 𝑖 · 𝐴 ) ( +g ‘ 𝐺 ) ( 𝑗 · 𝐴 ) ) = ( ( 𝑖 + 𝑗 ) · 𝐴 ) ) |
38 |
31 37
|
sylan9eqr |
⊢ ( ( ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ) ∧ 𝑖 ∈ ℕ0 ) ∧ 𝑗 ∈ ℕ0 ) ∧ ( 𝑏 = ( 𝑗 · 𝐴 ) ∧ 𝑎 = ( 𝑖 · 𝐴 ) ) ) → ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) = ( ( 𝑖 + 𝑗 ) · 𝐴 ) ) |
39 |
26 29 38
|
rspcedvd |
⊢ ( ( ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ) ∧ 𝑖 ∈ ℕ0 ) ∧ 𝑗 ∈ ℕ0 ) ∧ ( 𝑏 = ( 𝑗 · 𝐴 ) ∧ 𝑎 = ( 𝑖 · 𝐴 ) ) ) → ∃ 𝑘 ∈ ℕ0 ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) = ( 𝑘 · 𝐴 ) ) |
40 |
39
|
exp32 |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ) ∧ 𝑖 ∈ ℕ0 ) ∧ 𝑗 ∈ ℕ0 ) → ( 𝑏 = ( 𝑗 · 𝐴 ) → ( 𝑎 = ( 𝑖 · 𝐴 ) → ∃ 𝑘 ∈ ℕ0 ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) = ( 𝑘 · 𝐴 ) ) ) ) |
41 |
40
|
rexlimdva |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ) ∧ 𝑖 ∈ ℕ0 ) → ( ∃ 𝑗 ∈ ℕ0 𝑏 = ( 𝑗 · 𝐴 ) → ( 𝑎 = ( 𝑖 · 𝐴 ) → ∃ 𝑘 ∈ ℕ0 ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) = ( 𝑘 · 𝐴 ) ) ) ) |
42 |
41
|
com23 |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ) ∧ 𝑖 ∈ ℕ0 ) → ( 𝑎 = ( 𝑖 · 𝐴 ) → ( ∃ 𝑗 ∈ ℕ0 𝑏 = ( 𝑗 · 𝐴 ) → ∃ 𝑘 ∈ ℕ0 ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) = ( 𝑘 · 𝐴 ) ) ) ) |
43 |
42
|
rexlimdva |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ) → ( ∃ 𝑖 ∈ ℕ0 𝑎 = ( 𝑖 · 𝐴 ) → ( ∃ 𝑗 ∈ ℕ0 𝑏 = ( 𝑗 · 𝐴 ) → ∃ 𝑘 ∈ ℕ0 ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) = ( 𝑘 · 𝐴 ) ) ) ) |
44 |
43
|
impd |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ) → ( ( ∃ 𝑖 ∈ ℕ0 𝑎 = ( 𝑖 · 𝐴 ) ∧ ∃ 𝑗 ∈ ℕ0 𝑏 = ( 𝑗 · 𝐴 ) ) → ∃ 𝑘 ∈ ℕ0 ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) = ( 𝑘 · 𝐴 ) ) ) |
45 |
1 2 3 4
|
cycsubmel |
⊢ ( 𝑎 ∈ 𝐶 ↔ ∃ 𝑖 ∈ ℕ0 𝑎 = ( 𝑖 · 𝐴 ) ) |
46 |
1 2 3 4
|
cycsubmel |
⊢ ( 𝑏 ∈ 𝐶 ↔ ∃ 𝑗 ∈ ℕ0 𝑏 = ( 𝑗 · 𝐴 ) ) |
47 |
45 46
|
anbi12i |
⊢ ( ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ↔ ( ∃ 𝑖 ∈ ℕ0 𝑎 = ( 𝑖 · 𝐴 ) ∧ ∃ 𝑗 ∈ ℕ0 𝑏 = ( 𝑗 · 𝐴 ) ) ) |
48 |
1 2 3 4
|
cycsubmel |
⊢ ( ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ∈ 𝐶 ↔ ∃ 𝑘 ∈ ℕ0 ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) = ( 𝑘 · 𝐴 ) ) |
49 |
44 47 48
|
3imtr4g |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ) → ( ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) → ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ∈ 𝐶 ) ) |
50 |
49
|
ralrimivv |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ) → ∀ 𝑎 ∈ 𝐶 ∀ 𝑏 ∈ 𝐶 ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ∈ 𝐶 ) |
51 |
1 16 34
|
issubm |
⊢ ( 𝐺 ∈ Mnd → ( 𝐶 ∈ ( SubMnd ‘ 𝐺 ) ↔ ( 𝐶 ⊆ 𝐵 ∧ ( 0g ‘ 𝐺 ) ∈ 𝐶 ∧ ∀ 𝑎 ∈ 𝐶 ∀ 𝑏 ∈ 𝐶 ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ∈ 𝐶 ) ) ) |
52 |
51
|
adantr |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ) → ( 𝐶 ∈ ( SubMnd ‘ 𝐺 ) ↔ ( 𝐶 ⊆ 𝐵 ∧ ( 0g ‘ 𝐺 ) ∈ 𝐶 ∧ ∀ 𝑎 ∈ 𝐶 ∀ 𝑏 ∈ 𝐶 ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) ∈ 𝐶 ) ) ) |
53 |
10 22 50 52
|
mpbir3and |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ) → 𝐶 ∈ ( SubMnd ‘ 𝐺 ) ) |