Step |
Hyp |
Ref |
Expression |
1 |
|
rinvmod.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
rinvmod.0 |
⊢ 0 = ( 0g ‘ 𝐺 ) |
3 |
|
rinvmod.p |
⊢ + = ( +g ‘ 𝐺 ) |
4 |
|
rinvmod.m |
⊢ ( 𝜑 → 𝐺 ∈ CMnd ) |
5 |
|
rinvmod.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝐵 ) |
6 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐵 ) → 𝐺 ∈ CMnd ) |
7 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐵 ) → 𝑤 ∈ 𝐵 ) |
8 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐵 ) → 𝐴 ∈ 𝐵 ) |
9 |
1 3
|
cmncom |
⊢ ( ( 𝐺 ∈ CMnd ∧ 𝑤 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵 ) → ( 𝑤 + 𝐴 ) = ( 𝐴 + 𝑤 ) ) |
10 |
6 7 8 9
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐵 ) → ( 𝑤 + 𝐴 ) = ( 𝐴 + 𝑤 ) ) |
11 |
10
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝐴 + 𝑤 ) = 0 ) → ( 𝑤 + 𝐴 ) = ( 𝐴 + 𝑤 ) ) |
12 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝐴 + 𝑤 ) = 0 ) → ( 𝐴 + 𝑤 ) = 0 ) |
13 |
11 12
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝐴 + 𝑤 ) = 0 ) → ( 𝑤 + 𝐴 ) = 0 ) |
14 |
13 12
|
jca |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ 𝐵 ) ∧ ( 𝐴 + 𝑤 ) = 0 ) → ( ( 𝑤 + 𝐴 ) = 0 ∧ ( 𝐴 + 𝑤 ) = 0 ) ) |
15 |
14
|
ex |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐵 ) → ( ( 𝐴 + 𝑤 ) = 0 → ( ( 𝑤 + 𝐴 ) = 0 ∧ ( 𝐴 + 𝑤 ) = 0 ) ) ) |
16 |
15
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑤 ∈ 𝐵 ( ( 𝐴 + 𝑤 ) = 0 → ( ( 𝑤 + 𝐴 ) = 0 ∧ ( 𝐴 + 𝑤 ) = 0 ) ) ) |
17 |
|
cmnmnd |
⊢ ( 𝐺 ∈ CMnd → 𝐺 ∈ Mnd ) |
18 |
4 17
|
syl |
⊢ ( 𝜑 → 𝐺 ∈ Mnd ) |
19 |
1 2 3 18 5
|
mndinvmod |
⊢ ( 𝜑 → ∃* 𝑤 ∈ 𝐵 ( ( 𝑤 + 𝐴 ) = 0 ∧ ( 𝐴 + 𝑤 ) = 0 ) ) |
20 |
|
rmoim |
⊢ ( ∀ 𝑤 ∈ 𝐵 ( ( 𝐴 + 𝑤 ) = 0 → ( ( 𝑤 + 𝐴 ) = 0 ∧ ( 𝐴 + 𝑤 ) = 0 ) ) → ( ∃* 𝑤 ∈ 𝐵 ( ( 𝑤 + 𝐴 ) = 0 ∧ ( 𝐴 + 𝑤 ) = 0 ) → ∃* 𝑤 ∈ 𝐵 ( 𝐴 + 𝑤 ) = 0 ) ) |
21 |
16 19 20
|
sylc |
⊢ ( 𝜑 → ∃* 𝑤 ∈ 𝐵 ( 𝐴 + 𝑤 ) = 0 ) |