Step |
Hyp |
Ref |
Expression |
1 |
|
grpcominv.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
grpcominv.p |
⊢ + = ( +g ‘ 𝐺 ) |
3 |
|
grpcominv.n |
⊢ 𝑁 = ( invg ‘ 𝐺 ) |
4 |
|
grpcominv.g |
⊢ ( 𝜑 → 𝐺 ∈ Grp ) |
5 |
|
grpcominv.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
6 |
|
grpcominv.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
7 |
|
grpcominv.1 |
⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) = ( 𝑌 + 𝑋 ) ) |
8 |
1 3 4 6
|
grpinvcld |
⊢ ( 𝜑 → ( 𝑁 ‘ 𝑌 ) ∈ 𝐵 ) |
9 |
1 2 4 8 6 5
|
grpassd |
⊢ ( 𝜑 → ( ( ( 𝑁 ‘ 𝑌 ) + 𝑌 ) + 𝑋 ) = ( ( 𝑁 ‘ 𝑌 ) + ( 𝑌 + 𝑋 ) ) ) |
10 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
11 |
1 2 10 3 4 6
|
grplinvd |
⊢ ( 𝜑 → ( ( 𝑁 ‘ 𝑌 ) + 𝑌 ) = ( 0g ‘ 𝐺 ) ) |
12 |
11
|
oveq1d |
⊢ ( 𝜑 → ( ( ( 𝑁 ‘ 𝑌 ) + 𝑌 ) + 𝑋 ) = ( ( 0g ‘ 𝐺 ) + 𝑋 ) ) |
13 |
1 2 10 4 5
|
grplidd |
⊢ ( 𝜑 → ( ( 0g ‘ 𝐺 ) + 𝑋 ) = 𝑋 ) |
14 |
12 13
|
eqtr2d |
⊢ ( 𝜑 → 𝑋 = ( ( ( 𝑁 ‘ 𝑌 ) + 𝑌 ) + 𝑋 ) ) |
15 |
7
|
oveq2d |
⊢ ( 𝜑 → ( ( 𝑁 ‘ 𝑌 ) + ( 𝑋 + 𝑌 ) ) = ( ( 𝑁 ‘ 𝑌 ) + ( 𝑌 + 𝑋 ) ) ) |
16 |
9 14 15
|
3eqtr4rd |
⊢ ( 𝜑 → ( ( 𝑁 ‘ 𝑌 ) + ( 𝑋 + 𝑌 ) ) = 𝑋 ) |
17 |
1 2 4 8 5 6
|
grpassd |
⊢ ( 𝜑 → ( ( ( 𝑁 ‘ 𝑌 ) + 𝑋 ) + 𝑌 ) = ( ( 𝑁 ‘ 𝑌 ) + ( 𝑋 + 𝑌 ) ) ) |
18 |
1 2 3 4 5 6
|
grpasscan2d |
⊢ ( 𝜑 → ( ( 𝑋 + ( 𝑁 ‘ 𝑌 ) ) + 𝑌 ) = 𝑋 ) |
19 |
16 17 18
|
3eqtr4rd |
⊢ ( 𝜑 → ( ( 𝑋 + ( 𝑁 ‘ 𝑌 ) ) + 𝑌 ) = ( ( ( 𝑁 ‘ 𝑌 ) + 𝑋 ) + 𝑌 ) ) |
20 |
1 2 4 5 8
|
grpcld |
⊢ ( 𝜑 → ( 𝑋 + ( 𝑁 ‘ 𝑌 ) ) ∈ 𝐵 ) |
21 |
1 2 4 8 5
|
grpcld |
⊢ ( 𝜑 → ( ( 𝑁 ‘ 𝑌 ) + 𝑋 ) ∈ 𝐵 ) |
22 |
1 2
|
grprcan |
⊢ ( ( 𝐺 ∈ Grp ∧ ( ( 𝑋 + ( 𝑁 ‘ 𝑌 ) ) ∈ 𝐵 ∧ ( ( 𝑁 ‘ 𝑌 ) + 𝑋 ) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( ( 𝑋 + ( 𝑁 ‘ 𝑌 ) ) + 𝑌 ) = ( ( ( 𝑁 ‘ 𝑌 ) + 𝑋 ) + 𝑌 ) ↔ ( 𝑋 + ( 𝑁 ‘ 𝑌 ) ) = ( ( 𝑁 ‘ 𝑌 ) + 𝑋 ) ) ) |
23 |
4 20 21 6 22
|
syl13anc |
⊢ ( 𝜑 → ( ( ( 𝑋 + ( 𝑁 ‘ 𝑌 ) ) + 𝑌 ) = ( ( ( 𝑁 ‘ 𝑌 ) + 𝑋 ) + 𝑌 ) ↔ ( 𝑋 + ( 𝑁 ‘ 𝑌 ) ) = ( ( 𝑁 ‘ 𝑌 ) + 𝑋 ) ) ) |
24 |
19 23
|
mpbid |
⊢ ( 𝜑 → ( 𝑋 + ( 𝑁 ‘ 𝑌 ) ) = ( ( 𝑁 ‘ 𝑌 ) + 𝑋 ) ) |