Step |
Hyp |
Ref |
Expression |
1 |
|
ringacl.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
2 |
|
ringacl.p |
⊢ + = ( +g ‘ 𝑅 ) |
3 |
1 2
|
ringcomlem |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 + 𝑋 ) + ( 𝑌 + 𝑌 ) ) = ( ( 𝑋 + 𝑌 ) + ( 𝑋 + 𝑌 ) ) ) |
4 |
|
simp1 |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑅 ∈ Ring ) |
5 |
4
|
ringgrpd |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑅 ∈ Grp ) |
6 |
|
simp2 |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 ) |
7 |
1 2
|
ringacl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 + 𝑋 ) ∈ 𝐵 ) |
8 |
4 6 6 7
|
syl3anc |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑋 ) ∈ 𝐵 ) |
9 |
|
simp3 |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑌 ∈ 𝐵 ) |
10 |
1 2
|
grpass |
⊢ ( ( 𝑅 ∈ Grp ∧ ( ( 𝑋 + 𝑋 ) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( ( 𝑋 + 𝑋 ) + 𝑌 ) + 𝑌 ) = ( ( 𝑋 + 𝑋 ) + ( 𝑌 + 𝑌 ) ) ) |
11 |
5 8 9 9 10
|
syl13anc |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ( 𝑋 + 𝑋 ) + 𝑌 ) + 𝑌 ) = ( ( 𝑋 + 𝑋 ) + ( 𝑌 + 𝑌 ) ) ) |
12 |
1 2
|
ringacl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 ) |
13 |
1 2
|
grpass |
⊢ ( ( 𝑅 ∈ Grp ∧ ( ( 𝑋 + 𝑌 ) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( ( 𝑋 + 𝑌 ) + 𝑋 ) + 𝑌 ) = ( ( 𝑋 + 𝑌 ) + ( 𝑋 + 𝑌 ) ) ) |
14 |
5 12 6 9 13
|
syl13anc |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ( 𝑋 + 𝑌 ) + 𝑋 ) + 𝑌 ) = ( ( 𝑋 + 𝑌 ) + ( 𝑋 + 𝑌 ) ) ) |
15 |
3 11 14
|
3eqtr4d |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ( 𝑋 + 𝑋 ) + 𝑌 ) + 𝑌 ) = ( ( ( 𝑋 + 𝑌 ) + 𝑋 ) + 𝑌 ) ) |
16 |
1 2
|
ringacl |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑋 + 𝑋 ) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 + 𝑋 ) + 𝑌 ) ∈ 𝐵 ) |
17 |
4 8 9 16
|
syl3anc |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 + 𝑋 ) + 𝑌 ) ∈ 𝐵 ) |
18 |
1 2
|
ringacl |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑋 + 𝑌 ) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑋 + 𝑌 ) + 𝑋 ) ∈ 𝐵 ) |
19 |
4 12 6 18
|
syl3anc |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 + 𝑌 ) + 𝑋 ) ∈ 𝐵 ) |
20 |
1 2
|
grprcan |
⊢ ( ( 𝑅 ∈ Grp ∧ ( ( ( 𝑋 + 𝑋 ) + 𝑌 ) ∈ 𝐵 ∧ ( ( 𝑋 + 𝑌 ) + 𝑋 ) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( ( ( 𝑋 + 𝑋 ) + 𝑌 ) + 𝑌 ) = ( ( ( 𝑋 + 𝑌 ) + 𝑋 ) + 𝑌 ) ↔ ( ( 𝑋 + 𝑋 ) + 𝑌 ) = ( ( 𝑋 + 𝑌 ) + 𝑋 ) ) ) |
21 |
5 17 19 9 20
|
syl13anc |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ( ( 𝑋 + 𝑋 ) + 𝑌 ) + 𝑌 ) = ( ( ( 𝑋 + 𝑌 ) + 𝑋 ) + 𝑌 ) ↔ ( ( 𝑋 + 𝑋 ) + 𝑌 ) = ( ( 𝑋 + 𝑌 ) + 𝑋 ) ) ) |
22 |
15 21
|
mpbid |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 + 𝑋 ) + 𝑌 ) = ( ( 𝑋 + 𝑌 ) + 𝑋 ) ) |
23 |
1 2
|
grpass |
⊢ ( ( 𝑅 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑋 ) + 𝑌 ) = ( 𝑋 + ( 𝑋 + 𝑌 ) ) ) |
24 |
5 6 6 9 23
|
syl13anc |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 + 𝑋 ) + 𝑌 ) = ( 𝑋 + ( 𝑋 + 𝑌 ) ) ) |
25 |
1 2
|
grpass |
⊢ ( ( 𝑅 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) + 𝑋 ) = ( 𝑋 + ( 𝑌 + 𝑋 ) ) ) |
26 |
5 6 9 6 25
|
syl13anc |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 + 𝑌 ) + 𝑋 ) = ( 𝑋 + ( 𝑌 + 𝑋 ) ) ) |
27 |
22 24 26
|
3eqtr3d |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + ( 𝑋 + 𝑌 ) ) = ( 𝑋 + ( 𝑌 + 𝑋 ) ) ) |
28 |
1 2
|
ringacl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑌 + 𝑋 ) ∈ 𝐵 ) |
29 |
28
|
3com23 |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑌 + 𝑋 ) ∈ 𝐵 ) |
30 |
1 2
|
grplcan |
⊢ ( ( 𝑅 ∈ Grp ∧ ( ( 𝑋 + 𝑌 ) ∈ 𝐵 ∧ ( 𝑌 + 𝑋 ) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ) → ( ( 𝑋 + ( 𝑋 + 𝑌 ) ) = ( 𝑋 + ( 𝑌 + 𝑋 ) ) ↔ ( 𝑋 + 𝑌 ) = ( 𝑌 + 𝑋 ) ) ) |
31 |
5 12 29 6 30
|
syl13anc |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 + ( 𝑋 + 𝑌 ) ) = ( 𝑋 + ( 𝑌 + 𝑋 ) ) ↔ ( 𝑋 + 𝑌 ) = ( 𝑌 + 𝑋 ) ) ) |
32 |
27 31
|
mpbid |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) = ( 𝑌 + 𝑋 ) ) |