Step |
Hyp |
Ref |
Expression |
1 |
|
ringlz.1 |
⊢ 𝑍 = ( GId ‘ 𝐺 ) |
2 |
|
ringlz.2 |
⊢ 𝑋 = ran 𝐺 |
3 |
|
ringlz.3 |
⊢ 𝐺 = ( 1st ‘ 𝑅 ) |
4 |
|
ringlz.4 |
⊢ 𝐻 = ( 2nd ‘ 𝑅 ) |
5 |
3
|
rngogrpo |
⊢ ( 𝑅 ∈ RingOps → 𝐺 ∈ GrpOp ) |
6 |
2 1
|
grpoidcl |
⊢ ( 𝐺 ∈ GrpOp → 𝑍 ∈ 𝑋 ) |
7 |
2 1
|
grpolid |
⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝑍 ∈ 𝑋 ) → ( 𝑍 𝐺 𝑍 ) = 𝑍 ) |
8 |
5 6 7
|
syl2anc2 |
⊢ ( 𝑅 ∈ RingOps → ( 𝑍 𝐺 𝑍 ) = 𝑍 ) |
9 |
8
|
adantr |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( 𝑍 𝐺 𝑍 ) = 𝑍 ) |
10 |
9
|
oveq1d |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝑍 𝐺 𝑍 ) 𝐻 𝐴 ) = ( 𝑍 𝐻 𝐴 ) ) |
11 |
3 2 1
|
rngo0cl |
⊢ ( 𝑅 ∈ RingOps → 𝑍 ∈ 𝑋 ) |
12 |
11
|
adantr |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → 𝑍 ∈ 𝑋 ) |
13 |
|
simpr |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → 𝐴 ∈ 𝑋 ) |
14 |
12 12 13
|
3jca |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( 𝑍 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) |
15 |
3 4 2
|
rngodir |
⊢ ( ( 𝑅 ∈ RingOps ∧ ( 𝑍 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ) → ( ( 𝑍 𝐺 𝑍 ) 𝐻 𝐴 ) = ( ( 𝑍 𝐻 𝐴 ) 𝐺 ( 𝑍 𝐻 𝐴 ) ) ) |
16 |
14 15
|
syldan |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝑍 𝐺 𝑍 ) 𝐻 𝐴 ) = ( ( 𝑍 𝐻 𝐴 ) 𝐺 ( 𝑍 𝐻 𝐴 ) ) ) |
17 |
5
|
adantr |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → 𝐺 ∈ GrpOp ) |
18 |
|
simpl |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → 𝑅 ∈ RingOps ) |
19 |
3 4 2
|
rngocl |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑍 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( 𝑍 𝐻 𝐴 ) ∈ 𝑋 ) |
20 |
18 12 13 19
|
syl3anc |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( 𝑍 𝐻 𝐴 ) ∈ 𝑋 ) |
21 |
2 1
|
grporid |
⊢ ( ( 𝐺 ∈ GrpOp ∧ ( 𝑍 𝐻 𝐴 ) ∈ 𝑋 ) → ( ( 𝑍 𝐻 𝐴 ) 𝐺 𝑍 ) = ( 𝑍 𝐻 𝐴 ) ) |
22 |
21
|
eqcomd |
⊢ ( ( 𝐺 ∈ GrpOp ∧ ( 𝑍 𝐻 𝐴 ) ∈ 𝑋 ) → ( 𝑍 𝐻 𝐴 ) = ( ( 𝑍 𝐻 𝐴 ) 𝐺 𝑍 ) ) |
23 |
17 20 22
|
syl2anc |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( 𝑍 𝐻 𝐴 ) = ( ( 𝑍 𝐻 𝐴 ) 𝐺 𝑍 ) ) |
24 |
10 16 23
|
3eqtr3d |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝑍 𝐻 𝐴 ) 𝐺 ( 𝑍 𝐻 𝐴 ) ) = ( ( 𝑍 𝐻 𝐴 ) 𝐺 𝑍 ) ) |
25 |
2
|
grpolcan |
⊢ ( ( 𝐺 ∈ GrpOp ∧ ( ( 𝑍 𝐻 𝐴 ) ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ ( 𝑍 𝐻 𝐴 ) ∈ 𝑋 ) ) → ( ( ( 𝑍 𝐻 𝐴 ) 𝐺 ( 𝑍 𝐻 𝐴 ) ) = ( ( 𝑍 𝐻 𝐴 ) 𝐺 𝑍 ) ↔ ( 𝑍 𝐻 𝐴 ) = 𝑍 ) ) |
26 |
17 20 12 20 25
|
syl13anc |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( ( ( 𝑍 𝐻 𝐴 ) 𝐺 ( 𝑍 𝐻 𝐴 ) ) = ( ( 𝑍 𝐻 𝐴 ) 𝐺 𝑍 ) ↔ ( 𝑍 𝐻 𝐴 ) = 𝑍 ) ) |
27 |
24 26
|
mpbid |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( 𝑍 𝐻 𝐴 ) = 𝑍 ) |