Step |
Hyp |
Ref |
Expression |
1 |
|
ringnegmul.1 |
⊢ 𝐺 = ( 1st ‘ 𝑅 ) |
2 |
|
ringnegmul.2 |
⊢ 𝐻 = ( 2nd ‘ 𝑅 ) |
3 |
|
ringnegmul.3 |
⊢ 𝑋 = ran 𝐺 |
4 |
|
ringnegmul.4 |
⊢ 𝑁 = ( inv ‘ 𝐺 ) |
5 |
1
|
rneqi |
⊢ ran 𝐺 = ran ( 1st ‘ 𝑅 ) |
6 |
3 5
|
eqtri |
⊢ 𝑋 = ran ( 1st ‘ 𝑅 ) |
7 |
|
eqid |
⊢ ( GId ‘ 𝐻 ) = ( GId ‘ 𝐻 ) |
8 |
6 2 7
|
rngo1cl |
⊢ ( 𝑅 ∈ RingOps → ( GId ‘ 𝐻 ) ∈ 𝑋 ) |
9 |
1 3 4
|
rngonegcl |
⊢ ( ( 𝑅 ∈ RingOps ∧ ( GId ‘ 𝐻 ) ∈ 𝑋 ) → ( 𝑁 ‘ ( GId ‘ 𝐻 ) ) ∈ 𝑋 ) |
10 |
8 9
|
mpdan |
⊢ ( 𝑅 ∈ RingOps → ( 𝑁 ‘ ( GId ‘ 𝐻 ) ) ∈ 𝑋 ) |
11 |
1 2 3
|
rngoass |
⊢ ( ( 𝑅 ∈ RingOps ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ( 𝑁 ‘ ( GId ‘ 𝐻 ) ) ∈ 𝑋 ) ) → ( ( 𝐴 𝐻 𝐵 ) 𝐻 ( 𝑁 ‘ ( GId ‘ 𝐻 ) ) ) = ( 𝐴 𝐻 ( 𝐵 𝐻 ( 𝑁 ‘ ( GId ‘ 𝐻 ) ) ) ) ) |
12 |
11
|
3exp2 |
⊢ ( 𝑅 ∈ RingOps → ( 𝐴 ∈ 𝑋 → ( 𝐵 ∈ 𝑋 → ( ( 𝑁 ‘ ( GId ‘ 𝐻 ) ) ∈ 𝑋 → ( ( 𝐴 𝐻 𝐵 ) 𝐻 ( 𝑁 ‘ ( GId ‘ 𝐻 ) ) ) = ( 𝐴 𝐻 ( 𝐵 𝐻 ( 𝑁 ‘ ( GId ‘ 𝐻 ) ) ) ) ) ) ) ) |
13 |
12
|
com24 |
⊢ ( 𝑅 ∈ RingOps → ( ( 𝑁 ‘ ( GId ‘ 𝐻 ) ) ∈ 𝑋 → ( 𝐵 ∈ 𝑋 → ( 𝐴 ∈ 𝑋 → ( ( 𝐴 𝐻 𝐵 ) 𝐻 ( 𝑁 ‘ ( GId ‘ 𝐻 ) ) ) = ( 𝐴 𝐻 ( 𝐵 𝐻 ( 𝑁 ‘ ( GId ‘ 𝐻 ) ) ) ) ) ) ) ) |
14 |
13
|
com34 |
⊢ ( 𝑅 ∈ RingOps → ( ( 𝑁 ‘ ( GId ‘ 𝐻 ) ) ∈ 𝑋 → ( 𝐴 ∈ 𝑋 → ( 𝐵 ∈ 𝑋 → ( ( 𝐴 𝐻 𝐵 ) 𝐻 ( 𝑁 ‘ ( GId ‘ 𝐻 ) ) ) = ( 𝐴 𝐻 ( 𝐵 𝐻 ( 𝑁 ‘ ( GId ‘ 𝐻 ) ) ) ) ) ) ) ) |
15 |
10 14
|
mpd |
⊢ ( 𝑅 ∈ RingOps → ( 𝐴 ∈ 𝑋 → ( 𝐵 ∈ 𝑋 → ( ( 𝐴 𝐻 𝐵 ) 𝐻 ( 𝑁 ‘ ( GId ‘ 𝐻 ) ) ) = ( 𝐴 𝐻 ( 𝐵 𝐻 ( 𝑁 ‘ ( GId ‘ 𝐻 ) ) ) ) ) ) ) |
16 |
15
|
3imp |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝐴 𝐻 𝐵 ) 𝐻 ( 𝑁 ‘ ( GId ‘ 𝐻 ) ) ) = ( 𝐴 𝐻 ( 𝐵 𝐻 ( 𝑁 ‘ ( GId ‘ 𝐻 ) ) ) ) ) |
17 |
1 2 3
|
rngocl |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐻 𝐵 ) ∈ 𝑋 ) |
18 |
17
|
3expb |
⊢ ( ( 𝑅 ∈ RingOps ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( 𝐴 𝐻 𝐵 ) ∈ 𝑋 ) |
19 |
1 2 3 4 7
|
rngonegmn1r |
⊢ ( ( 𝑅 ∈ RingOps ∧ ( 𝐴 𝐻 𝐵 ) ∈ 𝑋 ) → ( 𝑁 ‘ ( 𝐴 𝐻 𝐵 ) ) = ( ( 𝐴 𝐻 𝐵 ) 𝐻 ( 𝑁 ‘ ( GId ‘ 𝐻 ) ) ) ) |
20 |
18 19
|
syldan |
⊢ ( ( 𝑅 ∈ RingOps ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( 𝑁 ‘ ( 𝐴 𝐻 𝐵 ) ) = ( ( 𝐴 𝐻 𝐵 ) 𝐻 ( 𝑁 ‘ ( GId ‘ 𝐻 ) ) ) ) |
21 |
20
|
3impb |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝑁 ‘ ( 𝐴 𝐻 𝐵 ) ) = ( ( 𝐴 𝐻 𝐵 ) 𝐻 ( 𝑁 ‘ ( GId ‘ 𝐻 ) ) ) ) |
22 |
1 2 3 4 7
|
rngonegmn1r |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐵 ∈ 𝑋 ) → ( 𝑁 ‘ 𝐵 ) = ( 𝐵 𝐻 ( 𝑁 ‘ ( GId ‘ 𝐻 ) ) ) ) |
23 |
22
|
3adant2 |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝑁 ‘ 𝐵 ) = ( 𝐵 𝐻 ( 𝑁 ‘ ( GId ‘ 𝐻 ) ) ) ) |
24 |
23
|
oveq2d |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐻 ( 𝑁 ‘ 𝐵 ) ) = ( 𝐴 𝐻 ( 𝐵 𝐻 ( 𝑁 ‘ ( GId ‘ 𝐻 ) ) ) ) ) |
25 |
16 21 24
|
3eqtr4d |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝑁 ‘ ( 𝐴 𝐻 𝐵 ) ) = ( 𝐴 𝐻 ( 𝑁 ‘ 𝐵 ) ) ) |