Step |
Hyp |
Ref |
Expression |
1 |
|
isringod.1 |
⊢ ( 𝜑 → 𝐺 ∈ AbelOp ) |
2 |
|
isringod.2 |
⊢ ( 𝜑 → 𝑋 = ran 𝐺 ) |
3 |
|
isringod.3 |
⊢ ( 𝜑 → 𝐻 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ) |
4 |
|
isringod.4 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ) ) → ( ( 𝑥 𝐻 𝑦 ) 𝐻 𝑧 ) = ( 𝑥 𝐻 ( 𝑦 𝐻 𝑧 ) ) ) |
5 |
|
isringod.5 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ) ) → ( 𝑥 𝐻 ( 𝑦 𝐺 𝑧 ) ) = ( ( 𝑥 𝐻 𝑦 ) 𝐺 ( 𝑥 𝐻 𝑧 ) ) ) |
6 |
|
isringod.6 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ) ) → ( ( 𝑥 𝐺 𝑦 ) 𝐻 𝑧 ) = ( ( 𝑥 𝐻 𝑧 ) 𝐺 ( 𝑦 𝐻 𝑧 ) ) ) |
7 |
|
isringod.7 |
⊢ ( 𝜑 → 𝑈 ∈ 𝑋 ) |
8 |
|
isringod.8 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑋 ) → ( 𝑈 𝐻 𝑦 ) = 𝑦 ) |
9 |
|
isringod.9 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑋 ) → ( 𝑦 𝐻 𝑈 ) = 𝑦 ) |
10 |
2
|
sqxpeqd |
⊢ ( 𝜑 → ( 𝑋 × 𝑋 ) = ( ran 𝐺 × ran 𝐺 ) ) |
11 |
10 2
|
feq23d |
⊢ ( 𝜑 → ( 𝐻 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ↔ 𝐻 : ( ran 𝐺 × ran 𝐺 ) ⟶ ran 𝐺 ) ) |
12 |
3 11
|
mpbid |
⊢ ( 𝜑 → 𝐻 : ( ran 𝐺 × ran 𝐺 ) ⟶ ran 𝐺 ) |
13 |
4 5 6
|
3jca |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ) ) → ( ( ( 𝑥 𝐻 𝑦 ) 𝐻 𝑧 ) = ( 𝑥 𝐻 ( 𝑦 𝐻 𝑧 ) ) ∧ ( 𝑥 𝐻 ( 𝑦 𝐺 𝑧 ) ) = ( ( 𝑥 𝐻 𝑦 ) 𝐺 ( 𝑥 𝐻 𝑧 ) ) ∧ ( ( 𝑥 𝐺 𝑦 ) 𝐻 𝑧 ) = ( ( 𝑥 𝐻 𝑧 ) 𝐺 ( 𝑦 𝐻 𝑧 ) ) ) ) |
14 |
13
|
ralrimivvva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( ( 𝑥 𝐻 𝑦 ) 𝐻 𝑧 ) = ( 𝑥 𝐻 ( 𝑦 𝐻 𝑧 ) ) ∧ ( 𝑥 𝐻 ( 𝑦 𝐺 𝑧 ) ) = ( ( 𝑥 𝐻 𝑦 ) 𝐺 ( 𝑥 𝐻 𝑧 ) ) ∧ ( ( 𝑥 𝐺 𝑦 ) 𝐻 𝑧 ) = ( ( 𝑥 𝐻 𝑧 ) 𝐺 ( 𝑦 𝐻 𝑧 ) ) ) ) |
15 |
2
|
raleqdv |
⊢ ( 𝜑 → ( ∀ 𝑧 ∈ 𝑋 ( ( ( 𝑥 𝐻 𝑦 ) 𝐻 𝑧 ) = ( 𝑥 𝐻 ( 𝑦 𝐻 𝑧 ) ) ∧ ( 𝑥 𝐻 ( 𝑦 𝐺 𝑧 ) ) = ( ( 𝑥 𝐻 𝑦 ) 𝐺 ( 𝑥 𝐻 𝑧 ) ) ∧ ( ( 𝑥 𝐺 𝑦 ) 𝐻 𝑧 ) = ( ( 𝑥 𝐻 𝑧 ) 𝐺 ( 𝑦 𝐻 𝑧 ) ) ) ↔ ∀ 𝑧 ∈ ran 𝐺 ( ( ( 𝑥 𝐻 𝑦 ) 𝐻 𝑧 ) = ( 𝑥 𝐻 ( 𝑦 𝐻 𝑧 ) ) ∧ ( 𝑥 𝐻 ( 𝑦 𝐺 𝑧 ) ) = ( ( 𝑥 𝐻 𝑦 ) 𝐺 ( 𝑥 𝐻 𝑧 ) ) ∧ ( ( 𝑥 𝐺 𝑦 ) 𝐻 𝑧 ) = ( ( 𝑥 𝐻 𝑧 ) 𝐺 ( 𝑦 𝐻 𝑧 ) ) ) ) ) |
16 |
2 15
|
raleqbidv |
⊢ ( 𝜑 → ( ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( ( 𝑥 𝐻 𝑦 ) 𝐻 𝑧 ) = ( 𝑥 𝐻 ( 𝑦 𝐻 𝑧 ) ) ∧ ( 𝑥 𝐻 ( 𝑦 𝐺 𝑧 ) ) = ( ( 𝑥 𝐻 𝑦 ) 𝐺 ( 𝑥 𝐻 𝑧 ) ) ∧ ( ( 𝑥 𝐺 𝑦 ) 𝐻 𝑧 ) = ( ( 𝑥 𝐻 𝑧 ) 𝐺 ( 𝑦 𝐻 𝑧 ) ) ) ↔ ∀ 𝑦 ∈ ran 𝐺 ∀ 𝑧 ∈ ran 𝐺 ( ( ( 𝑥 𝐻 𝑦 ) 𝐻 𝑧 ) = ( 𝑥 𝐻 ( 𝑦 𝐻 𝑧 ) ) ∧ ( 𝑥 𝐻 ( 𝑦 𝐺 𝑧 ) ) = ( ( 𝑥 𝐻 𝑦 ) 𝐺 ( 𝑥 𝐻 𝑧 ) ) ∧ ( ( 𝑥 𝐺 𝑦 ) 𝐻 𝑧 ) = ( ( 𝑥 𝐻 𝑧 ) 𝐺 ( 𝑦 𝐻 𝑧 ) ) ) ) ) |
17 |
2 16
|
raleqbidv |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( ( 𝑥 𝐻 𝑦 ) 𝐻 𝑧 ) = ( 𝑥 𝐻 ( 𝑦 𝐻 𝑧 ) ) ∧ ( 𝑥 𝐻 ( 𝑦 𝐺 𝑧 ) ) = ( ( 𝑥 𝐻 𝑦 ) 𝐺 ( 𝑥 𝐻 𝑧 ) ) ∧ ( ( 𝑥 𝐺 𝑦 ) 𝐻 𝑧 ) = ( ( 𝑥 𝐻 𝑧 ) 𝐺 ( 𝑦 𝐻 𝑧 ) ) ) ↔ ∀ 𝑥 ∈ ran 𝐺 ∀ 𝑦 ∈ ran 𝐺 ∀ 𝑧 ∈ ran 𝐺 ( ( ( 𝑥 𝐻 𝑦 ) 𝐻 𝑧 ) = ( 𝑥 𝐻 ( 𝑦 𝐻 𝑧 ) ) ∧ ( 𝑥 𝐻 ( 𝑦 𝐺 𝑧 ) ) = ( ( 𝑥 𝐻 𝑦 ) 𝐺 ( 𝑥 𝐻 𝑧 ) ) ∧ ( ( 𝑥 𝐺 𝑦 ) 𝐻 𝑧 ) = ( ( 𝑥 𝐻 𝑧 ) 𝐺 ( 𝑦 𝐻 𝑧 ) ) ) ) ) |
18 |
14 17
|
mpbid |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ran 𝐺 ∀ 𝑦 ∈ ran 𝐺 ∀ 𝑧 ∈ ran 𝐺 ( ( ( 𝑥 𝐻 𝑦 ) 𝐻 𝑧 ) = ( 𝑥 𝐻 ( 𝑦 𝐻 𝑧 ) ) ∧ ( 𝑥 𝐻 ( 𝑦 𝐺 𝑧 ) ) = ( ( 𝑥 𝐻 𝑦 ) 𝐺 ( 𝑥 𝐻 𝑧 ) ) ∧ ( ( 𝑥 𝐺 𝑦 ) 𝐻 𝑧 ) = ( ( 𝑥 𝐻 𝑧 ) 𝐺 ( 𝑦 𝐻 𝑧 ) ) ) ) |
19 |
8 9
|
jca |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑋 ) → ( ( 𝑈 𝐻 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐻 𝑈 ) = 𝑦 ) ) |
20 |
19
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝑋 ( ( 𝑈 𝐻 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐻 𝑈 ) = 𝑦 ) ) |
21 |
|
oveq1 |
⊢ ( 𝑥 = 𝑈 → ( 𝑥 𝐻 𝑦 ) = ( 𝑈 𝐻 𝑦 ) ) |
22 |
21
|
eqeq1d |
⊢ ( 𝑥 = 𝑈 → ( ( 𝑥 𝐻 𝑦 ) = 𝑦 ↔ ( 𝑈 𝐻 𝑦 ) = 𝑦 ) ) |
23 |
22
|
ovanraleqv |
⊢ ( 𝑥 = 𝑈 → ( ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 𝐻 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐻 𝑥 ) = 𝑦 ) ↔ ∀ 𝑦 ∈ 𝑋 ( ( 𝑈 𝐻 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐻 𝑈 ) = 𝑦 ) ) ) |
24 |
23
|
rspcev |
⊢ ( ( 𝑈 ∈ 𝑋 ∧ ∀ 𝑦 ∈ 𝑋 ( ( 𝑈 𝐻 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐻 𝑈 ) = 𝑦 ) ) → ∃ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 𝐻 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐻 𝑥 ) = 𝑦 ) ) |
25 |
7 20 24
|
syl2anc |
⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 𝐻 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐻 𝑥 ) = 𝑦 ) ) |
26 |
2
|
raleqdv |
⊢ ( 𝜑 → ( ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 𝐻 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐻 𝑥 ) = 𝑦 ) ↔ ∀ 𝑦 ∈ ran 𝐺 ( ( 𝑥 𝐻 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐻 𝑥 ) = 𝑦 ) ) ) |
27 |
2 26
|
rexeqbidv |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 𝐻 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐻 𝑥 ) = 𝑦 ) ↔ ∃ 𝑥 ∈ ran 𝐺 ∀ 𝑦 ∈ ran 𝐺 ( ( 𝑥 𝐻 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐻 𝑥 ) = 𝑦 ) ) ) |
28 |
25 27
|
mpbid |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ran 𝐺 ∀ 𝑦 ∈ ran 𝐺 ( ( 𝑥 𝐻 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐻 𝑥 ) = 𝑦 ) ) |
29 |
18 28
|
jca |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ran 𝐺 ∀ 𝑦 ∈ ran 𝐺 ∀ 𝑧 ∈ ran 𝐺 ( ( ( 𝑥 𝐻 𝑦 ) 𝐻 𝑧 ) = ( 𝑥 𝐻 ( 𝑦 𝐻 𝑧 ) ) ∧ ( 𝑥 𝐻 ( 𝑦 𝐺 𝑧 ) ) = ( ( 𝑥 𝐻 𝑦 ) 𝐺 ( 𝑥 𝐻 𝑧 ) ) ∧ ( ( 𝑥 𝐺 𝑦 ) 𝐻 𝑧 ) = ( ( 𝑥 𝐻 𝑧 ) 𝐺 ( 𝑦 𝐻 𝑧 ) ) ) ∧ ∃ 𝑥 ∈ ran 𝐺 ∀ 𝑦 ∈ ran 𝐺 ( ( 𝑥 𝐻 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐻 𝑥 ) = 𝑦 ) ) ) |
30 |
1 12 29
|
jca31 |
⊢ ( 𝜑 → ( ( 𝐺 ∈ AbelOp ∧ 𝐻 : ( ran 𝐺 × ran 𝐺 ) ⟶ ran 𝐺 ) ∧ ( ∀ 𝑥 ∈ ran 𝐺 ∀ 𝑦 ∈ ran 𝐺 ∀ 𝑧 ∈ ran 𝐺 ( ( ( 𝑥 𝐻 𝑦 ) 𝐻 𝑧 ) = ( 𝑥 𝐻 ( 𝑦 𝐻 𝑧 ) ) ∧ ( 𝑥 𝐻 ( 𝑦 𝐺 𝑧 ) ) = ( ( 𝑥 𝐻 𝑦 ) 𝐺 ( 𝑥 𝐻 𝑧 ) ) ∧ ( ( 𝑥 𝐺 𝑦 ) 𝐻 𝑧 ) = ( ( 𝑥 𝐻 𝑧 ) 𝐺 ( 𝑦 𝐻 𝑧 ) ) ) ∧ ∃ 𝑥 ∈ ran 𝐺 ∀ 𝑦 ∈ ran 𝐺 ( ( 𝑥 𝐻 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐻 𝑥 ) = 𝑦 ) ) ) ) |
31 |
|
rnexg |
⊢ ( 𝐺 ∈ AbelOp → ran 𝐺 ∈ V ) |
32 |
1 31
|
syl |
⊢ ( 𝜑 → ran 𝐺 ∈ V ) |
33 |
32 32
|
xpexd |
⊢ ( 𝜑 → ( ran 𝐺 × ran 𝐺 ) ∈ V ) |
34 |
12 33
|
fexd |
⊢ ( 𝜑 → 𝐻 ∈ V ) |
35 |
|
eqid |
⊢ ran 𝐺 = ran 𝐺 |
36 |
35
|
isrngo |
⊢ ( 𝐻 ∈ V → ( 〈 𝐺 , 𝐻 〉 ∈ RingOps ↔ ( ( 𝐺 ∈ AbelOp ∧ 𝐻 : ( ran 𝐺 × ran 𝐺 ) ⟶ ran 𝐺 ) ∧ ( ∀ 𝑥 ∈ ran 𝐺 ∀ 𝑦 ∈ ran 𝐺 ∀ 𝑧 ∈ ran 𝐺 ( ( ( 𝑥 𝐻 𝑦 ) 𝐻 𝑧 ) = ( 𝑥 𝐻 ( 𝑦 𝐻 𝑧 ) ) ∧ ( 𝑥 𝐻 ( 𝑦 𝐺 𝑧 ) ) = ( ( 𝑥 𝐻 𝑦 ) 𝐺 ( 𝑥 𝐻 𝑧 ) ) ∧ ( ( 𝑥 𝐺 𝑦 ) 𝐻 𝑧 ) = ( ( 𝑥 𝐻 𝑧 ) 𝐺 ( 𝑦 𝐻 𝑧 ) ) ) ∧ ∃ 𝑥 ∈ ran 𝐺 ∀ 𝑦 ∈ ran 𝐺 ( ( 𝑥 𝐻 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐻 𝑥 ) = 𝑦 ) ) ) ) ) |
37 |
34 36
|
syl |
⊢ ( 𝜑 → ( 〈 𝐺 , 𝐻 〉 ∈ RingOps ↔ ( ( 𝐺 ∈ AbelOp ∧ 𝐻 : ( ran 𝐺 × ran 𝐺 ) ⟶ ran 𝐺 ) ∧ ( ∀ 𝑥 ∈ ran 𝐺 ∀ 𝑦 ∈ ran 𝐺 ∀ 𝑧 ∈ ran 𝐺 ( ( ( 𝑥 𝐻 𝑦 ) 𝐻 𝑧 ) = ( 𝑥 𝐻 ( 𝑦 𝐻 𝑧 ) ) ∧ ( 𝑥 𝐻 ( 𝑦 𝐺 𝑧 ) ) = ( ( 𝑥 𝐻 𝑦 ) 𝐺 ( 𝑥 𝐻 𝑧 ) ) ∧ ( ( 𝑥 𝐺 𝑦 ) 𝐻 𝑧 ) = ( ( 𝑥 𝐻 𝑧 ) 𝐺 ( 𝑦 𝐻 𝑧 ) ) ) ∧ ∃ 𝑥 ∈ ran 𝐺 ∀ 𝑦 ∈ ran 𝐺 ( ( 𝑥 𝐻 𝑦 ) = 𝑦 ∧ ( 𝑦 𝐻 𝑥 ) = 𝑦 ) ) ) ) ) |
38 |
30 37
|
mpbird |
⊢ ( 𝜑 → 〈 𝐺 , 𝐻 〉 ∈ RingOps ) |