| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | df-br 5143 | . . . 4
⊢ (𝐺RingOps𝐻 ↔ 〈𝐺, 𝐻〉 ∈ RingOps) | 
| 2 |  | relrngo 37904 | . . . . 5
⊢ Rel
RingOps | 
| 3 | 2 | brrelex1i 5740 | . . . 4
⊢ (𝐺RingOps𝐻 → 𝐺 ∈ V) | 
| 4 | 1, 3 | sylbir 235 | . . 3
⊢
(〈𝐺, 𝐻〉 ∈ RingOps →
𝐺 ∈
V) | 
| 5 | 4 | a1i 11 | . 2
⊢ (𝐻 ∈ 𝐴 → (〈𝐺, 𝐻〉 ∈ RingOps → 𝐺 ∈ V)) | 
| 6 |  | elex 3500 | . . . 4
⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ V) | 
| 7 | 6 | ad2antrr 726 | . . 3
⊢ (((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))) → 𝐺 ∈ V) | 
| 8 | 7 | a1i 11 | . 2
⊢ (𝐻 ∈ 𝐴 → (((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))) → 𝐺 ∈ V)) | 
| 9 |  | df-rngo 37903 | . . . . 5
⊢ RingOps =
{〈𝑔, ℎ〉 ∣ ((𝑔 ∈ AbelOp ∧ ℎ:(ran 𝑔 × ran 𝑔)⟶ran 𝑔) ∧ (∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔∀𝑧 ∈ ran 𝑔(((𝑥ℎ𝑦)ℎ𝑧) = (𝑥ℎ(𝑦ℎ𝑧)) ∧ (𝑥ℎ(𝑦𝑔𝑧)) = ((𝑥ℎ𝑦)𝑔(𝑥ℎ𝑧)) ∧ ((𝑥𝑔𝑦)ℎ𝑧) = ((𝑥ℎ𝑧)𝑔(𝑦ℎ𝑧))) ∧ ∃𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔((𝑥ℎ𝑦) = 𝑦 ∧ (𝑦ℎ𝑥) = 𝑦)))} | 
| 10 | 9 | eleq2i 2832 | . . . 4
⊢
(〈𝐺, 𝐻〉 ∈ RingOps ↔
〈𝐺, 𝐻〉 ∈ {〈𝑔, ℎ〉 ∣ ((𝑔 ∈ AbelOp ∧ ℎ:(ran 𝑔 × ran 𝑔)⟶ran 𝑔) ∧ (∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔∀𝑧 ∈ ran 𝑔(((𝑥ℎ𝑦)ℎ𝑧) = (𝑥ℎ(𝑦ℎ𝑧)) ∧ (𝑥ℎ(𝑦𝑔𝑧)) = ((𝑥ℎ𝑦)𝑔(𝑥ℎ𝑧)) ∧ ((𝑥𝑔𝑦)ℎ𝑧) = ((𝑥ℎ𝑧)𝑔(𝑦ℎ𝑧))) ∧ ∃𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔((𝑥ℎ𝑦) = 𝑦 ∧ (𝑦ℎ𝑥) = 𝑦)))}) | 
| 11 |  | simpl 482 | . . . . . . . 8
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → 𝑔 = 𝐺) | 
| 12 | 11 | eleq1d 2825 | . . . . . . 7
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (𝑔 ∈ AbelOp ↔ 𝐺 ∈ AbelOp)) | 
| 13 |  | simpr 484 | . . . . . . . 8
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → ℎ = 𝐻) | 
| 14 | 11 | rneqd 5948 | . . . . . . . . . 10
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → ran 𝑔 = ran 𝐺) | 
| 15 |  | isring.1 | . . . . . . . . . 10
⊢ 𝑋 = ran 𝐺 | 
| 16 | 14, 15 | eqtr4di 2794 | . . . . . . . . 9
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → ran 𝑔 = 𝑋) | 
| 17 | 16 | sqxpeqd 5716 | . . . . . . . 8
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (ran 𝑔 × ran 𝑔) = (𝑋 × 𝑋)) | 
| 18 | 13, 17, 16 | feq123d 6724 | . . . . . . 7
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (ℎ:(ran 𝑔 × ran 𝑔)⟶ran 𝑔 ↔ 𝐻:(𝑋 × 𝑋)⟶𝑋)) | 
| 19 | 12, 18 | anbi12d 632 | . . . . . 6
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → ((𝑔 ∈ AbelOp ∧ ℎ:(ran 𝑔 × ran 𝑔)⟶ran 𝑔) ↔ (𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋))) | 
| 20 | 13 | oveqd 7449 | . . . . . . . . . . . . 13
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (𝑥ℎ𝑦) = (𝑥𝐻𝑦)) | 
| 21 |  | eqidd 2737 | . . . . . . . . . . . . 13
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → 𝑧 = 𝑧) | 
| 22 | 13, 20, 21 | oveq123d 7453 | . . . . . . . . . . . 12
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → ((𝑥ℎ𝑦)ℎ𝑧) = ((𝑥𝐻𝑦)𝐻𝑧)) | 
| 23 |  | eqidd 2737 | . . . . . . . . . . . . 13
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → 𝑥 = 𝑥) | 
| 24 | 13 | oveqd 7449 | . . . . . . . . . . . . 13
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (𝑦ℎ𝑧) = (𝑦𝐻𝑧)) | 
| 25 | 13, 23, 24 | oveq123d 7453 | . . . . . . . . . . . 12
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (𝑥ℎ(𝑦ℎ𝑧)) = (𝑥𝐻(𝑦𝐻𝑧))) | 
| 26 | 22, 25 | eqeq12d 2752 | . . . . . . . . . . 11
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (((𝑥ℎ𝑦)ℎ𝑧) = (𝑥ℎ(𝑦ℎ𝑧)) ↔ ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)))) | 
| 27 | 11 | oveqd 7449 | . . . . . . . . . . . . 13
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (𝑦𝑔𝑧) = (𝑦𝐺𝑧)) | 
| 28 | 13, 23, 27 | oveq123d 7453 | . . . . . . . . . . . 12
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (𝑥ℎ(𝑦𝑔𝑧)) = (𝑥𝐻(𝑦𝐺𝑧))) | 
| 29 | 13 | oveqd 7449 | . . . . . . . . . . . . 13
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (𝑥ℎ𝑧) = (𝑥𝐻𝑧)) | 
| 30 | 11, 20, 29 | oveq123d 7453 | . . . . . . . . . . . 12
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → ((𝑥ℎ𝑦)𝑔(𝑥ℎ𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧))) | 
| 31 | 28, 30 | eqeq12d 2752 | . . . . . . . . . . 11
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → ((𝑥ℎ(𝑦𝑔𝑧)) = ((𝑥ℎ𝑦)𝑔(𝑥ℎ𝑧)) ↔ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)))) | 
| 32 | 11 | oveqd 7449 | . . . . . . . . . . . . 13
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (𝑥𝑔𝑦) = (𝑥𝐺𝑦)) | 
| 33 | 13, 32, 21 | oveq123d 7453 | . . . . . . . . . . . 12
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → ((𝑥𝑔𝑦)ℎ𝑧) = ((𝑥𝐺𝑦)𝐻𝑧)) | 
| 34 | 11, 29, 24 | oveq123d 7453 | . . . . . . . . . . . 12
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → ((𝑥ℎ𝑧)𝑔(𝑦ℎ𝑧)) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) | 
| 35 | 33, 34 | eqeq12d 2752 | . . . . . . . . . . 11
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (((𝑥𝑔𝑦)ℎ𝑧) = ((𝑥ℎ𝑧)𝑔(𝑦ℎ𝑧)) ↔ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))) | 
| 36 | 26, 31, 35 | 3anbi123d 1437 | . . . . . . . . . 10
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → ((((𝑥ℎ𝑦)ℎ𝑧) = (𝑥ℎ(𝑦ℎ𝑧)) ∧ (𝑥ℎ(𝑦𝑔𝑧)) = ((𝑥ℎ𝑦)𝑔(𝑥ℎ𝑧)) ∧ ((𝑥𝑔𝑦)ℎ𝑧) = ((𝑥ℎ𝑧)𝑔(𝑦ℎ𝑧))) ↔ (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))))) | 
| 37 | 16, 36 | raleqbidv 3345 | . . . . . . . . 9
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (∀𝑧 ∈ ran 𝑔(((𝑥ℎ𝑦)ℎ𝑧) = (𝑥ℎ(𝑦ℎ𝑧)) ∧ (𝑥ℎ(𝑦𝑔𝑧)) = ((𝑥ℎ𝑦)𝑔(𝑥ℎ𝑧)) ∧ ((𝑥𝑔𝑦)ℎ𝑧) = ((𝑥ℎ𝑧)𝑔(𝑦ℎ𝑧))) ↔ ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))))) | 
| 38 | 16, 37 | raleqbidv 3345 | . . . . . . . 8
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (∀𝑦 ∈ ran 𝑔∀𝑧 ∈ ran 𝑔(((𝑥ℎ𝑦)ℎ𝑧) = (𝑥ℎ(𝑦ℎ𝑧)) ∧ (𝑥ℎ(𝑦𝑔𝑧)) = ((𝑥ℎ𝑦)𝑔(𝑥ℎ𝑧)) ∧ ((𝑥𝑔𝑦)ℎ𝑧) = ((𝑥ℎ𝑧)𝑔(𝑦ℎ𝑧))) ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))))) | 
| 39 | 16, 38 | raleqbidv 3345 | . . . . . . 7
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔∀𝑧 ∈ ran 𝑔(((𝑥ℎ𝑦)ℎ𝑧) = (𝑥ℎ(𝑦ℎ𝑧)) ∧ (𝑥ℎ(𝑦𝑔𝑧)) = ((𝑥ℎ𝑦)𝑔(𝑥ℎ𝑧)) ∧ ((𝑥𝑔𝑦)ℎ𝑧) = ((𝑥ℎ𝑧)𝑔(𝑦ℎ𝑧))) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))))) | 
| 40 | 20 | eqeq1d 2738 | . . . . . . . . . 10
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → ((𝑥ℎ𝑦) = 𝑦 ↔ (𝑥𝐻𝑦) = 𝑦)) | 
| 41 | 13 | oveqd 7449 | . . . . . . . . . . 11
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (𝑦ℎ𝑥) = (𝑦𝐻𝑥)) | 
| 42 | 41 | eqeq1d 2738 | . . . . . . . . . 10
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → ((𝑦ℎ𝑥) = 𝑦 ↔ (𝑦𝐻𝑥) = 𝑦)) | 
| 43 | 40, 42 | anbi12d 632 | . . . . . . . . 9
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (((𝑥ℎ𝑦) = 𝑦 ∧ (𝑦ℎ𝑥) = 𝑦) ↔ ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))) | 
| 44 | 16, 43 | raleqbidv 3345 | . . . . . . . 8
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (∀𝑦 ∈ ran 𝑔((𝑥ℎ𝑦) = 𝑦 ∧ (𝑦ℎ𝑥) = 𝑦) ↔ ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))) | 
| 45 | 16, 44 | rexeqbidv 3346 | . . . . . . 7
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (∃𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔((𝑥ℎ𝑦) = 𝑦 ∧ (𝑦ℎ𝑥) = 𝑦) ↔ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))) | 
| 46 | 39, 45 | anbi12d 632 | . . . . . 6
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → ((∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔∀𝑧 ∈ ran 𝑔(((𝑥ℎ𝑦)ℎ𝑧) = (𝑥ℎ(𝑦ℎ𝑧)) ∧ (𝑥ℎ(𝑦𝑔𝑧)) = ((𝑥ℎ𝑦)𝑔(𝑥ℎ𝑧)) ∧ ((𝑥𝑔𝑦)ℎ𝑧) = ((𝑥ℎ𝑧)𝑔(𝑦ℎ𝑧))) ∧ ∃𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔((𝑥ℎ𝑦) = 𝑦 ∧ (𝑦ℎ𝑥) = 𝑦)) ↔ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))) | 
| 47 | 19, 46 | anbi12d 632 | . . . . 5
⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (((𝑔 ∈ AbelOp ∧ ℎ:(ran 𝑔 × ran 𝑔)⟶ran 𝑔) ∧ (∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔∀𝑧 ∈ ran 𝑔(((𝑥ℎ𝑦)ℎ𝑧) = (𝑥ℎ(𝑦ℎ𝑧)) ∧ (𝑥ℎ(𝑦𝑔𝑧)) = ((𝑥ℎ𝑦)𝑔(𝑥ℎ𝑧)) ∧ ((𝑥𝑔𝑦)ℎ𝑧) = ((𝑥ℎ𝑧)𝑔(𝑦ℎ𝑧))) ∧ ∃𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔((𝑥ℎ𝑦) = 𝑦 ∧ (𝑦ℎ𝑥) = 𝑦))) ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))))) | 
| 48 | 47 | opelopabga 5537 | . . . 4
⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ 𝐴) → (〈𝐺, 𝐻〉 ∈ {〈𝑔, ℎ〉 ∣ ((𝑔 ∈ AbelOp ∧ ℎ:(ran 𝑔 × ran 𝑔)⟶ran 𝑔) ∧ (∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔∀𝑧 ∈ ran 𝑔(((𝑥ℎ𝑦)ℎ𝑧) = (𝑥ℎ(𝑦ℎ𝑧)) ∧ (𝑥ℎ(𝑦𝑔𝑧)) = ((𝑥ℎ𝑦)𝑔(𝑥ℎ𝑧)) ∧ ((𝑥𝑔𝑦)ℎ𝑧) = ((𝑥ℎ𝑧)𝑔(𝑦ℎ𝑧))) ∧ ∃𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔((𝑥ℎ𝑦) = 𝑦 ∧ (𝑦ℎ𝑥) = 𝑦)))} ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))))) | 
| 49 | 10, 48 | bitrid 283 | . . 3
⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ 𝐴) → (〈𝐺, 𝐻〉 ∈ RingOps ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))))) | 
| 50 | 49 | expcom 413 | . 2
⊢ (𝐻 ∈ 𝐴 → (𝐺 ∈ V → (〈𝐺, 𝐻〉 ∈ RingOps ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))))) | 
| 51 | 5, 8, 50 | pm5.21ndd 379 | 1
⊢ (𝐻 ∈ 𝐴 → (〈𝐺, 𝐻〉 ∈ RingOps ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))))) |