Theorem List for Intuitionistic Logic Explorer - 6001-6100 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | caovordd 6001* |
Convert an operation ordering law to class notation. (Contributed by
Mario Carneiro, 30-Dec-2014.)
|
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) & ⊢ (𝜑 → 𝐴 ∈ 𝑆)
& ⊢ (𝜑 → 𝐵 ∈ 𝑆)
& ⊢ (𝜑 → 𝐶 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))) |
|
Theorem | caovord2d 6002* |
Operation ordering law with commuted arguments. (Contributed by Mario
Carneiro, 30-Dec-2014.)
|
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) & ⊢ (𝜑 → 𝐴 ∈ 𝑆)
& ⊢ (𝜑 → 𝐵 ∈ 𝑆)
& ⊢ (𝜑 → 𝐶 ∈ 𝑆)
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) ⇒ ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶))) |
|
Theorem | caovord3d 6003* |
Ordering law. (Contributed by Mario Carneiro, 30-Dec-2014.)
|
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) & ⊢ (𝜑 → 𝐴 ∈ 𝑆)
& ⊢ (𝜑 → 𝐵 ∈ 𝑆)
& ⊢ (𝜑 → 𝐶 ∈ 𝑆)
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
& ⊢ (𝜑 → 𝐷 ∈ 𝑆) ⇒ ⊢ (𝜑 → ((𝐴𝐹𝐵) = (𝐶𝐹𝐷) → (𝐴𝑅𝐶 ↔ 𝐷𝑅𝐵))) |
|
Theorem | caovord 6004* |
Convert an operation ordering law to class notation. (Contributed by
NM, 19-Feb-1996.)
|
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ (𝑧 ∈ 𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) ⇒ ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))) |
|
Theorem | caovord2 6005* |
Operation ordering law with commuted arguments. (Contributed by NM,
27-Feb-1996.)
|
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ (𝑧 ∈ 𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) & ⊢ 𝐶 ∈ V & ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥) ⇒ ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶))) |
|
Theorem | caovord3 6006* |
Ordering law. (Contributed by NM, 29-Feb-1996.)
|
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ (𝑧 ∈ 𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) & ⊢ 𝐶 ∈ V & ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
& ⊢ 𝐷 ∈ V ⇒ ⊢ (((𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ (𝐴𝐹𝐵) = (𝐶𝐹𝐷)) → (𝐴𝑅𝐶 ↔ 𝐷𝑅𝐵)) |
|
Theorem | caovdig 6007* |
Convert an operation distributive law to class notation. (Contributed
by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 26-Jul-2014.)
|
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐻(𝑥𝐺𝑧))) ⇒ ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝐶))) |
|
Theorem | caovdid 6008* |
Convert an operation distributive law to class notation. (Contributed
by Mario Carneiro, 30-Dec-2014.)
|
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐻(𝑥𝐺𝑧))) & ⊢ (𝜑 → 𝐴 ∈ 𝐾)
& ⊢ (𝜑 → 𝐵 ∈ 𝑆)
& ⊢ (𝜑 → 𝐶 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝐶))) |
|
Theorem | caovdir2d 6009* |
Convert an operation distributive law to class notation. (Contributed
by Mario Carneiro, 30-Dec-2014.)
|
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧))) & ⊢ (𝜑 → 𝐴 ∈ 𝑆)
& ⊢ (𝜑 → 𝐵 ∈ 𝑆)
& ⊢ (𝜑 → 𝐶 ∈ 𝑆)
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥)) ⇒ ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶))) |
|
Theorem | caovdirg 6010* |
Convert an operation reverse distributive law to class notation.
(Contributed by Mario Carneiro, 19-Oct-2014.)
|
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝐾)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧))) ⇒ ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝐾)) → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶))) |
|
Theorem | caovdird 6011* |
Convert an operation distributive law to class notation. (Contributed
by Mario Carneiro, 30-Dec-2014.)
|
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝐾)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧))) & ⊢ (𝜑 → 𝐴 ∈ 𝑆)
& ⊢ (𝜑 → 𝐵 ∈ 𝑆)
& ⊢ (𝜑 → 𝐶 ∈ 𝐾) ⇒ ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶))) |
|
Theorem | caovdi 6012* |
Convert an operation distributive law to class notation. (Contributed
by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 28-Jun-2013.)
|
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧)) ⇒ ⊢ (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶)) |
|
Theorem | caov32d 6013* |
Rearrange arguments in a commutative, associative operation.
(Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro,
30-Dec-2014.)
|
⊢ (𝜑 → 𝐴 ∈ 𝑆)
& ⊢ (𝜑 → 𝐵 ∈ 𝑆)
& ⊢ (𝜑 → 𝐶 ∈ 𝑆)
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))) ⇒ ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐴𝐹𝐶)𝐹𝐵)) |
|
Theorem | caov12d 6014* |
Rearrange arguments in a commutative, associative operation.
(Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro,
30-Dec-2014.)
|
⊢ (𝜑 → 𝐴 ∈ 𝑆)
& ⊢ (𝜑 → 𝐵 ∈ 𝑆)
& ⊢ (𝜑 → 𝐶 ∈ 𝑆)
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))) ⇒ ⊢ (𝜑 → (𝐴𝐹(𝐵𝐹𝐶)) = (𝐵𝐹(𝐴𝐹𝐶))) |
|
Theorem | caov31d 6015* |
Rearrange arguments in a commutative, associative operation.
(Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro,
30-Dec-2014.)
|
⊢ (𝜑 → 𝐴 ∈ 𝑆)
& ⊢ (𝜑 → 𝐵 ∈ 𝑆)
& ⊢ (𝜑 → 𝐶 ∈ 𝑆)
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))) ⇒ ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐶𝐹𝐵)𝐹𝐴)) |
|
Theorem | caov13d 6016* |
Rearrange arguments in a commutative, associative operation.
(Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro,
30-Dec-2014.)
|
⊢ (𝜑 → 𝐴 ∈ 𝑆)
& ⊢ (𝜑 → 𝐵 ∈ 𝑆)
& ⊢ (𝜑 → 𝐶 ∈ 𝑆)
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))) ⇒ ⊢ (𝜑 → (𝐴𝐹(𝐵𝐹𝐶)) = (𝐶𝐹(𝐵𝐹𝐴))) |
|
Theorem | caov4d 6017* |
Rearrange arguments in a commutative, associative operation.
(Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro,
30-Dec-2014.)
|
⊢ (𝜑 → 𝐴 ∈ 𝑆)
& ⊢ (𝜑 → 𝐵 ∈ 𝑆)
& ⊢ (𝜑 → 𝐶 ∈ 𝑆)
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))) & ⊢ (𝜑 → 𝐷 ∈ 𝑆)
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) ⇒ ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷))) |
|
Theorem | caov411d 6018* |
Rearrange arguments in a commutative, associative operation.
(Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro,
30-Dec-2014.)
|
⊢ (𝜑 → 𝐴 ∈ 𝑆)
& ⊢ (𝜑 → 𝐵 ∈ 𝑆)
& ⊢ (𝜑 → 𝐶 ∈ 𝑆)
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))) & ⊢ (𝜑 → 𝐷 ∈ 𝑆)
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) ⇒ ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐶𝐹𝐵)𝐹(𝐴𝐹𝐷))) |
|
Theorem | caov42d 6019* |
Rearrange arguments in a commutative, associative operation.
(Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro,
30-Dec-2014.)
|
⊢ (𝜑 → 𝐴 ∈ 𝑆)
& ⊢ (𝜑 → 𝐵 ∈ 𝑆)
& ⊢ (𝜑 → 𝐶 ∈ 𝑆)
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))) & ⊢ (𝜑 → 𝐷 ∈ 𝑆)
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) ⇒ ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐷𝐹𝐵))) |
|
Theorem | caov32 6020* |
Rearrange arguments in a commutative, associative operation.
(Contributed by NM, 26-Aug-1995.)
|
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
& ⊢ ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)) ⇒ ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐴𝐹𝐶)𝐹𝐵) |
|
Theorem | caov12 6021* |
Rearrange arguments in a commutative, associative operation.
(Contributed by NM, 26-Aug-1995.)
|
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
& ⊢ ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)) ⇒ ⊢ (𝐴𝐹(𝐵𝐹𝐶)) = (𝐵𝐹(𝐴𝐹𝐶)) |
|
Theorem | caov31 6022* |
Rearrange arguments in a commutative, associative operation.
(Contributed by NM, 26-Aug-1995.)
|
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
& ⊢ ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)) ⇒ ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐶𝐹𝐵)𝐹𝐴) |
|
Theorem | caov13 6023* |
Rearrange arguments in a commutative, associative operation.
(Contributed by NM, 26-Aug-1995.)
|
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
& ⊢ ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)) ⇒ ⊢ (𝐴𝐹(𝐵𝐹𝐶)) = (𝐶𝐹(𝐵𝐹𝐴)) |
|
Theorem | caovdilemd 6024* |
Lemma used by real number construction. (Contributed by Jim Kingdon,
16-Sep-2019.)
|
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐹(𝑦𝐺𝑧))) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐺𝑦) ∈ 𝑆)
& ⊢ (𝜑 → 𝐴 ∈ 𝑆)
& ⊢ (𝜑 → 𝐵 ∈ 𝑆)
& ⊢ (𝜑 → 𝐶 ∈ 𝑆)
& ⊢ (𝜑 → 𝐷 ∈ 𝑆)
& ⊢ (𝜑 → 𝐻 ∈ 𝑆) ⇒ ⊢ (𝜑 → (((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻)))) |
|
Theorem | caovlem2d 6025* |
Rearrangement of expression involving multiplication (𝐺) and
addition (𝐹). (Contributed by Jim Kingdon,
3-Jan-2020.)
|
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐹(𝑦𝐺𝑧))) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐺𝑦) ∈ 𝑆)
& ⊢ (𝜑 → 𝐴 ∈ 𝑆)
& ⊢ (𝜑 → 𝐵 ∈ 𝑆)
& ⊢ (𝜑 → 𝐶 ∈ 𝑆)
& ⊢ (𝜑 → 𝐷 ∈ 𝑆)
& ⊢ (𝜑 → 𝐻 ∈ 𝑆)
& ⊢ (𝜑 → 𝑅 ∈ 𝑆)
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) ⇒ ⊢ (𝜑 → ((((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻)𝐹(((𝐴𝐺𝐷)𝐹(𝐵𝐺𝐶))𝐺𝑅)) = ((𝐴𝐺((𝐶𝐺𝐻)𝐹(𝐷𝐺𝑅)))𝐹(𝐵𝐺((𝐶𝐺𝑅)𝐹(𝐷𝐺𝐻))))) |
|
Theorem | caovimo 6026* |
Uniqueness of inverse element in commutative, associative operation with
identity. The identity element is 𝐵. (Contributed by Jim Kingdon,
18-Sep-2019.)
|
⊢ 𝐵 ∈ 𝑆
& ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
& ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))) & ⊢ (𝑥 ∈ 𝑆 → (𝑥𝐹𝐵) = 𝑥) ⇒ ⊢ (𝐴 ∈ 𝑆 → ∃*𝑤(𝑤 ∈ 𝑆 ∧ (𝐴𝐹𝑤) = 𝐵)) |
|
Theorem | grprinvlem 6027* |
Lemma for grprinvd 6028. (Contributed by NM, 9-Aug-2013.)
|
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
& ⊢ (𝜑 → 𝑂 ∈ 𝐵)
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑂 + 𝑥) = 𝑥)
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 𝑂)
& ⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ 𝐵)
& ⊢ ((𝜑 ∧ 𝜓) → (𝑋 + 𝑋) = 𝑋) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝑋 = 𝑂) |
|
Theorem | grprinvd 6028* |
Deduce right inverse from left inverse and left identity in an
associative structure (such as a group). (Contributed by NM,
10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)
|
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
& ⊢ (𝜑 → 𝑂 ∈ 𝐵)
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑂 + 𝑥) = 𝑥)
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 𝑂)
& ⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ 𝐵)
& ⊢ ((𝜑 ∧ 𝜓) → 𝑁 ∈ 𝐵)
& ⊢ ((𝜑 ∧ 𝜓) → (𝑁 + 𝑋) = 𝑂) ⇒ ⊢ ((𝜑 ∧ 𝜓) → (𝑋 + 𝑁) = 𝑂) |
|
Theorem | grpridd 6029* |
Deduce right identity from left inverse and left identity in an
associative structure (such as a group). (Contributed by NM,
10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)
|
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
& ⊢ (𝜑 → 𝑂 ∈ 𝐵)
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑂 + 𝑥) = 𝑥)
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 𝑂) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 𝑂) = 𝑥) |
|
2.6.12 Maps-to notation
|
|
Theorem | elmpocl 6030* |
If a two-parameter class is inhabited, constrain the implicit pair.
(Contributed by Stefan O'Rear, 7-Mar-2015.)
|
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐵)) |
|
Theorem | elmpocl1 6031* |
If a two-parameter class is inhabited, the first argument is in its
nominal domain. (Contributed by FL, 15-Oct-2012.) (Revised by Stefan
O'Rear, 7-Mar-2015.)
|
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → 𝑆 ∈ 𝐴) |
|
Theorem | elmpocl2 6032* |
If a two-parameter class is inhabited, the second argument is in its
nominal domain. (Contributed by FL, 15-Oct-2012.) (Revised by Stefan
O'Rear, 7-Mar-2015.)
|
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → 𝑇 ∈ 𝐵) |
|
Theorem | elovmpo 6033* |
Utility lemma for two-parameter classes. (Contributed by Stefan O'Rear,
21-Jan-2015.)
|
⊢ 𝐷 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)
& ⊢ 𝐶 ∈ V & ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → 𝐶 = 𝐸) ⇒ ⊢ (𝐹 ∈ (𝑋𝐷𝑌) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹 ∈ 𝐸)) |
|
Theorem | f1ocnvd 6034* |
Describe an implicit one-to-one onto function. (Contributed by Mario
Carneiro, 30-Apr-2015.)
|
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑊)
& ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝑋)
& ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷))) ⇒ ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐵 ∧ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ 𝐷))) |
|
Theorem | f1od 6035* |
Describe an implicit one-to-one onto function. (Contributed by Mario
Carneiro, 12-May-2014.)
|
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑊)
& ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝑋)
& ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷))) ⇒ ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) |
|
Theorem | f1ocnv2d 6036* |
Describe an implicit one-to-one onto function. (Contributed by Mario
Carneiro, 30-Apr-2015.)
|
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
& ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝐴)
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶)) ⇒ ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐵 ∧ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ 𝐷))) |
|
Theorem | f1o2d 6037* |
Describe an implicit one-to-one onto function. (Contributed by Mario
Carneiro, 12-May-2014.)
|
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
& ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝐴)
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶)) ⇒ ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) |
|
Theorem | f1opw2 6038* |
A one-to-one mapping induces a one-to-one mapping on power sets. This
version of f1opw 6039 avoids the Axiom of Replacement.
(Contributed by
Mario Carneiro, 26-Jun-2015.)
|
⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
& ⊢ (𝜑 → (◡𝐹 “ 𝑎) ∈ V) & ⊢ (𝜑 → (𝐹 “ 𝑏) ∈ V) ⇒ ⊢ (𝜑 → (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹 “ 𝑏)):𝒫 𝐴–1-1-onto→𝒫 𝐵) |
|
Theorem | f1opw 6039* |
A one-to-one mapping induces a one-to-one mapping on power sets.
(Contributed by Stefan O'Rear, 18-Nov-2014.) (Revised by Mario
Carneiro, 26-Jun-2015.)
|
⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝑏 ∈ 𝒫 𝐴 ↦ (𝐹 “ 𝑏)):𝒫 𝐴–1-1-onto→𝒫 𝐵) |
|
Theorem | suppssfv 6040* |
Formula building theorem for support restriction, on a function which
preserves zero. (Contributed by Stefan O'Rear, 9-Mar-2015.)
|
⊢ (𝜑 → (◡(𝑥 ∈ 𝐷 ↦ 𝐴) “ (V ∖ {𝑌})) ⊆ 𝐿)
& ⊢ (𝜑 → (𝐹‘𝑌) = 𝑍)
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → (◡(𝑥 ∈ 𝐷 ↦ (𝐹‘𝐴)) “ (V ∖ {𝑍})) ⊆ 𝐿) |
|
Theorem | suppssov1 6041* |
Formula building theorem for support restrictions: operator with left
annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)
|
⊢ (𝜑 → (◡(𝑥 ∈ 𝐷 ↦ 𝐴) “ (V ∖ {𝑌})) ⊆ 𝐿)
& ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑌𝑂𝑣) = 𝑍)
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝑉)
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ 𝑅) ⇒ ⊢ (𝜑 → (◡(𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) “ (V ∖ {𝑍})) ⊆ 𝐿) |
|
2.6.13 Function operation
|
|
Syntax | cof 6042 |
Extend class notation to include mapping of an operation to a function
operation.
|
class ∘𝑓 𝑅 |
|
Syntax | cofr 6043 |
Extend class notation to include mapping of a binary relation to a
function relation.
|
class ∘𝑟 𝑅 |
|
Definition | df-of 6044* |
Define the function operation map. The definition is designed so that
if 𝑅 is a binary operation, then ∘𝑓 𝑅 is the analogous operation
on functions which corresponds to applying 𝑅 pointwise to the values
of the functions. (Contributed by Mario Carneiro, 20-Jul-2014.)
|
⊢ ∘𝑓 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))) |
|
Definition | df-ofr 6045* |
Define the function relation map. The definition is designed so that if
𝑅 is a binary relation, then ∘𝑓 𝑅 is the analogous relation on
functions which is true when each element of the left function relates
to the corresponding element of the right function. (Contributed by
Mario Carneiro, 28-Jul-2014.)
|
⊢ ∘𝑟 𝑅 = {〈𝑓, 𝑔〉 ∣ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓‘𝑥)𝑅(𝑔‘𝑥)} |
|
Theorem | ofeq 6046 |
Equality theorem for function operation. (Contributed by Mario
Carneiro, 20-Jul-2014.)
|
⊢ (𝑅 = 𝑆 → ∘𝑓 𝑅 = ∘𝑓
𝑆) |
|
Theorem | ofreq 6047 |
Equality theorem for function relation. (Contributed by Mario Carneiro,
28-Jul-2014.)
|
⊢ (𝑅 = 𝑆 → ∘𝑟 𝑅 = ∘𝑟
𝑆) |
|
Theorem | ofexg 6048 |
A function operation restricted to a set is a set. (Contributed by NM,
28-Jul-2014.)
|
⊢ (𝐴 ∈ 𝑉 → ( ∘𝑓 𝑅 ↾ 𝐴) ∈ V) |
|
Theorem | nfof 6049 |
Hypothesis builder for function operation. (Contributed by Mario
Carneiro, 20-Jul-2014.)
|
⊢ Ⅎ𝑥𝑅 ⇒ ⊢ Ⅎ𝑥 ∘𝑓
𝑅 |
|
Theorem | nfofr 6050 |
Hypothesis builder for function relation. (Contributed by Mario
Carneiro, 28-Jul-2014.)
|
⊢ Ⅎ𝑥𝑅 ⇒ ⊢ Ⅎ𝑥 ∘𝑟
𝑅 |
|
Theorem | offval 6051* |
Value of an operation applied to two functions. (Contributed by Mario
Carneiro, 20-Jul-2014.)
|
⊢ (𝜑 → 𝐹 Fn 𝐴)
& ⊢ (𝜑 → 𝐺 Fn 𝐵)
& ⊢ (𝜑 → 𝐴 ∈ 𝑉)
& ⊢ (𝜑 → 𝐵 ∈ 𝑊)
& ⊢ (𝐴 ∩ 𝐵) = 𝑆
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶)
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐷) ⇒ ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑥 ∈ 𝑆 ↦ (𝐶𝑅𝐷))) |
|
Theorem | ofrfval 6052* |
Value of a relation applied to two functions. (Contributed by Mario
Carneiro, 28-Jul-2014.)
|
⊢ (𝜑 → 𝐹 Fn 𝐴)
& ⊢ (𝜑 → 𝐺 Fn 𝐵)
& ⊢ (𝜑 → 𝐴 ∈ 𝑉)
& ⊢ (𝜑 → 𝐵 ∈ 𝑊)
& ⊢ (𝐴 ∩ 𝐵) = 𝑆
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶)
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐷) ⇒ ⊢ (𝜑 → (𝐹 ∘𝑟 𝑅𝐺 ↔ ∀𝑥 ∈ 𝑆 𝐶𝑅𝐷)) |
|
Theorem | ofvalg 6053 |
Evaluate a function operation at a point. (Contributed by Mario
Carneiro, 20-Jul-2014.) (Revised by Jim Kingdon, 22-Nov-2023.)
|
⊢ (𝜑 → 𝐹 Fn 𝐴)
& ⊢ (𝜑 → 𝐺 Fn 𝐵)
& ⊢ (𝜑 → 𝐴 ∈ 𝑉)
& ⊢ (𝜑 → 𝐵 ∈ 𝑊)
& ⊢ (𝐴 ∩ 𝐵) = 𝑆
& ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶)
& ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) = 𝐷)
& ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → (𝐶𝑅𝐷) ∈ 𝑈) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷)) |
|
Theorem | ofrval 6054 |
Exhibit a function relation at a point. (Contributed by Mario
Carneiro, 28-Jul-2014.)
|
⊢ (𝜑 → 𝐹 Fn 𝐴)
& ⊢ (𝜑 → 𝐺 Fn 𝐵)
& ⊢ (𝜑 → 𝐴 ∈ 𝑉)
& ⊢ (𝜑 → 𝐵 ∈ 𝑊)
& ⊢ (𝐴 ∩ 𝐵) = 𝑆
& ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶)
& ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) = 𝐷) ⇒ ⊢ ((𝜑 ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝐶𝑅𝐷) |
|
Theorem | ofmresval 6055 |
Value of a restriction of the function operation map. (Contributed by
NM, 20-Oct-2014.)
|
⊢ (𝜑 → 𝐹 ∈ 𝐴)
& ⊢ (𝜑 → 𝐺 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐹( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵))𝐺) = (𝐹 ∘𝑓 𝑅𝐺)) |
|
Theorem | off 6056* |
The function operation produces a function. (Contributed by Mario
Carneiro, 20-Jul-2014.)
|
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)
& ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
& ⊢ (𝜑 → 𝐺:𝐵⟶𝑇)
& ⊢ (𝜑 → 𝐴 ∈ 𝑉)
& ⊢ (𝜑 → 𝐵 ∈ 𝑊)
& ⊢ (𝐴 ∩ 𝐵) = 𝐶 ⇒ ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺):𝐶⟶𝑈) |
|
Theorem | offeq 6057* |
Convert an identity of the operation to the analogous identity on
the function operation. (Contributed by Jim Kingdon,
26-Nov-2023.)
|
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)
& ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
& ⊢ (𝜑 → 𝐺:𝐵⟶𝑇)
& ⊢ (𝜑 → 𝐴 ∈ 𝑉)
& ⊢ (𝜑 → 𝐵 ∈ 𝑊)
& ⊢ (𝐴 ∩ 𝐵) = 𝐶
& ⊢ (𝜑 → 𝐻:𝐶⟶𝑈)
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐷)
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐸)
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝐷𝑅𝐸) = (𝐻‘𝑥)) ⇒ ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = 𝐻) |
|
Theorem | ofres 6058 |
Restrict the operands of a function operation to the same domain as that
of the operation itself. (Contributed by Mario Carneiro,
15-Sep-2014.)
|
⊢ (𝜑 → 𝐹 Fn 𝐴)
& ⊢ (𝜑 → 𝐺 Fn 𝐵)
& ⊢ (𝜑 → 𝐴 ∈ 𝑉)
& ⊢ (𝜑 → 𝐵 ∈ 𝑊)
& ⊢ (𝐴 ∩ 𝐵) = 𝐶 ⇒ ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = ((𝐹 ↾ 𝐶) ∘𝑓 𝑅(𝐺 ↾ 𝐶))) |
|
Theorem | offval2 6059* |
The function operation expressed as a mapping. (Contributed by Mario
Carneiro, 20-Jul-2014.)
|
⊢ (𝜑 → 𝐴 ∈ 𝑉)
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑋)
& ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶)) ⇒ ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶))) |
|
Theorem | ofrfval2 6060* |
The function relation acting on maps. (Contributed by Mario Carneiro,
20-Jul-2014.)
|
⊢ (𝜑 → 𝐴 ∈ 𝑉)
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑋)
& ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶)) ⇒ ⊢ (𝜑 → (𝐹 ∘𝑟 𝑅𝐺 ↔ ∀𝑥 ∈ 𝐴 𝐵𝑅𝐶)) |
|
Theorem | suppssof1 6061* |
Formula building theorem for support restrictions: vector operation with
left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)
|
⊢ (𝜑 → (◡𝐴 “ (V ∖ {𝑌})) ⊆ 𝐿)
& ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑌𝑂𝑣) = 𝑍)
& ⊢ (𝜑 → 𝐴:𝐷⟶𝑉)
& ⊢ (𝜑 → 𝐵:𝐷⟶𝑅)
& ⊢ (𝜑 → 𝐷 ∈ 𝑊) ⇒ ⊢ (𝜑 → (◡(𝐴 ∘𝑓 𝑂𝐵) “ (V ∖ {𝑍})) ⊆ 𝐿) |
|
Theorem | ofco 6062 |
The composition of a function operation with another function.
(Contributed by Mario Carneiro, 19-Dec-2014.)
|
⊢ (𝜑 → 𝐹 Fn 𝐴)
& ⊢ (𝜑 → 𝐺 Fn 𝐵)
& ⊢ (𝜑 → 𝐻:𝐷⟶𝐶)
& ⊢ (𝜑 → 𝐴 ∈ 𝑉)
& ⊢ (𝜑 → 𝐵 ∈ 𝑊)
& ⊢ (𝜑 → 𝐷 ∈ 𝑋)
& ⊢ (𝐴 ∩ 𝐵) = 𝐶 ⇒ ⊢ (𝜑 → ((𝐹 ∘𝑓 𝑅𝐺) ∘ 𝐻) = ((𝐹 ∘ 𝐻) ∘𝑓 𝑅(𝐺 ∘ 𝐻))) |
|
Theorem | offveqb 6063* |
Equivalent expressions for equality with a function operation.
(Contributed by NM, 9-Oct-2014.) (Proof shortened by Mario Carneiro,
5-Dec-2016.)
|
⊢ (𝜑 → 𝐴 ∈ 𝑉)
& ⊢ (𝜑 → 𝐹 Fn 𝐴)
& ⊢ (𝜑 → 𝐺 Fn 𝐴)
& ⊢ (𝜑 → 𝐻 Fn 𝐴)
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = 𝐶) ⇒ ⊢ (𝜑 → (𝐻 = (𝐹 ∘𝑓 𝑅𝐺) ↔ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐵𝑅𝐶))) |
|
Theorem | ofc12 6064 |
Function operation on two constant functions. (Contributed by Mario
Carneiro, 28-Jul-2014.)
|
⊢ (𝜑 → 𝐴 ∈ 𝑉)
& ⊢ (𝜑 → 𝐵 ∈ 𝑊)
& ⊢ (𝜑 → 𝐶 ∈ 𝑋) ⇒ ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 𝑅(𝐴 × {𝐶})) = (𝐴 × {(𝐵𝑅𝐶)})) |
|
Theorem | caofref 6065* |
Transfer a reflexive law to the function relation. (Contributed by
Mario Carneiro, 28-Jul-2014.)
|
⊢ (𝜑 → 𝐴 ∈ 𝑉)
& ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥𝑅𝑥) ⇒ ⊢ (𝜑 → 𝐹 ∘𝑟 𝑅𝐹) |
|
Theorem | caofinvl 6066* |
Transfer a left inverse law to the function operation. (Contributed
by NM, 22-Oct-2014.)
|
⊢ (𝜑 → 𝐴 ∈ 𝑉)
& ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
& ⊢ (𝜑 → 𝐵 ∈ 𝑊)
& ⊢ (𝜑 → 𝑁:𝑆⟶𝑆)
& ⊢ (𝜑 → 𝐺 = (𝑣 ∈ 𝐴 ↦ (𝑁‘(𝐹‘𝑣)))) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((𝑁‘𝑥)𝑅𝑥) = 𝐵) ⇒ ⊢ (𝜑 → (𝐺 ∘𝑓 𝑅𝐹) = (𝐴 × {𝐵})) |
|
Theorem | caofcom 6067* |
Transfer a commutative law to the function operation. (Contributed by
Mario Carneiro, 26-Jul-2014.)
|
⊢ (𝜑 → 𝐴 ∈ 𝑉)
& ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
& ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑅𝑦) = (𝑦𝑅𝑥)) ⇒ ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝐺 ∘𝑓 𝑅𝐹)) |
|
Theorem | caofrss 6068* |
Transfer a relation subset law to the function relation. (Contributed
by Mario Carneiro, 28-Jul-2014.)
|
⊢ (𝜑 → 𝐴 ∈ 𝑉)
& ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
& ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑅𝑦 → 𝑥𝑇𝑦)) ⇒ ⊢ (𝜑 → (𝐹 ∘𝑟 𝑅𝐺 → 𝐹 ∘𝑟 𝑇𝐺)) |
|
Theorem | caoftrn 6069* |
Transfer a transitivity law to the function relation. (Contributed by
Mario Carneiro, 28-Jul-2014.)
|
⊢ (𝜑 → 𝐴 ∈ 𝑉)
& ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
& ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
& ⊢ (𝜑 → 𝐻:𝐴⟶𝑆)
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝑅𝑦 ∧ 𝑦𝑇𝑧) → 𝑥𝑈𝑧)) ⇒ ⊢ (𝜑 → ((𝐹 ∘𝑟 𝑅𝐺 ∧ 𝐺 ∘𝑟 𝑇𝐻) → 𝐹 ∘𝑟 𝑈𝐻)) |
|
2.6.14 Functions (continued)
|
|
Theorem | resfunexgALT 6070 |
The restriction of a function to a set exists. Compare Proposition 6.17
of [TakeutiZaring] p. 28. This
version has a shorter proof than
resfunexg 5700 but requires ax-pow 4147 and ax-un 4405. (Contributed by NM,
7-Apr-1995.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) ∈ V) |
|
Theorem | cofunexg 6071 |
Existence of a composition when the first member is a function.
(Contributed by NM, 8-Oct-2007.)
|
⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ∘ 𝐵) ∈ V) |
|
Theorem | cofunex2g 6072 |
Existence of a composition when the second member is one-to-one.
(Contributed by NM, 8-Oct-2007.)
|
⊢ ((𝐴 ∈ 𝑉 ∧ Fun ◡𝐵) → (𝐴 ∘ 𝐵) ∈ V) |
|
Theorem | fnexALT 6073 |
If the domain of a function is a set, the function is a set. Theorem
6.16(1) of [TakeutiZaring] p. 28.
This theorem is derived using the Axiom
of Replacement in the form of funimaexg 5266. This version of fnex 5701
uses
ax-pow 4147 and ax-un 4405, whereas fnex 5701
does not. (Contributed by NM,
14-Aug-1994.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐹 ∈ V) |
|
Theorem | funrnex 6074 |
If the domain of a function exists, so does its range. Part of Theorem
4.15(v) of [Monk1] p. 46. This theorem is
derived using the Axiom of
Replacement in the form of funex 5702. (Contributed by NM, 11-Nov-1995.)
|
⊢ (dom 𝐹 ∈ 𝐵 → (Fun 𝐹 → ran 𝐹 ∈ V)) |
|
Theorem | fornex 6075 |
If the domain of an onto function exists, so does its codomain.
(Contributed by NM, 23-Jul-2004.)
|
⊢ (𝐴 ∈ 𝐶 → (𝐹:𝐴–onto→𝐵 → 𝐵 ∈ V)) |
|
Theorem | f1dmex 6076 |
If the codomain of a one-to-one function exists, so does its domain. This
can be thought of as a form of the Axiom of Replacement. (Contributed by
NM, 4-Sep-2004.)
|
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V) |
|
Theorem | abrexex 6077* |
Existence of a class abstraction of existentially restricted sets. 𝑥
is normally a free-variable parameter in the class expression
substituted for 𝐵, which can be thought of as 𝐵(𝑥). This
simple-looking theorem is actually quite powerful and appears to involve
the Axiom of Replacement in an intrinsic way, as can be seen by tracing
back through the path mptexg 5704, funex 5702, fnex 5701, resfunexg 5700, and
funimaexg 5266. See also abrexex2 6084. (Contributed by NM, 16-Oct-2003.)
(Proof shortened by Mario Carneiro, 31-Aug-2015.)
|
⊢ 𝐴 ∈ V ⇒ ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V |
|
Theorem | abrexexg 6078* |
Existence of a class abstraction of existentially restricted sets. 𝑥
is normally a free-variable parameter in 𝐵. The antecedent assures
us that 𝐴 is a set. (Contributed by NM,
3-Nov-2003.)
|
⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V) |
|
Theorem | iunexg 6079* |
The existence of an indexed union. 𝑥 is normally a free-variable
parameter in 𝐵. (Contributed by NM, 23-Mar-2006.)
|
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V) |
|
Theorem | abrexex2g 6080* |
Existence of an existentially restricted class abstraction.
(Contributed by Jeff Madsen, 2-Sep-2009.)
|
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ∈ 𝑊) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V) |
|
Theorem | opabex3d 6081* |
Existence of an ordered pair abstraction, deduction version.
(Contributed by Alexander van der Vekens, 19-Oct-2017.)
|
⊢ (𝜑 → 𝐴 ∈ V) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {𝑦 ∣ 𝜓} ∈ V) ⇒ ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} ∈ V) |
|
Theorem | opabex3 6082* |
Existence of an ordered pair abstraction. (Contributed by Jeff Madsen,
2-Sep-2009.)
|
⊢ 𝐴 ∈ V & ⊢ (𝑥 ∈ 𝐴 → {𝑦 ∣ 𝜑} ∈ V) ⇒ ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V |
|
Theorem | iunex 6083* |
The existence of an indexed union. 𝑥 is normally a free-variable
parameter in the class expression substituted for 𝐵, which can be
read informally as 𝐵(𝑥). (Contributed by NM, 13-Oct-2003.)
|
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈
V ⇒ ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V |
|
Theorem | abrexex2 6084* |
Existence of an existentially restricted class abstraction. 𝜑 is
normally has free-variable parameters 𝑥 and 𝑦. See
also
abrexex 6077. (Contributed by NM, 12-Sep-2004.)
|
⊢ 𝐴 ∈ V & ⊢ {𝑦 ∣ 𝜑} ∈ V ⇒ ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V |
|
Theorem | abexssex 6085* |
Existence of a class abstraction with an existentially quantified
expression. Both 𝑥 and 𝑦 can be free in 𝜑.
(Contributed
by NM, 29-Jul-2006.)
|
⊢ 𝐴 ∈ V & ⊢ {𝑦 ∣ 𝜑} ∈ V ⇒ ⊢ {𝑦 ∣ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝜑)} ∈ V |
|
Theorem | abexex 6086* |
A condition where a class builder continues to exist after its wff is
existentially quantified. (Contributed by NM, 4-Mar-2007.)
|
⊢ 𝐴 ∈ V & ⊢ (𝜑 → 𝑥 ∈ 𝐴)
& ⊢ {𝑦 ∣ 𝜑} ∈ V ⇒ ⊢ {𝑦 ∣ ∃𝑥𝜑} ∈ V |
|
Theorem | oprabexd 6087* |
Existence of an operator abstraction. (Contributed by Jeff Madsen,
2-Sep-2009.)
|
⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑧𝜓)
& ⊢ (𝜑 → 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)}) ⇒ ⊢ (𝜑 → 𝐹 ∈ V) |
|
Theorem | oprabex 6088* |
Existence of an operation class abstraction. (Contributed by NM,
19-Oct-2004.)
|
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃*𝑧𝜑)
& ⊢ 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} ⇒ ⊢ 𝐹 ∈ V |
|
Theorem | oprabex3 6089* |
Existence of an operation class abstraction (special case).
(Contributed by NM, 19-Oct-2004.)
|
⊢ 𝐻 ∈ V & ⊢ 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 𝑅))} ⇒ ⊢ 𝐹 ∈ V |
|
Theorem | oprabrexex2 6090* |
Existence of an existentially restricted operation abstraction.
(Contributed by Jeff Madsen, 11-Jun-2010.)
|
⊢ 𝐴 ∈ V & ⊢
{〈〈𝑥,
𝑦〉, 𝑧〉 ∣ 𝜑} ∈ V ⇒ ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ∃𝑤 ∈ 𝐴 𝜑} ∈ V |
|
Theorem | ab2rexex 6091* |
Existence of a class abstraction of existentially restricted sets.
Variables 𝑥 and 𝑦 are normally
free-variable parameters in the
class expression substituted for 𝐶, which can be thought of as
𝐶(𝑥, 𝑦). See comments for abrexex 6077. (Contributed by NM,
20-Sep-2011.)
|
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈
V ⇒ ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ∈ V |
|
Theorem | ab2rexex2 6092* |
Existence of an existentially restricted class abstraction. 𝜑
normally has free-variable parameters 𝑥, 𝑦, and 𝑧.
Compare abrexex2 6084. (Contributed by NM, 20-Sep-2011.)
|
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ {𝑧 ∣ 𝜑} ∈ V ⇒ ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑} ∈ V |
|
Theorem | xpexgALT 6093 |
The cross product of two sets is a set. Proposition 6.2 of
[TakeutiZaring] p. 23. This
version is proven using Replacement; see
xpexg 4712 for a version that uses the Power Set axiom
instead.
(Contributed by Mario Carneiro, 20-May-2013.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ∈ V) |
|
Theorem | offval3 6094* |
General value of (𝐹 ∘𝑓 𝑅𝐺) with no assumptions on
functionality
of 𝐹 and 𝐺. (Contributed by Stefan
O'Rear, 24-Jan-2015.)
|
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘𝑓 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
|
Theorem | offres 6095 |
Pointwise combination commutes with restriction. (Contributed by Stefan
O'Rear, 24-Jan-2015.)
|
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 ∘𝑓 𝑅𝐺) ↾ 𝐷) = ((𝐹 ↾ 𝐷) ∘𝑓 𝑅(𝐺 ↾ 𝐷))) |
|
Theorem | ofmres 6096* |
Equivalent expressions for a restriction of the function operation map.
Unlike ∘𝑓 𝑅 which is a proper class, ( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵)) can
be a set by ofmresex 6097, allowing it to be used as a function or
structure argument. By ofmresval 6055, the restricted operation map
values are the same as the original values, allowing theorems for
∘𝑓 𝑅 to be reused. (Contributed by NM,
20-Oct-2014.)
|
⊢ ( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵)) = (𝑓 ∈ 𝐴, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘𝑓 𝑅𝑔)) |
|
Theorem | ofmresex 6097 |
Existence of a restriction of the function operation map. (Contributed
by NM, 20-Oct-2014.)
|
⊢ (𝜑 → 𝐴 ∈ 𝑉)
& ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → ( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵)) ∈ V) |
|
2.6.15 First and second members of an ordered
pair
|
|
Syntax | c1st 6098 |
Extend the definition of a class to include the first member an ordered
pair function.
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class 1st |
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Syntax | c2nd 6099 |
Extend the definition of a class to include the second member an ordered
pair function.
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class 2nd |
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Definition | df-1st 6100 |
Define a function that extracts the first member, or abscissa, of an
ordered pair. Theorem op1st 6106 proves that it does this. For example,
(1st ‘〈 3 , 4 〉) = 3 . Equivalent to Definition 5.13 (i) of
[Monk1] p. 52 (compare op1sta 5079 and op1stb 4450). The notation is the same
as Monk's. (Contributed by NM, 9-Oct-2004.)
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⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥}) |