Theorem List for Intuitionistic Logic Explorer - 12801-12900 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | txvalex 12801 |
Existence of the binary topological product. If 𝑅 and 𝑆 are
known to be topologies, see txtop 12807. (Contributed by Jim Kingdon,
3-Aug-2023.)
|
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 ×t 𝑆) ∈ V) |
|
Theorem | txval 12802* |
Value of the binary topological product operation. (Contributed by Jeff
Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 30-Aug-2015.)
|
⊢ 𝐵 = ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 ×t 𝑆) = (topGen‘𝐵)) |
|
Theorem | txuni2 12803* |
The underlying set of the product of two topologies. (Contributed by
Mario Carneiro, 31-Aug-2015.)
|
⊢ 𝐵 = ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦))
& ⊢ 𝑋 = ∪ 𝑅 & ⊢ 𝑌 = ∪
𝑆
⇒ ⊢ (𝑋 × 𝑌) = ∪ 𝐵 |
|
Theorem | txbasex 12804* |
The basis for the product topology is a set. (Contributed by Mario
Carneiro, 2-Sep-2015.)
|
⊢ 𝐵 = ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝐵 ∈ V) |
|
Theorem | txbas 12805* |
The set of Cartesian products of elements from two topological bases is
a basis. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario
Carneiro, 31-Aug-2015.)
|
⊢ 𝐵 = ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)) ⇒ ⊢ ((𝑅 ∈ TopBases ∧ 𝑆 ∈ TopBases) → 𝐵 ∈ TopBases) |
|
Theorem | eltx 12806* |
A set in a product is open iff each point is surrounded by an open
rectangle. (Contributed by Stefan O'Rear, 25-Jan-2015.)
|
⊢ ((𝐽 ∈ 𝑉 ∧ 𝐾 ∈ 𝑊) → (𝑆 ∈ (𝐽 ×t 𝐾) ↔ ∀𝑝 ∈ 𝑆 ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐾 (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝑆))) |
|
Theorem | txtop 12807 |
The product of two topologies is a topology. (Contributed by Jeff
Madsen, 2-Sep-2009.)
|
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top) |
|
Theorem | txtopi 12808 |
The product of two topologies is a topology. (Contributed by Jeff
Madsen, 15-Jun-2010.)
|
⊢ 𝑅 ∈ Top & ⊢ 𝑆 ∈
Top ⇒ ⊢ (𝑅 ×t 𝑆) ∈ Top |
|
Theorem | txtopon 12809 |
The underlying set of the product of two topologies. (Contributed by
Mario Carneiro, 22-Aug-2015.) (Revised by Mario Carneiro,
2-Sep-2015.)
|
⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌))) |
|
Theorem | txuni 12810 |
The underlying set of the product of two topologies. (Contributed by
Jeff Madsen, 2-Sep-2009.)
|
⊢ 𝑋 = ∪ 𝑅 & ⊢ 𝑌 = ∪
𝑆
⇒ ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆)) |
|
Theorem | txunii 12811 |
The underlying set of the product of two topologies. (Contributed by
Jeff Madsen, 15-Jun-2010.)
|
⊢ 𝑅 ∈ Top & ⊢ 𝑆 ∈ Top & ⊢ 𝑋 = ∪
𝑅 & ⊢ 𝑌 = ∪
𝑆
⇒ ⊢ (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆) |
|
Theorem | txopn 12812 |
The product of two open sets is open in the product topology.
(Contributed by Jeff Madsen, 2-Sep-2009.)
|
⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴 × 𝐵) ∈ (𝑅 ×t 𝑆)) |
|
Theorem | txss12 12813 |
Subset property of the topological product. (Contributed by Mario
Carneiro, 2-Sep-2015.)
|
⊢ (((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)) → (𝐴 ×t 𝐶) ⊆ (𝐵 ×t 𝐷)) |
|
Theorem | txbasval 12814 |
It is sufficient to consider products of the bases for the topologies in
the topological product. (Contributed by Mario Carneiro,
25-Aug-2014.)
|
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ((topGen‘𝑅) ×t (topGen‘𝑆)) = (𝑅 ×t 𝑆)) |
|
Theorem | neitx 12815 |
The Cartesian product of two neighborhoods is a neighborhood in the
product topology. (Contributed by Thierry Arnoux, 13-Jan-2018.)
|
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝑌 = ∪
𝐾
⇒ ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐴 × 𝐵) ∈ ((nei‘(𝐽 ×t 𝐾))‘(𝐶 × 𝐷))) |
|
Theorem | tx1cn 12816 |
Continuity of the first projection map of a topological product.
(Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario
Carneiro, 22-Aug-2015.)
|
⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅)) |
|
Theorem | tx2cn 12817 |
Continuity of the second projection map of a topological product.
(Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario
Carneiro, 22-Aug-2015.)
|
⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆)) |
|
Theorem | txcnp 12818* |
If two functions are continuous at 𝐷, then the ordered pair of them
is continuous at 𝐷 into the product topology.
(Contributed by Mario
Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
|
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍)) & ⊢ (𝜑 → 𝐷 ∈ 𝑋)
& ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP 𝐾)‘𝐷)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ ((𝐽 CnP 𝐿)‘𝐷)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) ∈ ((𝐽 CnP (𝐾 ×t 𝐿))‘𝐷)) |
|
Theorem | upxp 12819* |
Universal property of the Cartesian product considered as a categorical
product in the category of sets. (Contributed by Jeff Madsen,
2-Sep-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)
|
⊢ 𝑃 = (1st ↾ (𝐵 × 𝐶)) & ⊢ 𝑄 = (2nd ↾
(𝐵 × 𝐶))
⇒ ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐴⟶𝐶) → ∃!ℎ(ℎ:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))) |
|
Theorem | txcnmpt 12820* |
A map into the product of two topological spaces is continuous if both
of its projections are continuous. (Contributed by Jeff Madsen,
2-Sep-2009.) (Revised by Mario Carneiro, 22-Aug-2015.)
|
⊢ 𝑊 = ∪ 𝑈 & ⊢ 𝐻 = (𝑥 ∈ 𝑊 ↦ 〈(𝐹‘𝑥), (𝐺‘𝑥)〉) ⇒ ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐻 ∈ (𝑈 Cn (𝑅 ×t 𝑆))) |
|
Theorem | uptx 12821* |
Universal property of the binary topological product. (Contributed by
Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro,
22-Aug-2015.)
|
⊢ 𝑇 = (𝑅 ×t 𝑆)
& ⊢ 𝑋 = ∪ 𝑅 & ⊢ 𝑌 = ∪
𝑆 & ⊢ 𝑍 = (𝑋 × 𝑌)
& ⊢ 𝑃 = (1st ↾ 𝑍) & ⊢ 𝑄 = (2nd ↾
𝑍) ⇒ ⊢ ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∃!ℎ ∈ (𝑈 Cn 𝑇)(𝐹 = (𝑃 ∘ ℎ) ∧ 𝐺 = (𝑄 ∘ ℎ))) |
|
Theorem | txcn 12822 |
A map into the product of two topological spaces is continuous iff both
of its projections are continuous. (Contributed by Jeff Madsen,
2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
|
⊢ 𝑋 = ∪ 𝑅 & ⊢ 𝑌 = ∪
𝑆 & ⊢ 𝑍 = (𝑋 × 𝑌)
& ⊢ 𝑊 = ∪ 𝑈 & ⊢ 𝑃 = (1st ↾
𝑍) & ⊢ 𝑄 = (2nd ↾
𝑍) ⇒ ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊⟶𝑍) → (𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ↔ ((𝑃 ∘ 𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (𝑈 Cn 𝑆)))) |
|
Theorem | txrest 12823 |
The subspace of a topological product space induced by a subset with a
Cartesian product representation is a topological product of the
subspaces induced by the subspaces of the terms of the products.
(Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario
Carneiro, 2-Sep-2015.)
|
⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝑅 ×t 𝑆) ↾t (𝐴 × 𝐵)) = ((𝑅 ↾t 𝐴) ×t (𝑆 ↾t 𝐵))) |
|
Theorem | txdis 12824 |
The topological product of discrete spaces is discrete. (Contributed by
Mario Carneiro, 14-Aug-2015.)
|
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝒫 𝐴 ×t 𝒫 𝐵) = 𝒫 (𝐴 × 𝐵)) |
|
Theorem | txdis1cn 12825* |
A function is jointly continuous on a discrete left topology iff it is
continuous as a function of its right argument, for each fixed left
value. (Contributed by Mario Carneiro, 19-Sep-2015.)
|
⊢ (𝜑 → 𝑋 ∈ 𝑉)
& ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → 𝐾 ∈ Top) & ⊢ (𝜑 → 𝐹 Fn (𝑋 × 𝑌)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ (𝑥𝐹𝑦)) ∈ (𝐽 Cn 𝐾)) ⇒ ⊢ (𝜑 → 𝐹 ∈ ((𝒫 𝑋 ×t 𝐽) Cn 𝐾)) |
|
Theorem | txlm 12826* |
Two sequences converge iff the sequence of their ordered pairs
converges. Proposition 14-2.6 of [Gleason] p. 230. (Contributed by
NM, 16-Jul-2007.) (Revised by Mario Carneiro, 5-May-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → 𝐹:𝑍⟶𝑋)
& ⊢ (𝜑 → 𝐺:𝑍⟶𝑌)
& ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ 〈(𝐹‘𝑛), (𝐺‘𝑛)〉) ⇒ ⊢ (𝜑 → ((𝐹(⇝𝑡‘𝐽)𝑅 ∧ 𝐺(⇝𝑡‘𝐾)𝑆) ↔ 𝐻(⇝𝑡‘(𝐽 ×t 𝐾))〈𝑅, 𝑆〉)) |
|
Theorem | lmcn2 12827* |
The image of a convergent sequence under a continuous map is convergent
to the image of the original point. Binary operation version.
(Contributed by Mario Carneiro, 15-May-2014.)
|
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → 𝐹:𝑍⟶𝑋)
& ⊢ (𝜑 → 𝐺:𝑍⟶𝑌)
& ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑅)
& ⊢ (𝜑 → 𝐺(⇝𝑡‘𝐾)𝑆)
& ⊢ (𝜑 → 𝑂 ∈ ((𝐽 ×t 𝐾) Cn 𝑁)) & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)𝑂(𝐺‘𝑛))) ⇒ ⊢ (𝜑 → 𝐻(⇝𝑡‘𝑁)(𝑅𝑂𝑆)) |
|
7.1.9 Continuous function-builders
|
|
Theorem | cnmptid 12828* |
The identity function is continuous. (Contributed by Mario Carneiro,
5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
|
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (𝐽 Cn 𝐽)) |
|
Theorem | cnmptc 12829* |
A constant function is continuous. (Contributed by Mario Carneiro,
5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
|
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → 𝑃 ∈ 𝑌) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝑃) ∈ (𝐽 Cn 𝐾)) |
|
Theorem | cnmpt11 12830* |
The composition of continuous functions is continuous. (Contributed
by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro,
22-Aug-2015.)
|
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ 𝐵) ∈ (𝐾 Cn 𝐿)) & ⊢ (𝑦 = 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐶) ∈ (𝐽 Cn 𝐿)) |
|
Theorem | cnmpt11f 12831* |
The composition of continuous functions is continuous. (Contributed
by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro,
22-Aug-2015.)
|
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝐹 ∈ (𝐾 Cn 𝐿)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐹‘𝐴)) ∈ (𝐽 Cn 𝐿)) |
|
Theorem | cnmpt1t 12832* |
The composition of continuous functions is continuous. (Contributed by
Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro,
22-Aug-2015.)
|
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 〈𝐴, 𝐵〉) ∈ (𝐽 Cn (𝐾 ×t 𝐿))) |
|
Theorem | cnmpt12f 12833* |
The composition of continuous functions is continuous. (Contributed
by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro,
22-Aug-2015.)
|
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿)) & ⊢ (𝜑 → 𝐹 ∈ ((𝐾 ×t 𝐿) Cn 𝑀)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝐽 Cn 𝑀)) |
|
Theorem | cnmpt12 12834* |
The composition of continuous functions is continuous. (Contributed by
Mario Carneiro, 12-Jun-2014.) (Revised by Mario Carneiro,
22-Aug-2015.)
|
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐿)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍)) & ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐶) ∈ ((𝐾 ×t 𝐿) Cn 𝑀)) & ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐷) ∈ (𝐽 Cn 𝑀)) |
|
Theorem | cnmpt1st 12835* |
The projection onto the first coordinate is continuous. (Contributed by
Mario Carneiro, 6-May-2014.) (Revised by Mario Carneiro,
22-Aug-2015.)
|
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑥) ∈ ((𝐽 ×t 𝐾) Cn 𝐽)) |
|
Theorem | cnmpt2nd 12836* |
The projection onto the second coordinate is continuous. (Contributed
by Mario Carneiro, 6-May-2014.) (Revised by Mario Carneiro,
22-Aug-2015.)
|
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦) ∈ ((𝐽 ×t 𝐾) Cn 𝐾)) |
|
Theorem | cnmpt2c 12837* |
A constant function is continuous. (Contributed by Mario Carneiro,
5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
|
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍)) & ⊢ (𝜑 → 𝑃 ∈ 𝑍) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) |
|
Theorem | cnmpt21 12838* |
The composition of continuous functions is continuous. (Contributed
by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro,
22-Aug-2015.)
|
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) & ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍)) & ⊢ (𝜑 → (𝑧 ∈ 𝑍 ↦ 𝐵) ∈ (𝐿 Cn 𝑀)) & ⊢ (𝑧 = 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) |
|
Theorem | cnmpt21f 12839* |
The composition of continuous functions is continuous. (Contributed
by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro,
22-Aug-2015.)
|
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) & ⊢ (𝜑 → 𝐹 ∈ (𝐿 Cn 𝑀)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐹‘𝐴)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) |
|
Theorem | cnmpt2t 12840* |
The composition of continuous functions is continuous. (Contributed by
Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro,
22-Aug-2015.)
|
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝐴, 𝐵〉) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ×t 𝑀))) |
|
Theorem | cnmpt22 12841* |
The composition of continuous functions is continuous. (Contributed
by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro,
22-Aug-2015.)
|
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) & ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍)) & ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑊)) & ⊢ (𝜑 → (𝑧 ∈ 𝑍, 𝑤 ∈ 𝑊 ↦ 𝐶) ∈ ((𝐿 ×t 𝑀) Cn 𝑁)) & ⊢ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐷) ∈ ((𝐽 ×t 𝐾) Cn 𝑁)) |
|
Theorem | cnmpt22f 12842* |
The composition of continuous functions is continuous. (Contributed by
Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro,
22-Aug-2015.)
|
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) & ⊢ (𝜑 → 𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴𝐹𝐵)) ∈ ((𝐽 ×t 𝐾) Cn 𝑁)) |
|
Theorem | cnmpt1res 12843* |
The restriction of a continuous function to a subset is continuous.
(Contributed by Mario Carneiro, 5-Jun-2014.)
|
⊢ 𝐾 = (𝐽 ↾t 𝑌)
& ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
& ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐽 Cn 𝐿)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐿)) |
|
Theorem | cnmpt2res 12844* |
The restriction of a continuous function to a subset is continuous.
(Contributed by Mario Carneiro, 6-Jun-2014.)
|
⊢ 𝐾 = (𝐽 ↾t 𝑌)
& ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
& ⊢ 𝑁 = (𝑀 ↾t 𝑊)
& ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑍)) & ⊢ (𝜑 → 𝑊 ⊆ 𝑍)
& ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑍 ↦ 𝐴) ∈ ((𝐽 ×t 𝑀) Cn 𝐿)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑊 ↦ 𝐴) ∈ ((𝐾 ×t 𝑁) Cn 𝐿)) |
|
Theorem | cnmptcom 12845* |
The argument converse of a continuous function is continuous.
(Contributed by Mario Carneiro, 6-Jun-2014.)
|
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) ⇒ ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐾 ×t 𝐽) Cn 𝐿)) |
|
Theorem | imasnopn 12846 |
If a relation graph is open, then an image set of a singleton is also
open. Corollary of Proposition 4 of [BourbakiTop1] p. I.26.
(Contributed by Thierry Arnoux, 14-Jan-2018.)
|
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ∈ (𝐽 ×t 𝐾) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) ∈ 𝐾) |
|
7.1.10 Homeomorphisms
|
|
Syntax | chmeo 12847 |
Extend class notation with the class of all homeomorphisms.
|
class Homeo |
|
Definition | df-hmeo 12848* |
Function returning all the homeomorphisms from topology 𝑗 to
topology 𝑘. (Contributed by FL, 14-Feb-2007.)
|
⊢ Homeo = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (𝑗 Cn 𝑘) ∣ ◡𝑓 ∈ (𝑘 Cn 𝑗)}) |
|
Theorem | hmeofn 12849 |
The set of homeomorphisms is a function on topologies. (Contributed by
Mario Carneiro, 23-Aug-2015.)
|
⊢ Homeo Fn (Top × Top) |
|
Theorem | hmeofvalg 12850* |
The set of all the homeomorphisms between two topologies. (Contributed
by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
|
⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ ◡𝑓 ∈ (𝐾 Cn 𝐽)}) |
|
Theorem | ishmeo 12851 |
The predicate F is a homeomorphism between topology 𝐽 and topology
𝐾. Proposition of [BourbakiTop1] p. I.2. (Contributed by FL,
14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
|
⊢ (𝐹 ∈ (𝐽Homeo𝐾) ↔ (𝐹 ∈ (𝐽 Cn 𝐾) ∧ ◡𝐹 ∈ (𝐾 Cn 𝐽))) |
|
Theorem | hmeocn 12852 |
A homeomorphism is continuous. (Contributed by Mario Carneiro,
22-Aug-2015.)
|
⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾)) |
|
Theorem | hmeocnvcn 12853 |
The converse of a homeomorphism is continuous. (Contributed by Mario
Carneiro, 22-Aug-2015.)
|
⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾 Cn 𝐽)) |
|
Theorem | hmeocnv 12854 |
The converse of a homeomorphism is a homeomorphism. (Contributed by FL,
5-Mar-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
|
⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾Homeo𝐽)) |
|
Theorem | hmeof1o2 12855 |
A homeomorphism is a 1-1-onto mapping. (Contributed by Mario Carneiro,
22-Aug-2015.)
|
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽Homeo𝐾)) → 𝐹:𝑋–1-1-onto→𝑌) |
|
Theorem | hmeof1o 12856 |
A homeomorphism is a 1-1-onto mapping. (Contributed by FL, 5-Mar-2007.)
(Revised by Mario Carneiro, 30-May-2014.)
|
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝑌 = ∪
𝐾
⇒ ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋–1-1-onto→𝑌) |
|
Theorem | hmeoima 12857 |
The image of an open set by a homeomorphism is an open set. (Contributed
by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
|
⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ∈ 𝐽) → (𝐹 “ 𝐴) ∈ 𝐾) |
|
Theorem | hmeoopn 12858 |
Homeomorphisms preserve openness. (Contributed by Jeff Madsen,
2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
|
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ 𝐽 ↔ (𝐹 “ 𝐴) ∈ 𝐾)) |
|
Theorem | hmeocld 12859 |
Homeomorphisms preserve closedness. (Contributed by Jeff Madsen,
2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
|
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (𝐹 “ 𝐴) ∈ (Clsd‘𝐾))) |
|
Theorem | hmeontr 12860 |
Homeomorphisms preserve interiors. (Contributed by Mario Carneiro,
25-Aug-2015.)
|
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐾)‘(𝐹 “ 𝐴)) = (𝐹 “ ((int‘𝐽)‘𝐴))) |
|
Theorem | hmeoimaf1o 12861* |
The function mapping open sets to their images under a homeomorphism is
a bijection of topologies. (Contributed by Mario Carneiro,
10-Sep-2015.)
|
⊢ 𝐺 = (𝑥 ∈ 𝐽 ↦ (𝐹 “ 𝑥)) ⇒ ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐺:𝐽–1-1-onto→𝐾) |
|
Theorem | hmeores 12862 |
The restriction of a homeomorphism is a homeomorphism. (Contributed by
Mario Carneiro, 14-Sep-2014.) (Proof shortened by Mario Carneiro,
22-Aug-2015.)
|
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → (𝐹 ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌)Homeo(𝐾 ↾t (𝐹 “ 𝑌)))) |
|
Theorem | hmeoco 12863 |
The composite of two homeomorphisms is a homeomorphism. (Contributed by
FL, 9-Mar-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
|
⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → (𝐺 ∘ 𝐹) ∈ (𝐽Homeo𝐿)) |
|
Theorem | idhmeo 12864 |
The identity function is a homeomorphism. (Contributed by FL,
14-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
|
⊢ (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽Homeo𝐽)) |
|
Theorem | hmeocnvb 12865 |
The converse of a homeomorphism is a homeomorphism. (Contributed by FL,
5-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
|
⊢ (Rel 𝐹 → (◡𝐹 ∈ (𝐽Homeo𝐾) ↔ 𝐹 ∈ (𝐾Homeo𝐽))) |
|
Theorem | txhmeo 12866* |
Lift a pair of homeomorphisms on the factors to a homeomorphism of
product topologies. (Contributed by Mario Carneiro, 2-Sep-2015.)
|
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝑌 = ∪
𝐾 & ⊢ (𝜑 → 𝐹 ∈ (𝐽Homeo𝐿)) & ⊢ (𝜑 → 𝐺 ∈ (𝐾Homeo𝑀)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈(𝐹‘𝑥), (𝐺‘𝑦)〉) ∈ ((𝐽 ×t 𝐾)Homeo(𝐿 ×t 𝑀))) |
|
Theorem | txswaphmeolem 12867* |
Show inverse for the "swap components" operation on a Cartesian
product.
(Contributed by Mario Carneiro, 21-Mar-2015.)
|
⊢ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 〈𝑥, 𝑦〉) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑦, 𝑥〉)) = ( I ↾ (𝑋 × 𝑌)) |
|
Theorem | txswaphmeo 12868* |
There is a homeomorphism from 𝑋 × 𝑌 to 𝑌 × 𝑋. (Contributed
by Mario Carneiro, 21-Mar-2015.)
|
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈𝑦, 𝑥〉) ∈ ((𝐽 ×t 𝐾)Homeo(𝐾 ×t 𝐽))) |
|
7.2 Metric spaces
|
|
7.2.1 Pseudometric spaces
|
|
Theorem | psmetrel 12869 |
The class of pseudometrics is a relation. (Contributed by Jim Kingdon,
24-Apr-2023.)
|
⊢ Rel PsMet |
|
Theorem | ispsmet 12870* |
Express the predicate "𝐷 is a pseudometric."
(Contributed by
Thierry Arnoux, 7-Feb-2018.)
|
⊢ (𝑋 ∈ 𝑉 → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧
∀𝑥 ∈ 𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))) |
|
Theorem | psmetdmdm 12871 |
Recover the base set from a pseudometric. (Contributed by Thierry
Arnoux, 7-Feb-2018.)
|
⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷) |
|
Theorem | psmetf 12872 |
The distance function of a pseudometric as a function. (Contributed by
Thierry Arnoux, 7-Feb-2018.)
|
⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) |
|
Theorem | psmetcl 12873 |
Closure of the distance function of a pseudometric space. (Contributed
by Thierry Arnoux, 7-Feb-2018.)
|
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈
ℝ*) |
|
Theorem | psmet0 12874 |
The distance function of a pseudometric space is zero if its arguments
are equal. (Contributed by Thierry Arnoux, 7-Feb-2018.)
|
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = 0) |
|
Theorem | psmettri2 12875 |
Triangle inequality for the distance function of a pseudometric.
(Contributed by Thierry Arnoux, 11-Feb-2018.)
|
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵))) |
|
Theorem | psmetsym 12876 |
The distance function of a pseudometric is symmetrical. (Contributed by
Thierry Arnoux, 7-Feb-2018.)
|
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴)) |
|
Theorem | psmettri 12877 |
Triangle inequality for the distance function of a pseudometric space.
(Contributed by Thierry Arnoux, 11-Feb-2018.)
|
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) +𝑒 (𝐶𝐷𝐵))) |
|
Theorem | psmetge0 12878 |
The distance function of a pseudometric space is nonnegative.
(Contributed by Thierry Arnoux, 7-Feb-2018.) (Revised by Jim Kingdon,
19-Apr-2023.)
|
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 0 ≤ (𝐴𝐷𝐵)) |
|
Theorem | psmetxrge0 12879 |
The distance function of a pseudometric space is a function into the
nonnegative extended real numbers. (Contributed by Thierry Arnoux,
24-Feb-2018.)
|
⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶(0[,]+∞)) |
|
Theorem | psmetres2 12880 |
Restriction of a pseudometric. (Contributed by Thierry Arnoux,
11-Feb-2018.)
|
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (PsMet‘𝑅)) |
|
Theorem | psmetlecl 12881 |
Real closure of an extended metric value that is upper bounded by a
real. (Contributed by Thierry Arnoux, 11-Mar-2018.)
|
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ ℝ ∧ (𝐴𝐷𝐵) ≤ 𝐶)) → (𝐴𝐷𝐵) ∈ ℝ) |
|
Theorem | distspace 12882 |
A set 𝑋 together with a (distance) function
𝐷
which is a
pseudometric is a distance space (according to E. Deza, M.M. Deza:
"Dictionary of Distances", Elsevier, 2006), i.e. a (base) set
𝑋
equipped with a distance 𝐷, which is a mapping of two elements
of
the base set to the (extended) reals and which is nonnegative, symmetric
and equal to 0 if the two elements are equal. (Contributed by AV,
15-Oct-2021.) (Revised by AV, 5-Jul-2022.)
|
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ (𝐴𝐷𝐴) = 0) ∧ (0 ≤ (𝐴𝐷𝐵) ∧ (𝐴𝐷𝐵) = (𝐵𝐷𝐴)))) |
|
7.2.2 Basic metric space
properties
|
|
Syntax | cxms 12883 |
Extend class notation with the class of extended metric spaces.
|
class ∞MetSp |
|
Syntax | cms 12884 |
Extend class notation with the class of metric spaces.
|
class MetSp |
|
Syntax | ctms 12885 |
Extend class notation with the function mapping a metric to the metric
space it defines.
|
class toMetSp |
|
Definition | df-xms 12886 |
Define the (proper) class of extended metric spaces. (Contributed by
Mario Carneiro, 2-Sep-2015.)
|
⊢ ∞MetSp = {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) =
(MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))} |
|
Definition | df-ms 12887 |
Define the (proper) class of metric spaces. (Contributed by NM,
27-Aug-2006.)
|
⊢ MetSp = {𝑓 ∈ ∞MetSp ∣
((dist‘𝑓) ↾
((Base‘𝑓) ×
(Base‘𝑓))) ∈
(Met‘(Base‘𝑓))} |
|
Definition | df-tms 12888 |
Define the function mapping a metric to the metric space which it defines.
(Contributed by Mario Carneiro, 2-Sep-2015.)
|
⊢ toMetSp = (𝑑 ∈ ∪ ran
∞Met ↦ ({〈(Base‘ndx), dom dom 𝑑〉, 〈(dist‘ndx), 𝑑〉} sSet
〈(TopSet‘ndx), (MetOpen‘𝑑)〉)) |
|
Theorem | metrel 12889 |
The class of metrics is a relation. (Contributed by Jim Kingdon,
20-Apr-2023.)
|
⊢ Rel Met |
|
Theorem | xmetrel 12890 |
The class of extended metrics is a relation. (Contributed by Jim
Kingdon, 20-Apr-2023.)
|
⊢ Rel ∞Met |
|
Theorem | ismet 12891* |
Express the predicate "𝐷 is a metric." (Contributed by
NM,
25-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
|
⊢ (𝑋 ∈ 𝐴 → (𝐷 ∈ (Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))))) |
|
Theorem | isxmet 12892* |
Express the predicate "𝐷 is an extended metric."
(Contributed by
Mario Carneiro, 20-Aug-2015.)
|
⊢ (𝑋 ∈ 𝐴 → (𝐷 ∈ (∞Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧
∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))) |
|
Theorem | ismeti 12893* |
Properties that determine a metric. (Contributed by NM, 17-Nov-2006.)
(Revised by Mario Carneiro, 14-Aug-2015.)
|
⊢ 𝑋 ∈ V & ⊢ 𝐷:(𝑋 × 𝑋)⟶ℝ & ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
& ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))) ⇒ ⊢ 𝐷 ∈ (Met‘𝑋) |
|
Theorem | isxmetd 12894* |
Properties that determine an extended metric. (Contributed by Mario
Carneiro, 20-Aug-2015.)
|
⊢ (𝜑 → 𝑋 ∈ V) & ⊢ (𝜑 → 𝐷:(𝑋 × 𝑋)⟶ℝ*) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))) ⇒ ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) |
|
Theorem | isxmet2d 12895* |
It is safe to only require the triangle inequality when the values are
real (so that we can use the standard addition over the reals), but in
this case the nonnegativity constraint cannot be deduced and must be
provided separately. (Counterexample:
𝐷(𝑥, 𝑦) = if(𝑥 = 𝑦, 0, -∞) satisfies all
hypotheses
except nonnegativity.) (Contributed by Mario Carneiro,
20-Aug-2015.)
|
⊢ (𝜑 → 𝑋 ∈ V) & ⊢ (𝜑 → 𝐷:(𝑋 × 𝑋)⟶ℝ*) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 0 ≤ (𝑥𝐷𝑦))
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥𝐷𝑦) ≤ 0 ↔ 𝑥 = 𝑦))
& ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))) ⇒ ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) |
|
Theorem | metflem 12896* |
Lemma for metf 12898 and others. (Contributed by NM,
30-Aug-2006.)
(Revised by Mario Carneiro, 14-Aug-2015.)
|
⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷:(𝑋 × 𝑋)⟶ℝ ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))))) |
|
Theorem | xmetf 12897 |
Mapping of the distance function of an extended metric. (Contributed by
Mario Carneiro, 20-Aug-2015.)
|
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) |
|
Theorem | metf 12898 |
Mapping of the distance function of a metric space. (Contributed by NM,
30-Aug-2006.)
|
⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ) |
|
Theorem | xmetcl 12899 |
Closure of the distance function of a metric space. Part of Property M1
of [Kreyszig] p. 3. (Contributed by
NM, 30-Aug-2006.)
|
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈
ℝ*) |
|
Theorem | metcl 12900 |
Closure of the distance function of a metric space. Part of Property M1
of [Kreyszig] p. 3. (Contributed by
NM, 30-Aug-2006.)
|
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ) |