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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | subfacval 31701* | The subfactorial is defined as the number of derangements (see derangval 31695) of the set (1...𝑁). (Contributed by Mario Carneiro, 21-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) ⇒ ⊢ (𝑁 ∈ ℕ0 → (𝑆‘𝑁) = (𝐷‘(1...𝑁))) | ||
Theorem | derangen2 31702* | Write the derangement number in terms of the subfactorial. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) ⇒ ⊢ (𝐴 ∈ Fin → (𝐷‘𝐴) = (𝑆‘(♯‘𝐴))) | ||
Theorem | subfacf 31703* | The subfactorial is a function from nonnegative integers to nonnegative integers. (Contributed by Mario Carneiro, 19-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) ⇒ ⊢ 𝑆:ℕ0⟶ℕ0 | ||
Theorem | subfaclefac 31704* | The subfactorial is less than the factorial. (Contributed by Mario Carneiro, 19-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) ⇒ ⊢ (𝑁 ∈ ℕ0 → (𝑆‘𝑁) ≤ (!‘𝑁)) | ||
Theorem | subfac0 31705* | The subfactorial at zero. (Contributed by Mario Carneiro, 19-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) ⇒ ⊢ (𝑆‘0) = 1 | ||
Theorem | subfac1 31706* | The subfactorial at one. (Contributed by Mario Carneiro, 19-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) ⇒ ⊢ (𝑆‘1) = 0 | ||
Theorem | subfacp1lem1 31707* | Lemma for subfacp1 31714. The set 𝐾 together with {1, 𝑀} partitions the set 1...(𝑁 + 1). (Contributed by Mario Carneiro, 23-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) & ⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ (2...(𝑁 + 1))) & ⊢ 𝑀 ∈ V & ⊢ 𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀}) ⇒ ⊢ (𝜑 → ((𝐾 ∩ {1, 𝑀}) = ∅ ∧ (𝐾 ∪ {1, 𝑀}) = (1...(𝑁 + 1)) ∧ (♯‘𝐾) = (𝑁 − 1))) | ||
Theorem | subfacp1lem2a 31708* | Lemma for subfacp1 31714. Properties of a bijection on 𝐾 augmented with the two-element flip to get a bijection on 𝐾 ∪ {1, 𝑀}. (Contributed by Mario Carneiro, 23-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) & ⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ (2...(𝑁 + 1))) & ⊢ 𝑀 ∈ V & ⊢ 𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀}) & ⊢ 𝐹 = (𝐺 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) & ⊢ (𝜑 → 𝐺:𝐾–1-1-onto→𝐾) ⇒ ⊢ (𝜑 → (𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ (𝐹‘1) = 𝑀 ∧ (𝐹‘𝑀) = 1)) | ||
Theorem | subfacp1lem2b 31709* | Lemma for subfacp1 31714. Properties of a bijection on 𝐾 augmented with the two-element flip to get a bijection on 𝐾 ∪ {1, 𝑀}. (Contributed by Mario Carneiro, 23-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) & ⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ (2...(𝑁 + 1))) & ⊢ 𝑀 ∈ V & ⊢ 𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀}) & ⊢ 𝐹 = (𝐺 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) & ⊢ (𝜑 → 𝐺:𝐾–1-1-onto→𝐾) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) = (𝐺‘𝑋)) | ||
Theorem | subfacp1lem3 31710* | Lemma for subfacp1 31714. In subfacp1lem6 31713 we cut up the set of all derangements on 1...(𝑁 + 1) first according to the value at 1, and then by whether or not (𝑓‘(𝑓‘1)) = 1. In this lemma, we show that the subset of all 𝑁 + 1 derangements that satisfy this for fixed 𝑀 = (𝑓‘1) is in bijection with 𝑁 − 1 derangements, by simply dropping the 𝑥 = 1 and 𝑥 = 𝑀 points from the function to get a derangement on 𝐾 = (1...(𝑁 − 1)) ∖ {1, 𝑀}. (Contributed by Mario Carneiro, 23-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) & ⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ (2...(𝑁 + 1))) & ⊢ 𝑀 ∈ V & ⊢ 𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀}) & ⊢ 𝐵 = {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = 𝑀 ∧ (𝑔‘𝑀) = 1)} & ⊢ 𝐶 = {𝑓 ∣ (𝑓:𝐾–1-1-onto→𝐾 ∧ ∀𝑦 ∈ 𝐾 (𝑓‘𝑦) ≠ 𝑦)} ⇒ ⊢ (𝜑 → (♯‘𝐵) = (𝑆‘(𝑁 − 1))) | ||
Theorem | subfacp1lem4 31711* | Lemma for subfacp1 31714. The function 𝐹, which swaps 1 with 𝑀 and leaves all other elements alone, is a bijection of order 2, i.e. it is its own inverse. (Contributed by Mario Carneiro, 19-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) & ⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ (2...(𝑁 + 1))) & ⊢ 𝑀 ∈ V & ⊢ 𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀}) & ⊢ 𝐵 = {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = 𝑀 ∧ (𝑔‘𝑀) ≠ 1)} & ⊢ 𝐹 = (( I ↾ 𝐾) ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) ⇒ ⊢ (𝜑 → ◡𝐹 = 𝐹) | ||
Theorem | subfacp1lem5 31712* | Lemma for subfacp1 31714. In subfacp1lem6 31713 we cut up the set of all derangements on 1...(𝑁 + 1) first according to the value at 1, and then by whether or not (𝑓‘(𝑓‘1)) = 1. In this lemma, we show that the subset of all 𝑁 + 1 derangements with (𝑓‘(𝑓‘1)) ≠ 1 for fixed 𝑀 = (𝑓‘1) is in bijection with derangements of 2...(𝑁 + 1), because pre-composing with the function 𝐹 swaps 1 and 𝑀 and turns the function into a bijection with (𝑓‘1) = 1 and (𝑓‘𝑥) ≠ 𝑥 for all other 𝑥, so dropping the point at 1 yields a derangement on the 𝑁 remaining points. (Contributed by Mario Carneiro, 23-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) & ⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ (2...(𝑁 + 1))) & ⊢ 𝑀 ∈ V & ⊢ 𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀}) & ⊢ 𝐵 = {𝑔 ∈ 𝐴 ∣ ((𝑔‘1) = 𝑀 ∧ (𝑔‘𝑀) ≠ 1)} & ⊢ 𝐹 = (( I ↾ 𝐾) ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) & ⊢ 𝐶 = {𝑓 ∣ (𝑓:(2...(𝑁 + 1))–1-1-onto→(2...(𝑁 + 1)) ∧ ∀𝑦 ∈ (2...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} ⇒ ⊢ (𝜑 → (♯‘𝐵) = (𝑆‘𝑁)) | ||
Theorem | subfacp1lem6 31713* | Lemma for subfacp1 31714. By induction, we cut up the set of all derangements on 𝑁 + 1 according to the 𝑁 possible values of (𝑓‘1) (since (𝑓‘1) ≠ 1), and for each set for fixed 𝑀 = (𝑓‘1), the subset of derangements with (𝑓‘𝑀) = 1 has size 𝑆(𝑁 − 1) (by subfacp1lem3 31710), while the subset with (𝑓‘𝑀) ≠ 1 has size 𝑆(𝑁) (by subfacp1lem5 31712). Adding it all up yields the desired equation 𝑁(𝑆(𝑁) + 𝑆(𝑁 − 1)) for the number of derangements on 𝑁 + 1. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) & ⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} ⇒ ⊢ (𝑁 ∈ ℕ → (𝑆‘(𝑁 + 1)) = (𝑁 · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))) | ||
Theorem | subfacp1 31714* | A two-term recurrence for the subfactorial. This theorem allows us to forget the combinatorial definition of the derangement number in favor of the recursive definition provided by this theorem and subfac0 31705, subfac1 31706. (Contributed by Mario Carneiro, 23-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) ⇒ ⊢ (𝑁 ∈ ℕ → (𝑆‘(𝑁 + 1)) = (𝑁 · ((𝑆‘𝑁) + (𝑆‘(𝑁 − 1))))) | ||
Theorem | subfacval2 31715* | A closed-form expression for the subfactorial. (Contributed by Mario Carneiro, 23-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) ⇒ ⊢ (𝑁 ∈ ℕ0 → (𝑆‘𝑁) = ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) / (!‘𝑘)))) | ||
Theorem | subfaclim 31716* | The subfactorial converges rapidly to 𝑁! / e. This is part of Metamath 100 proof #88. (Contributed by Mario Carneiro, 23-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) ⇒ ⊢ (𝑁 ∈ ℕ → (abs‘(((!‘𝑁) / e) − (𝑆‘𝑁))) < (1 / 𝑁)) | ||
Theorem | subfacval3 31717* | Another closed form expression for the subfactorial. The expression ⌊‘(𝑥 + 1 / 2) is a way of saying "rounded to the nearest integer". (Contributed by Mario Carneiro, 23-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) & ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) ⇒ ⊢ (𝑁 ∈ ℕ → (𝑆‘𝑁) = (⌊‘(((!‘𝑁) / e) + (1 / 2)))) | ||
Theorem | derangfmla 31718* | The derangements formula, which expresses the number of derangements of a finite nonempty set in terms of the factorial. The expression ⌊‘(𝑥 + 1 / 2) is a way of saying "rounded to the nearest integer". This is part of Metamath 100 proof #88. (Contributed by Mario Carneiro, 23-Jan-2015.) |
⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) ⇒ ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → (𝐷‘𝐴) = (⌊‘(((!‘(♯‘𝐴)) / e) + (1 / 2)))) | ||
Theorem | erdszelem1 31719* | Lemma for erdsze 31730. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ 𝑆 = {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)} ⇒ ⊢ (𝑋 ∈ 𝑆 ↔ (𝑋 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 “ 𝑋)) ∧ 𝐴 ∈ 𝑋)) | ||
Theorem | erdszelem2 31720* | Lemma for erdsze 31730. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ 𝑆 = {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)} ⇒ ⊢ ((♯ “ 𝑆) ∈ Fin ∧ (♯ “ 𝑆) ⊆ ℕ) | ||
Theorem | erdszelem3 31721* | Lemma for erdsze 31730. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ) & ⊢ 𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < )) ⇒ ⊢ (𝐴 ∈ (1...𝑁) → (𝐾‘𝐴) = sup((♯ “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}), ℝ, < )) | ||
Theorem | erdszelem4 31722* | Lemma for erdsze 31730. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ) & ⊢ 𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < )) & ⊢ 𝑂 Or ℝ ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ (1...𝑁)) → {𝐴} ∈ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}) | ||
Theorem | erdszelem5 31723* | Lemma for erdsze 31730. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ) & ⊢ 𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < )) & ⊢ 𝑂 Or ℝ ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ (1...𝑁)) → (𝐾‘𝐴) ∈ (♯ “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)})) | ||
Theorem | erdszelem6 31724* | Lemma for erdsze 31730. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ) & ⊢ 𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < )) & ⊢ 𝑂 Or ℝ ⇒ ⊢ (𝜑 → 𝐾:(1...𝑁)⟶ℕ) | ||
Theorem | erdszelem7 31725* | Lemma for erdsze 31730. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ) & ⊢ 𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < )) & ⊢ 𝑂 Or ℝ & ⊢ (𝜑 → 𝐴 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝑅 ∈ ℕ) & ⊢ (𝜑 → ¬ (𝐾‘𝐴) ∈ (1...(𝑅 − 1))) ⇒ ⊢ (𝜑 → ∃𝑠 ∈ 𝒫 (1...𝑁)(𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 “ 𝑠)))) | ||
Theorem | erdszelem8 31726* | Lemma for erdsze 31730. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ) & ⊢ 𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < )) & ⊢ 𝑂 Or ℝ & ⊢ (𝜑 → 𝐴 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐵 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → ((𝐾‘𝐴) = (𝐾‘𝐵) → ¬ (𝐹‘𝐴)𝑂(𝐹‘𝐵))) | ||
Theorem | erdszelem9 31727* | Lemma for erdsze 31730. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ) & ⊢ 𝐼 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < )) & ⊢ 𝐽 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < )) & ⊢ 𝑇 = (𝑛 ∈ (1...𝑁) ↦ 〈(𝐼‘𝑛), (𝐽‘𝑛)〉) ⇒ ⊢ (𝜑 → 𝑇:(1...𝑁)–1-1→(ℕ × ℕ)) | ||
Theorem | erdszelem10 31728* | Lemma for erdsze 31730. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ) & ⊢ 𝐼 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < )) & ⊢ 𝐽 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < )) & ⊢ 𝑇 = (𝑛 ∈ (1...𝑁) ↦ 〈(𝐼‘𝑛), (𝐽‘𝑛)〉) & ⊢ (𝜑 → 𝑅 ∈ ℕ) & ⊢ (𝜑 → 𝑆 ∈ ℕ) & ⊢ (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) < 𝑁) ⇒ ⊢ (𝜑 → ∃𝑚 ∈ (1...𝑁)(¬ (𝐼‘𝑚) ∈ (1...(𝑅 − 1)) ∨ ¬ (𝐽‘𝑚) ∈ (1...(𝑆 − 1)))) | ||
Theorem | erdszelem11 31729* | Lemma for erdsze 31730. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ) & ⊢ 𝐼 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < )) & ⊢ 𝐽 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , ◡ < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < )) & ⊢ 𝑇 = (𝑛 ∈ (1...𝑁) ↦ 〈(𝐼‘𝑛), (𝐽‘𝑛)〉) & ⊢ (𝜑 → 𝑅 ∈ ℕ) & ⊢ (𝜑 → 𝑆 ∈ ℕ) & ⊢ (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) < 𝑁) ⇒ ⊢ (𝜑 → ∃𝑠 ∈ 𝒫 (1...𝑁)((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠))))) | ||
Theorem | erdsze 31730* | The Erdős-Szekeres theorem. For any injective sequence 𝐹 on the reals of length at least (𝑅 − 1) · (𝑆 − 1) + 1, there is either a subsequence of length at least 𝑅 on which 𝐹 is increasing (i.e. a < , < order isomorphism) or a subsequence of length at least 𝑆 on which 𝐹 is decreasing (i.e. a < , ◡ < order isomorphism, recalling that ◡ < is the "greater than" relation). This is part of Metamath 100 proof #73. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ) & ⊢ (𝜑 → 𝑅 ∈ ℕ) & ⊢ (𝜑 → 𝑆 ∈ ℕ) & ⊢ (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) < 𝑁) ⇒ ⊢ (𝜑 → ∃𝑠 ∈ 𝒫 (1...𝑁)((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠))))) | ||
Theorem | erdsze2lem1 31731* | Lemma for erdsze2 31733. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ (𝜑 → 𝑅 ∈ ℕ) & ⊢ (𝜑 → 𝑆 ∈ ℕ) & ⊢ (𝜑 → 𝐹:𝐴–1-1→ℝ) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ 𝑁 = ((𝑅 − 1) · (𝑆 − 1)) & ⊢ (𝜑 → 𝑁 < (♯‘𝐴)) ⇒ ⊢ (𝜑 → ∃𝑓(𝑓:(1...(𝑁 + 1))–1-1→𝐴 ∧ 𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓))) | ||
Theorem | erdsze2lem2 31732* | Lemma for erdsze2 31733. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ (𝜑 → 𝑅 ∈ ℕ) & ⊢ (𝜑 → 𝑆 ∈ ℕ) & ⊢ (𝜑 → 𝐹:𝐴–1-1→ℝ) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ 𝑁 = ((𝑅 − 1) · (𝑆 − 1)) & ⊢ (𝜑 → 𝑁 < (♯‘𝐴)) & ⊢ (𝜑 → 𝐺:(1...(𝑁 + 1))–1-1→𝐴) & ⊢ (𝜑 → 𝐺 Isom < , < ((1...(𝑁 + 1)), ran 𝐺)) ⇒ ⊢ (𝜑 → ∃𝑠 ∈ 𝒫 𝐴((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠))))) | ||
Theorem | erdsze2 31733* | Generalize the statement of the Erdős-Szekeres theorem erdsze 31730 to "sequences" indexed by an arbitrary subset of ℝ, which can be infinite. This is part of Metamath 100 proof #73. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ (𝜑 → 𝑅 ∈ ℕ) & ⊢ (𝜑 → 𝑆 ∈ ℕ) & ⊢ (𝜑 → 𝐹:𝐴–1-1→ℝ) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) < (♯‘𝐴)) ⇒ ⊢ (𝜑 → ∃𝑠 ∈ 𝒫 𝐴((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ◡ < (𝑠, (𝐹 “ 𝑠))))) | ||
Theorem | kur14lem1 31734 | Lemma for kur14 31744. (Contributed by Mario Carneiro, 17-Feb-2015.) |
⊢ 𝐴 ⊆ 𝑋 & ⊢ (𝑋 ∖ 𝐴) ∈ 𝑇 & ⊢ (𝐾‘𝐴) ∈ 𝑇 ⇒ ⊢ (𝑁 = 𝐴 → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇)) | ||
Theorem | kur14lem2 31735 | Lemma for kur14 31744. Write interior in terms of closure and complement: 𝑖𝐴 = 𝑐𝑘𝑐𝐴 where 𝑐 is complement and 𝑘 is closure. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ 𝐽 ∈ Top & ⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (cls‘𝐽) & ⊢ 𝐼 = (int‘𝐽) & ⊢ 𝐴 ⊆ 𝑋 ⇒ ⊢ (𝐼‘𝐴) = (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝐴))) | ||
Theorem | kur14lem3 31736 | Lemma for kur14 31744. A closure is a subset of the base set. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ 𝐽 ∈ Top & ⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (cls‘𝐽) & ⊢ 𝐼 = (int‘𝐽) & ⊢ 𝐴 ⊆ 𝑋 ⇒ ⊢ (𝐾‘𝐴) ⊆ 𝑋 | ||
Theorem | kur14lem4 31737 | Lemma for kur14 31744. Complementation is an involution on the set of subsets of a topology. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ 𝐽 ∈ Top & ⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (cls‘𝐽) & ⊢ 𝐼 = (int‘𝐽) & ⊢ 𝐴 ⊆ 𝑋 ⇒ ⊢ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴 | ||
Theorem | kur14lem5 31738 | Lemma for kur14 31744. Closure is an idempotent operation in the set of subsets of a topology. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ 𝐽 ∈ Top & ⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (cls‘𝐽) & ⊢ 𝐼 = (int‘𝐽) & ⊢ 𝐴 ⊆ 𝑋 ⇒ ⊢ (𝐾‘(𝐾‘𝐴)) = (𝐾‘𝐴) | ||
Theorem | kur14lem6 31739 | Lemma for kur14 31744. If 𝑘 is the complementation operator and 𝑘 is the closure operator, this expresses the identity 𝑘𝑐𝑘𝐴 = 𝑘𝑐𝑘𝑐𝑘𝑐𝑘𝐴 for any subset 𝐴 of the topological space. This is the key result that lets us cut down long enough sequences of 𝑐𝑘𝑐𝑘... that arise when applying closure and complement repeatedly to 𝐴, and explains why we end up with a number as large as 14, yet no larger. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ 𝐽 ∈ Top & ⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (cls‘𝐽) & ⊢ 𝐼 = (int‘𝐽) & ⊢ 𝐴 ⊆ 𝑋 & ⊢ 𝐵 = (𝑋 ∖ (𝐾‘𝐴)) ⇒ ⊢ (𝐾‘(𝐼‘(𝐾‘𝐵))) = (𝐾‘𝐵) | ||
Theorem | kur14lem7 31740 | Lemma for kur14 31744: main proof. The set 𝑇 here contains all the distinct combinations of 𝑘 and 𝑐 that can arise, and we prove here that applying 𝑘 or 𝑐 to any element of 𝑇 yields another elemnt of 𝑇. In operator shorthand, we have 𝑇 = {𝐴, 𝑐𝐴, 𝑘𝐴 , 𝑐𝑘𝐴, 𝑘𝑐𝐴, 𝑐𝑘𝑐𝐴, 𝑘𝑐𝑘𝐴, 𝑐𝑘𝑐𝑘𝐴, 𝑘𝑐𝑘𝑐𝐴, 𝑐𝑘𝑐𝑘𝑐𝐴, 𝑘𝑐𝑘𝑐𝑘𝐴, 𝑐𝑘𝑐𝑘𝑐𝑘𝐴, 𝑘𝑐𝑘𝑐𝑘𝑐𝐴, 𝑐𝑘𝑐𝑘𝑐𝑘𝑐𝐴}. From the identities 𝑐𝑐𝐴 = 𝐴 and 𝑘𝑘𝐴 = 𝑘𝐴, we can reduce any operator combination containing two adjacent identical operators, which is why the list only contains alternating sequences. The reason the sequences don't keep going after a certain point is due to the identity 𝑘𝑐𝑘𝐴 = 𝑘𝑐𝑘𝑐𝑘𝑐𝑘𝐴, proved in kur14lem6 31739. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ 𝐽 ∈ Top & ⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (cls‘𝐽) & ⊢ 𝐼 = (int‘𝐽) & ⊢ 𝐴 ⊆ 𝑋 & ⊢ 𝐵 = (𝑋 ∖ (𝐾‘𝐴)) & ⊢ 𝐶 = (𝐾‘(𝑋 ∖ 𝐴)) & ⊢ 𝐷 = (𝐼‘(𝐾‘𝐴)) & ⊢ 𝑇 = ((({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ∪ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∪ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))})) ⇒ ⊢ (𝑁 ∈ 𝑇 → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇)) | ||
Theorem | kur14lem8 31741 | Lemma for kur14 31744. Show that the set 𝑇 contains at most 14 elements. (It could be less if some of the operators take the same value for a given set, but Kuratowski showed that this upper bound of 14 is tight in the sense that there exist topological spaces and subsets of these spaces for which all 14 generated sets are distinct, and indeed the real numbers form such a topological space.) (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ 𝐽 ∈ Top & ⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (cls‘𝐽) & ⊢ 𝐼 = (int‘𝐽) & ⊢ 𝐴 ⊆ 𝑋 & ⊢ 𝐵 = (𝑋 ∖ (𝐾‘𝐴)) & ⊢ 𝐶 = (𝐾‘(𝑋 ∖ 𝐴)) & ⊢ 𝐷 = (𝐼‘(𝐾‘𝐴)) & ⊢ 𝑇 = ((({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ∪ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∪ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))})) ⇒ ⊢ (𝑇 ∈ Fin ∧ (♯‘𝑇) ≤ ;14) | ||
Theorem | kur14lem9 31742* | Lemma for kur14 31744. Since the set 𝑇 is closed under closure and complement, it contains the minimal set 𝑆 as a subset, so 𝑆 also has at most 14 elements. (Indeed 𝑆 = 𝑇, and it's not hard to prove this, but we don't need it for this proof.) (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ 𝐽 ∈ Top & ⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (cls‘𝐽) & ⊢ 𝐼 = (int‘𝐽) & ⊢ 𝐴 ⊆ 𝑋 & ⊢ 𝐵 = (𝑋 ∖ (𝐾‘𝐴)) & ⊢ 𝐶 = (𝐾‘(𝑋 ∖ 𝐴)) & ⊢ 𝐷 = (𝐼‘(𝐾‘𝐴)) & ⊢ 𝑇 = ((({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ∪ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∪ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))})) & ⊢ 𝑆 = ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} ⇒ ⊢ (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ ;14) | ||
Theorem | kur14lem10 31743* | Lemma for kur14 31744. Discharge the set 𝑇. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ 𝐽 ∈ Top & ⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (cls‘𝐽) & ⊢ 𝑆 = ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} & ⊢ 𝐴 ⊆ 𝑋 ⇒ ⊢ (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ ;14) | ||
Theorem | kur14 31744* | Kuratowski's closure-complement theorem. There are at most 14 sets which can be obtained by the application of the closure and complement operations to a set in a topological space. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (cls‘𝐽) & ⊢ 𝑆 = ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ ;14)) | ||
Syntax | cretr 31745 | Extend class notation with the retract relation. |
class Retr | ||
Definition | df-retr 31746* | Define the set of retractions on two topological spaces. We say that 𝑅 is a retraction from 𝐽 to 𝐾. or 𝑅 ∈ (𝐽 Retr 𝐾) iff there is an 𝑆 such that 𝑅:𝐽⟶𝐾, 𝑆:𝐾⟶𝐽 are continuous functions called the retraction and section respectively, and their composite 𝑅 ∘ 𝑆 is homotopic to the identity map. If a retraction exists, we say 𝐽 is a retract of 𝐾. (This terminology is borrowed from HoTT and appears to be nonstandard, although it has similaries to the concept of retract in the category of topological spaces and to a deformation retract in general topology.) Two topological spaces that are retracts of each other are called homotopy equivalent. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ Retr = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑟 ∈ (𝑗 Cn 𝑘) ∣ ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗)) ≠ ∅}) | ||
Syntax | cpconn 31747 | Extend class notation with the class of path-connected topologies. |
class PConn | ||
Syntax | csconn 31748 | Extend class notation with the class of simply connected topologies. |
class SConn | ||
Definition | df-pconn 31749* | Define the class of path-connected topologies. A topology is path-connected if there is a path (a continuous function from the closed unit interval) that goes from 𝑥 to 𝑦 for any points 𝑥, 𝑦 in the space. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ PConn = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗∃𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)} | ||
Definition | df-sconn 31750* | Define the class of simply connected topologies. A topology is simply connected if it is path-connected and every loop (continuous path with identical start and endpoint) is contractible to a point (path-homotopic to a constant function). (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ SConn = {𝑗 ∈ PConn ∣ ∀𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝑗)((0[,]1) × {(𝑓‘0)}))} | ||
Theorem | ispconn 31751* | The property of being a path-connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐽 ∈ PConn ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))) | ||
Theorem | pconncn 31752* | The property of being a path-connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ PConn ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝐵)) | ||
Theorem | pconntop 31753 | A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ (𝐽 ∈ PConn → 𝐽 ∈ Top) | ||
Theorem | issconn 31754* | The property of being a simply connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)})))) | ||
Theorem | sconnpconn 31755 | A simply connected space is path-connected. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ (𝐽 ∈ SConn → 𝐽 ∈ PConn) | ||
Theorem | sconntop 31756 | A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ (𝐽 ∈ SConn → 𝐽 ∈ Top) | ||
Theorem | sconnpht 31757 | A closed path in a simply connected space is contractible to a point. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ ((𝐽 ∈ SConn ∧ 𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = (𝐹‘1)) → 𝐹( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)})) | ||
Theorem | cnpconn 31758 | An image of a path-connected space is path-connected. (Contributed by Mario Carneiro, 24-Mar-2015.) |
⊢ 𝑌 = ∪ 𝐾 ⇒ ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ PConn) | ||
Theorem | pconnconn 31759 | A path-connected space is connected. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ (𝐽 ∈ PConn → 𝐽 ∈ Conn) | ||
Theorem | txpconn 31760 | The topological product of two path-connected spaces is path-connected. (Contributed by Mario Carneiro, 12-Feb-2015.) |
⊢ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (𝑅 ×t 𝑆) ∈ PConn) | ||
Theorem | ptpconn 31761 | The topological product of a collection of path-connected spaces is path-connected. The proof uses the axiom of choice. (Contributed by Mario Carneiro, 17-Feb-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) → (∏t‘𝐹) ∈ PConn) | ||
Theorem | indispconn 31762 | The indiscrete topology (or trivial topology) on any set is path-connected. (Contributed by Mario Carneiro, 7-Jul-2015.) (Revised by Mario Carneiro, 14-Aug-2015.) |
⊢ {∅, 𝐴} ∈ PConn | ||
Theorem | connpconn 31763 | A connected and locally path-connected space is path-connected. (Contributed by Mario Carneiro, 7-Jul-2015.) |
⊢ ((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) → 𝐽 ∈ PConn) | ||
Theorem | qtoppconn 31764 | A quotient of a path-connected space is path-connected. (Contributed by Mario Carneiro, 24-Mar-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ PConn ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ PConn) | ||
Theorem | pconnpi1 31765 | All fundamental groups in a path-connected space are isomorphic. (Contributed by Mario Carneiro, 12-Feb-2015.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝑃 = (𝐽 π1 𝐴) & ⊢ 𝑄 = (𝐽 π1 𝐵) & ⊢ 𝑆 = (Base‘𝑃) & ⊢ 𝑇 = (Base‘𝑄) ⇒ ⊢ ((𝐽 ∈ PConn ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝑃 ≃𝑔 𝑄) | ||
Theorem | sconnpht2 31766 | Any two paths in a simply connected space with the same start and end point are path-homotopic. (Contributed by Mario Carneiro, 12-Feb-2015.) |
⊢ (𝜑 → 𝐽 ∈ SConn) & ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) & ⊢ (𝜑 → (𝐹‘0) = (𝐺‘0)) & ⊢ (𝜑 → (𝐹‘1) = (𝐺‘1)) ⇒ ⊢ (𝜑 → 𝐹( ≃ph‘𝐽)𝐺) | ||
Theorem | sconnpi1 31767 | A path-connected topological space is simply connected iff its fundamental group is trivial. (Contributed by Mario Carneiro, 12-Feb-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) → (𝐽 ∈ SConn ↔ (Base‘(𝐽 π1 𝑌)) ≈ 1o)) | ||
Theorem | txsconnlem 31768 | Lemma for txsconn 31769. (Contributed by Mario Carneiro, 9-Mar-2015.) |
⊢ (𝜑 → 𝑅 ∈ Top) & ⊢ (𝜑 → 𝑆 ∈ Top) & ⊢ (𝜑 → 𝐹 ∈ (II Cn (𝑅 ×t 𝑆))) & ⊢ 𝐴 = ((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹) & ⊢ 𝐵 = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹) & ⊢ (𝜑 → 𝐺 ∈ (𝐴(PHtpy‘𝑅)((0[,]1) × {(𝐴‘0)}))) & ⊢ (𝜑 → 𝐻 ∈ (𝐵(PHtpy‘𝑆)((0[,]1) × {(𝐵‘0)}))) ⇒ ⊢ (𝜑 → 𝐹( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)})) | ||
Theorem | txsconn 31769 | The topological product of two simply connected spaces is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.) |
⊢ ((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) → (𝑅 ×t 𝑆) ∈ SConn) | ||
Theorem | cvxpconn 31770* | A convex subset of the complex numbers is path-connected. (Contributed by Mario Carneiro, 12-Feb-2015.) |
⊢ (𝜑 → 𝑆 ⊆ ℂ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑡 ∈ (0[,]1))) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆) & ⊢ 𝐽 = (TopOpen‘ℂfld) & ⊢ 𝐾 = (𝐽 ↾t 𝑆) ⇒ ⊢ (𝜑 → 𝐾 ∈ PConn) | ||
Theorem | cvxsconn 31771* | A convex subset of the complex numbers is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.) |
⊢ (𝜑 → 𝑆 ⊆ ℂ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑡 ∈ (0[,]1))) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆) & ⊢ 𝐽 = (TopOpen‘ℂfld) & ⊢ 𝐾 = (𝐽 ↾t 𝑆) ⇒ ⊢ (𝜑 → 𝐾 ∈ SConn) | ||
Theorem | blsconn 31772 | An open ball in the complex numbers is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.) |
⊢ 𝐽 = (TopOpen‘ℂfld) & ⊢ 𝑆 = (𝑃(ball‘(abs ∘ − ))𝑅) & ⊢ 𝐾 = (𝐽 ↾t 𝑆) ⇒ ⊢ ((𝑃 ∈ ℂ ∧ 𝑅 ∈ ℝ*) → 𝐾 ∈ SConn) | ||
Theorem | cnllysconn 31773 | The topology of the complex numbers is locally simply connected. (Contributed by Mario Carneiro, 2-Mar-2015.) |
⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ 𝐽 ∈ Locally SConn | ||
Theorem | resconn 31774 | A subset of ℝ is simply connected iff it is connected. (Contributed by Mario Carneiro, 9-Mar-2015.) |
⊢ 𝐽 = ((topGen‘ran (,)) ↾t 𝐴) ⇒ ⊢ (𝐴 ⊆ ℝ → (𝐽 ∈ SConn ↔ 𝐽 ∈ Conn)) | ||
Theorem | ioosconn 31775 | An open interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.) |
⊢ ((topGen‘ran (,)) ↾t (𝐴(,)𝐵)) ∈ SConn | ||
Theorem | iccsconn 31776 | A closed interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ SConn) | ||
Theorem | retopsconn 31777 | The real numbers are simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.) |
⊢ (topGen‘ran (,)) ∈ SConn | ||
Theorem | iccllysconn 31778 | A closed interval is locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ Locally SConn) | ||
Theorem | rellysconn 31779 | The real numbers are locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.) |
⊢ (topGen‘ran (,)) ∈ Locally SConn | ||
Theorem | iisconn 31780 | The unit interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.) |
⊢ II ∈ SConn | ||
Theorem | iillysconn 31781 | The unit interval is locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.) |
⊢ II ∈ Locally SConn | ||
Theorem | iinllyconn 31782 | The unit interval is locally connected. (Contributed by Mario Carneiro, 6-Jul-2015.) |
⊢ II ∈ 𝑛-Locally Conn | ||
Syntax | ccvm 31783 | Extend class notation with the class of covering maps. |
class CovMap | ||
Definition | df-cvm 31784* | Define the class of covering maps on two topological spaces. A function 𝑓:𝑐⟶𝑗 is a covering map if it is continuous and for every point 𝑥 in the target space there is a neighborhood 𝑘 of 𝑥 and a decomposition 𝑠 of the preimage of 𝑘 as a disjoint union such that 𝑓 is a homeomorphism of each set 𝑢 ∈ 𝑠 onto 𝑘. (Contributed by Mario Carneiro, 13-Feb-2015.) |
⊢ CovMap = (𝑐 ∈ Top, 𝑗 ∈ Top ↦ {𝑓 ∈ (𝑐 Cn 𝑗) ∣ ∀𝑥 ∈ ∪ 𝑗∃𝑘 ∈ 𝑗 (𝑥 ∈ 𝑘 ∧ ∃𝑠 ∈ (𝒫 𝑐 ∖ {∅})(∪ 𝑠 = (◡𝑓 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝑓 ↾ 𝑢) ∈ ((𝑐 ↾t 𝑢)Homeo(𝑗 ↾t 𝑘)))))}) | ||
Theorem | fncvm 31785 | Lemma for covering maps. (Contributed by Mario Carneiro, 13-Feb-2015.) |
⊢ CovMap Fn (Top × Top) | ||
Theorem | cvmscbv 31786* | Change bound variables in the set of even coverings. (Contributed by Mario Carneiro, 17-Feb-2015.) |
⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) ⇒ ⊢ 𝑆 = (𝑎 ∈ 𝐽 ↦ {𝑏 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑏 = (◡𝐹 “ 𝑎) ∧ ∀𝑐 ∈ 𝑏 (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑎))))}) | ||
Theorem | iscvm 31787* | The property of being a covering map. (Contributed by Mario Carneiro, 13-Feb-2015.) |
⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) & ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) ↔ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top ∧ 𝐹 ∈ (𝐶 Cn 𝐽)) ∧ ∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅))) | ||
Theorem | cvmtop1 31788 | Reverse closure for a covering map. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top) | ||
Theorem | cvmtop2 31789 | Reverse closure for a covering map. (Contributed by Mario Carneiro, 13-Feb-2015.) |
⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top) | ||
Theorem | cvmcn 31790 | A covering map is a continuous function. (Contributed by Mario Carneiro, 13-Feb-2015.) |
⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽)) | ||
Theorem | cvmcov 31791* | Property of a covering map. In order to make the covering property more manageable, we define here the set 𝑆(𝑘) of all even coverings of an open set 𝑘 in the range. Then the covering property states that every point has a neighborhood which has an even covering. (Contributed by Mario Carneiro, 13-Feb-2015.) |
⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) & ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑃 ∈ 𝑋) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝑆‘𝑥) ≠ ∅)) | ||
Theorem | cvmsrcl 31792* | Reverse closure for an even covering. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) ⇒ ⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑈 ∈ 𝐽) | ||
Theorem | cvmsi 31793* | One direction of cvmsval 31794. (Contributed by Mario Carneiro, 13-Feb-2015.) |
⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) ⇒ ⊢ (𝑇 ∈ (𝑆‘𝑈) → (𝑈 ∈ 𝐽 ∧ (𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅) ∧ (∪ 𝑇 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈)))))) | ||
Theorem | cvmsval 31794* | Elementhood in the set 𝑆 of all even coverings of an open set in 𝐽. 𝑆 is an even covering of 𝑈 if it is a nonempty collection of disjoint open sets in 𝐶 whose union is the preimage of 𝑈, such that each set 𝑢 ∈ 𝑆 is homeomorphic under 𝐹 to 𝑈. (Contributed by Mario Carneiro, 13-Feb-2015.) |
⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) ⇒ ⊢ (𝐶 ∈ 𝑉 → (𝑇 ∈ (𝑆‘𝑈) ↔ (𝑈 ∈ 𝐽 ∧ (𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅) ∧ (∪ 𝑇 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))))) | ||
Theorem | cvmsss 31795* | An even covering is a subset of the topology of the domain (i.e. a collection of open sets). (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) ⇒ ⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑇 ⊆ 𝐶) | ||
Theorem | cvmsn0 31796* | An even covering is nonempty. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) ⇒ ⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑇 ≠ ∅) | ||
Theorem | cvmsuni 31797* | An even covering of 𝑈 has union equal to the preimage of 𝑈 by 𝐹. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) ⇒ ⊢ (𝑇 ∈ (𝑆‘𝑈) → ∪ 𝑇 = (◡𝐹 “ 𝑈)) | ||
Theorem | cvmsdisj 31798* | An even covering of 𝑈 is a disjoint union. (Contributed by Mario Carneiro, 13-Feb-2015.) |
⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) ⇒ ⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → (𝐴 = 𝐵 ∨ (𝐴 ∩ 𝐵) = ∅)) | ||
Theorem | cvmshmeo 31799* | Every element of an even covering of 𝑈 is homeomorphic to 𝑈 via 𝐹. (Contributed by Mario Carneiro, 13-Feb-2015.) |
⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) ⇒ ⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝐹 ↾ 𝐴) ∈ ((𝐶 ↾t 𝐴)Homeo(𝐽 ↾t 𝑈))) | ||
Theorem | cvmsf1o 31800* | 𝐹, localized to an element of an even covering of 𝑈, is a bijection. (Contributed by Mario Carneiro, 14-Feb-2015.) |
⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) ⇒ ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝐹 ↾ 𝐴):𝐴–1-1-onto→𝑈) |
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