P. 359 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 35801-35900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cvmliftlem14 35801*	Lemma for cvmlift 35803. Putting the results of cvmliftlem11 35799, cvmliftlem13 35800 and cvmliftmo 35788 together, we have that 𝐾 is a continuous function, satisfies 𝐹 ∘ 𝐾 = 𝐺 and 𝐾(0) = 𝑃, and is equal to any other function which also has these properties, so it follows that 𝐾 is the unique lift of 𝐺. (Contributed by Mario Carneiro, 16-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))))})    &   ⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑋 = ∪ 𝐽    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗)))    &   ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘)))    &   ⊢ 𝐿 = (topGen‘ran (,))    &   ⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))    &   ⊢ 𝐾 = ∪ 𝑘 ∈ (1...𝑁)(𝑄‘𝑘)    ⇒   ⊢ (𝜑 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))
Theorem	cvmliftlem15 35802*	Lemma for cvmlift 35803. Discharge the assumptions of cvmliftlem14 35801. The set of all open subsets 𝑢 of the unit interval such that 𝐺 “ 𝑢 is contained in an even covering of some open set in 𝐽 is a cover of II by the definition of a covering map, so by the Lebesgue number lemma lebnumii 25140, there is a subdivision of the closed unit interval into 𝑁 equal parts such that each part is entirely contained within one such open set of 𝐽. Then using finite choice ac6sfi 9246 to uniformly select one such subset and one even covering of each subset, we are ready to finish the proof with cvmliftlem14 35801. (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))))})    &   ⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑋 = ∪ 𝐽    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))    ⇒   ⊢ (𝜑 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))
Theorem	cvmlift 35803*	One of the important properties of covering maps is that any path 𝐺 in the base space "lifts" to a path 𝑓 in the covering space such that 𝐹 ∘ 𝑓 = 𝐺, and given a starting point 𝑃 in the covering space this lift is unique. The proof is contained in cvmliftlem1 35789 thru cvmliftlem15 35802. (Contributed by Mario Carneiro, 16-Feb-2015.)
⊢ 𝐵 = ∪ 𝐶    ⇒   ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐺 ∈ (II Cn 𝐽)) ∧ (𝑃 ∈ 𝐵 ∧ (𝐹‘𝑃) = (𝐺‘0))) → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))
Theorem	cvmfo 35804	A covering map is an onto function. (Contributed by Mario Carneiro, 13-Feb-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹:𝐵–onto→𝑋)
Theorem	cvmliftiota 35805*	Write out a function 𝐻 that is the unique lift of 𝐹. (Contributed by Mario Carneiro, 16-Feb-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))    ⇒   ⊢ (𝜑 → (𝐻 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐻) = 𝐺 ∧ (𝐻‘0) = 𝑃))
Theorem	cvmlift2lem1 35806*	Lemma for cvmlift2 35820. (Contributed by Mario Carneiro, 1-Jun-2015.)
⊢ (∀𝑦 ∈ (0[,]1)∃𝑢 ∈ ((nei‘II)‘{𝑦})((𝑢 × {𝑥}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀) → (((0[,]1) × {𝑥}) ⊆ 𝑀 → ((0[,]1) × {𝑡}) ⊆ 𝑀))
Theorem	cvmlift2lem9a 35807*	Lemma for cvmlift2 35820 and cvmlift3 35832. (Contributed by Mario Carneiro, 9-Jul-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))))})    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐻:𝑌⟶𝐵)    &   ⊢ (𝜑 → (𝐹 ∘ 𝐻) ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ Top)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑇 ∈ (𝑆‘𝐴))    &   ⊢ (𝜑 → (𝑊 ∈ 𝑇 ∧ (𝐻‘𝑋) ∈ 𝑊))    &   ⊢ (𝜑 → 𝑀 ⊆ 𝑌)    &   ⊢ (𝜑 → (𝐻 “ 𝑀) ⊆ 𝑊)    ⇒   ⊢ (𝜑 → (𝐻 ↾ 𝑀) ∈ ((𝐾 ↾t 𝑀) Cn 𝐶))
Theorem	cvmlift2lem2 35808*	Lemma for cvmlift2 35820. (Contributed by Mario Carneiro, 7-May-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))    &   ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    ⇒   ⊢ (𝜑 → (𝐻 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐻) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝐻‘0) = 𝑃))
Theorem	cvmlift2lem3 35809*	Lemma for cvmlift2 35820. (Contributed by Mario Carneiro, 7-May-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))    &   ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   ⊢ 𝐾 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑋)))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → (𝐾 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐾) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) ∧ (𝐾‘0) = (𝐻‘𝑋)))
Theorem	cvmlift2lem4 35810*	Lemma for cvmlift2 35820. (Contributed by Mario Carneiro, 1-Jun-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))    &   ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦))    ⇒   ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 ∈ (0[,]1)) → (𝑋𝐾𝑌) = ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑋)))‘𝑌))
Theorem	cvmlift2lem5 35811*	Lemma for cvmlift2 35820. (Contributed by Mario Carneiro, 7-May-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))    &   ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦))    ⇒   ⊢ (𝜑 → 𝐾:((0[,]1) × (0[,]1))⟶𝐵)
Theorem	cvmlift2lem6 35812*	Lemma for cvmlift2 35820. (Contributed by Mario Carneiro, 7-May-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))    &   ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → (𝐾 ↾ ({𝑋} × (0[,]1))) ∈ (((II ×t II) ↾t ({𝑋} × (0[,]1))) Cn 𝐶))
Theorem	cvmlift2lem7 35813*	Lemma for cvmlift2 35820. (Contributed by Mario Carneiro, 7-May-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))    &   ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦))    ⇒   ⊢ (𝜑 → (𝐹 ∘ 𝐾) = 𝐺)
Theorem	cvmlift2lem8 35814*	Lemma for cvmlift2 35820. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))    &   ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → (𝑋𝐾0) = (𝐻‘𝑋))
Theorem	cvmlift2lem9 35815*	Lemma for cvmlift2 35820. (Contributed by Mario Carneiro, 1-Jun-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))    &   ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦))    &   ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))))})    &   ⊢ (𝜑 → (𝑋𝐺𝑌) ∈ 𝑀)    &   ⊢ (𝜑 → 𝑇 ∈ (𝑆‘𝑀))    &   ⊢ (𝜑 → 𝑈 ∈ II)    &   ⊢ (𝜑 → 𝑉 ∈ II)    &   ⊢ (𝜑 → (II ↾t 𝑈) ∈ Conn)    &   ⊢ (𝜑 → (II ↾t 𝑉) ∈ Conn)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑈 × 𝑉) ⊆ (◡𝐺 “ 𝑀))    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → (𝐾 ↾ (𝑈 × {𝑍})) ∈ (((II ×t II) ↾t (𝑈 × {𝑍})) Cn 𝐶))    &   ⊢ 𝑊 = (℩𝑏 ∈ 𝑇 (𝑋𝐾𝑌) ∈ 𝑏)    ⇒   ⊢ (𝜑 → (𝐾 ↾ (𝑈 × 𝑉)) ∈ (((II ×t II) ↾t (𝑈 × 𝑉)) Cn 𝐶))
Theorem	cvmlift2lem10 35816*	Lemma for cvmlift2 35820. (Contributed by Mario Carneiro, 1-Jun-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))    &   ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦))    &   ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))))})    &   ⊢ (𝜑 → 𝑋 ∈ (0[,]1))    &   ⊢ (𝜑 → 𝑌 ∈ (0[,]1))    ⇒   ⊢ (𝜑 → ∃𝑢 ∈ II ∃𝑣 ∈ II (𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (∃𝑤 ∈ 𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶))))
Theorem	cvmlift2lem11 35817*	Lemma for cvmlift2 35820. (Contributed by Mario Carneiro, 1-Jun-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))    &   ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦))    &   ⊢ 𝑀 = {𝑧 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×t II) CnP 𝐶)‘𝑧)}    &   ⊢ (𝜑 → 𝑈 ∈ II)    &   ⊢ (𝜑 → 𝑉 ∈ II)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → (∃𝑤 ∈ 𝑉 (𝐾 ↾ (𝑈 × {𝑤})) ∈ (((II ×t II) ↾t (𝑈 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑈 × 𝑉)) ∈ (((II ×t II) ↾t (𝑈 × 𝑉)) Cn 𝐶)))    ⇒   ⊢ (𝜑 → ((𝑈 × {𝑌}) ⊆ 𝑀 → (𝑈 × {𝑍}) ⊆ 𝑀))
Theorem	cvmlift2lem12 35818*	Lemma for cvmlift2 35820. (Contributed by Mario Carneiro, 1-Jun-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))    &   ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦))    &   ⊢ 𝑀 = {𝑧 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×t II) CnP 𝐶)‘𝑧)}    &   ⊢ 𝐴 = {𝑎 ∈ (0[,]1) ∣ ((0[,]1) × {𝑎}) ⊆ 𝑀}    &   ⊢ 𝑆 = {⟨𝑟, 𝑡⟩ ∣ (𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀))}    ⇒   ⊢ (𝜑 → 𝐾 ∈ ((II ×t II) Cn 𝐶))
Theorem	cvmlift2lem13 35819*	Lemma for cvmlift2 35820. (Contributed by Mario Carneiro, 7-May-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))    &   ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦))    ⇒   ⊢ (𝜑 → ∃!𝑔 ∈ ((II ×t II) Cn 𝐶)((𝐹 ∘ 𝑔) = 𝐺 ∧ (0𝑔0) = 𝑃))
Theorem	cvmlift2 35820*	A two-dimensional version of cvmlift 35803. There is a unique lift of functions on the unit square II ×t II which commutes with the covering map. (Contributed by Mario Carneiro, 1-Jun-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0))    ⇒   ⊢ (𝜑 → ∃!𝑓 ∈ ((II ×t II) Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (0𝑓0) = 𝑃))
Theorem	cvmliftphtlem 35821*	Lemma for cvmliftpht 35822. (Contributed by Mario Carneiro, 6-Jul-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑀 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))    &   ⊢ 𝑁 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐻 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ (𝐺(PHtpy‘𝐽)𝐻))    &   ⊢ (𝜑 → 𝐴 ∈ ((II ×t II) Cn 𝐶))    &   ⊢ (𝜑 → (𝐹 ∘ 𝐴) = 𝐾)    &   ⊢ (𝜑 → (0𝐴0) = 𝑃)    ⇒   ⊢ (𝜑 → 𝐴 ∈ (𝑀(PHtpy‘𝐶)𝑁))
Theorem	cvmliftpht 35822*	If 𝐺 and 𝐻 are path-homotopic, then their lifts 𝑀 and 𝑁 are also path-homotopic. (Contributed by Mario Carneiro, 6-Jul-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑀 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))    &   ⊢ 𝑁 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))    &   ⊢ (𝜑 → 𝐺( ≃ph‘𝐽)𝐻)    ⇒   ⊢ (𝜑 → 𝑀( ≃ph‘𝐶)𝑁)
Theorem	cvmlift3lem1 35823*	Lemma for cvmlift3 35832. (Contributed by Mario Carneiro, 6-Jul-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ SConn)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))    &   ⊢ (𝜑 → 𝑀 ∈ (II Cn 𝐾))    &   ⊢ (𝜑 → (𝑀‘0) = 𝑂)    &   ⊢ (𝜑 → 𝑁 ∈ (II Cn 𝐾))    &   ⊢ (𝜑 → (𝑁‘0) = 𝑂)    &   ⊢ (𝜑 → (𝑀‘1) = (𝑁‘1))    ⇒   ⊢ (𝜑 → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑀) ∧ (𝑔‘0) = 𝑃))‘1) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑁) ∧ (𝑔‘0) = 𝑃))‘1))
Theorem	cvmlift3lem2 35824*	Lemma for cvmlift2 35820. (Contributed by Mario Carneiro, 6-Jul-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ SConn)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑌) → ∃!𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧))
Theorem	cvmlift3lem3 35825*	Lemma for cvmlift2 35820. (Contributed by Mario Carneiro, 6-Jul-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ SConn)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))    &   ⊢ 𝐻 = (𝑥 ∈ 𝑌 ↦ (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))    ⇒   ⊢ (𝜑 → 𝐻:𝑌⟶𝐵)
Theorem	cvmlift3lem4 35826*	Lemma for cvmlift2 35820. (Contributed by Mario Carneiro, 6-Jul-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ SConn)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))    &   ⊢ 𝐻 = (𝑥 ∈ 𝑌 ↦ (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑌) → ((𝐻‘𝑋) = 𝐴 ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝐴)))
Theorem	cvmlift3lem5 35827*	Lemma for cvmlift2 35820. (Contributed by Mario Carneiro, 6-Jul-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ SConn)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))    &   ⊢ 𝐻 = (𝑥 ∈ 𝑌 ↦ (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))    ⇒   ⊢ (𝜑 → (𝐹 ∘ 𝐻) = 𝐺)
Theorem	cvmlift3lem6 35828*	Lemma for cvmlift3 35832. (Contributed by Mario Carneiro, 9-Jul-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ SConn)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))    &   ⊢ 𝐻 = (𝑥 ∈ 𝑌 ↦ (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))    &   ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))))})    &   ⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐴)    &   ⊢ (𝜑 → 𝑇 ∈ (𝑆‘𝐴))    &   ⊢ (𝜑 → 𝑀 ⊆ (◡𝐺 “ 𝐴))    &   ⊢ 𝑊 = (℩𝑏 ∈ 𝑇 (𝐻‘𝑋) ∈ 𝑏)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑀)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑀)    &   ⊢ (𝜑 → 𝑄 ∈ (II Cn 𝐾))    &   ⊢ 𝑅 = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑄) ∧ (𝑔‘0) = 𝑃))    &   ⊢ (𝜑 → ((𝑄‘0) = 𝑂 ∧ (𝑄‘1) = 𝑋 ∧ (𝑅‘1) = (𝐻‘𝑋)))    &   ⊢ (𝜑 → 𝑁 ∈ (II Cn (𝐾 ↾t 𝑀)))    &   ⊢ (𝜑 → ((𝑁‘0) = 𝑋 ∧ (𝑁‘1) = 𝑍))    &   ⊢ 𝐼 = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑁) ∧ (𝑔‘0) = (𝐻‘𝑋)))    ⇒   ⊢ (𝜑 → (𝐻‘𝑍) ∈ 𝑊)
Theorem	cvmlift3lem7 35829*	Lemma for cvmlift3 35832. (Contributed by Mario Carneiro, 9-Jul-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ SConn)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))    &   ⊢ 𝐻 = (𝑥 ∈ 𝑌 ↦ (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))    &   ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))))})    &   ⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐴)    &   ⊢ (𝜑 → 𝑇 ∈ (𝑆‘𝐴))    &   ⊢ (𝜑 → 𝑀 ⊆ (◡𝐺 “ 𝐴))    &   ⊢ 𝑊 = (℩𝑏 ∈ 𝑇 (𝐻‘𝑋) ∈ 𝑏)    &   ⊢ (𝜑 → (𝐾 ↾t 𝑀) ∈ PConn)    &   ⊢ (𝜑 → 𝑉 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑉 ⊆ 𝑀)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑋))
Theorem	cvmlift3lem8 35830*	Lemma for cvmlift2 35820. (Contributed by Mario Carneiro, 6-Jul-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ SConn)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))    &   ⊢ 𝐻 = (𝑥 ∈ 𝑌 ↦ (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))    &   ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))))})    ⇒   ⊢ (𝜑 → 𝐻 ∈ (𝐾 Cn 𝐶))
Theorem	cvmlift3lem9 35831*	Lemma for cvmlift2 35820. (Contributed by Mario Carneiro, 7-May-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ SConn)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))    &   ⊢ 𝐻 = (𝑥 ∈ 𝑌 ↦ (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))    &   ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))))})    ⇒   ⊢ (𝜑 → ∃𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃))
Theorem	cvmlift3 35832*	A general version of cvmlift 35803. If 𝐾 is simply connected and weakly locally path-connected, then there is a unique lift of functions on 𝐾 which commutes with the covering map. (Contributed by Mario Carneiro, 9-Jul-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ SConn)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))    ⇒   ⊢ (𝜑 → ∃!𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃))
21.6.9  Normal numbers
Theorem	snmlff 35833*	The function 𝐹 from snmlval 35835 is a mapping from positive integers to real numbers in the range [0, 1]. (Contributed by Mario Carneiro, 6-Apr-2015.)
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) / 𝑛))    ⇒   ⊢ 𝐹:ℕ⟶(0[,]1)
Theorem	snmlfval 35834*	The function 𝐹 from snmlval 35835 maps 𝑁 to the relative density of 𝐵 in the first 𝑁 digits of the digit string of 𝐴 in base 𝑅. (Contributed by Mario Carneiro, 6-Apr-2015.)
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) / 𝑛))    ⇒   ⊢ (𝑁 ∈ ℕ → (𝐹‘𝑁) = ((♯‘{𝑘 ∈ (1...𝑁) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) / 𝑁))
Theorem	snmlval 35835*	The property "𝐴 is simply normal in base 𝑅". A number is simply normal if each digit 0 ≤ 𝑏 < 𝑅 occurs in the base- 𝑅 digit string of 𝐴 with frequency 1 / 𝑅 (which is consistent with the expectation in an infinite random string of numbers selected from 0...𝑅 − 1). (Contributed by Mario Carneiro, 6-Apr-2015.)
⊢ 𝑆 = (𝑟 ∈ (ℤ≥‘2) ↦ {𝑥 ∈ ℝ ∣ ∀𝑏 ∈ (0...(𝑟 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟↑𝑘)) mod 𝑟)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑟)})    ⇒   ⊢ (𝐴 ∈ (𝑆‘𝑅) ↔ (𝑅 ∈ (ℤ≥‘2) ∧ 𝐴 ∈ ℝ ∧ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅)))
Theorem	snmlflim 35836*	If 𝐴 is simply normal, then the function 𝐹 of relative density of 𝐵 in the digit string converges to 1 / 𝑅, i.e. the set of occurrences of 𝐵 in the digit string has natural density 1 / 𝑅. (Contributed by Mario Carneiro, 6-Apr-2015.)
⊢ 𝑆 = (𝑟 ∈ (ℤ≥‘2) ↦ {𝑥 ∈ ℝ ∣ ∀𝑏 ∈ (0...(𝑟 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟↑𝑘)) mod 𝑟)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑟)})    &   ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) / 𝑛))    ⇒   ⊢ ((𝐴 ∈ (𝑆‘𝑅) ∧ 𝐵 ∈ (0...(𝑅 − 1))) → 𝐹 ⇝ (1 / 𝑅))
21.6.10  Godel-sets of formulas - part 1
Syntax	cgoe 35837	The Godel-set of membership.
class ∈𝑔
Syntax	cgna 35838	The Godel-set for the Sheffer stroke.
class ⊼𝑔
Syntax	cgol 35839	The Godel-set of universal quantification. (Note that this is not a wff.)
class ∀𝑔𝑁𝑈
Syntax	csat 35840	The satisfaction function.
class Sat
Syntax	cfmla 35841	The formula set predicate.
class Fmla
Syntax	csate 35842	The ∈-satisfaction function.
class Sat∈
Syntax	cprv 35843	The "proves" relation.
class ⊧
Definition	df-goel 35844	Define the Godel-set of membership. Here the arguments 𝑥 = ⟨𝑁, 𝑃⟩ correspond to vN and vP , so (∅∈𝑔1o) actually means v0 ∈ v1 , not 0 ∈ 1. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ ∈𝑔 = (𝑥 ∈ (ω × ω) ↦ ⟨∅, 𝑥⟩)
Definition	df-gona 35845	Define the Godel-set for the Sheffer stroke NAND. Here the arguments 𝑥 = ⟨𝑈, 𝑉⟩ are also Godel-sets corresponding to smaller formulas. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ ⊼𝑔 = (𝑥 ∈ (V × V) ↦ ⟨1o, 𝑥⟩)
Definition	df-goal 35846	Define the Godel-set of universal quantification. Here 𝑁 ∈ ω corresponds to vN , and 𝑈 represents another formula, and this expression is [∀𝑥𝜑] = ∀𝑔𝑁𝑈 where 𝑥 is the 𝑁-th variable, 𝑈 = [𝜑] is the code for 𝜑. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ ∀𝑔𝑁𝑈 = ⟨2o, ⟨𝑁, 𝑈⟩⟩
Definition	df-sat 35847*	Define the satisfaction predicate. This recursive construction builds up a function over wff codes (see satff 35914) and simultaneously defines the set of assignments to all variables from 𝑀 that makes the coded wff true in the model 𝑀, where ∈ is interpreted as the binary relation 𝐸 on 𝑀. The interpretation of the statement 𝑆 ∈ (((𝑀 Sat 𝐸)‘𝑛)‘𝑈) is that for the model ⟨𝑀, 𝐸⟩, 𝑆:ω⟶𝑀 is a valuation of the variables (v0 = (𝑆‘∅), v1 = (𝑆‘1o), etc.) and 𝑈 is a code for a wff using ∈ , ⊼ , ∀ that is true under the assignment 𝑆. The function is defined by finite recursion; ((𝑀 Sat 𝐸)‘𝑛) only operates on wffs of depth at most 𝑛 ∈ ω, and ((𝑀 Sat 𝐸)‘ω) = ∪ 𝑛 ∈ ω((𝑀 Sat 𝐸)‘𝑛) operates on all wffs. The coding scheme for the wffs is defined so that vi ∈ vj is coded as ⟨∅, ⟨𝑖, 𝑗⟩⟩, (𝜑 ⊼ 𝜓) is coded as ⟨1o, ⟨𝜑, 𝜓⟩⟩, and ∀ vi 𝜑 is coded as ⟨2o, ⟨𝑖, 𝜑⟩⟩. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ Sat = (𝑚 ∈ V, 𝑒 ∈ V ↦ (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑚 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣ ∀𝑧 ∈ 𝑚 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣ (𝑎‘𝑖)𝑒(𝑎‘𝑗)})}) ↾ suc ω))
Definition	df-sate 35848*	A simplified version of the satisfaction predicate, using the standard membership relation and eliminating the extra variable 𝑛. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ Sat∈ = (𝑚 ∈ V, 𝑢 ∈ V ↦ (((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω)‘𝑢))
Definition	df-fmla 35849	Define the predicate which defines the set of valid Godel formulas. The parameter 𝑛 defines the maximum height of the formulas: the set (Fmla‘∅) is all formulas of the form 𝑥 ∈ 𝑦 (which in our coding scheme is the set ({∅} × (ω × ω)); see df-sat 35847 for the full coding scheme), see fmla0 35886, and each extra level adds to the complexity of the formulas in (Fmla‘𝑛), see fmlasuc 35890. Remark: it is sufficient to have atomic formulas of the form 𝑥 ∈ 𝑦 only, because equations (formulas of the form 𝑥 = 𝑦), which are required as (atomic) formulas, can be introduced as a defined notion in terms of ∈𝑔, see df-goeq 35948. (Fmla‘ω) = ∪ 𝑛 ∈ ω(Fmla‘𝑛) is the set of all valid formulas, see fmla 35885. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ Fmla = (𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛))
Definition	df-prv 35850*	Define the "proves" relation on a set. A wff is true in a model 𝑀 if for every valuation 𝑠 ∈ (𝑀 ↑m ω), the interpretation of the wff using the membership relation on 𝑀 is true. Since ⊧ is defined in terms of the interpretations making the given formula true, it is not defined on the empty "model" 𝑀 = ∅, since there are no interpretations. In particular, the empty set on the LHS of ⊧ should not be interpreted as the empty model. Statement prv0 35934 shows that our definition yields ∅⊧𝑈 for all formulas, though of course the formula ∃𝑥𝑥 = 𝑥 is not satisfied on the empty model. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊢ ⊧ = {⟨𝑚, 𝑢⟩ ∣ (𝑚 Sat∈ 𝑢) = (𝑚 ↑m ω)}
Theorem	goel 35851	A "Godel-set of membership". The variables are identified by their indices (which are natural numbers), and the membership vi ∈ vj is coded as ⟨∅, ⟨𝑖, 𝑗⟩⟩. (Contributed by AV, 15-Sep-2023.)
⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
Theorem	goelel3xp 35852	A "Godel-set of membership" is a member of a doubled Cartesian product. (Contributed by AV, 16-Sep-2023.)
⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) ∈ (ω × (ω × ω)))
Theorem	goeleq12bg 35853	Two "Godel-set of membership" codes for two variables are equal iff the two corresponding variables are equal. (Contributed by AV, 8-Oct-2023.)
⊢ (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → ((𝐼∈𝑔𝐽) = (𝑀∈𝑔𝑁) ↔ (𝐼 = 𝑀 ∧ 𝐽 = 𝑁)))
Theorem	gonafv 35854	The "Godel-set for the Sheffer stroke NAND" for two formulas 𝐴 and 𝐵. (Contributed by AV, 16-Oct-2023.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴⊼𝑔𝐵) = ⟨1o, ⟨𝐴, 𝐵⟩⟩)
Theorem	goaleq12d 35855	Equality of the "Godel-set of universal quantification". (Contributed by AV, 18-Sep-2023.)
⊢ (𝜑 → 𝑀 = 𝑁)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ∀𝑔𝑀𝐴 = ∀𝑔𝑁𝐵)
Theorem	gonanegoal 35856	The Godel-set for the Sheffer stroke NAND is not equal to the Godel-set of universal quantification. (Contributed by AV, 21-Oct-2023.)
⊢ (𝑎⊼𝑔𝑏) ≠ ∀𝑔𝑖𝑢
Theorem	satf 35857*	The satisfaction predicate as function over wff codes in the model 𝑀 and the binary relation 𝐸 on 𝑀. (Contributed by AV, 14-Sep-2023.)
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑀 Sat 𝐸) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}) ↾ suc ω))
Theorem	satfsucom 35858*	The satisfaction predicate for wff codes in the model 𝑀 and the binary relation 𝐸 on 𝑀 at an element of the successor of ω. (Contributed by AV, 22-Sep-2023.)
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ suc ω) → ((𝑀 Sat 𝐸)‘𝑁) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})})‘𝑁))
Theorem	satfn 35859	The satisfaction predicate for wff codes in the model 𝑀 and the binary relation 𝐸 on 𝑀 is a function over suc ω. (Contributed by AV, 6-Oct-2023.)
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑀 Sat 𝐸) Fn suc ω)
Theorem	satom 35860*	The satisfaction predicate for wff codes in the model 𝑀 and the binary relation 𝐸 on 𝑀 at omega (ω). (Contributed by AV, 6-Oct-2023.)
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ((𝑀 Sat 𝐸)‘ω) = ∪ 𝑛 ∈ ω ((𝑀 Sat 𝐸)‘𝑛))
Theorem	satfvsucom 35861*	The satisfaction predicate as function over wff codes at a successor of ω. (Contributed by AV, 22-Sep-2023.)
⊢ 𝑆 = (𝑀 Sat 𝐸)    ⇒   ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ suc ω) → (𝑆‘𝑁) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})})‘𝑁))
Theorem	satfv0 35862*	The value of the satisfaction predicate as function over wff codes at ∅. (Contributed by AV, 8-Oct-2023.)
⊢ 𝑆 = (𝑀 Sat 𝐸)    ⇒   ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘∅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})})
Theorem	satfvsuclem1 35863*	Lemma 1 for satfvsuc 35865. (Contributed by AV, 8-Oct-2023.)
⊢ 𝑆 = (𝑀 Sat 𝐸)    ⇒   ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)})) ∧ 𝑦 ∈ 𝒫 (𝑀 ↑m ω))} ∈ V)
Theorem	satfvsuclem2 35864*	Lemma 2 for satfvsuc 35865. (Contributed by AV, 8-Oct-2023.)
⊢ 𝑆 = (𝑀 Sat 𝐸)    ⇒   ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))} ∈ V)
Theorem	satfvsuc 35865*	The value of the satisfaction predicate as function over wff codes at a successor. (Contributed by AV, 10-Oct-2023.)
⊢ 𝑆 = (𝑀 Sat 𝐸)    ⇒   ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → (𝑆‘suc 𝑁) = ((𝑆‘𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))}))
Theorem	satfv1lem 35866*	Lemma for satfv1 35867. (Contributed by AV, 9-Nov-2023.)
⊢ ((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) → {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ {𝑏 ∈ (𝑀 ↑m ω) ∣ (𝑏‘𝐼)𝐸(𝑏‘𝐽)}} = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)), if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽)))})
Theorem	satfv1 35867*	The value of the satisfaction predicate as function over wff codes of height 1. (Contributed by AV, 9-Nov-2023.)
⊢ 𝑆 = (𝑀 Sat 𝐸)    ⇒   ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘1o) = ((𝑆‘∅) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ∨ ∃𝑛 ∈ ω (𝑥 = ∀𝑔𝑛(𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗)))}))}))
Theorem	satfsschain 35868	The binary relation of a satisfaction predicate as function over wff codes is an increasing chain (with respect to inclusion). (Contributed by AV, 15-Oct-2023.)
⊢ 𝑆 = (𝑀 Sat 𝐸)    ⇒   ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐵 ⊆ 𝐴 → (𝑆‘𝐵) ⊆ (𝑆‘𝐴)))
Theorem	satfvsucsuc 35869*	The satisfaction predicate as function over wff codes of height (𝑁 + 1), expressed by the minimally necessary satisfaction predicates as function over wff codes of height 𝑁. (Contributed by AV, 21-Oct-2023.)
⊢ 𝑆 = (𝑀 Sat 𝐸)    &   ⊢ 𝐴 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))    &   ⊢ 𝐵 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}    ⇒   ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → (𝑆‘suc suc 𝑁) = ((𝑆‘suc 𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = 𝐴))}))
Theorem	satfbrsuc 35870*	The binary relation of a satisfaction predicate as function over wff codes at a successor. (Contributed by AV, 13-Oct-2023.)
⊢ 𝑆 = (𝑀 Sat 𝐸)    &   ⊢ 𝑃 = (𝑆‘𝑁)    ⇒   ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑁 ∈ ω ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐴(𝑆‘suc 𝑁)𝐵 ↔ (𝐴𝑃𝐵 ∨ ∃𝑢 ∈ 𝑃 (∃𝑣 ∈ 𝑃 (𝐴 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝐵 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝐴 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝐵 = {𝑓 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)})))))
Theorem	satfrel 35871	The value of the satisfaction predicate as function over wff codes at a natural number is a relation. (Contributed by AV, 12-Oct-2023.)
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → Rel ((𝑀 Sat 𝐸)‘𝑁))
Theorem	satfdmlem 35872*	Lemma for satfdm 35873. (Contributed by AV, 12-Oct-2023.)
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)) → ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑌)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st ‘𝑎)⊼𝑔(1st ‘𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑎))))
Theorem	satfdm 35873*	The domain of the satisfaction predicate as function over wff codes does not depend on the model 𝑀 and the binary relation 𝐸 on 𝑀. (Contributed by AV, 13-Oct-2023.)
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌)) → ∀𝑛 ∈ ω dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((𝑁 Sat 𝐹)‘𝑛))
Theorem	satfrnmapom 35874	The range of the satisfaction predicate as function over wff codes in any model 𝑀 and any binary relation 𝐸 on 𝑀 for a natural number 𝑁 is a subset of the power set of all mappings from the natural numbers into the model 𝑀. (Contributed by AV, 13-Oct-2023.)
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → ran ((𝑀 Sat 𝐸)‘𝑁) ⊆ 𝒫 (𝑀 ↑m ω))
Theorem	satfv0fun 35875	The value of the satisfaction predicate as function over wff codes at ∅ is a function. (Contributed by AV, 15-Oct-2023.)
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘∅))
Theorem	satf0 35876*	The satisfaction predicate as function over wff codes in the empty model with an empty binary relation. (Contributed by AV, 14-Sep-2023.)
⊢ (∅ Sat ∅) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))}) ↾ suc ω)
Theorem	satf0sucom 35877*	The satisfaction predicate as function over wff codes in the empty model with an empty binary relation at a successor of ω. (Contributed by AV, 14-Sep-2023.)
⊢ (𝑁 ∈ suc ω → ((∅ Sat ∅)‘𝑁) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))})‘𝑁))
Theorem	satf00 35878*	The value of the satisfaction predicate as function over wff codes in the empty model with an empty binary relation at ∅. (Contributed by AV, 14-Sep-2023.)
⊢ ((∅ Sat ∅)‘∅) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))}
Theorem	satf0suclem 35879*	Lemma for satf0suc 35880, sat1el2xp 35883 and fmlasuc0 35888. (Contributed by AV, 19-Sep-2023.)
⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑋 (∃𝑣 ∈ 𝑌 𝑥 = 𝐵 ∨ ∃𝑤 ∈ 𝑍 𝑥 = 𝐶))} ∈ V)
Theorem	satf0suc 35880*	The value of the satisfaction predicate as function over wff codes in the empty model and the empty binary relation at a successor. (Contributed by AV, 19-Sep-2023.)
⊢ 𝑆 = (∅ Sat ∅)    ⇒   ⊢ (𝑁 ∈ ω → (𝑆‘suc 𝑁) = ((𝑆‘𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))}))
Theorem	satf0op 35881*	An element of a value of the satisfaction predicate as function over wff codes in the empty model and the empty binary relation expressed as ordered pair. (Contributed by AV, 19-Sep-2023.)
⊢ 𝑆 = (∅ Sat ∅)    ⇒   ⊢ (𝑁 ∈ ω → (𝑋 ∈ (𝑆‘𝑁) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆‘𝑁))))
Theorem	satf0n0 35882	The value of the satisfaction predicate as function over wff codes in the empty model and the empty binary relation does not contain the empty set. (Contributed by AV, 19-Sep-2023.)
⊢ (𝑁 ∈ ω → ∅ ∉ ((∅ Sat ∅)‘𝑁))
Theorem	sat1el2xp 35883*	The first component of an element of the value of the satisfaction predicate as function over wff codes in the empty model with an empty binary relation is a member of a doubled Cartesian product. (Contributed by AV, 17-Sep-2023.)
⊢ (𝑁 ∈ ω → ∀𝑤 ∈ ((∅ Sat ∅)‘𝑁)∃𝑎∃𝑏(1st ‘𝑤) ∈ (ω × (𝑎 × 𝑏)))
Theorem	fmlafv 35884	The valid Godel formulas of height 𝑁 is the domain of the value of the satisfaction predicate as function over wff codes in the empty model with an empty binary relation at 𝑁. (Contributed by AV, 15-Sep-2023.)
⊢ (𝑁 ∈ suc ω → (Fmla‘𝑁) = dom ((∅ Sat ∅)‘𝑁))
Theorem	fmla 35885	The set of all valid Godel formulas. (Contributed by AV, 20-Sep-2023.)
⊢ (Fmla‘ω) = ∪ 𝑛 ∈ ω (Fmla‘𝑛)
Theorem	fmla0 35886*	The valid Godel formulas of height 0 is the set of all formulas of the form vi ∈ vj ("Godel-set of membership") coded as ⟨∅, ⟨𝑖, 𝑗⟩⟩. (Contributed by AV, 14-Sep-2023.)
⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)}
Theorem	fmla0xp 35887	The valid Godel formulas of height 0 is the set of all formulas of the form vi ∈ vj ("Godel-set of membership") coded as ⟨∅, ⟨𝑖, 𝑗⟩⟩. (Contributed by AV, 15-Sep-2023.)
⊢ (Fmla‘∅) = ({∅} × (ω × ω))
Theorem	fmlasuc0 35888*	The valid Godel formulas of height (𝑁 + 1). (Contributed by AV, 18-Sep-2023.)
⊢ (𝑁 ∈ ω → (Fmla‘suc 𝑁) = ((Fmla‘𝑁) ∪ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑁)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢))}))
Theorem	fmlafvel 35889	A class is a valid Godel formula of height 𝑁 iff it is the first component of a member of the value of the satisfaction predicate as function over wff codes in the empty model with an empty binary relation at 𝑁. (Contributed by AV, 19-Sep-2023.)
⊢ (𝑁 ∈ ω → (𝐹 ∈ (Fmla‘𝑁) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
Theorem	fmlasuc 35890*	The valid Godel formulas of height (𝑁 + 1), expressed by the valid Godel formulas of height 𝑁. (Contributed by AV, 20-Sep-2023.)
⊢ (𝑁 ∈ ω → (Fmla‘suc 𝑁) = ((Fmla‘𝑁) ∪ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}))
Theorem	fmla1 35891*	The valid Godel formulas of height 1 is the set of all formulas of the form (𝑎⊼𝑔𝑏) and ∀𝑔𝑘𝑎 with atoms 𝑎, 𝑏 of the form 𝑥 ∈ 𝑦. (Contributed by AV, 20-Sep-2023.)
⊢ (Fmla‘1o) = (({∅} × (ω × ω)) ∪ {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗))})
Theorem	isfmlasuc 35892*	The characterization of a Godel formula of height at least 1. (Contributed by AV, 14-Oct-2023.)
⊢ ((𝑁 ∈ ω ∧ 𝐹 ∈ 𝑉) → (𝐹 ∈ (Fmla‘suc 𝑁) ↔ (𝐹 ∈ (Fmla‘𝑁) ∨ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝐹 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖𝑢))))
Theorem	fmlasssuc 35893	The Godel formulas of height 𝑁 are a subset of the Godel formulas of height 𝑁 + 1. (Contributed by AV, 20-Oct-2023.)
⊢ (𝑁 ∈ ω → (Fmla‘𝑁) ⊆ (Fmla‘suc 𝑁))
Theorem	fmlaomn0 35894	The empty set is not a Godel formula of any height. (Contributed by AV, 21-Oct-2023.)
⊢ (𝑁 ∈ ω → ∅ ∉ (Fmla‘𝑁))
Theorem	fmlan0 35895	The empty set is not a Godel formula. (Contributed by AV, 19-Nov-2023.)
⊢ ∅ ∉ (Fmla‘ω)
Theorem	gonan0 35896	The "Godel-set of NAND" is a Godel formula of at least height 1. (Contributed by AV, 21-Oct-2023.)
⊢ ((𝐴⊼𝑔𝐵) ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅)
Theorem	goaln0 35897*	The "Godel-set of universal quantification" is a Godel formula of at least height 1. (Contributed by AV, 22-Oct-2023.)
⊢ (∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅)
Theorem	gonarlem 35898*	Lemma for gonar 35899 (induction step). (Contributed by AV, 21-Oct-2023.)
⊢ (𝑁 ∈ ω → (((𝑎⊼𝑔𝑏) ∈ (Fmla‘suc 𝑁) → (𝑎 ∈ (Fmla‘suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc 𝑁))) → ((𝑎⊼𝑔𝑏) ∈ (Fmla‘suc suc 𝑁) → (𝑎 ∈ (Fmla‘suc suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc suc 𝑁)))))
Theorem	gonar 35899*	If the "Godel-set of NAND" applied to classes is a Godel formula, the classes are also Godel formulas. Remark: The reverse is not valid for 𝐴 or 𝐵 being of the same height as the "Godel-set of NAND". (Contributed by AV, 21-Oct-2023.)
⊢ ((𝑁 ∈ ω ∧ (𝑎⊼𝑔𝑏) ∈ (Fmla‘𝑁)) → (𝑎 ∈ (Fmla‘𝑁) ∧ 𝑏 ∈ (Fmla‘𝑁)))
Theorem	goalrlem 35900*	Lemma for goalr 35901 (induction step). (Contributed by AV, 22-Oct-2023.)
⊢ (𝑁 ∈ ω → ((∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑁) → 𝑎 ∈ (Fmla‘suc 𝑁)) → (∀𝑔𝑖𝑎 ∈ (Fmla‘suc suc 𝑁) → 𝑎 ∈ (Fmla‘suc suc 𝑁))))