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Theorem List for Metamath Proof Explorer - 30401-30500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcvmlift3lem2 30401* Lemma for cvmlift2 30397. (Contributed by Mario Carneiro, 6-Jul-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ SCon)    &   (𝜑𝐾 ∈ 𝑛-Locally PCon)    &   (𝜑𝑂𝑌)    &   (𝜑𝐺 ∈ (𝐾 Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺𝑂))       ((𝜑𝑋𝑌) → ∃!𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧))

Theoremcvmlift3lem3 30402* Lemma for cvmlift2 30397. (Contributed by Mario Carneiro, 6-Jul-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ SCon)    &   (𝜑𝐾 ∈ 𝑛-Locally PCon)    &   (𝜑𝑂𝑌)    &   (𝜑𝐺 ∈ (𝐾 Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺𝑂))    &   𝐻 = (𝑥𝑌 ↦ (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))       (𝜑𝐻:𝑌𝐵)

Theoremcvmlift3lem4 30403* Lemma for cvmlift2 30397. (Contributed by Mario Carneiro, 6-Jul-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ SCon)    &   (𝜑𝐾 ∈ 𝑛-Locally PCon)    &   (𝜑𝑂𝑌)    &   (𝜑𝐺 ∈ (𝐾 Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺𝑂))    &   𝐻 = (𝑥𝑌 ↦ (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))       ((𝜑𝑋𝑌) → ((𝐻𝑋) = 𝐴 ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝐴)))

Theoremcvmlift3lem5 30404* Lemma for cvmlift2 30397. (Contributed by Mario Carneiro, 6-Jul-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ SCon)    &   (𝜑𝐾 ∈ 𝑛-Locally PCon)    &   (𝜑𝑂𝑌)    &   (𝜑𝐺 ∈ (𝐾 Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺𝑂))    &   𝐻 = (𝑥𝑌 ↦ (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))       (𝜑 → (𝐹𝐻) = 𝐺)

Theoremcvmlift3lem6 30405* Lemma for cvmlift3 30409. (Contributed by Mario Carneiro, 9-Jul-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ SCon)    &   (𝜑𝐾 ∈ 𝑛-Locally PCon)    &   (𝜑𝑂𝑌)    &   (𝜑𝐺 ∈ (𝐾 Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺𝑂))    &   𝐻 = (𝑥𝑌 ↦ (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))    &   𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑐𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))))})    &   (𝜑 → (𝐺𝑋) ∈ 𝐴)    &   (𝜑𝑇 ∈ (𝑆𝐴))    &   (𝜑𝑀 ⊆ (𝐺𝐴))    &   𝑊 = (𝑏𝑇 (𝐻𝑋) ∈ 𝑏)    &   (𝜑𝑋𝑀)    &   (𝜑𝑍𝑀)    &   (𝜑𝑄 ∈ (II Cn 𝐾))    &   𝑅 = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑄) ∧ (𝑔‘0) = 𝑃))    &   (𝜑 → ((𝑄‘0) = 𝑂 ∧ (𝑄‘1) = 𝑋 ∧ (𝑅‘1) = (𝐻𝑋)))    &   (𝜑𝑁 ∈ (II Cn (𝐾t 𝑀)))    &   (𝜑 → ((𝑁‘0) = 𝑋 ∧ (𝑁‘1) = 𝑍))    &   𝐼 = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑁) ∧ (𝑔‘0) = (𝐻𝑋)))       (𝜑 → (𝐻𝑍) ∈ 𝑊)

Theoremcvmlift3lem7 30406* Lemma for cvmlift3 30409. (Contributed by Mario Carneiro, 9-Jul-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ SCon)    &   (𝜑𝐾 ∈ 𝑛-Locally PCon)    &   (𝜑𝑂𝑌)    &   (𝜑𝐺 ∈ (𝐾 Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺𝑂))    &   𝐻 = (𝑥𝑌 ↦ (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))    &   𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑐𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))))})    &   (𝜑 → (𝐺𝑋) ∈ 𝐴)    &   (𝜑𝑇 ∈ (𝑆𝐴))    &   (𝜑𝑀 ⊆ (𝐺𝐴))    &   𝑊 = (𝑏𝑇 (𝐻𝑋) ∈ 𝑏)    &   (𝜑 → (𝐾t 𝑀) ∈ PCon)    &   (𝜑𝑉𝐾)    &   (𝜑𝑉𝑀)    &   (𝜑𝑋𝑉)       (𝜑𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑋))

Theoremcvmlift3lem8 30407* Lemma for cvmlift2 30397. (Contributed by Mario Carneiro, 6-Jul-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ SCon)    &   (𝜑𝐾 ∈ 𝑛-Locally PCon)    &   (𝜑𝑂𝑌)    &   (𝜑𝐺 ∈ (𝐾 Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺𝑂))    &   𝐻 = (𝑥𝑌 ↦ (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))    &   𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑐𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))))})       (𝜑𝐻 ∈ (𝐾 Cn 𝐶))

Theoremcvmlift3lem9 30408* Lemma for cvmlift2 30397. (Contributed by Mario Carneiro, 7-May-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ SCon)    &   (𝜑𝐾 ∈ 𝑛-Locally PCon)    &   (𝜑𝑂𝑌)    &   (𝜑𝐺 ∈ (𝐾 Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺𝑂))    &   𝐻 = (𝑥𝑌 ↦ (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))    &   𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑐𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))))})       (𝜑 → ∃𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃))

Theoremcvmlift3 30409* A general version of cvmlift 30380. 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    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ SCon)    &   (𝜑𝐾 ∈ 𝑛-Locally PCon)    &   (𝜑𝑂𝑌)    &   (𝜑𝐺 ∈ (𝐾 Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺𝑂))       (𝜑 → ∃!𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃))

20.5.9  Normal numbers

Theoremsnmlff 30410* The function 𝐹 from snmlval 30412 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 · (𝑅𝑘)) mod 𝑅)) = 𝐵}) / 𝑛))       𝐹:ℕ⟶(0[,]1)

Theoremsnmlfval 30411* The function 𝐹 from snmlval 30412 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 · (𝑅𝑘)) mod 𝑅)) = 𝐵}) / 𝑛))       (𝑁 ∈ ℕ → (𝐹𝑁) = ((#‘{𝑘 ∈ (1...𝑁) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝐵}) / 𝑁))

Theoremsnmlval 30412* 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 · (𝑟𝑘)) mod 𝑟)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑟)})       (𝐴 ∈ (𝑆𝑅) ↔ (𝑅 ∈ (ℤ‘2) ∧ 𝐴 ∈ ℝ ∧ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅)))

Theoremsnmlflim 30413* 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 · (𝑟𝑘)) mod 𝑟)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑟)})    &   𝐹 = (𝑛 ∈ ℕ ↦ ((#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝐵}) / 𝑛))       ((𝐴 ∈ (𝑆𝑅) ∧ 𝐵 ∈ (0...(𝑅 − 1))) → 𝐹 ⇝ (1 / 𝑅))

20.5.10  Godel-sets of formulas

Syntaxcgoe 30414 The Godel-set of membership.
class 𝑔

Syntaxcgna 30415 The Godel-set for the Sheffer stroke.
class 𝑔

Syntaxcgol 30416 The Godel-set of universal quantification. (Note that this is not a wff.)
class 𝑔𝑁𝑈

Syntaxcsat 30417 The satisfaction function.
class Sat

Syntaxcfmla 30418 The formula set predicate.
class Fmla

Syntaxcsate 30419 The -satisfaction function.
class Sat

Syntaxcprv 30420 The "proves" relation.
class

Definitiondf-goel 30421 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⟩)

Definitiondf-gona 30422 Define the Godel-set for the Sheffer stroke NAND. Here the arguments 𝑥 = ⟨𝑈, 𝑉 are also Godel-sets corresponding to smaller formulae. (Contributed by Mario Carneiro, 14-Jul-2013.)
𝑔 = (𝑥 ∈ (V × V) ↦ ⟨1𝑜, 𝑥⟩)

Definitiondf-goal 30423 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⟩⟩

Definitiondf-sat 30424* Define the satisfaction predicate. This recursive construction builds up a function over wff codes 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⟩⟩,
• (𝜑𝜓) 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⟩ ∣ ∃𝑢𝑓 (∃𝑣𝑓 (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑚𝑚 ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚𝑚 ω) ∣ ∀𝑧𝑚 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑚𝑚 ω) ∣ (𝑎𝑖)𝑒(𝑎𝑗)})}) ↾ suc ω))

Definitiondf-sate 30425* 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))

Definitiondf-fmla 30426 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 or 𝑥𝑦 (which in our coding scheme is the set ({∅, 1𝑜} × (ω × ω)); see df-sat 30424 for the full coding scheme), and each extra level adds to the complexity of the formulas in (Fmla‘𝑛). (Fmla‘ω) = 𝑛 ∈ ω(Fmla‘𝑛) is the set of all valid formulas. (Contributed by Mario Carneiro, 14-Jul-2013.)
Fmla = (𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛))

Syntaxcgon 30427 The Godel-set of negation. (Note that this is not a wff.)
class ¬𝑔𝑈

Syntaxcgoa 30428 The Godel-set of conjunction.
class 𝑔

Syntaxcgoi 30429 The Godel-set of implication.
class 𝑔

Syntaxcgoo 30430 The Godel-set of disjunction.
class 𝑔

Syntaxcgob 30431 The Godel-set of equivalence.
class 𝑔

Syntaxcgoq 30432 The Godel-set of equality.
class =𝑔

Syntaxcgox 30433 The Godel-set of existential quantification. (Note that this is not a wff.)
class 𝑔𝑁𝑈

Definitiondf-gonot 30434 Define the Godel-set of negation. Here the argument 𝑈 is also a Godel-set corresponding to smaller formulae. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)
¬𝑔𝑈 = (𝑈𝑔𝑈)

Definitiondf-goan 30435* Define the Godel-set of conjunction. Here the arguments 𝑈 and 𝑉 are also Godel-sets corresponding to smaller formulae. (Contributed by Mario Carneiro, 14-Jul-2013.)
𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ ¬𝑔(𝑢𝑔𝑣))

Definitiondf-goim 30436* Define the Godel-set of implication. Here the arguments 𝑈 and 𝑉 are also Godel-sets corresponding to smaller formulae. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)
𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑢𝑔¬𝑔𝑣))

Definitiondf-goor 30437* Define the Godel-set of disjunction. Here the arguments 𝑈 and 𝑉 are also Godel-sets corresponding to smaller formulae. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)
𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ (¬𝑔𝑢𝑔 𝑣))

Definitiondf-gobi 30438* Define the Godel-set of equivalence. Here the arguments 𝑈 and 𝑉 are also Godel-sets corresponding to smaller formulae. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)
𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ ((𝑢𝑔 𝑣)∧𝑔(𝑣𝑔 𝑢)))

Definitiondf-goeq 30439* Define the Godel-set of equality. Here the arguments 𝑥 = ⟨𝑁, 𝑃 correspond to vN and vP , so (∅=𝑔1𝑜) actually means v0 = v1 , not 0 = 1. Here we use the trick mentioned in ax-ext 2494 to introduce equality as a defined notion in terms of 𝑔. The expression suc (𝑢𝑣) = max (𝑢, 𝑣) + 1 here is a convenient way of getting a dummy variable distinct from 𝑢 and 𝑣. (Contributed by Mario Carneiro, 14-Jul-2013.)
=𝑔 = (𝑢 ∈ ω, 𝑣 ∈ ω ↦ suc (𝑢𝑣) / 𝑤𝑔𝑤((𝑤𝑔𝑢) ↔𝑔 (𝑤𝑔𝑣)))

Definitiondf-goex 30440 Define the Godel-set of existential 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 = ¬𝑔𝑔𝑁¬𝑔𝑈

Definitiondf-prv 30441* Define the "proves" relation on a set. A wff is true in a model 𝑀 if for every valuation 𝑠 ∈ (𝑀𝑚 ω), the interpretation of the wff using the membership relation on 𝑀 is true. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊧ = {⟨𝑚, 𝑢⟩ ∣ (𝑚 Sat 𝑢) = (𝑚𝑚 ω)}

20.5.11  Models of ZF

Syntaxcgze 30442 The Axiom of Extensionality.
class AxExt

Syntaxcgzr 30443 The Axiom Scheme of Replacement.
class AxRep

Syntaxcgzp 30444 The Axiom of Power Sets.
class AxPow

Syntaxcgzu 30445 The Axiom of Unions.
class AxUn

Syntaxcgzg 30446 The Axiom of Regularity.
class AxReg

Syntaxcgzi 30447 The Axiom of Infinity.
class AxInf

Syntaxcgzf 30448 The set of models of ZF.
class ZF

Definitiondf-gzext 30449 The Godel-set version of the Axiom of Extensionality. (Contributed by Mario Carneiro, 14-Jul-2013.)
AxExt = (∀𝑔2𝑜((2𝑜𝑔∅) ↔𝑔 (2𝑜𝑔1𝑜)) →𝑔 (∅=𝑔1𝑜))

Definitiondf-gzrep 30450 The Godel-set version of the Axiom Scheme of Replacement. Since this is a scheme and not a single axiom, it manifests as a function on wffs, each giving rise to a different axiom. (Contributed by Mario Carneiro, 14-Jul-2013.)
AxRep = (𝑢 ∈ (Fmla‘ω) ↦ (∀𝑔3𝑜𝑔1𝑜𝑔2𝑜(∀𝑔1𝑜𝑢𝑔 (2𝑜=𝑔1𝑜)) →𝑔𝑔1𝑜𝑔2𝑜((2𝑜𝑔1𝑜) ↔𝑔𝑔3𝑜((3𝑜𝑔∅)∧𝑔𝑔1𝑜𝑢))))

Definitiondf-gzpow 30451 The Godel-set version of the Axiom of Power Sets. (Contributed by Mario Carneiro, 14-Jul-2013.)
AxPow = ∃𝑔1𝑜𝑔2𝑜(∀𝑔1𝑜((1𝑜𝑔2𝑜) ↔𝑔 (1𝑜𝑔∅)) →𝑔 (2𝑜𝑔1𝑜))

Definitiondf-gzun 30452 The Godel-set version of the Axiom of Unions. (Contributed by Mario Carneiro, 14-Jul-2013.)
AxUn = ∃𝑔1𝑜𝑔2𝑜(∃𝑔1𝑜((2𝑜𝑔1𝑜)∧𝑔(1𝑜𝑔∅)) →𝑔 (2𝑜𝑔1𝑜))

Definitiondf-gzreg 30453 The Godel-set version of the Axiom of Regularity. (Contributed by Mario Carneiro, 14-Jul-2013.)
AxReg = (∃𝑔1𝑜(1𝑜𝑔∅) →𝑔𝑔1𝑜((1𝑜𝑔∅)∧𝑔𝑔2𝑜((2𝑜𝑔1𝑜) →𝑔 ¬𝑔(2𝑜𝑔∅))))

Definitiondf-gzinf 30454 The Godel-set version of the Axiom of Infinity. (Contributed by Mario Carneiro, 14-Jul-2013.)
AxInf = ∃𝑔1𝑜((∅∈𝑔1𝑜)∧𝑔𝑔2𝑜((2𝑜𝑔1𝑜) →𝑔𝑔∅((2𝑜𝑔∅)∧𝑔(∅∈𝑔1𝑜))))

Definitiondf-gzf 30455* Define the class of all (transitive) models of ZF. (Contributed by Mario Carneiro, 14-Jul-2013.)
ZF = {𝑚 ∣ ((Tr 𝑚𝑚⊧AxExt ∧ 𝑚⊧AxPow) ∧ (𝑚⊧AxUn ∧ 𝑚⊧AxReg ∧ 𝑚⊧AxInf) ∧ ∀𝑢 ∈ (Fmla‘ω)𝑚⊧(AxRep‘𝑢))}

20.5.12  Metamath formal systems

This is a formalization of Appendix C of the Metamath book, which describes the mathematical representation of a formal system, of which set.mm (this file) is one.

Syntaxcmcn 30456 The set of constants.
class mCN

Syntaxcmvar 30457 The set of variables.
class mVR

Syntaxcmty 30458 The type function.
class mType

Syntaxcmvt 30459 The set of variable typecodes.
class mVT

Syntaxcmtc 30460 The set of typecodes.
class mTC

Syntaxcmax 30461 The set of axioms.
class mAx

Syntaxcmrex 30462 The set of raw expressions.
class mREx

Syntaxcmex 30463 The set of expressions.
class mEx

Syntaxcmdv 30464 The set of distinct variables.
class mDV

Syntaxcmvrs 30465 The variables in an expression.
class mVars

Syntaxcmrsub 30466 The set of raw substitutions.
class mRSubst

Syntaxcmsub 30467 The set of substitutions.
class mSubst

Syntaxcmvh 30468 The set of variable hypotheses.
class mVH

Syntaxcmpst 30469 The set of pre-statements.
class mPreSt

Syntaxcmsr 30470 The reduct of a pre-statement.
class mStRed

Syntaxcmsta 30471 The set of statements.
class mStat

Syntaxcmfs 30472 The set of formal systems.
class mFS

Syntaxcmcls 30473 The closure of a set of statements.
class mCls

Syntaxcmpps 30474 The set of provable pre-statements.
class mPPSt

Syntaxcmthm 30475 The set of theorems.
class mThm

Definitiondf-mcn 30476 Define the set of constants in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.)
mCN = Slot 1

Definitiondf-mvar 30477 Define the set of variables in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.)
mVR = Slot 2

Definitiondf-mty 30478 Define the type function in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.)
mType = Slot 3

Definitiondf-mtc 30479 Define the set of typecodes in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.)
mTC = Slot 4

Definitiondf-mmax 30480 Define the set of axioms in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.)
mAx = Slot 5

Definitiondf-mvt 30481 Define the set of variable typecodes in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.)
mVT = (𝑡 ∈ V ↦ ran (mType‘𝑡))

Definitiondf-mrex 30482 Define the set of "raw expressions", which are expressions without a typecode attached. (Contributed by Mario Carneiro, 14-Jul-2016.)
mREx = (𝑡 ∈ V ↦ Word ((mCN‘𝑡) ∪ (mVR‘𝑡)))

Definitiondf-mex 30483 Define the set of expressions, which are strings of constants and variables headed by a typecode constant. (Contributed by Mario Carneiro, 14-Jul-2016.)
mEx = (𝑡 ∈ V ↦ ((mTC‘𝑡) × (mREx‘𝑡)))

Definitiondf-mdv 30484 Define the set of distinct variable conditions, which are pairs of distinct variables. (Contributed by Mario Carneiro, 14-Jul-2016.)
mDV = (𝑡 ∈ V ↦ (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I ))

Definitiondf-mvrs 30485* Define the set of variables in an expression. (Contributed by Mario Carneiro, 14-Jul-2016.)
mVars = (𝑡 ∈ V ↦ (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd𝑒) ∩ (mVR‘𝑡))))

Definitiondf-mrsub 30486* Define a substitution of raw expressions given a mapping from variables to expressions. (Contributed by Mario Carneiro, 14-Jul-2016.)
mRSubst = (𝑡 ∈ V ↦ (𝑓 ∈ ((mREx‘𝑡) ↑pm (mVR‘𝑡)) ↦ (𝑒 ∈ (mREx‘𝑡) ↦ ((freeMnd‘((mCN‘𝑡) ∪ (mVR‘𝑡))) Σg ((𝑣 ∈ ((mCN‘𝑡) ∪ (mVR‘𝑡)) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))))

Definitiondf-msub 30487* Define a substitution of expressions given a mapping from variables to expressions. (Contributed by Mario Carneiro, 14-Jul-2016.)
mSubst = (𝑡 ∈ V ↦ (𝑓 ∈ ((mREx‘𝑡) ↑pm (mVR‘𝑡)) ↦ (𝑒 ∈ (mEx‘𝑡) ↦ ⟨(1st𝑒), (((mRSubst‘𝑡)‘𝑓)‘(2nd𝑒))⟩)))

Definitiondf-mvh 30488* Define the mapping from variables to their variable hypothesis. (Contributed by Mario Carneiro, 14-Jul-2016.)
mVH = (𝑡 ∈ V ↦ (𝑣 ∈ (mVR‘𝑡) ↦ ⟨((mType‘𝑡)‘𝑣), ⟨“𝑣”⟩⟩))

Definitiondf-mpst 30489* Define the set of all pre-statements. (Contributed by Mario Carneiro, 14-Jul-2016.)
mPreSt = (𝑡 ∈ V ↦ (({𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ 𝑑 = 𝑑} × (𝒫 (mEx‘𝑡) ∩ Fin)) × (mEx‘𝑡)))

Definitiondf-msr 30490* Define the reduct of a pre-statement. (Contributed by Mario Carneiro, 14-Jul-2016.)
mStRed = (𝑡 ∈ V ↦ (𝑠 ∈ (mPreSt‘𝑡) ↦ (2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ ((mVars‘𝑡) “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩))

Definitiondf-msta 30491 Define the set of all statements. (Contributed by Mario Carneiro, 14-Jul-2016.)
mStat = (𝑡 ∈ V ↦ ran (mStRed‘𝑡))

Definitiondf-mfs 30492* Define the set of all formal systems. (Contributed by Mario Carneiro, 14-Jul-2016.)
mFS = {𝑡 ∣ ((((mCN‘𝑡) ∩ (mVR‘𝑡)) = ∅ ∧ (mType‘𝑡):(mVR‘𝑡)⟶(mTC‘𝑡)) ∧ ((mAx‘𝑡) ⊆ (mStat‘𝑡) ∧ ∀𝑣 ∈ (mVT‘𝑡) ¬ ((mType‘𝑡) “ {𝑣}) ∈ Fin))}

Definitiondf-mcls 30493* Define the closure of a set of statements relative to a set of disjointness constraints. (Contributed by Mario Carneiro, 14-Jul-2016.)
mCls = (𝑡 ∈ V ↦ (𝑑 ∈ 𝒫 (mDV‘𝑡), ∈ 𝒫 (mEx‘𝑡) ↦ {𝑐 ∣ (( ∪ ran (mVH‘𝑡)) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐)))}))

Definitiondf-mpps 30494* Define the set of provable pre-statements. (Contributed by Mario Carneiro, 14-Jul-2016.)
mPPSt = (𝑡 ∈ V ↦ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ (mPreSt‘𝑡) ∧ 𝑎 ∈ (𝑑(mCls‘𝑡)))})

Definitiondf-mthm 30495 Define the set of theorems. (Contributed by Mario Carneiro, 14-Jul-2016.)
mThm = (𝑡 ∈ V ↦ ((mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))))

Theoremmvtval 30496 The set of variable typecodes. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑉 = (mVT‘𝑇)    &   𝑌 = (mType‘𝑇)       𝑉 = ran 𝑌

Theoremmrexval 30497 The set of "raw expressions", which are expressions without a typecode, that is, just sequences of constants and variables. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐶 = (mCN‘𝑇)    &   𝑉 = (mVR‘𝑇)    &   𝑅 = (mREx‘𝑇)       (𝑇𝑊𝑅 = Word (𝐶𝑉))

Theoremmexval 30498 The set of expressions, which are pairs whose first element is a typecode, and whose second element is a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐾 = (mTC‘𝑇)    &   𝐸 = (mEx‘𝑇)    &   𝑅 = (mREx‘𝑇)       𝐸 = (𝐾 × 𝑅)

Theoremmexval2 30499 The set of expressions, which are pairs whose first element is a typecode, and whose second element is a list of constants and variables. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐾 = (mTC‘𝑇)    &   𝐸 = (mEx‘𝑇)    &   𝐶 = (mCN‘𝑇)    &   𝑉 = (mVR‘𝑇)       𝐸 = (𝐾 × Word (𝐶𝑉))

Theoremmdvval 30500 The set of disjoint variable conditions, which are pairs of distinct variables. (This definition differs from appendix C, which uses unordered pairs instead. We use ordered pairs, but all sets of dv conditions of interest will be symmetric, so it does not matter.) (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑉 = (mVR‘𝑇)    &   𝐷 = (mDV‘𝑇)       𝐷 = ((𝑉 × 𝑉) ∖ I )

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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42426
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