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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | infeuti 6501* | An infimum is unique. (Contributed by Jim Kingdon, 19-Dec-2021.) |
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) ⇒ ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) | ||
Theorem | infsnti 6502* | The infimum of a singleton. (Contributed by Jim Kingdon, 19-Dec-2021.) |
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) ⇒ ⊢ (𝜑 → inf({𝐵}, 𝐴, 𝑅) = 𝐵) | ||
Theorem | inf00 6503 | The infimum regarding an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.) |
⊢ inf(𝐵, ∅, 𝑅) = ∅ | ||
Theorem | infisoti 6504* | Image of an infimum under an isomorphism. (Contributed by Jim Kingdon, 19-Dec-2021.) |
⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵)) & ⊢ (𝜑 → 𝐶 ⊆ 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐶 𝑧𝑅𝑦))) & ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))) ⇒ ⊢ (𝜑 → inf((𝐹 “ 𝐶), 𝐵, 𝑆) = (𝐹‘inf(𝐶, 𝐴, 𝑅))) | ||
Theorem | ordiso2 6505 | Generalize ordiso 6506 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.) |
⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → 𝐴 = 𝐵) | ||
Theorem | ordiso 6506* | Order-isomorphic ordinal numbers are equal. (Contributed by Jeff Hankins, 16-Oct-2009.) (Proof shortened by Mario Carneiro, 24-Jun-2015.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 = 𝐵 ↔ ∃𝑓 𝑓 Isom E , E (𝐴, 𝐵))) | ||
Syntax | ccrd 6507 | Extend class definition to include the cardinal size function. |
class card | ||
Definition | df-card 6508* | Define the cardinal number function. The cardinal number of a set is the least ordinal number equinumerous to it. In other words, it is the "size" of the set. Definition of [Enderton] p. 197. Our notation is from Enderton. Other textbooks often use a double bar over the set to express this function. (Contributed by NM, 21-Oct-2003.) |
⊢ card = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥}) | ||
Theorem | cardcl 6509* | The cardinality of a well-orderable set is an ordinal. (Contributed by Jim Kingdon, 30-Aug-2021.) |
⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → (card‘𝐴) ∈ On) | ||
Theorem | isnumi 6510 | A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → 𝐵 ∈ dom card) | ||
Theorem | finnum 6511 | Every finite set is numerable. (Contributed by Mario Carneiro, 4-Feb-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) |
⊢ (𝐴 ∈ Fin → 𝐴 ∈ dom card) | ||
Theorem | onenon 6512 | Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
⊢ (𝐴 ∈ On → 𝐴 ∈ dom card) | ||
Theorem | cardval3ex 6513* | The value of (card‘𝐴). (Contributed by Jim Kingdon, 30-Aug-2021.) |
⊢ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴}) | ||
Theorem | oncardval 6514* | The value of the cardinal number function with an ordinal number as its argument. (Contributed by NM, 24-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.) |
⊢ (𝐴 ∈ On → (card‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴}) | ||
Theorem | cardonle 6515 | The cardinal of an ordinal number is less than or equal to the ordinal number. Proposition 10.6(3) of [TakeutiZaring] p. 85. (Contributed by NM, 22-Oct-2003.) |
⊢ (𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴) | ||
Theorem | card0 6516 | The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.) |
⊢ (card‘∅) = ∅ | ||
Theorem | carden2bex 6517* | If two numerable sets are equinumerous, then they have equal cardinalities. (Contributed by Jim Kingdon, 30-Aug-2021.) |
⊢ ((𝐴 ≈ 𝐵 ∧ ∃𝑥 ∈ On 𝑥 ≈ 𝐴) → (card‘𝐴) = (card‘𝐵)) | ||
Theorem | pm54.43 6518 | Theorem *54.43 of [WhiteheadRussell] p. 360. (Contributed by NM, 4-Apr-2007.) |
⊢ ((𝐴 ≈ 1_{𝑜} ∧ 𝐵 ≈ 1_{𝑜}) → ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐴 ∪ 𝐵) ≈ 2_{𝑜})) | ||
Theorem | pr2nelem 6519 | Lemma for pr2ne 6520. (Contributed by FL, 17-Aug-2008.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≈ 2_{𝑜}) | ||
Theorem | pr2ne 6520 | If an unordered pair has two elements they are different. (Contributed by FL, 14-Feb-2010.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ({𝐴, 𝐵} ≈ 2_{𝑜} ↔ 𝐴 ≠ 𝐵)) | ||
Theorem | en2eleq 6521 | Express a set of pair cardinality as the unordered pair of a given element and the other element. (Contributed by Stefan O'Rear, 22-Aug-2015.) |
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2_{𝑜}) → 𝑃 = {𝑋, ∪ (𝑃 ∖ {𝑋})}) | ||
Theorem | en2other2 6522 | Taking the other element twice in a pair gets back to the original element. (Contributed by Stefan O'Rear, 22-Aug-2015.) |
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2_{𝑜}) → ∪ (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) = 𝑋) | ||
Theorem | infpwfidom 6523 | The collection of finite subsets of a set dominates the set. (We use the weaker sethood assumption (𝒫 𝐴 ∩ Fin) ∈ V because this theorem also implies that 𝐴 is a set if 𝒫 𝐴 ∩ Fin is.) (Contributed by Mario Carneiro, 17-May-2015.) |
⊢ ((𝒫 𝐴 ∩ Fin) ∈ V → 𝐴 ≼ (𝒫 𝐴 ∩ Fin)) | ||
This section derives the basics of real and complex numbers. To construct the real numbers constructively, we follow two main sources. The first is Metamath Proof Explorer, which has the advantage of being already formalized in metamath. Its disadvantage, for our purposes, is that it assumes the law of the excluded middle throughout. Since we have already developed natural numbers ( for example, nna0 6118 and similar theorems ), going from there to positive integers (df-ni 6556) and then positive rational numbers (df-nqqs 6600) does not involve a major change in approach compared with the Metamath Proof Explorer. It is when we proceed to Dedekind cuts that we bring in more material from Section 11.2 of [HoTT], which focuses on the aspects of Dedekind cuts which are different without excluded middle. With excluded middle, it is natural to define the cut as the lower set only (as Metamath Proof Explorer does), but we define the cut as a pair of both the lower and upper sets, as [HoTT] does. There are also differences in how we handle order and replacing "not equal to zero" with "apart from zero". | ||
Syntax | cnpi 6524 |
The set of positive integers, which is the set of natural numbers ω
with 0 removed.
Note: This is the start of the Dedekind-cut construction of real and _complex numbers. |
class N | ||
Syntax | cpli 6525 | Positive integer addition. |
class +_{N} | ||
Syntax | cmi 6526 | Positive integer multiplication. |
class ·_{N} | ||
Syntax | clti 6527 | Positive integer ordering relation. |
class <_{N} | ||
Syntax | cplpq 6528 | Positive pre-fraction addition. |
class +_{pQ} | ||
Syntax | cmpq 6529 | Positive pre-fraction multiplication. |
class ·_{pQ} | ||
Syntax | cltpq 6530 | Positive pre-fraction ordering relation. |
class <_{pQ} | ||
Syntax | ceq 6531 | Equivalence class used to construct positive fractions. |
class ~_{Q} | ||
Syntax | cnq 6532 | Set of positive fractions. |
class Q | ||
Syntax | c1q 6533 | The positive fraction constant 1. |
class 1_{Q} | ||
Syntax | cplq 6534 | Positive fraction addition. |
class +_{Q} | ||
Syntax | cmq 6535 | Positive fraction multiplication. |
class ·_{Q} | ||
Syntax | crq 6536 | Positive fraction reciprocal operation. |
class *_{Q} | ||
Syntax | cltq 6537 | Positive fraction ordering relation. |
class <_{Q} | ||
Syntax | ceq0 6538 | Equivalence class used to construct non-negative fractions. |
class ~_{Q0} | ||
Syntax | cnq0 6539 | Set of non-negative fractions. |
class Q_{0} | ||
Syntax | c0q0 6540 | The non-negative fraction constant 0. |
class 0_{Q0} | ||
Syntax | cplq0 6541 | Non-negative fraction addition. |
class +_{Q0} | ||
Syntax | cmq0 6542 | Non-negative fraction multiplication. |
class ·_{Q0} | ||
Syntax | cnp 6543 | Set of positive reals. |
class P | ||
Syntax | c1p 6544 | Positive real constant 1. |
class 1_{P} | ||
Syntax | cpp 6545 | Positive real addition. |
class +_{P} | ||
Syntax | cmp 6546 | Positive real multiplication. |
class ·_{P} | ||
Syntax | cltp 6547 | Positive real ordering relation. |
class <_{P} | ||
Syntax | cer 6548 | Equivalence class used to construct signed reals. |
class ~_{R} | ||
Syntax | cnr 6549 | Set of signed reals. |
class R | ||
Syntax | c0r 6550 | The signed real constant 0. |
class 0_{R} | ||
Syntax | c1r 6551 | The signed real constant 1. |
class 1_{R} | ||
Syntax | cm1r 6552 | The signed real constant -1. |
class -1_{R} | ||
Syntax | cplr 6553 | Signed real addition. |
class +_{R} | ||
Syntax | cmr 6554 | Signed real multiplication. |
class ·_{R} | ||
Syntax | cltr 6555 | Signed real ordering relation. |
class <_{R} | ||
Definition | df-ni 6556 | Define the class of positive integers. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 15-Aug-1995.) |
⊢ N = (ω ∖ {∅}) | ||
Definition | df-pli 6557 | Define addition on positive integers. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.) |
⊢ +_{N} = ( +_{𝑜} ↾ (N × N)) | ||
Definition | df-mi 6558 | Define multiplication on positive integers. This is a "temporary" set used in the construction of complex numbers and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.) |
⊢ ·_{N} = ( ·_{𝑜} ↾ (N × N)) | ||
Definition | df-lti 6559 | Define 'less than' on positive integers. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 6-Feb-1996.) |
⊢ <_{N} = ( E ∩ (N × N)) | ||
Theorem | elni 6560 | Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.) |
⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅)) | ||
Theorem | pinn 6561 | A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.) |
⊢ (𝐴 ∈ N → 𝐴 ∈ ω) | ||
Theorem | pion 6562 | A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.) |
⊢ (𝐴 ∈ N → 𝐴 ∈ On) | ||
Theorem | piord 6563 | A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.) |
⊢ (𝐴 ∈ N → Ord 𝐴) | ||
Theorem | niex 6564 | The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.) |
⊢ N ∈ V | ||
Theorem | 0npi 6565 | The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.) |
⊢ ¬ ∅ ∈ N | ||
Theorem | elni2 6566 | Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.) |
⊢ (𝐴 ∈ N ↔ (𝐴 ∈ ω ∧ ∅ ∈ 𝐴)) | ||
Theorem | 1pi 6567 | Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.) |
⊢ 1_{𝑜} ∈ N | ||
Theorem | addpiord 6568 | Positive integer addition in terms of ordinal addition. (Contributed by NM, 27-Aug-1995.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +_{N} 𝐵) = (𝐴 +_{𝑜} 𝐵)) | ||
Theorem | mulpiord 6569 | Positive integer multiplication in terms of ordinal multiplication. (Contributed by NM, 27-Aug-1995.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·_{N} 𝐵) = (𝐴 ·_{𝑜} 𝐵)) | ||
Theorem | mulidpi 6570 | 1 is an identity element for multiplication on positive integers. (Contributed by NM, 4-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐴 ∈ N → (𝐴 ·_{N} 1_{𝑜}) = 𝐴) | ||
Theorem | ltpiord 6571 | Positive integer 'less than' in terms of ordinal membership. (Contributed by NM, 6-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <_{N} 𝐵 ↔ 𝐴 ∈ 𝐵)) | ||
Theorem | ltsopi 6572 | Positive integer 'less than' is a strict ordering. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Mario Carneiro, 10-Jul-2014.) |
⊢ <_{N} Or N | ||
Theorem | pitric 6573 | Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <_{N} 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <_{N} 𝐴))) | ||
Theorem | pitri3or 6574 | Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <_{N} 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 <_{N} 𝐴)) | ||
Theorem | ltdcpi 6575 | Less-than for positive integers is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → DECID 𝐴 <_{N} 𝐵) | ||
Theorem | ltrelpi 6576 | Positive integer 'less than' is a relation on positive integers. (Contributed by NM, 8-Feb-1996.) |
⊢ <_{N} ⊆ (N × N) | ||
Theorem | dmaddpi 6577 | Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.) |
⊢ dom +_{N} = (N × N) | ||
Theorem | dmmulpi 6578 | Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.) |
⊢ dom ·_{N} = (N × N) | ||
Theorem | addclpi 6579 | Closure of addition of positive integers. (Contributed by NM, 18-Oct-1995.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +_{N} 𝐵) ∈ N) | ||
Theorem | mulclpi 6580 | Closure of multiplication of positive integers. (Contributed by NM, 18-Oct-1995.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·_{N} 𝐵) ∈ N) | ||
Theorem | addcompig 6581 | Addition of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 +_{N} 𝐵) = (𝐵 +_{N} 𝐴)) | ||
Theorem | addasspig 6582 | Addition of positive integers is associative. (Contributed by Jim Kingdon, 26-Aug-2019.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → ((𝐴 +_{N} 𝐵) +_{N} 𝐶) = (𝐴 +_{N} (𝐵 +_{N} 𝐶))) | ||
Theorem | mulcompig 6583 | Multiplication of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·_{N} 𝐵) = (𝐵 ·_{N} 𝐴)) | ||
Theorem | mulasspig 6584 | Multiplication of positive integers is associative. (Contributed by Jim Kingdon, 26-Aug-2019.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → ((𝐴 ·_{N} 𝐵) ·_{N} 𝐶) = (𝐴 ·_{N} (𝐵 ·_{N} 𝐶))) | ||
Theorem | distrpig 6585 | Multiplication of positive integers is distributive. (Contributed by Jim Kingdon, 26-Aug-2019.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐴 ·_{N} (𝐵 +_{N} 𝐶)) = ((𝐴 ·_{N} 𝐵) +_{N} (𝐴 ·_{N} 𝐶))) | ||
Theorem | addcanpig 6586 | Addition cancellation law for positive integers. (Contributed by Jim Kingdon, 27-Aug-2019.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → ((𝐴 +_{N} 𝐵) = (𝐴 +_{N} 𝐶) ↔ 𝐵 = 𝐶)) | ||
Theorem | mulcanpig 6587 | Multiplication cancellation law for positive integers. (Contributed by Jim Kingdon, 29-Aug-2019.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → ((𝐴 ·_{N} 𝐵) = (𝐴 ·_{N} 𝐶) ↔ 𝐵 = 𝐶)) | ||
Theorem | addnidpig 6588 | There is no identity element for addition on positive integers. (Contributed by NM, 28-Nov-1995.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ¬ (𝐴 +_{N} 𝐵) = 𝐴) | ||
Theorem | ltexpi 6589* | Ordering on positive integers in terms of existence of sum. (Contributed by NM, 15-Mar-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <_{N} 𝐵 ↔ ∃𝑥 ∈ N (𝐴 +_{N} 𝑥) = 𝐵)) | ||
Theorem | ltapig 6590 | Ordering property of addition for positive integers. (Contributed by Jim Kingdon, 31-Aug-2019.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐴 <_{N} 𝐵 ↔ (𝐶 +_{N} 𝐴) <_{N} (𝐶 +_{N} 𝐵))) | ||
Theorem | ltmpig 6591 | Ordering property of multiplication for positive integers. (Contributed by Jim Kingdon, 31-Aug-2019.) |
⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N ∧ 𝐶 ∈ N) → (𝐴 <_{N} 𝐵 ↔ (𝐶 ·_{N} 𝐴) <_{N} (𝐶 ·_{N} 𝐵))) | ||
Theorem | 1lt2pi 6592 | One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) |
⊢ 1_{𝑜} <_{N} (1_{𝑜} +_{N} 1_{𝑜}) | ||
Theorem | nlt1pig 6593 | No positive integer is less than one. (Contributed by Jim Kingdon, 31-Aug-2019.) |
⊢ (𝐴 ∈ N → ¬ 𝐴 <_{N} 1_{𝑜}) | ||
Theorem | indpi 6594* | Principle of Finite Induction on positive integers. (Contributed by NM, 23-Mar-1996.) |
⊢ (𝑥 = 1_{𝑜} → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = (𝑦 +_{N} 1_{𝑜}) → (𝜑 ↔ 𝜃)) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) & ⊢ 𝜓 & ⊢ (𝑦 ∈ N → (𝜒 → 𝜃)) ⇒ ⊢ (𝐴 ∈ N → 𝜏) | ||
Theorem | nnppipi 6595 | A natural number plus a positive integer is a positive integer. (Contributed by Jim Kingdon, 10-Nov-2019.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ N) → (𝐴 +_{𝑜} 𝐵) ∈ N) | ||
Definition | df-plpq 6596* | Define pre-addition on positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. This "pre-addition" operation works directly with ordered pairs of integers. The actual positive fraction addition +_{Q} (df-plqqs 6601) works with the equivalence classes of these ordered pairs determined by the equivalence relation ~_{Q} (df-enq 6599). (Analogous remarks apply to the other "pre-" operations in the complex number construction that follows.) From Proposition 9-2.3 of [Gleason] p. 117. (Contributed by NM, 28-Aug-1995.) |
⊢ +_{pQ} = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨(((1^{st} ‘𝑥) ·_{N} (2^{nd} ‘𝑦)) +_{N} ((1^{st} ‘𝑦) ·_{N} (2^{nd} ‘𝑥))), ((2^{nd} ‘𝑥) ·_{N} (2^{nd} ‘𝑦))⟩) | ||
Definition | df-mpq 6597* | Define pre-multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.4 of [Gleason] p. 119. (Contributed by NM, 28-Aug-1995.) |
⊢ ·_{pQ} = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1^{st} ‘𝑥) ·_{N} (1^{st} ‘𝑦)), ((2^{nd} ‘𝑥) ·_{N} (2^{nd} ‘𝑦))⟩) | ||
Definition | df-ltpq 6598* | Define pre-ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. Similar to Definition 5 of [Suppes] p. 162. (Contributed by NM, 28-Aug-1995.) |
⊢ <_{pQ} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1^{st} ‘𝑥) ·_{N} (2^{nd} ‘𝑦)) <_{N} ((1^{st} ‘𝑦) ·_{N} (2^{nd} ‘𝑥)))} | ||
Definition | df-enq 6599* | Define equivalence relation for positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.1 of [Gleason] p. 117. (Contributed by NM, 27-Aug-1995.) |
⊢ ~_{Q} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·_{N} 𝑢) = (𝑤 ·_{N} 𝑣)))} | ||
Definition | df-nqqs 6600 | Define class of positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.2 of [Gleason] p. 117. (Contributed by NM, 16-Aug-1995.) |
⊢ Q = ((N × N) / ~_{Q} ) |
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