P. 73 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7201-7300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	exmidfodomrlemr 7201*	The existence of a mapping from any set onto any inhabited set that it dominates implies excluded middle. Proposition 1.2 of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 1-Jul-2022.)
|- ( A. x A. y ( ( E. z z e. y /\ y ~<_ x ) -> E. f f : x -onto-> y ) -> EXMID )
Theorem	exmidfodomrlemrALT 7202*	The existence of a mapping from any set onto any inhabited set that it dominates implies excluded middle. Proposition 1.2 of [PradicBrown2022], p. 2. An alternative proof of exmidfodomrlemr 7201. In particular, this proof uses eldju 7067 instead of djur 7068 and avoids djulclb 7054. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Jim Kingdon, 9-Jul-2022.)
|- ( A. x A. y ( ( E. z z e. y /\ y ~<_ x ) -> E. f f : x -onto-> y ) -> EXMID )
Theorem	exmidfodomr 7203*	Excluded middle is equivalent to the existence of a mapping from any set onto any inhabited set that it dominates. (Contributed by Jim Kingdon, 1-Jul-2022.)
|- (EXMID <-> A. x A. y ( ( E. z z e. y /\ y ~<_ x ) -> E. f f : x -onto-> y ) )
2.6.42  Axiom of Choice equivalents
Syntax	wac 7204	Formula for an abbreviation of the axiom of choice.
wff CHOICE
Definition	df-ac 7205*	The expression CHOICE will be used as a readable shorthand for any form of the axiom of choice; all concrete forms are long, cryptic, have dummy variables, or all three, making it useful to have a short name. Similar to the Axiom of Choice (first form) of [Enderton] p. 49. There are some decisions about how to write this definition especially around whether ax-setind 4537 is needed to show equivalence to other ways of stating choice, and about whether choice functions are available for nonempty sets or inhabited sets. (Contributed by Mario Carneiro, 22-Feb-2015.)
|- (CHOICE <-> A. x E. f ( f C_ x /\ f Fn dom x ) )
Theorem	acfun 7206*	A convenient form of choice. The goal here is to state choice as the existence of a choice function on a set of inhabited sets, while making full use of our notation around functions and function values. (Contributed by Jim Kingdon, 20-Nov-2023.)
|- ( ph -> CHOICE )   &    |- ( ph -> A e. V )   &    |- ( ph -> A. x e. A E. w w e. x )   =>    |- ( ph -> E. f ( f Fn A /\ A. x e. A ( f ` x ) e. x ) )
Theorem	exmidaclem 7207*	Lemma for exmidac 7208. The result, with a few hypotheses to break out commonly used expressions. (Contributed by Jim Kingdon, 21-Nov-2023.)
|- A = { x e. { (/) , { (/) } } | ( x = (/) \/ y = { (/) } ) }   &    |- B = { x e. { (/) , { (/) } } | ( x = { (/) } \/ y = { (/) } ) }   &    |- C = { A , B }   =>    |- (CHOICE -> EXMID )
Theorem	exmidac 7208	The axiom of choice implies excluded middle. See acexmid 5874 for more discussion of this theorem and a way of stating it without using CHOICE or EXMID. (Contributed by Jim Kingdon, 21-Nov-2023.)
|- (CHOICE -> EXMID )
2.6.43  Cardinal number arithmetic
Theorem	endjudisj 7209	Equinumerosity of a disjoint union and a union of two disjoint sets. (Contributed by Jim Kingdon, 30-Jul-2023.)
|- ( ( A e. V /\ B e. W /\ ( A i^i B ) = (/) ) -> ( A ⊔ B ) ~~ ( A u. B ) )
Theorem	djuen 7210	Disjoint unions of equinumerous sets are equinumerous. (Contributed by Jim Kingdon, 30-Jul-2023.)
|- ( ( A ~~ B /\ C ~~ D ) -> ( A ⊔ C ) ~~ ( B ⊔ D ) )
Theorem	djuenun 7211	Disjoint union is equinumerous to union for disjoint sets. (Contributed by Mario Carneiro, 29-Apr-2015.) (Revised by Jim Kingdon, 19-Aug-2023.)
|- ( ( A ~~ B /\ C ~~ D /\ ( B i^i D ) = (/) ) -> ( A ⊔ C ) ~~ ( B u. D ) )
Theorem	dju1en 7212	Cardinal addition with cardinal one (which is the same as ordinal one). Used in proof of Theorem 6J of [Enderton] p. 143. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
|- ( ( A e. V /\ -. A e. A ) -> ( A ⊔ 1o ) ~~ suc A )
Theorem	dju0en 7213	Cardinal addition with cardinal zero (the empty set). Part (a1) of proof of Theorem 6J of [Enderton] p. 143. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
|- ( A e. V -> ( A ⊔ (/) ) ~~ A )
Theorem	xp2dju 7214	Two times a cardinal number. Exercise 4.56(g) of [Mendelson] p. 258. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
|- ( 2o X. A ) = ( A ⊔ A )
Theorem	djucomen 7215	Commutative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 24-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
|- ( ( A e. V /\ B e. W ) -> ( A ⊔ B ) ~~ ( B ⊔ A ) )
Theorem	djuassen 7216	Associative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( A ⊔ B ) ⊔ C ) ~~ ( A ⊔ ( B ⊔ C ) ) )
Theorem	xpdjuen 7217	Cardinal multiplication distributes over cardinal addition. Theorem 6I(3) of [Enderton] p. 142. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( A X. ( B ⊔ C ) ) ~~ ( ( A X. B ) ⊔ ( A X. C ) ) )
Theorem	djudoml 7218	A set is dominated by its disjoint union with another. (Contributed by Jim Kingdon, 11-Jul-2023.)
|- ( ( A e. V /\ B e. W ) -> A ~<_ ( A ⊔ B ) )
Theorem	djudomr 7219	A set is dominated by its disjoint union with another. (Contributed by Jim Kingdon, 11-Jul-2023.)
|- ( ( A e. V /\ B e. W ) -> B ~<_ ( A ⊔ B ) )
2.6.44  Ordinal trichotomy
Theorem	exmidontriimlem1 7220	Lemma for exmidontriim 7224. A variation of r19.30dc 2624. (Contributed by Jim Kingdon, 12-Aug-2024.)
|- ( ( A. x e. A ( ph \/ ps \/ ch ) /\ EXMID ) -> ( E. x e. A ph \/ E. x e. A ps \/ A. x e. A ch ) )
Theorem	exmidontriimlem2 7221*	Lemma for exmidontriim 7224. (Contributed by Jim Kingdon, 12-Aug-2024.)
|- ( ph -> B e. On )   &    |- ( ph -> EXMID )   &    |- ( ph -> A. y e. B ( A e. y \/ A = y \/ y e. A ) )   =>    |- ( ph -> ( A e. B \/ A. y e. B y e. A ) )
Theorem	exmidontriimlem3 7222*	Lemma for exmidontriim 7224. What we get to do based on induction on both A and B. (Contributed by Jim Kingdon, 10-Aug-2024.)
|- ( ph -> A e. On )   &    |- ( ph -> B e. On )   &    |- ( ph -> EXMID )   &    |- ( ph -> A. z e. A A. y e. On ( z e. y \/ z = y \/ y e. z ) )   &    |- ( ph -> A. y e. B ( A e. y \/ A = y \/ y e. A ) )   =>    |- ( ph -> ( A e. B \/ A = B \/ B e. A ) )
Theorem	exmidontriimlem4 7223*	Lemma for exmidontriim 7224. The induction step for the induction on A. (Contributed by Jim Kingdon, 10-Aug-2024.)
|- ( ph -> A e. On )   &    |- ( ph -> B e. On )   &    |- ( ph -> EXMID )   &    |- ( ph -> A. z e. A A. y e. On ( z e. y \/ z = y \/ y e. z ) )   =>    |- ( ph -> ( A e. B \/ A = B \/ B e. A ) )
Theorem	exmidontriim 7224*	Excluded middle implies ordinal trichotomy. Lemma 10.4.1 of [HoTT], p. (varies). The proof follows the proof from the HoTT book fairly closely. (Contributed by Jim Kingdon, 10-Aug-2024.)
|- (EXMID -> A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) )
2.6.45  Excluded middle and the power set of a singleton
Theorem	pw1on 7225	The power set of 1o is an ordinal. (Contributed by Jim Kingdon, 29-Jul-2024.)
|- ~P 1o e. On
Theorem	pw1dom2 7226	The power set of 1o dominates 2o. Also see pwpw0ss 3805 which is similar. (Contributed by Jim Kingdon, 21-Sep-2022.)
|- 2o ~<_ ~P 1o
Theorem	pw1ne0 7227	The power set of 1o is not zero. (Contributed by Jim Kingdon, 30-Jul-2024.)
|- ~P 1o =/= (/)
Theorem	pw1ne1 7228	The power set of 1o is not one. (Contributed by Jim Kingdon, 30-Jul-2024.)
|- ~P 1o =/= 1o
Theorem	pw1ne3 7229	The power set of 1o is not three. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
|- ~P 1o =/= 3o
Theorem	pw1nel3 7230	Negated excluded middle implies that the power set of 1o is not an element of 3o. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
|- ( -. EXMID -> -. ~P 1o e. 3o )
Theorem	sucpw1ne3 7231	Negated excluded middle implies that the successor of the power set of 1o is not three . (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
|- ( -. EXMID -> suc ~P 1o =/= 3o )
Theorem	sucpw1nel3 7232	The successor of the power set of 1o is not an element of 3o. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
|- -. suc ~P 1o e. 3o
Theorem	3nelsucpw1 7233	Three is not an element of the successor of the power set of 1o. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
|- -. 3o e. suc ~P 1o
Theorem	sucpw1nss3 7234	Negated excluded middle implies that the successor of the power set of 1o is not a subset of 3o. (Contributed by James E. Hanson and Jim Kingdon, 31-Jul-2024.)
|- ( -. EXMID -> -. suc ~P 1o C_ 3o )
Theorem	3nsssucpw1 7235	Negated excluded middle implies that 3o is not a subset of the successor of the power set of 1o. (Contributed by James E. Hanson and Jim Kingdon, 31-Jul-2024.)
|- ( -. EXMID -> -. 3o C_ suc ~P 1o )
Theorem	onntri35 7236*	Double negated ordinal trichotomy. There are five equivalent statements: (1) -. -. A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ), (2) -. -. A. x e. On A. y e. On ( x C_ y \/ y C_ x ), (3) A. x e. On A. y e. On -. -. ( x e. y \/ x = y \/ y e. x ), (4) A. x e. On A. y e. On -. -. ( x C_ y \/ y C_ x ), and (5) -. -. EXMID. That these are all equivalent is expressed by (1) implies (3) (onntri13 7237), (3) implies (5) (onntri35 7236), (5) implies (1) (onntri51 7239), (2) implies (4) (onntri24 7241), (4) implies (5) (onntri45 7240), and (5) implies (2) (onntri52 7243). Another way of stating this is that EXMID is equivalent to trichotomy, either the x e. y \/ x = y \/ y e. x or the x C_ y \/ y C_ x form, as shown in exmidontri 7238 and exmidontri2or 7242, respectively. Thus -. -. EXMID is equivalent to (1) or (2). In addition, -. -. EXMID is equivalent to (3) by onntri3or 7244 and (4) by onntri2or 7245. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
|- ( A. x e. On A. y e. On -. -. ( x e. y \/ x = y \/ y e. x ) -> -. -. EXMID )
Theorem	onntri13 7237	Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
|- ( -. -. A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) -> A. x e. On A. y e. On -. -. ( x e. y \/ x = y \/ y e. x ) )
Theorem	exmidontri 7238*	Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.)
|- (EXMID <-> A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) )
Theorem	onntri51 7239*	Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
|- ( -. -. EXMID -> -. -. A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) )
Theorem	onntri45 7240*	Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
|- ( A. x e. On A. y e. On -. -. ( x C_ y \/ y C_ x ) -> -. -. EXMID )
Theorem	onntri24 7241	Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
|- ( -. -. A. x e. On A. y e. On ( x C_ y \/ y C_ x ) -> A. x e. On A. y e. On -. -. ( x C_ y \/ y C_ x ) )
Theorem	exmidontri2or 7242*	Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.)
|- (EXMID <-> A. x e. On A. y e. On ( x C_ y \/ y C_ x ) )
Theorem	onntri52 7243*	Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
|- ( -. -. EXMID -> -. -. A. x e. On A. y e. On ( x C_ y \/ y C_ x ) )
Theorem	onntri3or 7244*	Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.)
|- ( -. -. EXMID <-> A. x e. On A. y e. On -. -. ( x e. y \/ x = y \/ y e. x ) )
Theorem	onntri2or 7245*	Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.)
|- ( -. -. EXMID <-> A. x e. On A. y e. On -. -. ( x C_ y \/ y C_ x ) )
2.6.46  Apartness relations
Syntax	wap 7246	Apartness predicate symbol.
wff R Ap A
Definition	df-pap 7247*	Apartness predicate. A relation R is an apartness if it is irreflexive, symmetric, and cotransitive. (Contributed by Jim Kingdon, 14-Feb-2025.)
|- ( R Ap A <-> ( ( R C_ ( A X. A ) /\ A. x e. A -. x R x ) /\ ( A. x e. A A. y e. A ( x R y -> y R x ) /\ A. x e. A A. y e. A A. z e. A ( x R y -> ( x R z \/ y R z ) ) ) ) )
Syntax	wtap 7248	Tight apartness predicate symbol.
wff R TAp A
Definition	df-tap 7249*	Tight apartness predicate. A relation R is a tight apartness if it is irreflexive, symmetric, cotransitive, and tight. (Contributed by Jim Kingdon, 5-Feb-2025.)
|- ( R TAp A <-> ( R Ap A /\ A. x e. A A. y e. A ( -. x R y -> x = y ) ) )
Theorem	dftap2 7250*	Tight apartness with the apartness properties from df-pap 7247 expanded. (Contributed by Jim Kingdon, 21-Feb-2025.)
|- ( R TAp A <-> ( R C_ ( A X. A ) /\ ( A. x e. A -. x R x /\ A. x e. A A. y e. A ( x R y -> y R x ) ) /\ ( A. x e. A A. y e. A A. z e. A ( x R y -> ( x R z \/ y R z ) ) /\ A. x e. A A. y e. A ( -. x R y -> x = y ) ) ) )
Theorem	tapeq1 7251	Equality theorem for tight apartness predicate. (Contributed by Jim Kingdon, 8-Feb-2025.)
|- ( R = S -> ( R TAp A <-> S TAp A ) )
Theorem	tapeq2 7252	Equality theorem for tight apartness predicate. (Contributed by Jim Kingdon, 15-Feb-2025.)
|- ( A = B -> ( R TAp A <-> R TAp B ) )
Theorem	netap 7253*	Negated equality on a set with decidable equality is a tight apartness. (Contributed by Jim Kingdon, 5-Feb-2025.)
|- ( A. x e. A A. y e. A DECID x = y -> { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } TAp A )
Theorem	2onetap 7254*	Negated equality is a tight apartness on 2o. (Contributed by Jim Kingdon, 6-Feb-2025.)
|- { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ u =/= v ) } TAp 2o
Theorem	2oneel 7255*	(/) and 1o are two unequal elements of 2o. (Contributed by Jim Kingdon, 8-Feb-2025.)
|- <. (/) , 1o >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ u =/= v ) }
Theorem	2omotaplemap 7256*	Lemma for 2omotap 7258. (Contributed by Jim Kingdon, 6-Feb-2025.)
|- ( -. -. ph -> { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } TAp 2o )
Theorem	2omotaplemst 7257*	Lemma for 2omotap 7258. (Contributed by Jim Kingdon, 6-Feb-2025.)
|- ( ( E* r r TAp 2o /\ -. -. ph ) -> ph )
Theorem	2omotap 7258	If there is at most one tight apartness on 2o, excluded middle follows. Based on online discussions by Tom de Jong, Andrew W Swan, and Martin Escardo. (Contributed by Jim Kingdon, 6-Feb-2025.)
|- ( E* r r TAp 2o -> EXMID )
Theorem	exmidapne 7259*	Excluded middle implies there is only one tight apartness on any class, namely negated equality. (Contributed by Jim Kingdon, 14-Feb-2025.)
|- (EXMID -> ( R TAp A <-> R = { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } ) )
Theorem	exmidmotap 7260*	The proposition that every class has at most one tight apartness is equivalent to excluded middle. (Contributed by Jim Kingdon, 14-Feb-2025.)
|- (EXMID <-> A. x E* r r TAp x )
PART 3  CHOICE PRINCIPLES We have already introduced the full Axiom of Choice df-ac 7205 but since it implies excluded middle as shown at exmidac 7208, it is not especially relevant to us. In this section we define countable choice and dependent choice, which are not as strong as thus often considered in mathematics which seeks to avoid full excluded middle.
3.1  Countable Choice and Dependent Choice
3.1.1  Introduce Countable Choice
Syntax	wacc 7261	Formula for an abbreviation of countable choice.
wff CCHOICE
Definition	df-cc 7262*	The expression CCHOICE will be used as a readable shorthand for any form of countable choice, analogous to df-ac 7205 for full choice. (Contributed by Jim Kingdon, 27-Nov-2023.)
|- (CCHOICE <-> A. x ( dom x ~~ om -> E. f ( f C_ x /\ f Fn dom x ) ) )
Theorem	ccfunen 7263*	Existence of a choice function for a countably infinite set. (Contributed by Jim Kingdon, 28-Nov-2023.)
|- ( ph -> CCHOICE )   &    |- ( ph -> A ~~ om )   &    |- ( ph -> A. x e. A E. w w e. x )   =>    |- ( ph -> E. f ( f Fn A /\ A. x e. A ( f ` x ) e. x ) )
Theorem	cc1 7264*	Countable choice in terms of a choice function on a countably infinite set of inhabited sets. (Contributed by Jim Kingdon, 27-Apr-2024.)
|- (CCHOICE -> A. x ( ( x ~~ om /\ A. z e. x E. w w e. z ) -> E. f A. z e. x ( f ` z ) e. z ) )
Theorem	cc2lem 7265*	Lemma for cc2 7266. (Contributed by Jim Kingdon, 27-Apr-2024.)
|- ( ph -> CCHOICE )   &    |- ( ph -> F Fn om )   &    |- ( ph -> A. x e. om E. w w e. ( F ` x ) )   &    |- A = ( n e. om |-> ( { n } X. ( F ` n ) ) )   &    |- G = ( n e. om |-> ( 2nd ` ( f ` ( A ` n ) ) ) )   =>    |- ( ph -> E. g ( g Fn om /\ A. n e. om ( g ` n ) e. ( F ` n ) ) )
Theorem	cc2 7266*	Countable choice using sequences instead of countable sets. (Contributed by Jim Kingdon, 27-Apr-2024.)
|- ( ph -> CCHOICE )   &    |- ( ph -> F Fn om )   &    |- ( ph -> A. x e. om E. w w e. ( F ` x ) )   =>    |- ( ph -> E. g ( g Fn om /\ A. n e. om ( g ` n ) e. ( F ` n ) ) )
Theorem	cc3 7267*	Countable choice using a sequence F(n) . (Contributed by Mario Carneiro, 8-Feb-2013.) (Revised by Jim Kingdon, 29-Apr-2024.)
|- ( ph -> CCHOICE )   &    |- ( ph -> A. n e. N F e. _V )   &    |- ( ph -> A. n e. N E. w w e. F )   &    |- ( ph -> N ~~ om )   =>    |- ( ph -> E. f ( f Fn N /\ A. n e. N ( f ` n ) e. F ) )
Theorem	cc4f 7268*	Countable choice by showing the existence of a function f which can choose a value at each index n such that ch holds. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 3-May-2024.)
|- ( ph -> CCHOICE )   &    |- ( ph -> A e. V )   &    |- F/_ n A   &    |- ( ph -> N ~~ om )   &    |- ( x = ( f ` n ) -> ( ps <-> ch ) )   &    |- ( ph -> A. n e. N E. x e. A ps )   =>    |- ( ph -> E. f ( f : N --> A /\ A. n e. N ch ) )
Theorem	cc4 7269*	Countable choice by showing the existence of a function f which can choose a value at each index n such that ch holds. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 1-May-2024.)
|- ( ph -> CCHOICE )   &    |- ( ph -> A e. V )   &    |- ( ph -> N ~~ om )   &    |- ( x = ( f ` n ) -> ( ps <-> ch ) )   &    |- ( ph -> A. n e. N E. x e. A ps )   =>    |- ( ph -> E. f ( f : N --> A /\ A. n e. N ch ) )
Theorem	cc4n 7270*	Countable choice with a simpler restriction on how every set in the countable collection needs to be inhabited. That is, compared with cc4 7269, the hypotheses only require an A(n) for each value of n, not a single set A which suffices for every n e. om. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 3-May-2024.)
|- ( ph -> CCHOICE )   &    |- ( ph -> A. n e. N { x e. A | ps } e. V )   &    |- ( ph -> N ~~ om )   &    |- ( x = ( f ` n ) -> ( ps <-> ch ) )   &    |- ( ph -> A. n e. N E. x e. A ps )   =>    |- ( ph -> E. f ( f Fn N /\ A. n e. N ch ) )
PART 4  REAL AND COMPLEX NUMBERS 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 6475 and similar theorems ), going from there to positive integers (df-ni 7303) and then positive rational numbers (df-nqqs 7347) 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 or choice principles. With excluded middle, it is natural to define a cut as the lower set only (as Metamath Proof Explorer does), but here 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". When working constructively, there are several possible definitions of real numbers. Here we adopt the most common definition, as two-sided Dedekind cuts with the properties described at df-inp 7465. The Cauchy reals (without countable choice) fail to satisfy ax-caucvg 7931 and the MacNeille reals fail to satisfy axltwlin 8025, and we do not develop them here. For more on differing definitions of the reals, see the introduction to Chapter 11 in [HoTT] or Section 1.2 of [BauerHanson].
4.1  Construction and axiomatization of real and complex numbers
4.1.1  Dedekind-cut construction of real and complex numbers
Syntax	cnpi 7271	The set of positive integers, which is the set of natural numbers om with 0 removed. Note: This is the start of the Dedekind-cut construction of real and complex numbers.
class N.
Syntax	cpli 7272	Positive integer addition.
class +N
Syntax	cmi 7273	Positive integer multiplication.
class .N
Syntax	clti 7274	Positive integer ordering relation.
class <N
Syntax	cplpq 7275	Positive pre-fraction addition.
class +pQ
Syntax	cmpq 7276	Positive pre-fraction multiplication.
class .pQ
Syntax	cltpq 7277	Positive pre-fraction ordering relation.
class <pQ
Syntax	ceq 7278	Equivalence class used to construct positive fractions.
class ~Q
Syntax	cnq 7279	Set of positive fractions.
class Q.
Syntax	c1q 7280	The positive fraction constant 1.
class 1Q
Syntax	cplq 7281	Positive fraction addition.
class +Q
Syntax	cmq 7282	Positive fraction multiplication.
class .Q
Syntax	crq 7283	Positive fraction reciprocal operation.
class *Q
Syntax	cltq 7284	Positive fraction ordering relation.
class <Q
Syntax	ceq0 7285	Equivalence class used to construct nonnegative fractions.
class ~Q0
Syntax	cnq0 7286	Set of nonnegative fractions.
class Q0
Syntax	c0q0 7287	The nonnegative fraction constant 0.
class 0Q0
Syntax	cplq0 7288	Nonnegative fraction addition.
class +Q0
Syntax	cmq0 7289	Nonnegative fraction multiplication.
class ·Q0
Syntax	cnp 7290	Set of positive reals.
class P.
Syntax	c1p 7291	Positive real constant 1.
class 1P
Syntax	cpp 7292	Positive real addition.
class +P.
Syntax	cmp 7293	Positive real multiplication.
class .P.
Syntax	cltp 7294	Positive real ordering relation.
class <P
Syntax	cer 7295	Equivalence class used to construct signed reals.
class ~R
Syntax	cnr 7296	Set of signed reals.
class R.
Syntax	c0r 7297	The signed real constant 0.
class 0R
Syntax	c1r 7298	The signed real constant 1.
class 1R
Syntax	cm1r 7299	The signed real constant -1.
class -1R
Syntax	cplr 7300	Signed real addition.
class +R