P. 76 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7501-7600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	onntri51 7501*	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 7502*	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 7503	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 7504*	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 7505*	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 7506*	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 7507*	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 ) )
Theorem	fmelpw1o 7508	With a formula ph one can associate an element of ~P 1o, which can therefore be thought of as the set of "truth values" (but recall that there are no other genuine truth values than T. and F., by nndc 859, which translate to 1o and (/) respectively by iftrue 3614 and iffalse 3617, giving pwtrufal 16702). As proved in if0ab 3610, the associated element of ~P 1o is the extension, in ~P 1o, of the formula ph. (Contributed by BJ, 15-Aug-2024.) (Proof shortened by BJ, 5-May-2026.)
|- if ( ph , 1o , (/) ) e. ~P 1o
2.6.47  Apartness relations
Syntax	wap 7509	Apartness predicate symbol.
wff R Ap A
Definition	df-pap 7510*	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 7511	Tight apartness predicate symbol.
wff R TAp A
Definition	df-tap 7512*	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 7513*	Tight apartness with the apartness properties from df-pap 7510 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 7514	Equality theorem for tight apartness predicate. (Contributed by Jim Kingdon, 8-Feb-2025.)
|- ( R = S -> ( R TAp A <-> S TAp A ) )
Theorem	tapeq2 7515	Equality theorem for tight apartness predicate. (Contributed by Jim Kingdon, 15-Feb-2025.)
|- ( A = B -> ( R TAp A <-> R TAp B ) )
Theorem	netap 7516*	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 7517*	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 7518*	(/) 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 7519*	Lemma for 2omotap 7521. (Contributed by Jim Kingdon, 6-Feb-2025.)
|- ( -. -. ph -> { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } TAp 2o )
Theorem	2omotaplemst 7520*	Lemma for 2omotap 7521. (Contributed by Jim Kingdon, 6-Feb-2025.)
|- ( ( E* r r TAp 2o /\ -. -. ph ) -> ph )
Theorem	2omotap 7521	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 7522*	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 7523*	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 7464 but since it implies excluded middle as shown at exmidac 7467, 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 7524	Formula for an abbreviation of countable choice.
wff CCHOICE
Definition	df-cc 7525*	The expression CCHOICE will be used as a readable shorthand for any form of countable choice, analogous to df-ac 7464 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 7526*	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 7527*	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 7528*	Lemma for cc2 7529. (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 7529*	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 7530*	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 7531*	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 7532*	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 7533*	Countable choice with a simpler restriction on how every set in the countable collection needs to be inhabited. That is, compared with cc4 7532, 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 ) )
Theorem	acnccim 7534	Given countable choice, every set has choice sets of length om. (Contributed by Mario Carneiro, 31-Aug-2015.)
|- (CCHOICE -> AC om = _V )
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 6685 and similar theorems ), going from there to positive integers (df-ni 7567) and then positive rational numbers (df-nqqs 7611) 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 7729. The Cauchy reals (without countable choice) fail to satisfy ax-caucvg 8195 and the MacNeille reals fail to satisfy axltwlin 8289, 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 7535	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 7536	Positive integer addition.
class +N
Syntax	cmi 7537	Positive integer multiplication.
class .N
Syntax	clti 7538	Positive integer ordering relation.
class <N
Syntax	cplpq 7539	Positive pre-fraction addition.
class +pQ
Syntax	cmpq 7540	Positive pre-fraction multiplication.
class .pQ
Syntax	cltpq 7541	Positive pre-fraction ordering relation.
class <pQ
Syntax	ceq 7542	Equivalence class used to construct positive fractions.
class ~Q
Syntax	cnq 7543	Set of positive fractions.
class Q.
Syntax	c1q 7544	The positive fraction constant 1.
class 1Q
Syntax	cplq 7545	Positive fraction addition.
class +Q
Syntax	cmq 7546	Positive fraction multiplication.
class .Q
Syntax	crq 7547	Positive fraction reciprocal operation.
class *Q
Syntax	cltq 7548	Positive fraction ordering relation.
class <Q
Syntax	ceq0 7549	Equivalence class used to construct nonnegative fractions.
class ~Q0
Syntax	cnq0 7550	Set of nonnegative fractions.
class Q0
Syntax	c0q0 7551	The nonnegative fraction constant 0.
class 0Q0
Syntax	cplq0 7552	Nonnegative fraction addition.
class +Q0
Syntax	cmq0 7553	Nonnegative fraction multiplication.
class ·Q0
Syntax	cnp 7554	Set of positive reals.
class P.
Syntax	c1p 7555	Positive real constant 1.
class 1P
Syntax	cpp 7556	Positive real addition.
class +P.
Syntax	cmp 7557	Positive real multiplication.
class .P.
Syntax	cltp 7558	Positive real ordering relation.
class <P
Syntax	cer 7559	Equivalence class used to construct signed reals.
class ~R
Syntax	cnr 7560	Set of signed reals.
class R.
Syntax	c0r 7561	The signed real constant 0.
class 0R
Syntax	c1r 7562	The signed real constant 1.
class 1R
Syntax	cm1r 7563	The signed real constant -1.
class -1R
Syntax	cplr 7564	Signed real addition.
class +R
Syntax	cmr 7565	Signed real multiplication.
class .R
Syntax	cltr 7566	Signed real ordering relation.
class <R
Definition	df-ni 7567	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. = ( om \ { (/) } )
Definition	df-pli 7568	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. X. N. ) )
Definition	df-mi 7569	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. X. N. ) )
Definition	df-lti 7570	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 i^i ( N. X. N. ) )
Theorem	elni 7571	Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.)
|- ( A e. N. <-> ( A e. om /\ A =/= (/) ) )
Theorem	pinn 7572	A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.)
|- ( A e. N. -> A e. om )
Theorem	pion 7573	A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.)
|- ( A e. N. -> A e. On )
Theorem	piord 7574	A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.)
|- ( A e. N. -> Ord A )
Theorem	niex 7575	The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.)
|- N. e. _V
Theorem	0npi 7576	The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.)
|- -. (/) e. N.
Theorem	elni2 7577	Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.)
|- ( A e. N. <-> ( A e. om /\ (/) e. A ) )
Theorem	1pi 7578	Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.)
|- 1o e. N.
Theorem	addpiord 7579	Positive integer addition in terms of ordinal addition. (Contributed by NM, 27-Aug-1995.)
|- ( ( A e. N. /\ B e. N. ) -> ( A +N B ) = ( A +o B ) )
Theorem	mulpiord 7580	Positive integer multiplication in terms of ordinal multiplication. (Contributed by NM, 27-Aug-1995.)
|- ( ( A e. N. /\ B e. N. ) -> ( A .N B ) = ( A .o B ) )
Theorem	mulidpi 7581	1 is an identity element for multiplication on positive integers. (Contributed by NM, 4-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.)
|- ( A e. N. -> ( A .N 1o ) = A )
Theorem	ltpiord 7582	Positive integer 'less than' in terms of ordinal membership. (Contributed by NM, 6-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2015.)
|- ( ( A e. N. /\ B e. N. ) -> ( A <N B <-> A e. B ) )
Theorem	ltsopi 7583	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 7584	Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.)
|- ( ( A e. N. /\ B e. N. ) -> ( A <N B <-> -. ( A = B \/ B <N A ) ) )
Theorem	pitri3or 7585	Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.)
|- ( ( A e. N. /\ B e. N. ) -> ( A <N B \/ A = B \/ B <N A ) )
Theorem	ltdcpi 7586	Less-than for positive integers is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.)
|- ( ( A e. N. /\ B e. N. ) -> DECID A <N B )
Theorem	ltrelpi 7587	Positive integer 'less than' is a relation on positive integers. (Contributed by NM, 8-Feb-1996.)
|- <N C_ ( N. X. N. )
Theorem	dmaddpi 7588	Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.)
|- dom +N = ( N. X. N. )
Theorem	dmmulpi 7589	Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.)
|- dom .N = ( N. X. N. )
Theorem	addclpi 7590	Closure of addition of positive integers. (Contributed by NM, 18-Oct-1995.)
|- ( ( A e. N. /\ B e. N. ) -> ( A +N B ) e. N. )
Theorem	mulclpi 7591	Closure of multiplication of positive integers. (Contributed by NM, 18-Oct-1995.)
|- ( ( A e. N. /\ B e. N. ) -> ( A .N B ) e. N. )
Theorem	addcompig 7592	Addition of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.)
|- ( ( A e. N. /\ B e. N. ) -> ( A +N B ) = ( B +N A ) )
Theorem	addasspig 7593	Addition of positive integers is associative. (Contributed by Jim Kingdon, 26-Aug-2019.)
|- ( ( A e. N. /\ B e. N. /\ C e. N. ) -> ( ( A +N B ) +N C ) = ( A +N ( B +N C ) ) )
Theorem	mulcompig 7594	Multiplication of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.)
|- ( ( A e. N. /\ B e. N. ) -> ( A .N B ) = ( B .N A ) )
Theorem	mulasspig 7595	Multiplication of positive integers is associative. (Contributed by Jim Kingdon, 26-Aug-2019.)
|- ( ( A e. N. /\ B e. N. /\ C e. N. ) -> ( ( A .N B ) .N C ) = ( A .N ( B .N C ) ) )
Theorem	distrpig 7596	Multiplication of positive integers is distributive. (Contributed by Jim Kingdon, 26-Aug-2019.)
|- ( ( A e. N. /\ B e. N. /\ C e. N. ) -> ( A .N ( B +N C ) ) = ( ( A .N B ) +N ( A .N C ) ) )
Theorem	addcanpig 7597	Addition cancellation law for positive integers. (Contributed by Jim Kingdon, 27-Aug-2019.)
|- ( ( A e. N. /\ B e. N. /\ C e. N. ) -> ( ( A +N B ) = ( A +N C ) <-> B = C ) )
Theorem	mulcanpig 7598	Multiplication cancellation law for positive integers. (Contributed by Jim Kingdon, 29-Aug-2019.)
|- ( ( A e. N. /\ B e. N. /\ C e. N. ) -> ( ( A .N B ) = ( A .N C ) <-> B = C ) )
Theorem	addnidpig 7599	There is no identity element for addition on positive integers. (Contributed by NM, 28-Nov-1995.)
|- ( ( A e. N. /\ B e. N. ) -> -. ( A +N B ) = A )
Theorem	ltexpi 7600*	Ordering on positive integers in terms of existence of sum. (Contributed by NM, 15-Mar-1996.) (Revised by Mario Carneiro, 14-Jun-2013.)
|- ( ( A e. N. /\ B e. N. ) -> ( A <N B <-> E. x e. N. ( A +N x ) = B ) )