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	exmidontriimlem4 7201*	Lemma for exmidontriim 7202. 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 7202*	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 7203	The power set of 1o is an ordinal. (Contributed by Jim Kingdon, 29-Jul-2024.)
|- ~P 1o e. On
Theorem	pw1dom2 7204	The power set of 1o dominates 2o. Also see pwpw0ss 3791 which is similar. (Contributed by Jim Kingdon, 21-Sep-2022.)
|- 2o ~<_ ~P 1o
Theorem	pw1ne0 7205	The power set of 1o is not zero. (Contributed by Jim Kingdon, 30-Jul-2024.)
|- ~P 1o =/= (/)
Theorem	pw1ne1 7206	The power set of 1o is not one. (Contributed by Jim Kingdon, 30-Jul-2024.)
|- ~P 1o =/= 1o
Theorem	pw1ne3 7207	The power set of 1o is not three. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
|- ~P 1o =/= 3o
Theorem	pw1nel3 7208	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 7209	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 7210	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 7211	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 7212	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 7213	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 7214*	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 7215), (3) implies (5) (onntri35 7214), (5) implies (1) (onntri51 7217), (2) implies (4) (onntri24 7219), (4) implies (5) (onntri45 7218), and (5) implies (2) (onntri52 7221). 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 7216 and exmidontri2or 7220, respectively. Thus -. -. EXMID is equivalent to (1) or (2). In addition, -. -. EXMID is equivalent to (3) by onntri3or 7222 and (4) by onntri2or 7223. (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 7215	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 7216*	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 7217*	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 7218*	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 7219	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 7220*	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 7221*	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 7222*	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 7223*	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 ) )
PART 3  CHOICE PRINCIPLES We have already introduced the full Axiom of Choice df-ac 7183 but since it implies excluded middle as shown at exmidac 7186, 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 7224	Formula for an abbreviation of countable choice.
wff CCHOICE
Definition	df-cc 7225*	The expression CCHOICE will be used as a readable shorthand for any form of countable choice, analogous to df-ac 7183 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 7226*	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 7227*	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 7228*	Lemma for cc2 7229. (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 7229*	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 7230*	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 7231*	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 7232*	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 7233*	Countable choice with a simpler restriction on how every set in the countable collection needs to be inhabited. That is, compared with cc4 7232, 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 6453 and similar theorems ), going from there to positive integers (df-ni 7266) and then positive rational numbers (df-nqqs 7310) 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 7428. The Cauchy reals (without countable choice) fail to satisfy ax-caucvg 7894 and the MacNeille reals fail to satisfy axltwlin 7987, 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 7234	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 7235	Positive integer addition.
class +N
Syntax	cmi 7236	Positive integer multiplication.
class .N
Syntax	clti 7237	Positive integer ordering relation.
class <N
Syntax	cplpq 7238	Positive pre-fraction addition.
class +pQ
Syntax	cmpq 7239	Positive pre-fraction multiplication.
class .pQ
Syntax	cltpq 7240	Positive pre-fraction ordering relation.
class <pQ
Syntax	ceq 7241	Equivalence class used to construct positive fractions.
class ~Q
Syntax	cnq 7242	Set of positive fractions.
class Q.
Syntax	c1q 7243	The positive fraction constant 1.
class 1Q
Syntax	cplq 7244	Positive fraction addition.
class +Q
Syntax	cmq 7245	Positive fraction multiplication.
class .Q
Syntax	crq 7246	Positive fraction reciprocal operation.
class *Q
Syntax	cltq 7247	Positive fraction ordering relation.
class <Q
Syntax	ceq0 7248	Equivalence class used to construct nonnegative fractions.
class ~Q0
Syntax	cnq0 7249	Set of nonnegative fractions.
class Q0
Syntax	c0q0 7250	The nonnegative fraction constant 0.
class 0Q0
Syntax	cplq0 7251	Nonnegative fraction addition.
class +Q0
Syntax	cmq0 7252	Nonnegative fraction multiplication.
class ·Q0
Syntax	cnp 7253	Set of positive reals.
class P.
Syntax	c1p 7254	Positive real constant 1.
class 1P
Syntax	cpp 7255	Positive real addition.
class +P.
Syntax	cmp 7256	Positive real multiplication.
class .P.
Syntax	cltp 7257	Positive real ordering relation.
class <P
Syntax	cer 7258	Equivalence class used to construct signed reals.
class ~R
Syntax	cnr 7259	Set of signed reals.
class R.
Syntax	c0r 7260	The signed real constant 0.
class 0R
Syntax	c1r 7261	The signed real constant 1.
class 1R
Syntax	cm1r 7262	The signed real constant -1.
class -1R
Syntax	cplr 7263	Signed real addition.
class +R
Syntax	cmr 7264	Signed real multiplication.
class .R
Syntax	cltr 7265	Signed real ordering relation.
class <R
Definition	df-ni 7266	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 7267	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 7268	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 7269	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 7270	Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.)
|- ( A e. N. <-> ( A e. om /\ A =/= (/) ) )
Theorem	pinn 7271	A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.)
|- ( A e. N. -> A e. om )
Theorem	pion 7272	A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.)
|- ( A e. N. -> A e. On )
Theorem	piord 7273	A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.)
|- ( A e. N. -> Ord A )
Theorem	niex 7274	The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.)
|- N. e. _V
Theorem	0npi 7275	The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.)
|- -. (/) e. N.
Theorem	elni2 7276	Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.)
|- ( A e. N. <-> ( A e. om /\ (/) e. A ) )
Theorem	1pi 7277	Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.)
|- 1o e. N.
Theorem	addpiord 7278	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 7279	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 7280	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 7281	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 7282	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 7283	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 7284	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 7285	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 7286	Positive integer 'less than' is a relation on positive integers. (Contributed by NM, 8-Feb-1996.)
|- <N C_ ( N. X. N. )
Theorem	dmaddpi 7287	Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.)
|- dom +N = ( N. X. N. )
Theorem	dmmulpi 7288	Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.)
|- dom .N = ( N. X. N. )
Theorem	addclpi 7289	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 7290	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 7291	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 7292	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 7293	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 7294	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 7295	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 7296	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 7297	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 7298	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 7299*	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 ) )
Theorem	ltapig 7300	Ordering property of addition for positive integers. (Contributed by Jim Kingdon, 31-Aug-2019.)
|- ( ( A e. N. /\ B e. N. /\ C e. N. ) -> ( A <N B <-> ( C +N A ) <N ( C +N B ) ) )