P. 77 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7601-7700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	lenegcon1i 7601	Contraposition of negative in 'less than or equal to'. (Contributed by NM, 6-Apr-2005.)
|- A e. RR   &    |- B e. RR   =>    |- ( -u A <_ B <-> -u B <_ A )
Theorem	subge0i 7602	Nonnegative subtraction. (Contributed by NM, 13-Aug-2000.)
|- A e. RR   &    |- B e. RR   =>    |- ( 0 <_ ( A - B ) <-> B <_ A )
Theorem	ltadd1i 7603	Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( A < B <-> ( A + C ) < ( B + C ) )
Theorem	leadd1i 7604	Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( A <_ B <-> ( A + C ) <_ ( B + C ) )
Theorem	leadd2i 7605	Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( A <_ B <-> ( C + A ) <_ ( C + B ) )
Theorem	ltsubaddi 7606	'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( ( A - B ) < C <-> A < ( C + B ) )
Theorem	lesubaddi 7607	'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 30-Sep-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( ( A - B ) <_ C <-> A <_ ( C + B ) )
Theorem	ltsubadd2i 7608	'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( ( A - B ) < C <-> A < ( B + C ) )
Theorem	lesubadd2i 7609	'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 3-Aug-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( ( A - B ) <_ C <-> A <_ ( B + C ) )
Theorem	ltaddsubi 7610	'Less than' relationship between subtraction and addition. (Contributed by NM, 14-May-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( ( A + B ) < C <-> A < ( C - B ) )
Theorem	lt2addi 7611	Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   &    |- D e. RR   =>    |- ( ( A < C /\ B < D ) -> ( A + B ) < ( C + D ) )
Theorem	le2addi 7612	Adding both side of two inequalities. (Contributed by NM, 16-Sep-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   &    |- D e. RR   =>    |- ( ( A <_ C /\ B <_ D ) -> ( A + B ) <_ ( C + D ) )
Theorem	gt0ne0d 7613	Positive implies nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> 0 < A )   =>    |- ( ph -> A =/= 0 )
Theorem	lt0ne0d 7614	Something less than zero is not zero. Deduction form. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A < 0 )   =>    |- ( ph -> A =/= 0 )
Theorem	leidd 7615	'Less than or equal to' is reflexive. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> A <_ A )
Theorem	lt0neg1d 7616	Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( A < 0 <-> 0 < -u A ) )
Theorem	lt0neg2d 7617	Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( 0 < A <-> -u A < 0 ) )
Theorem	le0neg1d 7618	Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( A <_ 0 <-> 0 <_ -u A ) )
Theorem	le0neg2d 7619	Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( 0 <_ A <-> -u A <_ 0 ) )
Theorem	addgegt0d 7620	Addition of nonnegative and positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> 0 < B )   =>    |- ( ph -> 0 < ( A + B ) )
Theorem	addgt0d 7621	Addition of 2 positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> 0 < A )   &    |- ( ph -> 0 < B )   =>    |- ( ph -> 0 < ( A + B ) )
Theorem	addge0d 7622	Addition of 2 nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> 0 <_ B )   =>    |- ( ph -> 0 <_ ( A + B ) )
Theorem	ltnegd 7623	Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( A < B <-> -u B < -u A ) )
Theorem	lenegd 7624	Negative of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( A <_ B <-> -u B <_ -u A ) )
Theorem	ltnegcon1d 7625	Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> -u A < B )   =>    |- ( ph -> -u B < A )
Theorem	ltnegcon2d 7626	Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A < -u B )   =>    |- ( ph -> B < -u A )
Theorem	lenegcon1d 7627	Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> -u A <_ B )   =>    |- ( ph -> -u B <_ A )
Theorem	lenegcon2d 7628	Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A <_ -u B )   =>    |- ( ph -> B <_ -u A )
Theorem	ltaddposd 7629	Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( 0 < A <-> B < ( B + A ) ) )
Theorem	ltaddpos2d 7630	Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( 0 < A <-> B < ( A + B ) ) )
Theorem	ltsubposd 7631	Subtracting a positive number from another number decreases it. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( 0 < A <-> ( B - A ) < B ) )
Theorem	posdifd 7632	Comparison of two numbers whose difference is positive. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( A < B <-> 0 < ( B - A ) ) )
Theorem	addge01d 7633	A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( 0 <_ B <-> A <_ ( A + B ) ) )
Theorem	addge02d 7634	A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( 0 <_ B <-> A <_ ( B + A ) ) )
Theorem	subge0d 7635	Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( 0 <_ ( A - B ) <-> B <_ A ) )
Theorem	suble0d 7636	Nonpositive subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( ( A - B ) <_ 0 <-> A <_ B ) )
Theorem	subge02d 7637	Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( 0 <_ B <-> ( A - B ) <_ A ) )
Theorem	ltadd1d 7638	Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( A < B <-> ( A + C ) < ( B + C ) ) )
Theorem	leadd1d 7639	Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( A <_ B <-> ( A + C ) <_ ( B + C ) ) )
Theorem	leadd2d 7640	Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( A <_ B <-> ( C + A ) <_ ( C + B ) ) )
Theorem	ltsubaddd 7641	'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( ( A - B ) < C <-> A < ( C + B ) ) )
Theorem	lesubaddd 7642	'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( ( A - B ) <_ C <-> A <_ ( C + B ) ) )
Theorem	ltsubadd2d 7643	'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( ( A - B ) < C <-> A < ( B + C ) ) )
Theorem	lesubadd2d 7644	'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( ( A - B ) <_ C <-> A <_ ( B + C ) ) )
Theorem	ltaddsubd 7645	'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( ( A + B ) < C <-> A < ( C - B ) ) )
Theorem	ltaddsub2d 7646	'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 29-Dec-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( ( A + B ) < C <-> B < ( C - A ) ) )
Theorem	leaddsub2d 7647	'Less than or equal to' relationship between and addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( ( A + B ) <_ C <-> B <_ ( C - A ) ) )
Theorem	subled 7648	Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> ( A - B ) <_ C )   =>    |- ( ph -> ( A - C ) <_ B )
Theorem	lesubd 7649	Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> A <_ ( B - C ) )   =>    |- ( ph -> C <_ ( B - A ) )
Theorem	ltsub23d 7650	'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> ( A - B ) < C )   =>    |- ( ph -> ( A - C ) < B )
Theorem	ltsub13d 7651	'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> A < ( B - C ) )   =>    |- ( ph -> C < ( B - A ) )
Theorem	lesub1d 7652	Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( A <_ B <-> ( A - C ) <_ ( B - C ) ) )
Theorem	lesub2d 7653	Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( A <_ B <-> ( C - B ) <_ ( C - A ) ) )
Theorem	ltsub1d 7654	Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( A < B <-> ( A - C ) < ( B - C ) ) )
Theorem	ltsub2d 7655	Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( A < B <-> ( C - B ) < ( C - A ) ) )
Theorem	ltadd1dd 7656	Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> A < B )   =>    |- ( ph -> ( A + C ) < ( B + C ) )
Theorem	ltsub1dd 7657	Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> A < B )   =>    |- ( ph -> ( A - C ) < ( B - C ) )
Theorem	ltsub2dd 7658	Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> A < B )   =>    |- ( ph -> ( C - B ) < ( C - A ) )
Theorem	leadd1dd 7659	Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> A <_ B )   =>    |- ( ph -> ( A + C ) <_ ( B + C ) )
Theorem	leadd2dd 7660	Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> A <_ B )   =>    |- ( ph -> ( C + A ) <_ ( C + B ) )
Theorem	lesub1dd 7661	Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> A <_ B )   =>    |- ( ph -> ( A - C ) <_ ( B - C ) )
Theorem	lesub2dd 7662	Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> A <_ B )   =>    |- ( ph -> ( C - B ) <_ ( C - A ) )
Theorem	le2addd 7663	Adding both side of two inequalities. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> D e. RR )   &    |- ( ph -> A <_ C )   &    |- ( ph -> B <_ D )   =>    |- ( ph -> ( A + B ) <_ ( C + D ) )
Theorem	le2subd 7664	Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> D e. RR )   &    |- ( ph -> A <_ C )   &    |- ( ph -> B <_ D )   =>    |- ( ph -> ( A - D ) <_ ( C - B ) )
Theorem	ltleaddd 7665	Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> D e. RR )   &    |- ( ph -> A < C )   &    |- ( ph -> B <_ D )   =>    |- ( ph -> ( A + B ) < ( C + D ) )
Theorem	leltaddd 7666	Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> D e. RR )   &    |- ( ph -> A <_ C )   &    |- ( ph -> B < D )   =>    |- ( ph -> ( A + B ) < ( C + D ) )
Theorem	lt2addd 7667	Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> D e. RR )   &    |- ( ph -> A < C )   &    |- ( ph -> B < D )   =>    |- ( ph -> ( A + B ) < ( C + D ) )
Theorem	lt2subd 7668	Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> D e. RR )   &    |- ( ph -> A < C )   &    |- ( ph -> B < D )   =>    |- ( ph -> ( A - D ) < ( C - B ) )
Theorem	possumd 7669	Condition for a positive sum. (Contributed by Scott Fenton, 16-Dec-2017.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( 0 < ( A + B ) <-> -u B < A ) )
Theorem	sublt0d 7670	When a subtraction gives a negative result. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( ( A - B ) < 0 <-> A < B ) )
Theorem	ltaddsublt 7671	Addition and subtraction on one side of 'less than'. (Contributed by AV, 24-Nov-2018.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B < C <-> ( ( A + B ) - C ) < A ) )
Theorem	1le1 7672	1 <_ 1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)
|- 1 <_ 1
Theorem	gt0add 7673	A positive sum must have a positive addend. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 26-Jan-2020.)
|- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> ( 0 < A \/ 0 < B ) )
3.3.5  Real Apartness
Syntax	creap 7674	Class of real apartness relation.
class #ℝ
Definition	df-reap 7675*	Define real apartness. Definition in Section 11.2.1 of [HoTT], p. (varies). Although #ℝ is an apartness relation on the reals (see df-ap 7682 for more discussion of apartness relations), for our purposes it is just a stepping stone to defining # which is an apartness relation on complex numbers. On the reals, #ℝ and # agree (apreap 7687). (Contributed by Jim Kingdon, 26-Jan-2020.)
|- #ℝ = { <. x , y >. | ( ( x e. RR /\ y e. RR ) /\ ( x < y \/ y < x ) ) }
Theorem	reapval 7676	Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 7688 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 29-Jan-2020.)
|- ( ( A e. RR /\ B e. RR ) -> ( A #ℝ B <-> ( A < B \/ B < A ) ) )
Theorem	reapirr 7677	Real apartness is irreflexive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). Beyond the development of # itself, proofs should use apirr 7705 instead. (Contributed by Jim Kingdon, 26-Jan-2020.)
|- ( A e. RR -> -. A #ℝ A )
Theorem	recexre 7678*	Existence of reciprocal of real number. (Contributed by Jim Kingdon, 29-Jan-2020.)
|- ( ( A e. RR /\ A #ℝ 0 ) -> E. x e. RR ( A x. x ) = 1 )
Theorem	reapti 7679	Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 7722. (Contributed by Jim Kingdon, 30-Jan-2020.) (New usage is discouraged.)
|- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> -. A #ℝ B ) )
Theorem	recexgt0 7680*	Existence of reciprocal of positive real number. (Contributed by Jim Kingdon, 6-Feb-2020.)
|- ( ( A e. RR /\ 0 < A ) -> E. x e. RR ( 0 < x /\ ( A x. x ) = 1 ) )
3.3.6  Complex Apartness
Syntax	cap 7681	Class of complex apartness relation.
class #
Definition	df-ap 7682*	Define complex apartness. Definition 6.1 of [Geuvers], p. 17. Two numbers are considered apart if it is possible to separate them. One common usage is that we can divide by a number if it is apart from zero (see for example recclap 7767 which says that a number apart from zero has a reciprocal). The defining characteristics of an apartness are irreflexivity (apirr 7705), symmetry (apsym 7706), and cotransitivity (apcotr 7707). Apartness implies negated equality, as seen at apne 7723, and the converse would also follow if we assumed excluded middle. In addition, apartness of complex numbers is tight, which means that two numbers which are not apart are equal (apti 7722). (Contributed by Jim Kingdon, 26-Jan-2020.)
|- # = { <. x , y >. | E. r e. RR E. s e. RR E. t e. RR E. u e. RR ( ( x = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) ) }
Theorem	ixi 7683	_i times itself is minus 1. (Contributed by NM, 6-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- ( _i x. _i ) = -u 1
Theorem	inelr 7684	The imaginary unit _i is not a real number. (Contributed by NM, 6-May-1999.)
|- -. _i e. RR
Theorem	rimul 7685	A real number times the imaginary unit is real only if the number is 0. (Contributed by NM, 28-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( A e. RR /\ ( _i x. A ) e. RR ) -> A = 0 )
Theorem	rereim 7686	Decomposition of a real number into real part (itself) and imaginary part (zero). (Contributed by Jim Kingdon, 30-Jan-2020.)
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> ( B = A /\ C = 0 ) )
Theorem	apreap 7687	Complex apartness and real apartness agree on the real numbers. (Contributed by Jim Kingdon, 31-Jan-2020.)
|- ( ( A e. RR /\ B e. RR ) -> ( A # B <-> A #ℝ B ) )
Theorem	reaplt 7688	Real apartness in terms of less than. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 1-Feb-2020.)
|- ( ( A e. RR /\ B e. RR ) -> ( A # B <-> ( A < B \/ B < A ) ) )
Theorem	reapltxor 7689	Real apartness in terms of less than (exclusive-or version). (Contributed by Jim Kingdon, 23-Mar-2020.)
|- ( ( A e. RR /\ B e. RR ) -> ( A # B <-> ( A < B \/_ B < A ) ) )
Theorem	1ap0 7690	One is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
|- 1 # 0
Theorem	ltmul1a 7691	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 15-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) /\ A < B ) -> ( A x. C ) < ( B x. C ) )
Theorem	ltmul1 7692	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A < B <-> ( A x. C ) < ( B x. C ) ) )
Theorem	lemul1 7693	Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 21-Feb-2005.)
|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A <_ B <-> ( A x. C ) <_ ( B x. C ) ) )
Theorem	reapmul1lem 7694	Lemma for reapmul1 7695. (Contributed by Jim Kingdon, 8-Feb-2020.)
|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A # B <-> ( A x. C ) # ( B x. C ) ) )
Theorem	reapmul1 7695	Multiplication of both sides of real apartness by a real number apart from zero. Special case of apmul1 7876. (Contributed by Jim Kingdon, 8-Feb-2020.)
|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ C # 0 ) ) -> ( A # B <-> ( A x. C ) # ( B x. C ) ) )
Theorem	reapadd1 7696	Real addition respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A # B <-> ( A + C ) # ( B + C ) ) )
Theorem	reapneg 7697	Real negation respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
|- ( ( A e. RR /\ B e. RR ) -> ( A # B <-> -u A # -u B ) )
Theorem	reapcotr 7698	Real apartness is cotransitive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A # B -> ( A # C \/ B # C ) ) )
Theorem	remulext1 7699	Left extensionality for multiplication. (Contributed by Jim Kingdon, 19-Feb-2020.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A x. C ) # ( B x. C ) -> A # B ) )
Theorem	remulext2 7700	Right extensionality for real multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C x. A ) # ( C x. B ) -> A # B ) )