P. 85 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 8401-8500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	leaddsub2d 8401	'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 8402	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 8403	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 8404	'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 8405	'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 8406	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 8407	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 8408	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 8409	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 8410	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 8411	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 8412	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 8413	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 8414	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 8415	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 8416	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 8417	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 8418	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 8419	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 8420	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 8421	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 8422	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 8423	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 8424	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 8425	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 8426	1 <_ 1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)
|- 1 <_ 1
Theorem	gt0add 8427	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 ) )
4.3.5  Real Apartness
Syntax	creap 8428	Class of real apartness relation.
class #ℝ
Definition	df-reap 8429*	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 8436 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 8441). (Contributed by Jim Kingdon, 26-Jan-2020.)
|- #ℝ = { <. x , y >. | ( ( x e. RR /\ y e. RR ) /\ ( x < y \/ y < x ) ) }
Theorem	reapval 8430	Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 8442 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 8431	Real apartness is irreflexive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). Beyond the development of # itself, proofs should use apirr 8459 instead. (Contributed by Jim Kingdon, 26-Jan-2020.)
|- ( A e. RR -> -. A #ℝ A )
Theorem	recexre 8432*	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 8433	Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 8476. (Contributed by Jim Kingdon, 30-Jan-2020.) (New usage is discouraged.)
|- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> -. A #ℝ B ) )
Theorem	recexgt0 8434*	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 ) )
4.3.6  Complex Apartness
Syntax	cap 8435	Class of complex apartness relation.
class #
Definition	df-ap 8436*	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 8531 which says that a number apart from zero has a reciprocal). The defining characteristics of an apartness are irreflexivity (apirr 8459), symmetry (apsym 8460), and cotransitivity (apcotr 8461). Apartness implies negated equality, as seen at apne 8477, 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 8476). (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 8437	_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 8438	The imaginary unit _i is not a real number. (Contributed by NM, 6-May-1999.)
|- -. _i e. RR
Theorem	rimul 8439	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 8440	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 8441	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 8442	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 8443	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 8444	One is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
|- 1 # 0
Theorem	ltmul1a 8445	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 8446	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 8447	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 8448	Lemma for reapmul1 8449. (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 8449	Multiplication of both sides of real apartness by a real number apart from zero. Special case of apmul1 8640. (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 8450	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 8451	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 8452	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 8453	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 8454	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 ) )
Theorem	apsqgt0 8455	The square of a real number apart from zero is positive. (Contributed by Jim Kingdon, 7-Feb-2020.)
|- ( ( A e. RR /\ A # 0 ) -> 0 < ( A x. A ) )
Theorem	cru 8456	The representation of complex numbers in terms of real and imaginary parts is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A + ( _i x. B ) ) = ( C + ( _i x. D ) ) <-> ( A = C /\ B = D ) ) )
Theorem	apreim 8457	Complex apartness in terms of real and imaginary parts. (Contributed by Jim Kingdon, 12-Feb-2020.)
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A + ( _i x. B ) ) # ( C + ( _i x. D ) ) <-> ( A # C \/ B # D ) ) )
Theorem	mulreim 8458	Complex multiplication in terms of real and imaginary parts. (Contributed by Jim Kingdon, 23-Feb-2020.)
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A + ( _i x. B ) ) x. ( C + ( _i x. D ) ) ) = ( ( ( A x. C ) + -u ( B x. D ) ) + ( _i x. ( ( C x. B ) + ( D x. A ) ) ) ) )
Theorem	apirr 8459	Apartness is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2020.)
|- ( A e. CC -> -. A # A )
Theorem	apsym 8460	Apartness is symmetric. This theorem for real numbers is part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.)
|- ( ( A e. CC /\ B e. CC ) -> ( A # B <-> B # A ) )
Theorem	apcotr 8461	Apartness is cotransitive. (Contributed by Jim Kingdon, 16-Feb-2020.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A # B -> ( A # C \/ B # C ) ) )
Theorem	apadd1 8462	Addition respects apartness. Analogue of addcan 8034 for apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A # B <-> ( A + C ) # ( B + C ) ) )
Theorem	apadd2 8463	Addition respects apartness. (Contributed by Jim Kingdon, 16-Feb-2020.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A # B <-> ( C + A ) # ( C + B ) ) )
Theorem	addext 8464	Strong extensionality for addition. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5823. For us, it is proved a different way. (Contributed by Jim Kingdon, 15-Feb-2020.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) # ( C + D ) -> ( A # C \/ B # D ) ) )
Theorem	apneg 8465	Negation respects apartness. (Contributed by Jim Kingdon, 14-Feb-2020.)
|- ( ( A e. CC /\ B e. CC ) -> ( A # B <-> -u A # -u B ) )
Theorem	mulext1 8466	Left extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. C ) # ( B x. C ) -> A # B ) )
Theorem	mulext2 8467	Right extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( C x. A ) # ( C x. B ) -> A # B ) )
Theorem	mulext 8468	Strong extensionality for multiplication. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5823. For us, it is proved a different way. (Contributed by Jim Kingdon, 23-Feb-2020.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A x. B ) # ( C x. D ) -> ( A # C \/ B # D ) ) )
Theorem	mulap0r 8469	A product apart from zero. Lemma 2.13 of [Geuvers], p. 6. (Contributed by Jim Kingdon, 24-Feb-2020.)
|- ( ( A e. CC /\ B e. CC /\ ( A x. B ) # 0 ) -> ( A # 0 /\ B # 0 ) )
Theorem	msqge0 8470	A square is nonnegative. Lemma 2.35 of [Geuvers], p. 9. (Contributed by NM, 23-May-2007.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( A e. RR -> 0 <_ ( A x. A ) )
Theorem	msqge0i 8471	A square is nonnegative. (Contributed by NM, 14-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- A e. RR   =>    |- 0 <_ ( A x. A )
Theorem	msqge0d 8472	A square is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> 0 <_ ( A x. A ) )
Theorem	mulge0 8473	The product of two nonnegative numbers is nonnegative. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> 0 <_ ( A x. B ) )
Theorem	mulge0i 8474	The product of two nonnegative numbers is nonnegative. (Contributed by NM, 30-Jul-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( ( 0 <_ A /\ 0 <_ B ) -> 0 <_ ( A x. B ) )
Theorem	mulge0d 8475	The product of two 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 x. B ) )
Theorem	apti 8476	Complex apartness is tight. (Contributed by Jim Kingdon, 21-Feb-2020.)
|- ( ( A e. CC /\ B e. CC ) -> ( A = B <-> -. A # B ) )
Theorem	apne 8477	Apartness implies negated equality. We cannot in general prove the converse (as shown at neapmkv 13579), which is the whole point of having separate notations for apartness and negated equality. (Contributed by Jim Kingdon, 21-Feb-2020.)
|- ( ( A e. CC /\ B e. CC ) -> ( A # B -> A =/= B ) )
Theorem	apcon4bid 8478	Contrapositive law deduction for apartness. (Contributed by Jim Kingdon, 31-Jul-2023.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> D e. CC )   &    |- ( ph -> ( A # B <-> C # D ) )   =>    |- ( ph -> ( A = B <-> C = D ) )
Theorem	leltap 8479	<_ implies 'less than' is 'apart'. (Contributed by Jim Kingdon, 13-Aug-2021.)
|- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( A < B <-> B # A ) )
Theorem	gt0ap0 8480	Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
|- ( ( A e. RR /\ 0 < A ) -> A # 0 )
Theorem	gt0ap0i 8481	Positive means apart from zero (useful for ordering theorems involving division). (Contributed by Jim Kingdon, 27-Feb-2020.)
|- A e. RR   =>    |- ( 0 < A -> A # 0 )
Theorem	gt0ap0ii 8482	Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
|- A e. RR   &    |- 0 < A   =>    |- A # 0
Theorem	gt0ap0d 8483	Positive implies apart from zero. Because of the way we define #, A must be an element of RR, not just RR*. (Contributed by Jim Kingdon, 27-Feb-2020.)
|- ( ph -> A e. RR )   &    |- ( ph -> 0 < A )   =>    |- ( ph -> A # 0 )
Theorem	negap0 8484	A number is apart from zero iff its negative is apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
|- ( A e. CC -> ( A # 0 <-> -u A # 0 ) )
Theorem	negap0d 8485	The negative of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 25-Feb-2024.)
|- ( ph -> A e. CC )   &    |- ( ph -> A # 0 )   =>    |- ( ph -> -u A # 0 )
Theorem	ltleap 8486	Less than in terms of non-strict order and apartness. (Contributed by Jim Kingdon, 28-Feb-2020.)
|- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> ( A <_ B /\ A # B ) ) )
Theorem	ltap 8487	'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
|- ( ( A e. RR /\ B e. RR /\ A < B ) -> B # A )
Theorem	gtapii 8488	'Greater than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
|- A e. RR   &    |- B e. RR   &    |- A < B   =>    |- B # A
Theorem	ltapii 8489	'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
|- A e. RR   &    |- B e. RR   &    |- A < B   =>    |- A # B
Theorem	ltapi 8490	'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
|- A e. RR   &    |- B e. RR   =>    |- ( A < B -> B # A )
Theorem	gtapd 8491	'Greater than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A < B )   =>    |- ( ph -> B # A )
Theorem	ltapd 8492	'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A < B )   =>    |- ( ph -> A # B )
Theorem	leltapd 8493	<_ implies 'less than' is 'apart'. (Contributed by Jim Kingdon, 13-Aug-2021.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A <_ B )   =>    |- ( ph -> ( A < B <-> B # A ) )
Theorem	ap0gt0 8494	A nonnegative number is apart from zero if and only if it is positive. (Contributed by Jim Kingdon, 11-Aug-2021.)
|- ( ( A e. RR /\ 0 <_ A ) -> ( A # 0 <-> 0 < A ) )
Theorem	ap0gt0d 8495	A nonzero nonnegative number is positive. (Contributed by Jim Kingdon, 11-Aug-2021.)
|- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> A # 0 )   =>    |- ( ph -> 0 < A )
Theorem	apsub1 8496	Subtraction respects apartness. Analogue of subcan2 8079 for apartness. (Contributed by Jim Kingdon, 6-Jan-2022.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A # B <-> ( A - C ) # ( B - C ) ) )
Theorem	subap0 8497	Two numbers being apart is equivalent to their difference being apart from zero. (Contributed by Jim Kingdon, 25-Dec-2022.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) # 0 <-> A # B ) )
Theorem	subap0d 8498	Two numbers apart from each other have difference apart from zero. (Contributed by Jim Kingdon, 12-Aug-2021.) (Proof shortened by BJ, 15-Aug-2024.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> A # B )   =>    |- ( ph -> ( A - B ) # 0 )
Theorem	cnstab 8499	Equality of complex numbers is stable. Stability here means -. -. A = B -> A = B as defined at df-stab 817. This theorem for real numbers is Proposition 5.2 of [BauerHanson], p. 27. (Contributed by Jim Kingdon, 1-Aug-2023.) (Proof shortened by BJ, 15-Aug-2024.)
|- ( ( A e. CC /\ B e. CC ) -> STAB A = B )
Theorem	aprcl 8500	Reverse closure for apartness. (Contributed by Jim Kingdon, 19-Dec-2023.)
|- ( A # B -> ( A e. CC /\ B e. CC ) )