P. 81 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 8001-8100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	subled 8001	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 8002	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 8003	'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 8004	'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 8005	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 8006	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 8007	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 8008	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 8009	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 8010	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 8011	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 8012	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 8013	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 8014	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 8015	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 8016	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 8017	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 8018	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 8019	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 8020	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 8021	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 8022	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 8023	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 8024	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 8025	1 <_ 1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)
|- 1 <_ 1
Theorem	gt0add 8026	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 8027	Class of real apartness relation.
class #ℝ
Definition	df-reap 8028*	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 8035 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 8040). (Contributed by Jim Kingdon, 26-Jan-2020.)
|- #ℝ = { <. x , y >. | ( ( x e. RR /\ y e. RR ) /\ ( x < y \/ y < x ) ) }
Theorem	reapval 8029	Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 8041 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 8030	Real apartness is irreflexive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). Beyond the development of # itself, proofs should use apirr 8058 instead. (Contributed by Jim Kingdon, 26-Jan-2020.)
|- ( A e. RR -> -. A #ℝ A )
Theorem	recexre 8031*	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 8032	Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 8075. (Contributed by Jim Kingdon, 30-Jan-2020.) (New usage is discouraged.)
|- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> -. A #ℝ B ) )
Theorem	recexgt0 8033*	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 8034	Class of complex apartness relation.
class #
Definition	df-ap 8035*	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 8120 which says that a number apart from zero has a reciprocal). The defining characteristics of an apartness are irreflexivity (apirr 8058), symmetry (apsym 8059), and cotransitivity (apcotr 8060). Apartness implies negated equality, as seen at apne 8076, 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 8075). (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 8036	_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 8037	The imaginary unit _i is not a real number. (Contributed by NM, 6-May-1999.)
|- -. _i e. RR
Theorem	rimul 8038	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 8039	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 8040	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 8041	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 8042	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 8043	One is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
|- 1 # 0
Theorem	ltmul1a 8044	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 8045	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 8046	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 8047	Lemma for reapmul1 8048. (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 8048	Multiplication of both sides of real apartness by a real number apart from zero. Special case of apmul1 8229. (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 8049	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 8050	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 8051	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 8052	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 8053	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 8054	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 8055	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 8056	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 8057	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 8058	Apartness is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2020.)
|- ( A e. CC -> -. A # A )
Theorem	apsym 8059	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 8060	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 8061	Addition respects apartness. Analogue of addcan 7641 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 8062	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 8063	Strong extensionality for addition. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5643. 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 8064	Negation respects apartness. (Contributed by Jim Kingdon, 14-Feb-2020.)
|- ( ( A e. CC /\ B e. CC ) -> ( A # B <-> -u A # -u B ) )
Theorem	mulext1 8065	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 8066	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 8067	Strong extensionality for multiplication. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5643. 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 8068	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 8069	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 8070	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 8071	A square is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> 0 <_ ( A x. A ) )
Theorem	mulge0 8072	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 8073	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 8074	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 8075	Complex apartness is tight. (Contributed by Jim Kingdon, 21-Feb-2020.)
|- ( ( A e. CC /\ B e. CC ) -> ( A = B <-> -. A # B ) )
Theorem	apne 8076	Apartness implies negated equality. We cannot in general prove the converse, 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	leltap 8077	'<_' 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 8078	Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
|- ( ( A e. RR /\ 0 < A ) -> A # 0 )
Theorem	gt0ap0i 8079	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 8080	Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
|- A e. RR   &    |- 0 < A   =>    |- A # 0
Theorem	gt0ap0d 8081	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 8082	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	ltleap 8083	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 8084	'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
|- ( ( A e. RR /\ B e. RR /\ A < B ) -> B # A )
Theorem	gtapii 8085	'Greater than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
|- A e. RR   &    |- B e. RR   &    |- A < B   =>    |- B # A
Theorem	ltapii 8086	'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
|- A e. RR   &    |- B e. RR   &    |- A < B   =>    |- A # B
Theorem	ltapi 8087	'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
|- A e. RR   &    |- B e. RR   =>    |- ( A < B -> B # A )
Theorem	gtapd 8088	'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 8089	'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 8090	'<_' 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 8091	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 8092	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	subap0d 8093	Two numbers apart from each other have difference apart from zero. (Contributed by Jim Kingdon, 12-Aug-2021.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> A # B )   =>    |- ( ph -> ( A - B ) # 0 )
3.3.7  Reciprocals
Theorem	recextlem1 8094	Lemma for recexap 8096. (Contributed by Eric Schmidt, 23-May-2007.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( A + ( _i x. B ) ) x. ( A - ( _i x. B ) ) ) = ( ( A x. A ) + ( B x. B ) ) )
Theorem	recexaplem2 8095	Lemma for recexap 8096. (Contributed by Jim Kingdon, 20-Feb-2020.)
|- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> ( ( A x. A ) + ( B x. B ) ) # 0 )
Theorem	recexap 8096*	Existence of reciprocal of nonzero complex number. (Contributed by Jim Kingdon, 20-Feb-2020.)
|- ( ( A e. CC /\ A # 0 ) -> E. x e. CC ( A x. x ) = 1 )
Theorem	mulap0 8097	The product of two numbers apart from zero is apart from zero. Lemma 2.15 of [Geuvers], p. 6. (Contributed by Jim Kingdon, 22-Feb-2020.)
|- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> ( A x. B ) # 0 )
Theorem	mulap0b 8098	The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( A # 0 /\ B # 0 ) <-> ( A x. B ) # 0 ) )
Theorem	mulap0i 8099	The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 23-Feb-2020.)
|- A e. CC   &    |- B e. CC   &    |- A # 0   &    |- B # 0   =>    |- ( A x. B ) # 0
Theorem	mulap0bd 8100	The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( ( A # 0 /\ B # 0 ) <-> ( A x. B ) # 0 ) )