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	ltnrd 8401	'Less than' is irreflexive. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> -. A < A )
Theorem	gtned 8402	'Less than' implies not equal. See also gtapd 8928 which is the same but for apartness. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> A < B )   =>    |- ( ph -> B =/= A )
Theorem	ltned 8403	'Greater than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> A < B )   =>    |- ( ph -> A =/= B )
Theorem	lttri3d 8404	Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
Theorem	letri3d 8405	Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( A = B <-> ( A <_ B /\ B <_ A ) ) )
Theorem	eqleltd 8406	Equality in terms of 'less than or equal to', 'less than'. (Contributed by NM, 7-Apr-2001.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( A = B <-> ( A <_ B /\ -. A < B ) ) )
Theorem	lenltd 8407	'Less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( A <_ B <-> -. B < A ) )
Theorem	ltled 8408	'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A < B )   =>    |- ( ph -> A <_ B )
Theorem	ltnsymd 8409	'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A < B )   =>    |- ( ph -> -. B < A )
Theorem	nltled 8410	'Not less than ' implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> -. B < A )   =>    |- ( ph -> A <_ B )
Theorem	lensymd 8411	'Less than or equal to' implies 'not less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A <_ B )   =>    |- ( ph -> -. B < A )
Theorem	mulgt0d 8412	The product of two 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 x. B ) )
Theorem	letrd 8413	Transitive law deduction for 'less than or equal to'. (Contributed by NM, 20-May-2005.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> A <_ B )   &    |- ( ph -> B <_ C )   =>    |- ( ph -> A <_ C )
Theorem	lelttrd 8414	Transitive law deduction for 'less than or equal to', 'less than'. (Contributed by NM, 8-Jan-2006.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> A <_ B )   &    |- ( ph -> B < C )   =>    |- ( ph -> A < C )
Theorem	lttrd 8415	Transitive law deduction for 'less than'. (Contributed by NM, 9-Jan-2006.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> B < C )   =>    |- ( ph -> A < C )
Theorem	0lt1 8416	0 is less than 1. Theorem I.21 of [Apostol] p. 20. Part of definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 17-Jan-1997.)
|- 0 < 1
Theorem	ltntri 8417	Negative trichotomy property for real numbers. It is well known that we cannot prove real number trichotomy, A < B \/ A = B \/ B < A. Does that mean there is a pair of real numbers where none of those hold (that is, where we can refute each of those three relationships)? Actually, no, as shown here. This is another example of distinguishing between being unable to prove something, or being able to refute it. (Contributed by Jim Kingdon, 13-Aug-2023.)
|- ( ( A e. RR /\ B e. RR ) -> -. ( -. A < B /\ -. A = B /\ -. B < A ) )
4.2.5  Initial properties of the complex numbers
Theorem	mul12 8418	Commutative/associative law for multiplication. (Contributed by NM, 30-Apr-2005.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( B x. C ) ) = ( B x. ( A x. C ) ) )
Theorem	mul32 8419	Commutative/associative law. (Contributed by NM, 8-Oct-1999.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) x. C ) = ( ( A x. C ) x. B ) )
Theorem	mul31 8420	Commutative/associative law. (Contributed by Scott Fenton, 3-Jan-2013.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) x. C ) = ( ( C x. B ) x. A ) )
Theorem	mul4 8421	Rearrangement of 4 factors. (Contributed by NM, 8-Oct-1999.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A x. B ) x. ( C x. D ) ) = ( ( A x. C ) x. ( B x. D ) ) )
Theorem	muladd11 8422	A simple product of sums expansion. (Contributed by NM, 21-Feb-2005.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. ( 1 + B ) ) = ( ( 1 + A ) + ( B + ( A x. B ) ) ) )
Theorem	1p1times 8423	Two times a number. (Contributed by NM, 18-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( A e. CC -> ( ( 1 + 1 ) x. A ) = ( A + A ) )
Theorem	peano2cn 8424	A theorem for complex numbers analogous the second Peano postulate peano2 4722. (Contributed by NM, 17-Aug-2005.)
|- ( A e. CC -> ( A + 1 ) e. CC )
Theorem	peano2re 8425	A theorem for reals analogous the second Peano postulate peano2 4722. (Contributed by NM, 5-Jul-2005.)
|- ( A e. RR -> ( A + 1 ) e. RR )
Theorem	addcom 8426	Addition commutes. (Contributed by Jim Kingdon, 17-Jan-2020.)
|- ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) )
Theorem	addrid 8427	0 is an additive identity. (Contributed by Jim Kingdon, 16-Jan-2020.)
|- ( A e. CC -> ( A + 0 ) = A )
Theorem	addlid 8428	0 is a left identity for addition. (Contributed by Scott Fenton, 3-Jan-2013.)
|- ( A e. CC -> ( 0 + A ) = A )
Theorem	readdcan 8429	Cancellation law for addition over the reals. (Contributed by Scott Fenton, 3-Jan-2013.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C + A ) = ( C + B ) <-> A = B ) )
Theorem	00id 8430	0 is its own additive identity. (Contributed by Scott Fenton, 3-Jan-2013.)
|- ( 0 + 0 ) = 0
Theorem	addridi 8431	0 is an additive identity. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)
|- A e. CC   =>    |- ( A + 0 ) = A
Theorem	addlidi 8432	0 is a left identity for addition. (Contributed by NM, 3-Jan-2013.)
|- A e. CC   =>    |- ( 0 + A ) = A
Theorem	addcomi 8433	Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)
|- A e. CC   &    |- B e. CC   =>    |- ( A + B ) = ( B + A )
Theorem	addcomli 8434	Addition commutes. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- A e. CC   &    |- B e. CC   &    |- ( A + B ) = C   =>    |- ( B + A ) = C
Theorem	mul12i 8435	Commutative/associative law that swaps the first two factors in a triple product. (Contributed by NM, 11-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   =>    |- ( A x. ( B x. C ) ) = ( B x. ( A x. C ) )
Theorem	mul32i 8436	Commutative/associative law that swaps the last two factors in a triple product. (Contributed by NM, 11-May-1999.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   =>    |- ( ( A x. B ) x. C ) = ( ( A x. C ) x. B )
Theorem	mul4i 8437	Rearrangement of 4 factors. (Contributed by NM, 16-Feb-1995.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   &    |- D e. CC   =>    |- ( ( A x. B ) x. ( C x. D ) ) = ( ( A x. C ) x. ( B x. D ) )
Theorem	addridd 8438	0 is an additive identity. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A + 0 ) = A )
Theorem	addlidd 8439	0 is a left identity for addition. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( 0 + A ) = A )
Theorem	addcomd 8440	Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( A + B ) = ( B + A ) )
Theorem	mul12d 8441	Commutative/associative law that swaps the first two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( A x. ( B x. C ) ) = ( B x. ( A x. C ) ) )
Theorem	mul32d 8442	Commutative/associative law that swaps the last two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( ( A x. B ) x. C ) = ( ( A x. C ) x. B ) )
Theorem	mul31d 8443	Commutative/associative law. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( ( A x. B ) x. C ) = ( ( C x. B ) x. A ) )
Theorem	mul4d 8444	Rearrangement of 4 factors. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> D e. CC )   =>    |- ( ph -> ( ( A x. B ) x. ( C x. D ) ) = ( ( A x. C ) x. ( B x. D ) ) )
Theorem	muladd11r 8445	A simple product of sums expansion. (Contributed by AV, 30-Jul-2021.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( A + 1 ) x. ( B + 1 ) ) = ( ( ( A x. B ) + ( A + B ) ) + 1 ) )
Theorem	comraddd 8446	Commute RHS addition, in deduction form. (Contributed by David A. Wheeler, 11-Oct-2018.)
|- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> A = ( B + C ) )   =>    |- ( ph -> A = ( C + B ) )
4.3  Real and complex numbers - basic operations
4.3.1  Addition
Theorem	add12 8447	Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 11-May-2004.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( B + C ) ) = ( B + ( A + C ) ) )
Theorem	add32 8448	Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 13-Nov-1999.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( ( A + C ) + B ) )
Theorem	add32r 8449	Commutative/associative law that swaps the last two terms in a triple sum, rearranging the parentheses. (Contributed by Paul Chapman, 18-May-2007.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( B + C ) ) = ( ( A + C ) + B ) )
Theorem	add4 8450	Rearrangement of 4 terms in a sum. (Contributed by NM, 13-Nov-1999.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( B + D ) ) )
Theorem	add42 8451	Rearrangement of 4 terms in a sum. (Contributed by NM, 12-May-2005.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( D + B ) ) )
Theorem	add12i 8452	Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   =>    |- ( A + ( B + C ) ) = ( B + ( A + C ) )
Theorem	add32i 8453	Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   =>    |- ( ( A + B ) + C ) = ( ( A + C ) + B )
Theorem	add4i 8454	Rearrangement of 4 terms in a sum. (Contributed by NM, 9-May-1999.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   &    |- D e. CC   =>    |- ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( B + D ) )
Theorem	add42i 8455	Rearrangement of 4 terms in a sum. (Contributed by NM, 22-Aug-1999.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   &    |- D e. CC   =>    |- ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( D + B ) )
Theorem	add12d 8456	Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( A + ( B + C ) ) = ( B + ( A + C ) ) )
Theorem	add32d 8457	Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( ( A + B ) + C ) = ( ( A + C ) + B ) )
Theorem	add4d 8458	Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> D e. CC )   =>    |- ( ph -> ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( B + D ) ) )
Theorem	add42d 8459	Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> D e. CC )   =>    |- ( ph -> ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( D + B ) ) )
4.3.2  Subtraction
Syntax	cmin 8460	Extend class notation to include subtraction.
class -
Syntax	cneg 8461	Extend class notation to include unary minus. The symbol -u is not a class by itself but part of a compound class definition. We do this rather than making it a formal function since it is so commonly used. Note: We use different symbols for unary minus ( -u) and subtraction cmin 8460 ( -) to prevent syntax ambiguity. For example, looking at the syntax definition co 6058, if we used the same symbol then " ( - A - B ) " could mean either " - A " minus " B", or it could represent the (meaningless) operation of classes " - " and " - B " connected with "operation" " A". On the other hand, " ( -u A - B ) " is unambiguous.
class -u A
Definition	df-sub 8462*	Define subtraction. Theorem subval 8481 shows its value (and describes how this definition works), Theorem subaddi 8576 relates it to addition, and Theorems subcli 8565 and resubcli 8552 prove its closure laws. (Contributed by NM, 26-Nov-1994.)
|- - = ( x e. CC , y e. CC |-> ( iota_ z e. CC ( y + z ) = x ) )
Definition	df-neg 8463	Define the negative of a number (unary minus). We use different symbols for unary minus ( -u) and subtraction ( -) to prevent syntax ambiguity. See cneg 8461 for a discussion of this. (Contributed by NM, 10-Feb-1995.)
|- -u A = ( 0 - A )
Theorem	cnegexlem1 8464	Addition cancellation of a real number from two complex numbers. Lemma for cnegex 8467. (Contributed by Eric Schmidt, 22-May-2007.)
|- ( ( A e. RR /\ B e. CC /\ C e. CC ) -> ( ( A + B ) = ( A + C ) <-> B = C ) )
Theorem	cnegexlem2 8465	Existence of a real number which produces a real number when multiplied by _i. (Hint: zero is such a number, although we don't need to prove that yet). Lemma for cnegex 8467. (Contributed by Eric Schmidt, 22-May-2007.)
|- E. y e. RR ( _i x. y ) e. RR
Theorem	cnegexlem3 8466*	Existence of real number difference. Lemma for cnegex 8467. (Contributed by Eric Schmidt, 22-May-2007.)
|- ( ( b e. RR /\ y e. RR ) -> E. c e. RR ( b + c ) = y )
Theorem	cnegex 8467*	Existence of the negative of a complex number. (Contributed by Eric Schmidt, 21-May-2007.)
|- ( A e. CC -> E. x e. CC ( A + x ) = 0 )
Theorem	cnegex2 8468*	Existence of a left inverse for addition. (Contributed by Scott Fenton, 3-Jan-2013.)
|- ( A e. CC -> E. x e. CC ( x + A ) = 0 )
Theorem	addcan 8469	Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) = ( A + C ) <-> B = C ) )
Theorem	addcan2 8470	Cancellation law for addition. (Contributed by NM, 30-Jul-2004.) (Revised by Scott Fenton, 3-Jan-2013.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + C ) = ( B + C ) <-> A = B ) )
Theorem	addcani 8471	Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 27-Oct-1999.) (Revised by Scott Fenton, 3-Jan-2013.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   =>    |- ( ( A + B ) = ( A + C ) <-> B = C )
Theorem	addcan2i 8472	Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 14-May-2003.) (Revised by Scott Fenton, 3-Jan-2013.)
|- A e. CC   &    |- B e. CC   &    |- C e. CC   =>    |- ( ( A + C ) = ( B + C ) <-> A = B )
Theorem	addcand 8473	Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( ( A + B ) = ( A + C ) <-> B = C ) )
Theorem	addcan2d 8474	Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( ( A + C ) = ( B + C ) <-> A = B ) )
Theorem	addcanad 8475	Cancelling a term on the left-hand side of a sum in an equality. Consequence of addcand 8473. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> ( A + B ) = ( A + C ) )   =>    |- ( ph -> B = C )
Theorem	addcan2ad 8476	Cancelling a term on the right-hand side of a sum in an equality. Consequence of addcan2d 8474. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> ( A + C ) = ( B + C ) )   =>    |- ( ph -> A = B )
Theorem	addneintrd 8477	Introducing a term on the left-hand side of a sum in a negated equality. Contrapositive of addcanad 8475. Consequence of addcand 8473. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> B =/= C )   =>    |- ( ph -> ( A + B ) =/= ( A + C ) )
Theorem	addneintr2d 8478	Introducing a term on the right-hand side of a sum in a negated equality. Contrapositive of addcan2ad 8476. Consequence of addcan2d 8474. (Contributed by David Moews, 28-Feb-2017.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> A =/= B )   =>    |- ( ph -> ( A + C ) =/= ( B + C ) )
Theorem	0cnALT 8479	Alternate proof of 0cn 8282. (Contributed by NM, 19-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
|- 0 e. CC
Theorem	negeu 8480*	Existential uniqueness of negatives. Theorem I.2 of [Apostol] p. 18. (Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC ) -> E! x e. CC ( A + x ) = B )
Theorem	subval 8481*	Value of subtraction, which is the (unique) element x such that B + x = A. (Contributed by NM, 4-Aug-2007.) (Revised by Mario Carneiro, 2-Nov-2013.)
|- ( ( A e. CC /\ B e. CC ) -> ( A - B ) = ( iota_ x e. CC ( B + x ) = A ) )
Theorem	negeq 8482	Equality theorem for negatives. (Contributed by NM, 10-Feb-1995.)
|- ( A = B -> -u A = -u B )
Theorem	negeqi 8483	Equality inference for negatives. (Contributed by NM, 14-Feb-1995.)
|- A = B   =>    |- -u A = -u B
Theorem	negeqd 8484	Equality deduction for negatives. (Contributed by NM, 14-May-1999.)
|- ( ph -> A = B )   =>    |- ( ph -> -u A = -u B )
Theorem	nfnegd 8485	Deduction version of nfneg 8486. (Contributed by NM, 29-Feb-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- ( ph -> F/_ x A )   =>    |- ( ph -> F/_ x -u A )
Theorem	nfneg 8486	Bound-variable hypothesis builder for the negative of a complex number. (Contributed by NM, 12-Jun-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- F/_ x A   =>    |- F/_ x -u A
Theorem	csbnegg 8487	Move class substitution in and out of the negative of a number. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|- ( A e. V -> [_ A / x ]_ -u B = -u [_ A / x ]_ B )
Theorem	subcl 8488	Closure law for subtraction. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 21-Dec-2013.)
|- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
Theorem	negcl 8489	Closure law for negative. (Contributed by NM, 6-Aug-2003.)
|- ( A e. CC -> -u A e. CC )
Theorem	negicn 8490	-u _i is a complex number (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
|- -u _i e. CC
Theorem	subf 8491	Subtraction is an operation on the complex numbers. (Contributed by NM, 4-Aug-2007.) (Revised by Mario Carneiro, 16-Nov-2013.)
|- - : ( CC X. CC ) --> CC
Theorem	subadd 8492	Relationship between subtraction and addition. (Contributed by NM, 20-Jan-1997.) (Revised by Mario Carneiro, 21-Dec-2013.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) = C <-> ( B + C ) = A ) )
Theorem	subadd2 8493	Relationship between subtraction and addition. (Contributed by Scott Fenton, 5-Jul-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) = C <-> ( C + B ) = A ) )
Theorem	subsub23 8494	Swap subtrahend and result of subtraction. (Contributed by NM, 14-Dec-2007.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) = C <-> ( A - C ) = B ) )
Theorem	pncan 8495	Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) - B ) = A )
Theorem	pncan2 8496	Cancellation law for subtraction. (Contributed by NM, 17-Apr-2005.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) - A ) = B )
Theorem	pncan3 8497	Subtraction and addition of equals. (Contributed by NM, 14-Mar-2005.)
|- ( ( A e. CC /\ B e. CC ) -> ( A + ( B - A ) ) = B )
Theorem	npcan 8498	Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) + B ) = A )
Theorem	addsubass 8499	Associative-type law for addition and subtraction. (Contributed by NM, 6-Aug-2003.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - C ) = ( A + ( B - C ) ) )
Theorem	addsub 8500	Law for addition and subtraction. (Contributed by NM, 19-Aug-2001.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - C ) = ( ( A - C ) + B ) )