P. 84 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 8301-8400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	lttri3 8301	Tightness of real apartness. (Contributed by NM, 5-May-1999.)
|- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
Theorem	letri3 8302	Tightness of real apartness. (Contributed by NM, 14-May-1999.)
|- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> ( A <_ B /\ B <_ A ) ) )
Theorem	ltleletr 8303	Transitive law, weaker form of ( A < B /\ B <_ C ) -> A < C. (Contributed by AV, 14-Oct-2018.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < B /\ B <_ C ) -> A <_ C ) )
Theorem	letr 8304	Transitive law. (Contributed by NM, 12-Nov-1999.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A <_ B /\ B <_ C ) -> A <_ C ) )
Theorem	leid 8305	'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)
|- ( A e. RR -> A <_ A )
Theorem	ltne 8306	'Less than' implies not equal. See also ltap 8855 which is the same but for apartness. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 16-Sep-2015.)
|- ( ( A e. RR /\ A < B ) -> B =/= A )
Theorem	ltnsym 8307	'Less than' is not symmetric. (Contributed by NM, 8-Jan-2002.)
|- ( ( A e. RR /\ B e. RR ) -> ( A < B -> -. B < A ) )
Theorem	eqlelt 8308	Equality in terms of 'less than or equal to', 'less than'. (Contributed by NM, 7-Apr-2001.)
|- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> ( A <_ B /\ -. A < B ) ) )
Theorem	ltle 8309	'Less than' implies 'less than or equal to'. (Contributed by NM, 25-Aug-1999.)
|- ( ( A e. RR /\ B e. RR ) -> ( A < B -> A <_ B ) )
Theorem	lelttr 8310	Transitive law. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 23-May-1999.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A <_ B /\ B < C ) -> A < C ) )
Theorem	ltletr 8311	Transitive law. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 25-Aug-1999.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < B /\ B <_ C ) -> A < C ) )
Theorem	ltnsym2 8312	'Less than' is antisymmetric and irreflexive. (Contributed by NM, 13-Aug-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
|- ( ( A e. RR /\ B e. RR ) -> -. ( A < B /\ B < A ) )
Theorem	eqle 8313	Equality implies 'less than or equal to'. (Contributed by NM, 4-Apr-2005.)
|- ( ( A e. RR /\ A = B ) -> A <_ B )
Theorem	ltnri 8314	'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
|- A e. RR   =>    |- -. A < A
Theorem	eqlei 8315	Equality implies 'less than or equal to'. (Contributed by NM, 23-May-1999.) (Revised by Alexander van der Vekens, 20-Mar-2018.)
|- A e. RR   =>    |- ( A = B -> A <_ B )
Theorem	eqlei2 8316	Equality implies 'less than or equal to'. (Contributed by Alexander van der Vekens, 20-Mar-2018.)
|- A e. RR   =>    |- ( B = A -> B <_ A )
Theorem	gtneii 8317	'Less than' implies not equal. See also gtapii 8856 which is the same for apartness. (Contributed by Mario Carneiro, 30-Sep-2013.)
|- A e. RR   &    |- A < B   =>    |- B =/= A
Theorem	ltneii 8318	'Greater than' implies not equal. (Contributed by Mario Carneiro, 16-Sep-2015.)
|- A e. RR   &    |- A < B   =>    |- A =/= B
Theorem	lttri3i 8319	Tightness of real apartness. (Contributed by NM, 14-May-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( A = B <-> ( -. A < B /\ -. B < A ) )
Theorem	letri3i 8320	Tightness of real apartness. (Contributed by NM, 14-May-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( A = B <-> ( A <_ B /\ B <_ A ) )
Theorem	ltnsymi 8321	'Less than' is not symmetric. (Contributed by NM, 6-May-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( A < B -> -. B < A )
Theorem	lenlti 8322	'Less than or equal to' in terms of 'less than'. (Contributed by NM, 24-May-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( A <_ B <-> -. B < A )
Theorem	ltlei 8323	'Less than' implies 'less than or equal to'. (Contributed by NM, 14-May-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( A < B -> A <_ B )
Theorem	ltleii 8324	'Less than' implies 'less than or equal to' (inference). (Contributed by NM, 22-Aug-1999.)
|- A e. RR   &    |- B e. RR   &    |- A < B   =>    |- A <_ B
Theorem	ltnei 8325	'Less than' implies not equal. (Contributed by NM, 28-Jul-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( A < B -> B =/= A )
Theorem	lttri 8326	'Less than' is transitive. Theorem I.17 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( ( A < B /\ B < C ) -> A < C )
Theorem	lelttri 8327	'Less than or equal to', 'less than' transitive law. (Contributed by NM, 14-May-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( ( A <_ B /\ B < C ) -> A < C )
Theorem	ltletri 8328	'Less than', 'less than or equal to' transitive law. (Contributed by NM, 14-May-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( ( A < B /\ B <_ C ) -> A < C )
Theorem	letri 8329	'Less than or equal to' is transitive. (Contributed by NM, 14-May-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( ( A <_ B /\ B <_ C ) -> A <_ C )
Theorem	le2tri3i 8330	Extended trichotomy law for 'less than or equal to'. (Contributed by NM, 14-Aug-2000.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( ( A <_ B /\ B <_ C /\ C <_ A ) <-> ( A = B /\ B = C /\ C = A ) )
Theorem	mulgt0i 8331	The product of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( ( 0 < A /\ 0 < B ) -> 0 < ( A x. B ) )
Theorem	mulgt0ii 8332	The product of two positive numbers is positive. (Contributed by NM, 18-May-1999.)
|- A e. RR   &    |- B e. RR   &    |- 0 < A   &    |- 0 < B   =>    |- 0 < ( A x. B )
Theorem	ltnrd 8333	'Less than' is irreflexive. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> -. A < A )
Theorem	gtned 8334	'Less than' implies not equal. See also gtapd 8859 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 8335	'Greater than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> A < B )   =>    |- ( ph -> A =/= B )
Theorem	lttri3d 8336	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 8337	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 8338	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 8339	'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 8340	'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 8341	'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 8342	'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 8343	'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 8344	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 8345	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 8346	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 8347	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 8348	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 8349	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 8350	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 8351	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 8352	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 8353	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 8354	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 8355	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 8356	A theorem for complex numbers analogous the second Peano postulate peano2 4699. (Contributed by NM, 17-Aug-2005.)
|- ( A e. CC -> ( A + 1 ) e. CC )
Theorem	peano2re 8357	A theorem for reals analogous the second Peano postulate peano2 4699. (Contributed by NM, 5-Jul-2005.)
|- ( A e. RR -> ( A + 1 ) e. RR )
Theorem	addcom 8358	Addition commutes. (Contributed by Jim Kingdon, 17-Jan-2020.)
|- ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) )
Theorem	addrid 8359	0 is an additive identity. (Contributed by Jim Kingdon, 16-Jan-2020.)
|- ( A e. CC -> ( A + 0 ) = A )
Theorem	addlid 8360	0 is a left identity for addition. (Contributed by Scott Fenton, 3-Jan-2013.)
|- ( A e. CC -> ( 0 + A ) = A )
Theorem	readdcan 8361	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 8362	0 is its own additive identity. (Contributed by Scott Fenton, 3-Jan-2013.)
|- ( 0 + 0 ) = 0
Theorem	addridi 8363	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 8364	0 is a left identity for addition. (Contributed by NM, 3-Jan-2013.)
|- A e. CC   =>    |- ( 0 + A ) = A
Theorem	addcomi 8365	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 8366	Addition commutes. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- A e. CC   &    |- B e. CC   &    |- ( A + B ) = C   =>    |- ( B + A ) = C
Theorem	mul12i 8367	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 8368	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 8369	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 8370	0 is an additive identity. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A + 0 ) = A )
Theorem	addlidd 8371	0 is a left identity for addition. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( 0 + A ) = A )
Theorem	addcomd 8372	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 8373	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 8374	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 8375	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 8376	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 8377	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 8378	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 8379	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 8380	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 8381	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 8382	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 8383	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 8384	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 8385	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 8386	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 8387	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 8388	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 8389	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 8390	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 8391	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 8392	Extend class notation to include subtraction.
class -
Syntax	cneg 8393	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 8392 ( -) to prevent syntax ambiguity. For example, looking at the syntax definition co 6028, 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 8394*	Define subtraction. Theorem subval 8413 shows its value (and describes how this definition works), Theorem subaddi 8508 relates it to addition, and Theorems subcli 8497 and resubcli 8484 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 8395	Define the negative of a number (unary minus). We use different symbols for unary minus ( -u) and subtraction ( -) to prevent syntax ambiguity. See cneg 8393 for a discussion of this. (Contributed by NM, 10-Feb-1995.)
|- -u A = ( 0 - A )
Theorem	cnegexlem1 8396	Addition cancellation of a real number from two complex numbers. Lemma for cnegex 8399. (Contributed by Eric Schmidt, 22-May-2007.)
|- ( ( A e. RR /\ B e. CC /\ C e. CC ) -> ( ( A + B ) = ( A + C ) <-> B = C ) )
Theorem	cnegexlem2 8397	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 8399. (Contributed by Eric Schmidt, 22-May-2007.)
|- E. y e. RR ( _i x. y ) e. RR
Theorem	cnegexlem3 8398*	Existence of real number difference. Lemma for cnegex 8399. (Contributed by Eric Schmidt, 22-May-2007.)
|- ( ( b e. RR /\ y e. RR ) -> E. c e. RR ( b + c ) = y )
Theorem	cnegex 8399*	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 8400*	Existence of a left inverse for addition. (Contributed by Scott Fenton, 3-Jan-2013.)
|- ( A e. CC -> E. x e. CC ( x + A ) = 0 )