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	axltadd 8401	Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-ltadd 8301 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B -> ( C + A ) < ( C + B ) ) )
Theorem	axapti 8402	Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-apti 8300 with ordering on the extended reals.) (Contributed by Jim Kingdon, 29-Jan-2020.)
|- ( ( A e. RR /\ B e. RR /\ -. ( A < B \/ B < A ) ) -> A = B )
Theorem	axmulgt0 8403	The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-mulgt0 8302 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)
|- ( ( A e. RR /\ B e. RR ) -> ( ( 0 < A /\ 0 < B ) -> 0 < ( A x. B ) ) )
Theorem	axsuploc 8404*	An inhabited, bounded-above, located set of reals has a supremum. Axiom for real and complex numbers, derived from ZF set theory. (This restates ax-pre-suploc 8306 with ordering on the extended reals.) (Contributed by Jim Kingdon, 30-Jan-2024.)
|- ( ( ( A C_ RR /\ E. x x e. A ) /\ ( E. x e. RR A. y e. A y < x /\ A. x e. RR A. y e. RR ( x < y -> ( E. z e. A x < z \/ A. z e. A z < y ) ) ) ) -> E. x e. RR ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. z e. A y < z ) ) )
4.2.4  Ordering on reals
Theorem	lttr 8405	Alias for axlttrn 8400, for naming consistency with lttri 8437. New proofs should generally use this instead of ax-pre-lttrn 8299. (Contributed by NM, 10-Mar-2008.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < B /\ B < C ) -> A < C ) )
Theorem	mulgt0 8406	The product of two positive numbers is positive. (Contributed by NM, 10-Mar-2008.)
|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 < ( A x. B ) )
Theorem	lenlt 8407	'Less than or equal to' expressed in terms of 'less than'. Part of definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 13-May-1999.)
|- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> -. B < A ) )
Theorem	ltnr 8408	'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
|- ( A e. RR -> -. A < A )
Theorem	ltso 8409	'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.)
|- < Or RR
Theorem	gtso 8410	'Greater than' is a strict ordering. (Contributed by JJ, 11-Oct-2018.)
|- `' < Or RR
Theorem	lttri3 8411	Tightness of real apartness. (Contributed by NM, 5-May-1999.)
|- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
Theorem	letri3 8412	Tightness of real apartness. (Contributed by NM, 14-May-1999.)
|- ( ( A e. RR /\ B e. RR ) -> ( A = B <-> ( A <_ B /\ B <_ A ) ) )
Theorem	ltleletr 8413	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 8414	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 8415	'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)
|- ( A e. RR -> A <_ A )
Theorem	ltne 8416	'Less than' implies not equal. See also ltap 8969 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 8417	'Less than' is not symmetric. (Contributed by NM, 8-Jan-2002.)
|- ( ( A e. RR /\ B e. RR ) -> ( A < B -> -. B < A ) )
Theorem	eqlelt 8418	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 8419	'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 8420	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 8421	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 8422	'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 8423	Equality implies 'less than or equal to'. (Contributed by NM, 4-Apr-2005.)
|- ( ( A e. RR /\ A = B ) -> A <_ B )
Theorem	eqled 8424	Equality implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. RR )   &    |- ( ph -> A = B )   =>    |- ( ph -> A <_ B )
Theorem	ltnri 8425	'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
|- A e. RR   =>    |- -. A < A
Theorem	eqlei 8426	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 8427	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 8428	'Less than' implies not equal. See also gtapii 8970 which is the same for apartness. (Contributed by Mario Carneiro, 30-Sep-2013.)
|- A e. RR   &    |- A < B   =>    |- B =/= A
Theorem	ltneii 8429	'Greater than' implies not equal. (Contributed by Mario Carneiro, 16-Sep-2015.)
|- A e. RR   &    |- A < B   =>    |- A =/= B
Theorem	lttri3i 8430	Tightness of real apartness. (Contributed by NM, 14-May-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( A = B <-> ( -. A < B /\ -. B < A ) )
Theorem	letri3i 8431	Tightness of real apartness. (Contributed by NM, 14-May-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( A = B <-> ( A <_ B /\ B <_ A ) )
Theorem	ltnsymi 8432	'Less than' is not symmetric. (Contributed by NM, 6-May-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( A < B -> -. B < A )
Theorem	lenlti 8433	'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 8434	'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 8435	'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 8436	'Less than' implies not equal. (Contributed by NM, 28-Jul-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( A < B -> B =/= A )
Theorem	lttri 8437	'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 8438	'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 8439	'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 8440	'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 8441	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 8442	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 8443	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 8444	'Less than' is irreflexive. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> -. A < A )
Theorem	gtned 8445	'Less than' implies not equal. See also gtapd 8973 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 8446	'Greater than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> A < B )   =>    |- ( ph -> A =/= B )
Theorem	lttri3d 8447	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 8448	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	letrid 8449	Tightness of real apartness. (Contributed by Matthew House, 28-Jun-2026.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A <_ B )   &    |- ( ph -> B <_ A )   =>    |- ( ph -> A = B )
Theorem	eqleltd 8450	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 8451	'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 8452	'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 8453	'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 8454	'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 8455	'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 8456	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 8457	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 8458	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 8459	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 8460	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 8461	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 8462	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 8463	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 8464	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 8465	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 8466	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 8467	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 8468	A theorem for complex numbers analogous the second Peano postulate peano2 4744. (Contributed by NM, 17-Aug-2005.)
|- ( A e. CC -> ( A + 1 ) e. CC )
Theorem	peano2re 8469	A theorem for reals analogous the second Peano postulate peano2 4744. (Contributed by NM, 5-Jul-2005.)
|- ( A e. RR -> ( A + 1 ) e. RR )
Theorem	addcom 8470	Addition is commutative. (Contributed by Jim Kingdon, 17-Jan-2020.)
|- ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) )
Theorem	addrid 8471	0 is an additive identity. (Contributed by Jim Kingdon, 16-Jan-2020.)
|- ( A e. CC -> ( A + 0 ) = A )
Theorem	addlid 8472	0 is a left identity for addition. (Contributed by Scott Fenton, 3-Jan-2013.)
|- ( A e. CC -> ( 0 + A ) = A )
Theorem	readdcan 8473	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 8474	0 is its own additive identity. (Contributed by Scott Fenton, 3-Jan-2013.)
|- ( 0 + 0 ) = 0
Theorem	addridi 8475	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 8476	0 is a left identity for addition. (Contributed by NM, 3-Jan-2013.)
|- A e. CC   =>    |- ( 0 + A ) = A
Theorem	addcomi 8477	Addition is commutative. 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 8478	Addition is commutative. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- A e. CC   &    |- B e. CC   &    |- ( A + B ) = C   =>    |- ( B + A ) = C
Theorem	mul12i 8479	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 8480	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 8481	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 8482	0 is an additive identity. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A + 0 ) = A )
Theorem	addlidd 8483	0 is a left identity for addition. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( 0 + A ) = A )
Theorem	addcomd 8484	Addition is commutative. 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 8485	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 8486	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 8487	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 8488	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 8489	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 8490	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 8491	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 8492	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 8493	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 8494	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 8495	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 8496	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 8497	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 8498	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 8499	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 8500	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 ) ) )