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Theorem List for Intuitionistic Logic Explorer - 7801-7900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremaxlttrn 7801 Ordering on reals is transitive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-lttrn 7702 with ordering on the extended reals. New proofs should use lttr 7806 instead for naming consistency. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  <  B 
 /\  B  <  C )  ->  A  <  C ) )
 
Theoremaxltadd 7802 Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-ltadd 7704 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 ) ) )
 
Theoremaxapti 7803 Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-apti 7703 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 )
 
Theoremaxmulgt0 7804 The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-mulgt0 7705 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 ) ) )
 
Theoremaxsuploc 7805* 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 7709 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
 
Theoremlttr 7806 Alias for axlttrn 7801, for naming consistency with lttri 7836. New proofs should generally use this instead of ax-pre-lttrn 7702. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  <  B 
 /\  B  <  C )  ->  A  <  C ) )
 
Theoremmulgt0 7807 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 ) )
 
Theoremlenlt 7808 '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 ) )
 
Theoremltnr 7809 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
 |-  ( A  e.  RR  ->  -.  A  <  A )
 
Theoremltso 7810 'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.)
 |- 
 <  Or  RR
 
Theoremgtso 7811 'Greater than' is a strict ordering. (Contributed by JJ, 11-Oct-2018.)
 |-  `'  <  Or  RR
 
Theoremlttri3 7812 Tightness of real apartness. (Contributed by NM, 5-May-1999.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =  B 
 <->  ( -.  A  <  B 
 /\  -.  B  <  A ) ) )
 
Theoremletri3 7813 Tightness of real apartness. (Contributed by NM, 14-May-1999.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =  B 
 <->  ( A  <_  B  /\  B  <_  A )
 ) )
 
Theoremltleletr 7814 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 ) )
 
Theoremletr 7815 Transitive law. (Contributed by NM, 12-Nov-1999.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  <_  B 
 /\  B  <_  C )  ->  A  <_  C ) )
 
Theoremleid 7816 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)
 |-  ( A  e.  RR  ->  A  <_  A )
 
Theoremltne 7817 'Less than' implies not equal. See also ltap 8363 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 )
 
Theoremltnsym 7818 'Less than' is not symmetric. (Contributed by NM, 8-Jan-2002.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B 
 ->  -.  B  <  A ) )
 
Theoremltle 7819 'Less than' implies 'less than or equal to'. (Contributed by NM, 25-Aug-1999.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B 
 ->  A  <_  B )
 )
 
Theoremlelttr 7820 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 ) )
 
Theoremltletr 7821 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 ) )
 
Theoremltnsym2 7822 '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 ) )
 
Theoremeqle 7823 Equality implies 'less than or equal to'. (Contributed by NM, 4-Apr-2005.)
 |-  ( ( A  e.  RR  /\  A  =  B )  ->  A  <_  B )
 
Theoremltnri 7824 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
 |-  A  e.  RR   =>    |-  -.  A  <  A
 
Theoremeqlei 7825 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 )
 
Theoremeqlei2 7826 Equality implies 'less than or equal to'. (Contributed by Alexander van der Vekens, 20-Mar-2018.)
 |-  A  e.  RR   =>    |-  ( B  =  A  ->  B  <_  A )
 
Theoremgtneii 7827 'Less than' implies not equal. See also gtapii 8364 which is the same for apartness. (Contributed by Mario Carneiro, 30-Sep-2013.)
 |-  A  e.  RR   &    |-  A  <  B   =>    |-  B  =/=  A
 
Theoremltneii 7828 'Greater than' implies not equal. (Contributed by Mario Carneiro, 16-Sep-2015.)
 |-  A  e.  RR   &    |-  A  <  B   =>    |-  A  =/=  B
 
Theoremlttri3i 7829 Tightness of real apartness. (Contributed by NM, 14-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  =  B  <->  ( -.  A  <  B  /\  -.  B  <  A ) )
 
Theoremletri3i 7830 Tightness of real apartness. (Contributed by NM, 14-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  =  B  <->  ( A  <_  B  /\  B  <_  A ) )
 
Theoremltnsymi 7831 'Less than' is not symmetric. (Contributed by NM, 6-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  <  B  ->  -.  B  <  A )
 
Theoremlenlti 7832 '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 )
 
Theoremltlei 7833 'Less than' implies 'less than or equal to'. (Contributed by NM, 14-May-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  <  B  ->  A  <_  B )
 
Theoremltleii 7834 '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
 
Theoremltnei 7835 'Less than' implies not equal. (Contributed by NM, 28-Jul-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  <  B  ->  B  =/=  A )
 
Theoremlttri 7836 '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 )
 
Theoremlelttri 7837 '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 )
 
Theoremltletri 7838 '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 )
 
Theoremletri 7839 '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 )
 
Theoremle2tri3i 7840 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 ) )
 
Theoremmulgt0i 7841 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 ) )
 
Theoremmulgt0ii 7842 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 )
 
Theoremltnrd 7843 'Less than' is irreflexive. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  -.  A  <  A )
 
Theoremgtned 7844 'Less than' implies not equal. See also gtapd 8367 which is the same but for apartness. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  B  =/=  A )
 
Theoremltned 7845 'Greater than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  A  =/=  B )
 
Theoremlttri3d 7846 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 ) ) )
 
Theoremletri3d 7847 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 ) ) )
 
Theoremlenltd 7848 '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 ) )
 
Theoremltled 7849 '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 )
 
Theoremltnsymd 7850 '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 )
 
Theoremnltled 7851 '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 )
 
Theoremlensymd 7852 '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 )
 
Theoremmulgt0d 7853 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 ) )
 
Theoremletrd 7854 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 )
 
Theoremlelttrd 7855 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 )
 
Theoremlttrd 7856 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 )
 
Theorem0lt1 7857 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
 
Theoremltntri 7858 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
 
Theoremmul12 7859 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 ) ) )
 
Theoremmul32 7860 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 ) )
 
Theoremmul31 7861 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 ) )
 
Theoremmul4 7862 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 ) ) )
 
Theoremmuladd11 7863 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 ) ) ) )
 
Theorem1p1times 7864 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 ) )
 
Theorempeano2cn 7865 A theorem for complex numbers analogous the second Peano postulate peano2 4479. (Contributed by NM, 17-Aug-2005.)
 |-  ( A  e.  CC  ->  ( A  +  1 )  e.  CC )
 
Theorempeano2re 7866 A theorem for reals analogous the second Peano postulate peano2 4479. (Contributed by NM, 5-Jul-2005.)
 |-  ( A  e.  RR  ->  ( A  +  1 )  e.  RR )
 
Theoremaddcom 7867 Addition commutes. (Contributed by Jim Kingdon, 17-Jan-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B )  =  ( B  +  A )
 )
 
Theoremaddid1 7868  0 is an additive identity. (Contributed by Jim Kingdon, 16-Jan-2020.)
 |-  ( A  e.  CC  ->  ( A  +  0 )  =  A )
 
Theoremaddid2 7869  0 is a left identity for addition. (Contributed by Scott Fenton, 3-Jan-2013.)
 |-  ( A  e.  CC  ->  ( 0  +  A )  =  A )
 
Theoremreaddcan 7870 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 ) )
 
Theorem00id 7871  0 is its own additive identity. (Contributed by Scott Fenton, 3-Jan-2013.)
 |-  ( 0  +  0 )  =  0
 
Theoremaddid1i 7872  0 is an additive identity. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)
 |-  A  e.  CC   =>    |-  ( A  +  0 )  =  A
 
Theoremaddid2i 7873  0 is a left identity for addition. (Contributed by NM, 3-Jan-2013.)
 |-  A  e.  CC   =>    |-  ( 0  +  A )  =  A
 
Theoremaddcomi 7874 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 )
 
Theoremaddcomli 7875 Addition commutes. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  ( A  +  B )  =  C   =>    |-  ( B  +  A )  =  C
 
Theoremmul12i 7876 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 ) )
 
Theoremmul32i 7877 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 )
 
Theoremmul4i 7878 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 ) )
 
Theoremaddid1d 7879  0 is an additive identity. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  +  0 )  =  A )
 
Theoremaddid2d 7880  0 is a left identity for addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 0  +  A )  =  A )
 
Theoremaddcomd 7881 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 ) )
 
Theoremmul12d 7882 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 ) ) )
 
Theoremmul32d 7883 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 ) )
 
Theoremmul31d 7884 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 ) )
 
Theoremmul4d 7885 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 ) ) )
 
Theoremmuladd11r 7886 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 ) )
 
Theoremcomraddd 7887 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
 
Theoremadd12 7888 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 ) ) )
 
Theoremadd32 7889 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 ) )
 
Theoremadd32r 7890 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 ) )
 
Theoremadd4 7891 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 ) ) )
 
Theoremadd42 7892 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 ) ) )
 
Theoremadd12i 7893 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 )
 )
 
Theoremadd32i 7894 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 )
 
Theoremadd4i 7895 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 ) )
 
Theoremadd42i 7896 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 ) )
 
Theoremadd12d 7897 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 )
 ) )
 
Theoremadd32d 7898 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 ) )
 
Theoremadd4d 7899 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 )
 ) )
 
Theoremadd42d 7900 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 )
 ) )
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