Theorem List for Intuitionistic Logic Explorer - 8101-8200 *Has distinct variable
group(s)
Type | Label | Description |
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Theorem | nnncan2d 8101 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | npncan3d 8102 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | pnpcand 8103 |
Cancellation law for mixed addition and subtraction. (Contributed by
Mario Carneiro, 27-May-2016.)
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Theorem | pnpcan2d 8104 |
Cancellation law for mixed addition and subtraction. (Contributed by
Mario Carneiro, 27-May-2016.)
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Theorem | pnncand 8105 |
Cancellation law for mixed addition and subtraction. (Contributed by
Mario Carneiro, 27-May-2016.)
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Theorem | ppncand 8106 |
Cancellation law for mixed addition and subtraction. (Contributed by
Mario Carneiro, 27-May-2016.)
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Theorem | subcand 8107 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | subcan2d 8108 |
Cancellation law for subtraction. (Contributed by Mario Carneiro,
22-Sep-2016.)
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Theorem | subcanad 8109 |
Cancellation law for subtraction. Deduction form of subcan 8010.
Generalization of subcand 8107. (Contributed by David Moews,
28-Feb-2017.)
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Theorem | subneintrd 8110 |
Introducing subtraction on both sides of a statement of inequality.
Contrapositive of subcand 8107. (Contributed by David Moews,
28-Feb-2017.)
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Theorem | subcan2ad 8111 |
Cancellation law for subtraction. Deduction form of subcan2 7980.
Generalization of subcan2d 8108. (Contributed by David Moews,
28-Feb-2017.)
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Theorem | subneintr2d 8112 |
Introducing subtraction on both sides of a statement of inequality.
Contrapositive of subcan2d 8108. (Contributed by David Moews,
28-Feb-2017.)
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Theorem | addsub4d 8113 |
Rearrangement of 4 terms in a mixed addition and subtraction.
(Contributed by Mario Carneiro, 27-May-2016.)
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Theorem | subadd4d 8114 |
Rearrangement of 4 terms in a mixed addition and subtraction.
(Contributed by Mario Carneiro, 27-May-2016.)
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Theorem | sub4d 8115 |
Rearrangement of 4 terms in a subtraction. (Contributed by Mario
Carneiro, 27-May-2016.)
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Theorem | 2addsubd 8116 |
Law for subtraction and addition. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | addsubeq4d 8117 |
Relation between sums and differences. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | subeqxfrd 8118 |
Transfer two terms of a subtraction in an equality. (Contributed by
Thierry Arnoux, 2-Feb-2020.)
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Theorem | mvlraddd 8119 |
Move LHS right addition to RHS. (Contributed by David A. Wheeler,
15-Oct-2018.)
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Theorem | mvlladdd 8120 |
Move LHS left addition to RHS. (Contributed by David A. Wheeler,
15-Oct-2018.)
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Theorem | mvrraddd 8121 |
Move RHS right addition to LHS. (Contributed by David A. Wheeler,
15-Oct-2018.)
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Theorem | mvrladdd 8122 |
Move RHS left addition to LHS. (Contributed by David A. Wheeler,
11-Oct-2018.)
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Theorem | assraddsubd 8123 |
Associate RHS addition-subtraction. (Contributed by David A. Wheeler,
15-Oct-2018.)
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Theorem | subaddeqd 8124 |
Transfer two terms of a subtraction to an addition in an equality.
(Contributed by Thierry Arnoux, 2-Feb-2020.)
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Theorem | addlsub 8125 |
Left-subtraction: Subtraction of the left summand from the result of an
addition. (Contributed by BJ, 6-Jun-2019.)
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Theorem | addrsub 8126 |
Right-subtraction: Subtraction of the right summand from the result of
an addition. (Contributed by BJ, 6-Jun-2019.)
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Theorem | subexsub 8127 |
A subtraction law: Exchanging the subtrahend and the result of the
subtraction. (Contributed by BJ, 6-Jun-2019.)
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Theorem | addid0 8128 |
If adding a number to a another number yields the other number, the added
number must be .
This shows that is the
unique (right)
identity of the complex numbers. (Contributed by AV, 17-Jan-2021.)
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Theorem | addn0nid 8129 |
Adding a nonzero number to a complex number does not yield the complex
number. (Contributed by AV, 17-Jan-2021.)
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Theorem | pnpncand 8130 |
Addition/subtraction cancellation law. (Contributed by Scott Fenton,
14-Dec-2017.)
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Theorem | subeqrev 8131 |
Reverse the order of subtraction in an equality. (Contributed by Scott
Fenton, 8-Jul-2013.)
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Theorem | pncan1 8132 |
Cancellation law for addition and subtraction with 1. (Contributed by
Alexander van der Vekens, 3-Oct-2018.)
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Theorem | npcan1 8133 |
Cancellation law for subtraction and addition with 1. (Contributed by
Alexander van der Vekens, 5-Oct-2018.)
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Theorem | subeq0bd 8134 |
If two complex numbers are equal, their difference is zero. Consequence
of subeq0ad 8076. Converse of subeq0d 8074. Contrapositive of subne0ad 8077.
(Contributed by David Moews, 28-Feb-2017.)
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Theorem | renegcld 8135 |
Closure law for negative of reals. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | resubcld 8136 |
Closure law for subtraction of reals. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | negf1o 8137* |
Negation is an isomorphism of a subset of the real numbers to the
negated elements of the subset. (Contributed by AV, 9-Aug-2020.)
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4.3.3 Multiplication
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Theorem | kcnktkm1cn 8138 |
k times k minus 1 is a complex number if k is a complex number.
(Contributed by Alexander van der Vekens, 11-Mar-2018.)
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Theorem | muladd 8139 |
Product of two sums. (Contributed by NM, 14-Jan-2006.) (Proof shortened
by Andrew Salmon, 19-Nov-2011.)
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Theorem | subdi 8140 |
Distribution of multiplication over subtraction. Theorem I.5 of [Apostol]
p. 18. (Contributed by NM, 18-Nov-2004.)
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Theorem | subdir 8141 |
Distribution of multiplication over subtraction. Theorem I.5 of [Apostol]
p. 18. (Contributed by NM, 30-Dec-2005.)
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Theorem | mul02 8142 |
Multiplication by .
Theorem I.6 of [Apostol] p. 18. (Contributed
by NM, 10-Aug-1999.)
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Theorem | mul02lem2 8143 |
Zero times a real is zero. Although we prove it as a corollary of
mul02 8142, the name is for consistency with the
Metamath Proof Explorer
which proves it before mul02 8142. (Contributed by Scott Fenton,
3-Jan-2013.)
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Theorem | mul01 8144 |
Multiplication by .
Theorem I.6 of [Apostol] p. 18. (Contributed
by NM, 15-May-1999.) (Revised by Scott Fenton, 3-Jan-2013.)
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Theorem | mul02i 8145 |
Multiplication by 0. Theorem I.6 of [Apostol]
p. 18. (Contributed by
NM, 23-Nov-1994.)
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Theorem | mul01i 8146 |
Multiplication by .
Theorem I.6 of [Apostol] p. 18. (Contributed
by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)
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Theorem | mul02d 8147 |
Multiplication by 0. Theorem I.6 of [Apostol]
p. 18. (Contributed by
Mario Carneiro, 27-May-2016.)
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Theorem | mul01d 8148 |
Multiplication by .
Theorem I.6 of [Apostol] p. 18. (Contributed
by Mario Carneiro, 27-May-2016.)
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Theorem | ine0 8149 |
The imaginary unit
is not zero. (Contributed by NM,
6-May-1999.)
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Theorem | mulneg1 8150 |
Product with negative is negative of product. Theorem I.12 of [Apostol]
p. 18. (Contributed by NM, 14-May-1999.) (Proof shortened by Mario
Carneiro, 27-May-2016.)
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Theorem | mulneg2 8151 |
The product with a negative is the negative of the product. (Contributed
by NM, 30-Jul-2004.)
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Theorem | mulneg12 8152 |
Swap the negative sign in a product. (Contributed by NM, 30-Jul-2004.)
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Theorem | mul2neg 8153 |
Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed
by NM, 30-Jul-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
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Theorem | submul2 8154 |
Convert a subtraction to addition using multiplication by a negative.
(Contributed by NM, 2-Feb-2007.)
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Theorem | mulm1 8155 |
Product with minus one is negative. (Contributed by NM, 16-Nov-1999.)
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Theorem | mulsub 8156 |
Product of two differences. (Contributed by NM, 14-Jan-2006.)
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Theorem | mulsub2 8157 |
Swap the order of subtraction in a multiplication. (Contributed by Scott
Fenton, 24-Jun-2013.)
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Theorem | mulm1i 8158 |
Product with minus one is negative. (Contributed by NM,
31-Jul-1999.)
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Theorem | mulneg1i 8159 |
Product with negative is negative of product. Theorem I.12 of [Apostol]
p. 18. (Contributed by NM, 10-Feb-1995.) (Revised by Mario Carneiro,
27-May-2016.)
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Theorem | mulneg2i 8160 |
Product with negative is negative of product. (Contributed by NM,
31-Jul-1999.) (Revised by Mario Carneiro, 27-May-2016.)
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Theorem | mul2negi 8161 |
Product of two negatives. Theorem I.12 of [Apostol] p. 18.
(Contributed by NM, 14-Feb-1995.) (Revised by Mario Carneiro,
27-May-2016.)
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Theorem | subdii 8162 |
Distribution of multiplication over subtraction. Theorem I.5 of
[Apostol] p. 18. (Contributed by NM,
26-Nov-1994.)
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Theorem | subdiri 8163 |
Distribution of multiplication over subtraction. Theorem I.5 of
[Apostol] p. 18. (Contributed by NM,
8-May-1999.)
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Theorem | muladdi 8164 |
Product of two sums. (Contributed by NM, 17-May-1999.)
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Theorem | mulm1d 8165 |
Product with minus one is negative. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | mulneg1d 8166 |
Product with negative is negative of product. Theorem I.12 of [Apostol]
p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
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Theorem | mulneg2d 8167 |
Product with negative is negative of product. (Contributed by Mario
Carneiro, 27-May-2016.)
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Theorem | mul2negd 8168 |
Product of two negatives. Theorem I.12 of [Apostol] p. 18.
(Contributed by Mario Carneiro, 27-May-2016.)
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Theorem | subdid 8169 |
Distribution of multiplication over subtraction. Theorem I.5 of
[Apostol] p. 18. (Contributed by Mario
Carneiro, 27-May-2016.)
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Theorem | subdird 8170 |
Distribution of multiplication over subtraction. Theorem I.5 of
[Apostol] p. 18. (Contributed by Mario
Carneiro, 27-May-2016.)
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Theorem | muladdd 8171 |
Product of two sums. (Contributed by Mario Carneiro, 27-May-2016.)
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Theorem | mulsubd 8172 |
Product of two differences. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | mulsubfacd 8173 |
Multiplication followed by the subtraction of a factor. (Contributed by
Alexander van der Vekens, 28-Aug-2018.)
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4.3.4 Ordering on reals (cont.)
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Theorem | ltadd2 8174 |
Addition to both sides of 'less than'. (Contributed by NM,
12-Nov-1999.) (Revised by Mario Carneiro, 27-May-2016.)
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Theorem | ltadd2i 8175 |
Addition to both sides of 'less than'. (Contributed by NM,
21-Jan-1997.)
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Theorem | ltadd2d 8176 |
Addition to both sides of 'less than'. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | ltadd2dd 8177 |
Addition to both sides of 'less than'. (Contributed by Mario
Carneiro, 30-May-2016.)
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Theorem | ltletrd 8178 |
Transitive law deduction for 'less than', 'less than or equal to'.
(Contributed by NM, 9-Jan-2006.)
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Theorem | ltaddneg 8179 |
Adding a negative number to another number decreases it. (Contributed by
Glauco Siliprandi, 11-Dec-2019.)
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Theorem | ltaddnegr 8180 |
Adding a negative number to another number decreases it. (Contributed by
AV, 19-Mar-2021.)
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Theorem | lelttrdi 8181 |
If a number is less than another number, and the other number is less
than or equal to a third number, the first number is less than the third
number. (Contributed by Alexander van der Vekens, 24-Mar-2018.)
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Theorem | gt0ne0 8182 |
Positive implies nonzero. (Contributed by NM, 3-Oct-1999.) (Proof
shortened by Mario Carneiro, 27-May-2016.)
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Theorem | lt0ne0 8183 |
A number which is less than zero is not zero. See also lt0ap0 8403 which is
similar but for apartness. (Contributed by Stefan O'Rear,
13-Sep-2014.)
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Theorem | ltadd1 8184 |
Addition to both sides of 'less than'. Part of definition 11.2.7(vi) of
[HoTT], p. (varies). (Contributed by NM,
12-Nov-1999.) (Proof shortened
by Mario Carneiro, 27-May-2016.)
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Theorem | leadd1 8185 |
Addition to both sides of 'less than or equal to'. Part of definition
11.2.7(vi) of [HoTT], p. (varies).
(Contributed by NM, 18-Oct-1999.)
(Proof shortened by Mario Carneiro, 27-May-2016.)
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Theorem | leadd2 8186 |
Addition to both sides of 'less than or equal to'. (Contributed by NM,
26-Oct-1999.)
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Theorem | ltsubadd 8187 |
'Less than' relationship between subtraction and addition. (Contributed
by NM, 21-Jan-1997.) (Proof shortened by Mario Carneiro, 27-May-2016.)
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Theorem | ltsubadd2 8188 |
'Less than' relationship between subtraction and addition. (Contributed
by NM, 21-Jan-1997.)
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Theorem | lesubadd 8189 |
'Less than or equal to' relationship between subtraction and addition.
(Contributed by NM, 17-Nov-2004.) (Proof shortened by Mario Carneiro,
27-May-2016.)
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Theorem | lesubadd2 8190 |
'Less than or equal to' relationship between subtraction and addition.
(Contributed by NM, 10-Aug-1999.)
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Theorem | ltaddsub 8191 |
'Less than' relationship between addition and subtraction. (Contributed
by NM, 17-Nov-2004.)
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Theorem | ltaddsub2 8192 |
'Less than' relationship between addition and subtraction. (Contributed
by NM, 17-Nov-2004.)
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Theorem | leaddsub 8193 |
'Less than or equal to' relationship between addition and subtraction.
(Contributed by NM, 6-Apr-2005.)
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Theorem | leaddsub2 8194 |
'Less than or equal to' relationship between and addition and subtraction.
(Contributed by NM, 6-Apr-2005.)
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Theorem | suble 8195 |
Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.)
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Theorem | lesub 8196 |
Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.)
(Proof shortened by Andrew Salmon, 19-Nov-2011.)
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Theorem | ltsub23 8197 |
'Less than' relationship between subtraction and addition. (Contributed
by NM, 4-Oct-1999.)
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Theorem | ltsub13 8198 |
'Less than' relationship between subtraction and addition. (Contributed
by NM, 17-Nov-2004.)
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Theorem | le2add 8199 |
Adding both sides of two 'less than or equal to' relations. (Contributed
by NM, 17-Apr-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
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Theorem | lt2add 8200 |
Adding both sides of two 'less than' relations. Theorem I.25 of [Apostol]
p. 20. (Contributed by NM, 15-Aug-1999.) (Proof shortened by Mario
Carneiro, 27-May-2016.)
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