P. 93 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9201-9300   *Has distinct variable group(s)

Type	Label	Description
Statement
4.4.2  Principle of mathematical induction
Theorem	nnind 9201*	Principle of Mathematical Induction (inference schema). The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. See nnaddcl 9205 for an example of its use. This is an alternative for Metamath 100 proof #74. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)
|- ( x = 1 -> ( ph <-> ps ) )   &    |- ( x = y -> ( ph <-> ch ) )   &    |- ( x = ( y + 1 ) -> ( ph <-> th ) )   &    |- ( x = A -> ( ph <-> ta ) )   &    |- ps   &    |- ( y e. NN -> ( ch -> th ) )   =>    |- ( A e. NN -> ta )
Theorem	nnindALT 9202*	Principle of Mathematical Induction (inference schema). The last four hypotheses give us the substitution instances we need; the first two are the induction step and the basis. This ALT version of nnind 9201 has a different hypothesis order. It may be easier to use with the metamath program's Proof Assistant, because "MM-PA> assign last" will be applied to the substitution instances first. We may eventually use this one as the official version. You may use either version. After the proof is complete, the ALT version can be changed to the non-ALT version with "MM-PA> minimize nnind /allow". (Contributed by NM, 7-Dec-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ( y e. NN -> ( ch -> th ) )   &    |- ps   &    |- ( x = 1 -> ( ph <-> ps ) )   &    |- ( x = y -> ( ph <-> ch ) )   &    |- ( x = ( y + 1 ) -> ( ph <-> th ) )   &    |- ( x = A -> ( ph <-> ta ) )   =>    |- ( A e. NN -> ta )
Theorem	nn1m1nn 9203	Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.)
|- ( A e. NN -> ( A = 1 \/ ( A - 1 ) e. NN ) )
Theorem	nn1suc 9204*	If a statement holds for 1 and also holds for a successor, it holds for all positive integers. The first three hypotheses give us the substitution instances we need; the last two show that it holds for 1 and for a successor. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 16-May-2014.)
|- ( x = 1 -> ( ph <-> ps ) )   &    |- ( x = ( y + 1 ) -> ( ph <-> ch ) )   &    |- ( x = A -> ( ph <-> th ) )   &    |- ps   &    |- ( y e. NN -> ch )   =>    |- ( A e. NN -> th )
Theorem	nnaddcl 9205	Closure of addition of positive integers, proved by induction on the second addend. (Contributed by NM, 12-Jan-1997.)
|- ( ( A e. NN /\ B e. NN ) -> ( A + B ) e. NN )
Theorem	nnmulcl 9206	Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.)
|- ( ( A e. NN /\ B e. NN ) -> ( A x. B ) e. NN )
Theorem	nnmulcli 9207	Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- A e. NN   &    |- B e. NN   =>    |- ( A x. B ) e. NN
Theorem	nnge1 9208	A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.)
|- ( A e. NN -> 1 <_ A )
Theorem	nnle1eq1 9209	A positive integer is less than or equal to one iff it is equal to one. (Contributed by NM, 3-Apr-2005.)
|- ( A e. NN -> ( A <_ 1 <-> A = 1 ) )
Theorem	nngt0 9210	A positive integer is positive. (Contributed by NM, 26-Sep-1999.)
|- ( A e. NN -> 0 < A )
Theorem	nnnlt1 9211	A positive integer is not less than one. (Contributed by NM, 18-Jan-2004.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( A e. NN -> -. A < 1 )
Theorem	0nnn 9212	Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.)
|- -. 0 e. NN
Theorem	nnne0 9213	A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.)
|- ( A e. NN -> A =/= 0 )
Theorem	nnap0 9214	A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.)
|- ( A e. NN -> A # 0 )
Theorem	nngt0i 9215	A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.)
|- A e. NN   =>    |- 0 < A
Theorem	nnap0i 9216	A positive integer is apart from zero (inference version). (Contributed by Jim Kingdon, 1-Jan-2023.)
|- A e. NN   =>    |- A # 0
Theorem	nnne0i 9217	A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)
|- A e. NN   =>    |- A =/= 0
Theorem	nn2ge 9218*	There exists a positive integer greater than or equal to any two others. (Contributed by NM, 18-Aug-1999.)
|- ( ( A e. NN /\ B e. NN ) -> E. x e. NN ( A <_ x /\ B <_ x ) )
Theorem	nn1gt1 9219	A positive integer is either one or greater than one. This is for NN; 0elnn 4723 is a similar theorem for om (the natural numbers as ordinals). (Contributed by Jim Kingdon, 7-Mar-2020.)
|- ( A e. NN -> ( A = 1 \/ 1 < A ) )
Theorem	nngt1ne1 9220	A positive integer is greater than one iff it is not equal to one. (Contributed by NM, 7-Oct-2004.)
|- ( A e. NN -> ( 1 < A <-> A =/= 1 ) )
Theorem	nndivre 9221	The quotient of a real and a positive integer is real. (Contributed by NM, 28-Nov-2008.)
|- ( ( A e. RR /\ N e. NN ) -> ( A / N ) e. RR )
Theorem	nnrecre 9222	The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.)
|- ( N e. NN -> ( 1 / N ) e. RR )
Theorem	nnrecgt0 9223	The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.)
|- ( A e. NN -> 0 < ( 1 / A ) )
Theorem	nnsub 9224	Subtraction of positive integers. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 16-May-2014.)
|- ( ( A e. NN /\ B e. NN ) -> ( A < B <-> ( B - A ) e. NN ) )
Theorem	nnsubi 9225	Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.)
|- A e. NN   &    |- B e. NN   =>    |- ( A < B <-> ( B - A ) e. NN )
Theorem	nndiv 9226*	Two ways to express " A divides B " for positive integers. (Contributed by NM, 3-Feb-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
|- ( ( A e. NN /\ B e. NN ) -> ( E. x e. NN ( A x. x ) = B <-> ( B / A ) e. NN ) )
Theorem	nndivtr 9227	Transitive property of divisibility: if A divides B and B divides C, then A divides C. Typically, C would be an integer, although the theorem holds for complex C. (Contributed by NM, 3-May-2005.)
|- ( ( ( A e. NN /\ B e. NN /\ C e. CC ) /\ ( ( B / A ) e. NN /\ ( C / B ) e. NN ) ) -> ( C / A ) e. NN )
Theorem	nnge1d 9228	A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> 1 <_ A )
Theorem	nngt0d 9229	A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> 0 < A )
Theorem	nnne0d 9230	A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A =/= 0 )
Theorem	nnap0d 9231	A positive integer is apart from zero. (Contributed by Jim Kingdon, 25-Aug-2021.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A # 0 )
Theorem	nnrecred 9232	The reciprocal of a positive integer is real. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> ( 1 / A ) e. RR )
Theorem	nnaddcld 9233	Closure of addition of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   &    |- ( ph -> B e. NN )   =>    |- ( ph -> ( A + B ) e. NN )
Theorem	nnmulcld 9234	Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   &    |- ( ph -> B e. NN )   =>    |- ( ph -> ( A x. B ) e. NN )
Theorem	nndivred 9235	A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. NN )   =>    |- ( ph -> ( A / B ) e. RR )
4.4.3  Decimal representation of numbers The decimal representation of numbers/integers is based on the decimal digits 0 through 9 (df-0 8082 through df-9 9251), which are explicitly defined in the following. Note that the numbers 0 and 1 are constants defined as primitives of the complex number axiom system (see df-0 8082 and df-1 8083). Integers can also be exhibited as sums of powers of 10 (e.g., the number 103 can be expressed as ( (; 1 0 ^ 2 ) + 3 )) or as some other expression built from operations on the numbers 0 through 9. For example, the prime number 823541 can be expressed as ( 7 ^ 7 ) - 2. Most abstract math rarely requires numbers larger than 4. Even in Wiles' proof of Fermat's Last Theorem, the largest number used appears to be 12.
Syntax	c2 9236	Extend class notation to include the number 2.
class 2
Syntax	c3 9237	Extend class notation to include the number 3.
class 3
Syntax	c4 9238	Extend class notation to include the number 4.
class 4
Syntax	c5 9239	Extend class notation to include the number 5.
class 5
Syntax	c6 9240	Extend class notation to include the number 6.
class 6
Syntax	c7 9241	Extend class notation to include the number 7.
class 7
Syntax	c8 9242	Extend class notation to include the number 8.
class 8
Syntax	c9 9243	Extend class notation to include the number 9.
class 9
Definition	df-2 9244	Define the number 2. (Contributed by NM, 27-May-1999.)
|- 2 = ( 1 + 1 )
Definition	df-3 9245	Define the number 3. (Contributed by NM, 27-May-1999.)
|- 3 = ( 2 + 1 )
Definition	df-4 9246	Define the number 4. (Contributed by NM, 27-May-1999.)
|- 4 = ( 3 + 1 )
Definition	df-5 9247	Define the number 5. (Contributed by NM, 27-May-1999.)
|- 5 = ( 4 + 1 )
Definition	df-6 9248	Define the number 6. (Contributed by NM, 27-May-1999.)
|- 6 = ( 5 + 1 )
Definition	df-7 9249	Define the number 7. (Contributed by NM, 27-May-1999.)
|- 7 = ( 6 + 1 )
Definition	df-8 9250	Define the number 8. (Contributed by NM, 27-May-1999.)
|- 8 = ( 7 + 1 )
Definition	df-9 9251	Define the number 9. (Contributed by NM, 27-May-1999.)
|- 9 = ( 8 + 1 )
Theorem	0ne1 9252	0 =/= 1 (common case). See aso 1ap0 8812. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 0 =/= 1
Theorem	1ne0 9253	1 =/= 0. See aso 1ap0 8812. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- 1 =/= 0
Theorem	1m1e0 9254	( 1 - 1 ) = 0 (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
|- ( 1 - 1 ) = 0
Theorem	2re 9255	The number 2 is real. (Contributed by NM, 27-May-1999.)
|- 2 e. RR
Theorem	2cn 9256	The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.)
|- 2 e. CC
Theorem	2ex 9257	2 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 2 e. _V
Theorem	2cnd 9258	2 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( ph -> 2 e. CC )
Theorem	3re 9259	The number 3 is real. (Contributed by NM, 27-May-1999.)
|- 3 e. RR
Theorem	3cn 9260	The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.)
|- 3 e. CC
Theorem	3ex 9261	3 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 3 e. _V
Theorem	4re 9262	The number 4 is real. (Contributed by NM, 27-May-1999.)
|- 4 e. RR
Theorem	4cn 9263	The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)
|- 4 e. CC
Theorem	5re 9264	The number 5 is real. (Contributed by NM, 27-May-1999.)
|- 5 e. RR
Theorem	5cn 9265	The number 5 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 5 e. CC
Theorem	6re 9266	The number 6 is real. (Contributed by NM, 27-May-1999.)
|- 6 e. RR
Theorem	6cn 9267	The number 6 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 6 e. CC
Theorem	7re 9268	The number 7 is real. (Contributed by NM, 27-May-1999.)
|- 7 e. RR
Theorem	7cn 9269	The number 7 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 7 e. CC
Theorem	8re 9270	The number 8 is real. (Contributed by NM, 27-May-1999.)
|- 8 e. RR
Theorem	8cn 9271	The number 8 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 8 e. CC
Theorem	9re 9272	The number 9 is real. (Contributed by NM, 27-May-1999.)
|- 9 e. RR
Theorem	9cn 9273	The number 9 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 9 e. CC
Theorem	0le0 9274	Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)
|- 0 <_ 0
Theorem	0le2 9275	0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.)
|- 0 <_ 2
Theorem	2pos 9276	The number 2 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 2
Theorem	2ne0 9277	The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)
|- 2 =/= 0
Theorem	2ap0 9278	The number 2 is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- 2 # 0
Theorem	3pos 9279	The number 3 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 3
Theorem	3ne0 9280	The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)
|- 3 =/= 0
Theorem	3ap0 9281	The number 3 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
|- 3 # 0
Theorem	4pos 9282	The number 4 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 4
Theorem	4ne0 9283	The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.)
|- 4 =/= 0
Theorem	4ap0 9284	The number 4 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
|- 4 # 0
Theorem	5pos 9285	The number 5 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 5
Theorem	6pos 9286	The number 6 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 6
Theorem	7pos 9287	The number 7 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 7
Theorem	8pos 9288	The number 8 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 8
Theorem	9pos 9289	The number 9 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 9
4.4.4  Some properties of specific numbers This includes adding two pairs of values 1..10 (where the right is less than the left) and where the left is less than the right for the values 1..10.
Theorem	neg1cn 9290	-1 is a complex number (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
|- -u 1 e. CC
Theorem	neg1rr 9291	-1 is a real number (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
|- -u 1 e. RR
Theorem	neg1ne0 9292	-1 is nonzero (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u 1 =/= 0
Theorem	neg1lt0 9293	-1 is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u 1 < 0
Theorem	neg1ap0 9294	-1 is apart from zero. (Contributed by Jim Kingdon, 9-Jun-2020.)
|- -u 1 # 0
Theorem	negneg1e1 9295	-u -u 1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u -u 1 = 1
Theorem	1pneg1e0 9296	1 + -u 1 is 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 + -u 1 ) = 0
Theorem	0m0e0 9297	0 minus 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 0 - 0 ) = 0
Theorem	1m0e1 9298	1 - 0 = 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 - 0 ) = 1
Theorem	0p1e1 9299	0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
|- ( 0 + 1 ) = 1
Theorem	fv0p1e1 9300	Function value at N + 1 with N replaced by 0. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 13-Aug-2022.)
|- ( N = 0 -> ( F ` ( N + 1 ) ) = ( F ` 1 ) )