P. 98 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9701-9800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	5t3e15 9701	5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
|- ( 5 x. 3 ) = ; 1 5
Theorem	5t4e20 9702	5 times 4 equals 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
|- ( 5 x. 4 ) = ; 2 0
Theorem	5t5e25 9703	5 times 5 equals 25. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
|- ( 5 x. 5 ) = ; 2 5
Theorem	6t2e12 9704	6 times 2 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 6 x. 2 ) = ; 1 2
Theorem	6t3e18 9705	6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 6 x. 3 ) = ; 1 8
Theorem	6t4e24 9706	6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 6 x. 4 ) = ; 2 4
Theorem	6t5e30 9707	6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
|- ( 6 x. 5 ) = ; 3 0
Theorem	6t6e36 9708	6 times 6 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
|- ( 6 x. 6 ) = ; 3 6
Theorem	7t2e14 9709	7 times 2 equals 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 7 x. 2 ) = ; 1 4
Theorem	7t3e21 9710	7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 7 x. 3 ) = ; 2 1
Theorem	7t4e28 9711	7 times 4 equals 28. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 7 x. 4 ) = ; 2 8
Theorem	7t5e35 9712	7 times 5 equals 35. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 7 x. 5 ) = ; 3 5
Theorem	7t6e42 9713	7 times 6 equals 42. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 7 x. 6 ) = ; 4 2
Theorem	7t7e49 9714	7 times 7 equals 49. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 7 x. 7 ) = ; 4 9
Theorem	8t2e16 9715	8 times 2 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 8 x. 2 ) = ; 1 6
Theorem	8t3e24 9716	8 times 3 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 8 x. 3 ) = ; 2 4
Theorem	8t4e32 9717	8 times 4 equals 32. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 8 x. 4 ) = ; 3 2
Theorem	8t5e40 9718	8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
|- ( 8 x. 5 ) = ; 4 0
Theorem	8t6e48 9719	8 times 6 equals 48. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
|- ( 8 x. 6 ) = ; 4 8
Theorem	8t7e56 9720	8 times 7 equals 56. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 8 x. 7 ) = ; 5 6
Theorem	8t8e64 9721	8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 8 x. 8 ) = ; 6 4
Theorem	9t2e18 9722	9 times 2 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 9 x. 2 ) = ; 1 8
Theorem	9t3e27 9723	9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 9 x. 3 ) = ; 2 7
Theorem	9t4e36 9724	9 times 4 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 9 x. 4 ) = ; 3 6
Theorem	9t5e45 9725	9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 9 x. 5 ) = ; 4 5
Theorem	9t6e54 9726	9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 9 x. 6 ) = ; 5 4
Theorem	9t7e63 9727	9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 9 x. 7 ) = ; 6 3
Theorem	9t8e72 9728	9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 9 x. 8 ) = ; 7 2
Theorem	9t9e81 9729	9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 9 x. 9 ) = ; 8 1
Theorem	9t11e99 9730	9 times 11 equals 99. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 6-Sep-2021.)
|- ( 9 x. ; 1 1 ) = ; 9 9
Theorem	9lt10 9731	9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.)
|- 9 < ; 1 0
Theorem	8lt10 9732	8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.)
|- 8 < ; 1 0
Theorem	7lt10 9733	7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
|- 7 < ; 1 0
Theorem	6lt10 9734	6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
|- 6 < ; 1 0
Theorem	5lt10 9735	5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
|- 5 < ; 1 0
Theorem	4lt10 9736	4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
|- 4 < ; 1 0
Theorem	3lt10 9737	3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
|- 3 < ; 1 0
Theorem	2lt10 9738	2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
|- 2 < ; 1 0
Theorem	1lt10 9739	1 is less than 10. (Contributed by NM, 7-Nov-2012.) (Revised by Mario Carneiro, 9-Mar-2015.) (Revised by AV, 8-Sep-2021.)
|- 1 < ; 1 0
Theorem	decbin0 9740	Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- A e. NN0   =>    |- ( 4 x. A ) = ( 2 x. ( 2 x. A ) )
Theorem	decbin2 9741	Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- A e. NN0   =>    |- ( ( 4 x. A ) + 2 ) = ( 2 x. ( ( 2 x. A ) + 1 ) )
Theorem	decbin3 9742	Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- A e. NN0   =>    |- ( ( 4 x. A ) + 3 ) = ( ( 2 x. ( ( 2 x. A ) + 1 ) ) + 1 )
Theorem	halfthird 9743	Half minus a third. (Contributed by Scott Fenton, 8-Jul-2015.)
|- ( ( 1 / 2 ) - ( 1 / 3 ) ) = ( 1 / 6 )
Theorem	5recm6rec 9744	One fifth minus one sixth. (Contributed by Scott Fenton, 9-Jan-2017.)
|- ( ( 1 / 5 ) - ( 1 / 6 ) ) = ( 1 / ; 3 0 )
4.4.11  Upper sets of integers
Syntax	cuz 9745	Extend class notation with the upper integer function. Read " ZZ>= ` M " as "the set of integers greater than or equal to M".
class ZZ>=
Definition	df-uz 9746*	Define a function whose value at j is the semi-infinite set of contiguous integers starting at j, which we will also call the upper integers starting at j. Read " ZZ>= ` M " as "the set of integers greater than or equal to M". See uzval 9747 for its value, uzssz 9766 for its relationship to ZZ, nnuz 9782 and nn0uz 9781 for its relationships to NN and NN0, and eluz1 9749 and eluz2 9751 for its membership relations. (Contributed by NM, 5-Sep-2005.)
|- ZZ>= = ( j e. ZZ |-> { k e. ZZ | j <_ k } )
Theorem	uzval 9747*	The value of the upper integers function. (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
|- ( N e. ZZ -> ( ZZ>= ` N ) = { k e. ZZ | N <_ k } )
Theorem	uzf 9748	The domain and codomain of the upper integers function. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 3-Nov-2013.)
|- ZZ>= : ZZ --> ~P ZZ
Theorem	eluz1 9749	Membership in the upper set of integers starting at M. (Contributed by NM, 5-Sep-2005.)
|- ( M e. ZZ -> ( N e. ( ZZ>= ` M ) <-> ( N e. ZZ /\ M <_ N ) ) )
Theorem	eluzel2 9750	Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
|- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
Theorem	eluz2 9751	Membership in an upper set of integers. We use the fact that a function's value (under our function value definition) is empty outside of its domain to show M e. ZZ. (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
|- ( N e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ N e. ZZ /\ M <_ N ) )
Theorem	eluzmn 9752	Membership in an earlier upper set of integers. (Contributed by Thierry Arnoux, 8-Oct-2018.)
|- ( ( M e. ZZ /\ N e. NN0 ) -> M e. ( ZZ>= ` ( M - N ) ) )
Theorem	eluz1i 9753	Membership in an upper set of integers. (Contributed by NM, 5-Sep-2005.)
|- M e. ZZ   =>    |- ( N e. ( ZZ>= ` M ) <-> ( N e. ZZ /\ M <_ N ) )
Theorem	eluzuzle 9754	An integer in an upper set of integers is an element of an upper set of integers with a smaller bound. (Contributed by Alexander van der Vekens, 17-Jun-2018.)
|- ( ( B e. ZZ /\ B <_ A ) -> ( C e. ( ZZ>= ` A ) -> C e. ( ZZ>= ` B ) ) )
Theorem	eluzelz 9755	A member of an upper set of integers is an integer. (Contributed by NM, 6-Sep-2005.)
|- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
Theorem	eluzelre 9756	A member of an upper set of integers is a real. (Contributed by Mario Carneiro, 31-Aug-2013.)
|- ( N e. ( ZZ>= ` M ) -> N e. RR )
Theorem	eluzelcn 9757	A member of an upper set of integers is a complex number. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
|- ( N e. ( ZZ>= ` M ) -> N e. CC )
Theorem	eluzle 9758	Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.)
|- ( N e. ( ZZ>= ` M ) -> M <_ N )
Theorem	eluz 9759	Membership in an upper set of integers. (Contributed by NM, 2-Oct-2005.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( N e. ( ZZ>= ` M ) <-> M <_ N ) )
Theorem	uzid 9760	Membership of the least member in an upper set of integers. (Contributed by NM, 2-Sep-2005.)
|- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
Theorem	uzidd 9761	Membership of the least member in an upper set of integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
|- ( ph -> M e. ZZ )   =>    |- ( ph -> M e. ( ZZ>= ` M ) )
Theorem	uzn0 9762	The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.)
|- ( M e. ran ZZ>= -> M =/= (/) )
Theorem	uztrn 9763	Transitive law for sets of upper integers. (Contributed by NM, 20-Sep-2005.)
|- ( ( M e. ( ZZ>= ` K ) /\ K e. ( ZZ>= ` N ) ) -> M e. ( ZZ>= ` N ) )
Theorem	uztrn2 9764	Transitive law for sets of upper integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
|- Z = ( ZZ>= ` K )   =>    |- ( ( N e. Z /\ M e. ( ZZ>= ` N ) ) -> M e. Z )
Theorem	uzneg 9765	Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.)
|- ( N e. ( ZZ>= ` M ) -> -u M e. ( ZZ>= ` -u N ) )
Theorem	uzssz 9766	An upper set of integers is a subset of all integers. (Contributed by NM, 2-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
|- ( ZZ>= ` M ) C_ ZZ
Theorem	uzss 9767	Subset relationship for two sets of upper integers. (Contributed by NM, 5-Sep-2005.)
|- ( N e. ( ZZ>= ` M ) -> ( ZZ>= ` N ) C_ ( ZZ>= ` M ) )
Theorem	uztric 9768	Trichotomy of the ordering relation on integers, stated in terms of upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jun-2013.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( N e. ( ZZ>= ` M ) \/ M e. ( ZZ>= ` N ) ) )
Theorem	uz11 9769	The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005.)
|- ( M e. ZZ -> ( ( ZZ>= ` M ) = ( ZZ>= ` N ) <-> M = N ) )
Theorem	eluzp1m1 9770	Membership in the next upper set of integers. (Contributed by NM, 12-Sep-2005.)
|- ( ( M e. ZZ /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> ( N - 1 ) e. ( ZZ>= ` M ) )
Theorem	eluzp1l 9771	Strict ordering implied by membership in the next upper set of integers. (Contributed by NM, 12-Sep-2005.)
|- ( ( M e. ZZ /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> M < N )
Theorem	eluzp1p1 9772	Membership in the next upper set of integers. (Contributed by NM, 5-Oct-2005.)
|- ( N e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( ZZ>= ` ( M + 1 ) ) )
Theorem	eluzaddi 9773	Membership in a later upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.)
|- M e. ZZ   &    |- K e. ZZ   =>    |- ( N e. ( ZZ>= ` M ) -> ( N + K ) e. ( ZZ>= ` ( M + K ) ) )
Theorem	eluzsubi 9774	Membership in an earlier upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.)
|- M e. ZZ   &    |- K e. ZZ   =>    |- ( N e. ( ZZ>= ` ( M + K ) ) -> ( N - K ) e. ( ZZ>= ` M ) )
Theorem	eluzadd 9775	Membership in a later upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( N e. ( ZZ>= ` M ) /\ K e. ZZ ) -> ( N + K ) e. ( ZZ>= ` ( M + K ) ) )
Theorem	eluzsub 9776	Membership in an earlier upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( M e. ZZ /\ K e. ZZ /\ N e. ( ZZ>= ` ( M + K ) ) ) -> ( N - K ) e. ( ZZ>= ` M ) )
Theorem	uzm1 9777	Choices for an element of an upper interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( N e. ( ZZ>= ` M ) -> ( N = M \/ ( N - 1 ) e. ( ZZ>= ` M ) ) )
Theorem	uznn0sub 9778	The nonnegative difference of integers is a nonnegative integer. (Contributed by NM, 4-Sep-2005.)
|- ( N e. ( ZZ>= ` M ) -> ( N - M ) e. NN0 )
Theorem	uzin 9779	Intersection of two upper intervals of integers. (Contributed by Mario Carneiro, 24-Dec-2013.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( ZZ>= ` M ) i^i ( ZZ>= ` N ) ) = ( ZZ>= ` if ( M <_ N , N , M ) ) )
Theorem	uzp1 9780	Choices for an element of an upper interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( N e. ( ZZ>= ` M ) -> ( N = M \/ N e. ( ZZ>= ` ( M + 1 ) ) ) )
Theorem	nn0uz 9781	Nonnegative integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
|- NN0 = ( ZZ>= ` 0 )
Theorem	nnuz 9782	Positive integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
|- NN = ( ZZ>= ` 1 )
Theorem	elnnuz 9783	A positive integer expressed as a member of an upper set of integers. (Contributed by NM, 6-Jun-2006.)
|- ( N e. NN <-> N e. ( ZZ>= ` 1 ) )
Theorem	elnn0uz 9784	A nonnegative integer expressed as a member an upper set of integers. (Contributed by NM, 6-Jun-2006.)
|- ( N e. NN0 <-> N e. ( ZZ>= ` 0 ) )
Theorem	5eluz3 9785	5 is an integer greater than or equal to 3. (Contributed by AV, 7-Sep-2025.)
|- 5 e. ( ZZ>= ` 3 )
Theorem	uzuzle23 9786	An integer in the upper set of integers starting at 3 is element of the upper set of integers starting at 2. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
|- ( A e. ( ZZ>= ` 3 ) -> A e. ( ZZ>= ` 2 ) )
Theorem	uzuzle24 9787	An integer greater than or equal to 4 is an integer greater than or equal to 2. (Contributed by AV, 30-May-2023.)
|- ( X e. ( ZZ>= ` 4 ) -> X e. ( ZZ>= ` 2 ) )
Theorem	uzuzle34 9788	An integer greater than or equal to 4 is an integer greater than or equal to 3. (Contributed by AV, 5-Sep-2025.)
|- ( X e. ( ZZ>= ` 4 ) -> X e. ( ZZ>= ` 3 ) )
Theorem	uzuzle35 9789	An integer greater than or equal to 5 is an integer greater than or equal to 3. (Contributed by AV, 15-Nov-2025.)
|- ( A e. ( ZZ>= ` 5 ) -> A e. ( ZZ>= ` 3 ) )
Theorem	eluz2nn 9790	An integer is greater than or equal to 2 is a positive integer. (Contributed by AV, 3-Nov-2018.)
|- ( A e. ( ZZ>= ` 2 ) -> A e. NN )
Theorem	eluz3nn 9791	An integer greater than or equal to 3 is a positive integer. (Contributed by Alexander van der Vekens, 17-Sep-2018.) (Proof shortened by AV, 30-Nov-2025.)
|- ( N e. ( ZZ>= ` 3 ) -> N e. NN )
Theorem	eluz4eluz2 9792	An integer greater than or equal to 4 is an integer greater than or equal to 2. (Contributed by AV, 30-May-2023.)
|- ( X e. ( ZZ>= ` 4 ) -> X e. ( ZZ>= ` 2 ) )
Theorem	eluz4nn 9793	An integer greater than or equal to 4 is a positive integer. (Contributed by AV, 30-May-2023.)
|- ( X e. ( ZZ>= ` 4 ) -> X e. NN )
Theorem	eluzge2nn0 9794	If an integer is greater than or equal to 2, then it is a nonnegative integer. (Contributed by AV, 27-Aug-2018.) (Proof shortened by AV, 3-Nov-2018.)
|- ( N e. ( ZZ>= ` 2 ) -> N e. NN0 )
Theorem	eluz2n0 9795	An integer greater than or equal to 2 is not 0. (Contributed by AV, 25-May-2020.)
|- ( N e. ( ZZ>= ` 2 ) -> N =/= 0 )
Theorem	eluzge3nn 9796	If an integer is greater than 3, then it is a positive integer. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
|- ( N e. ( ZZ>= ` 3 ) -> N e. NN )
Theorem	uz3m2nn 9797	An integer greater than or equal to 3 decreased by 2 is a positive integer. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
|- ( N e. ( ZZ>= ` 3 ) -> ( N - 2 ) e. NN )
Theorem	1eluzge0 9798	1 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
|- 1 e. ( ZZ>= ` 0 )
Theorem	2eluzge0 9799	2 is an integer greater than or equal to 0. (Contributed by Alexander van der Vekens, 8-Jun-2018.) (Proof shortened by OpenAI, 25-Mar-2020.)
|- 2 e. ( ZZ>= ` 0 )
Theorem	2eluzge1 9800	2 is an integer greater than or equal to 1. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
|- 2 e. ( ZZ>= ` 1 )