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	9p7e16 9701	9 + 7 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 9 + 7 ) = ; 1 6
Theorem	9p8e17 9702	9 + 8 = 17. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 9 + 8 ) = ; 1 7
Theorem	9p9e18 9703	9 + 9 = 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 9 + 9 ) = ; 1 8
Theorem	10p10e20 9704	10 + 10 = 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
|- (; 1 0 + ; 1 0 ) = ; 2 0
Theorem	10m1e9 9705	10 - 1 = 9. (Contributed by AV, 6-Sep-2021.)
|- (; 1 0 - 1 ) = 9
Theorem	4t3lem 9706	Lemma for 4t3e12 9707 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- A e. NN0   &    |- B e. NN0   &    |- C = ( B + 1 )   &    |- ( A x. B ) = D   &    |- ( D + A ) = E   =>    |- ( A x. C ) = E
Theorem	4t3e12 9707	4 times 3 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 4 x. 3 ) = ; 1 2
Theorem	4t4e16 9708	4 times 4 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 4 x. 4 ) = ; 1 6
Theorem	5t2e10 9709	5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 4-Sep-2021.)
|- ( 5 x. 2 ) = ; 1 0
Theorem	5t3e15 9710	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 9711	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 9712	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 9713	6 times 2 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 6 x. 2 ) = ; 1 2
Theorem	6t3e18 9714	6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 6 x. 3 ) = ; 1 8
Theorem	6t4e24 9715	6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 6 x. 4 ) = ; 2 4
Theorem	6t5e30 9716	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 9717	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 9718	7 times 2 equals 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 7 x. 2 ) = ; 1 4
Theorem	7t3e21 9719	7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 7 x. 3 ) = ; 2 1
Theorem	7t4e28 9720	7 times 4 equals 28. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 7 x. 4 ) = ; 2 8
Theorem	7t5e35 9721	7 times 5 equals 35. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 7 x. 5 ) = ; 3 5
Theorem	7t6e42 9722	7 times 6 equals 42. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 7 x. 6 ) = ; 4 2
Theorem	7t7e49 9723	7 times 7 equals 49. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 7 x. 7 ) = ; 4 9
Theorem	8t2e16 9724	8 times 2 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 8 x. 2 ) = ; 1 6
Theorem	8t3e24 9725	8 times 3 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 8 x. 3 ) = ; 2 4
Theorem	8t4e32 9726	8 times 4 equals 32. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 8 x. 4 ) = ; 3 2
Theorem	8t5e40 9727	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 9728	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 9729	8 times 7 equals 56. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 8 x. 7 ) = ; 5 6
Theorem	8t8e64 9730	8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 8 x. 8 ) = ; 6 4
Theorem	9t2e18 9731	9 times 2 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 9 x. 2 ) = ; 1 8
Theorem	9t3e27 9732	9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 9 x. 3 ) = ; 2 7
Theorem	9t4e36 9733	9 times 4 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 9 x. 4 ) = ; 3 6
Theorem	9t5e45 9734	9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 9 x. 5 ) = ; 4 5
Theorem	9t6e54 9735	9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 9 x. 6 ) = ; 5 4
Theorem	9t7e63 9736	9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 9 x. 7 ) = ; 6 3
Theorem	9t8e72 9737	9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 9 x. 8 ) = ; 7 2
Theorem	9t9e81 9738	9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 9 x. 9 ) = ; 8 1
Theorem	9t11e99 9739	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 9740	9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.)
|- 9 < ; 1 0
Theorem	8lt10 9741	8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.)
|- 8 < ; 1 0
Theorem	7lt10 9742	7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
|- 7 < ; 1 0
Theorem	6lt10 9743	6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
|- 6 < ; 1 0
Theorem	5lt10 9744	5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
|- 5 < ; 1 0
Theorem	4lt10 9745	4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
|- 4 < ; 1 0
Theorem	3lt10 9746	3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
|- 3 < ; 1 0
Theorem	2lt10 9747	2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
|- 2 < ; 1 0
Theorem	1lt10 9748	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 9749	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 9750	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 9751	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 9752	Half minus a third. (Contributed by Scott Fenton, 8-Jul-2015.)
|- ( ( 1 / 2 ) - ( 1 / 3 ) ) = ( 1 / 6 )
Theorem	5recm6rec 9753	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 9754	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 9755*	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 9756 for its value, uzssz 9775 for its relationship to ZZ, nnuz 9791 and nn0uz 9790 for its relationships to NN and NN0, and eluz1 9758 and eluz2 9760 for its membership relations. (Contributed by NM, 5-Sep-2005.)
|- ZZ>= = ( j e. ZZ |-> { k e. ZZ | j <_ k } )
Theorem	uzval 9756*	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 9757	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 9758	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 9759	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 9760	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 9761	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 9762	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 9763	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 9764	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 9765	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 9766	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 9767	Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.)
|- ( N e. ( ZZ>= ` M ) -> M <_ N )
Theorem	eluz 9768	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 9769	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 9770	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 9771	The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.)
|- ( M e. ran ZZ>= -> M =/= (/) )
Theorem	uztrn 9772	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 9773	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 9774	Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.)
|- ( N e. ( ZZ>= ` M ) -> -u M e. ( ZZ>= ` -u N ) )
Theorem	uzssz 9775	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 9776	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 9777	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 9778	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 9779	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 9780	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 9781	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 9782	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 9783	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 9784	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 9785	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 9786	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 9787	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 9788	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 9789	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 9790	Nonnegative integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
|- NN0 = ( ZZ>= ` 0 )
Theorem	nnuz 9791	Positive integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
|- NN = ( ZZ>= ` 1 )
Theorem	elnnuz 9792	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 9793	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 9794	5 is an integer greater than or equal to 3. (Contributed by AV, 7-Sep-2025.)
|- 5 e. ( ZZ>= ` 3 )
Theorem	uzuzle23 9795	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 9796	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 9797	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 9798	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 9799	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 9800	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 )