P. 97 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9601-9700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	4t4e16 9601	4 times 4 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 4 x. 4 ) = ; 1 6
Theorem	5t2e10 9602	5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.) (Revised by AV, 4-Sep-2021.)
|- ( 5 x. 2 ) = ; 1 0
Theorem	5t3e15 9603	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 9604	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 9605	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 9606	6 times 2 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 6 x. 2 ) = ; 1 2
Theorem	6t3e18 9607	6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 6 x. 3 ) = ; 1 8
Theorem	6t4e24 9608	6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 6 x. 4 ) = ; 2 4
Theorem	6t5e30 9609	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 9610	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 9611	7 times 2 equals 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 7 x. 2 ) = ; 1 4
Theorem	7t3e21 9612	7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 7 x. 3 ) = ; 2 1
Theorem	7t4e28 9613	7 times 4 equals 28. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 7 x. 4 ) = ; 2 8
Theorem	7t5e35 9614	7 times 5 equals 35. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 7 x. 5 ) = ; 3 5
Theorem	7t6e42 9615	7 times 6 equals 42. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 7 x. 6 ) = ; 4 2
Theorem	7t7e49 9616	7 times 7 equals 49. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 7 x. 7 ) = ; 4 9
Theorem	8t2e16 9617	8 times 2 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 8 x. 2 ) = ; 1 6
Theorem	8t3e24 9618	8 times 3 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 8 x. 3 ) = ; 2 4
Theorem	8t4e32 9619	8 times 4 equals 32. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 8 x. 4 ) = ; 3 2
Theorem	8t5e40 9620	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 9621	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 9622	8 times 7 equals 56. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 8 x. 7 ) = ; 5 6
Theorem	8t8e64 9623	8 times 8 equals 64. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 8 x. 8 ) = ; 6 4
Theorem	9t2e18 9624	9 times 2 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 9 x. 2 ) = ; 1 8
Theorem	9t3e27 9625	9 times 3 equals 27. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 9 x. 3 ) = ; 2 7
Theorem	9t4e36 9626	9 times 4 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 9 x. 4 ) = ; 3 6
Theorem	9t5e45 9627	9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 9 x. 5 ) = ; 4 5
Theorem	9t6e54 9628	9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 9 x. 6 ) = ; 5 4
Theorem	9t7e63 9629	9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 9 x. 7 ) = ; 6 3
Theorem	9t8e72 9630	9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 9 x. 8 ) = ; 7 2
Theorem	9t9e81 9631	9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- ( 9 x. 9 ) = ; 8 1
Theorem	9t11e99 9632	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 9633	9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.)
|- 9 < ; 1 0
Theorem	8lt10 9634	8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.)
|- 8 < ; 1 0
Theorem	7lt10 9635	7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
|- 7 < ; 1 0
Theorem	6lt10 9636	6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
|- 6 < ; 1 0
Theorem	5lt10 9637	5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
|- 5 < ; 1 0
Theorem	4lt10 9638	4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
|- 4 < ; 1 0
Theorem	3lt10 9639	3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
|- 3 < ; 1 0
Theorem	2lt10 9640	2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.)
|- 2 < ; 1 0
Theorem	1lt10 9641	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 9642	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 9643	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 9644	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 9645	Half minus a third. (Contributed by Scott Fenton, 8-Jul-2015.)
|- ( ( 1 / 2 ) - ( 1 / 3 ) ) = ( 1 / 6 )
Theorem	5recm6rec 9646	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 9647	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 9648*	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 9649 for its value, uzssz 9667 for its relationship to ZZ, nnuz 9683 and nn0uz 9682 for its relationships to NN and NN0, and eluz1 9651 and eluz2 9653 for its membership relations. (Contributed by NM, 5-Sep-2005.)
|- ZZ>= = ( j e. ZZ |-> { k e. ZZ | j <_ k } )
Theorem	uzval 9649*	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 9650	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 9651	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 9652	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 9653	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	eluz1i 9654	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 9655	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 9656	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 9657	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 9658	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 9659	Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.)
|- ( N e. ( ZZ>= ` M ) -> M <_ N )
Theorem	eluz 9660	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 9661	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 9662	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 9663	The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.)
|- ( M e. ran ZZ>= -> M =/= (/) )
Theorem	uztrn 9664	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 9665	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 9666	Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.)
|- ( N e. ( ZZ>= ` M ) -> -u M e. ( ZZ>= ` -u N ) )
Theorem	uzssz 9667	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 9668	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 9669	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 9670	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 9671	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 9672	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 9673	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 9674	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 9675	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 9676	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 9677	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 9678	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 9679	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 9680	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 9681	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 9682	Nonnegative integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
|- NN0 = ( ZZ>= ` 0 )
Theorem	nnuz 9683	Positive integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
|- NN = ( ZZ>= ` 1 )
Theorem	elnnuz 9684	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 9685	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	eluz2nn 9686	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	eluz4eluz2 9687	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 9688	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 9689	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 9690	An integer greater than or equal to 2 is not 0. (Contributed by AV, 25-May-2020.)
|- ( N e. ( ZZ>= ` 2 ) -> N =/= 0 )
Theorem	uzuzle23 9691	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	eluzge3nn 9692	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 9693	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 9694	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 9695	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 9696	2 is an integer greater than or equal to 1. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
|- 2 e. ( ZZ>= ` 1 )
Theorem	uznnssnn 9697	The upper integers starting from a natural are a subset of the naturals. (Contributed by Scott Fenton, 29-Jun-2013.)
|- ( N e. NN -> ( ZZ>= ` N ) C_ NN )
Theorem	raluz 9698*	Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
|- ( M e. ZZ -> ( A. n e. ( ZZ>= ` M ) ph <-> A. n e. ZZ ( M <_ n -> ph ) ) )
Theorem	raluz2 9699*	Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
|- ( A. n e. ( ZZ>= ` M ) ph <-> ( M e. ZZ -> A. n e. ZZ ( M <_ n -> ph ) ) )
Theorem	rexuz 9700*	Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
|- ( M e. ZZ -> ( E. n e. ( ZZ>= ` M ) ph <-> E. n e. ZZ ( M <_ n /\ ph ) ) )