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	decbin0 9601	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 9602	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 9603	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 9604	Half minus a third. (Contributed by Scott Fenton, 8-Jul-2015.)
|- ( ( 1 / 2 ) - ( 1 / 3 ) ) = ( 1 / 6 )
Theorem	5recm6rec 9605	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 9606	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 9607*	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 9608 for its value, uzssz 9626 for its relationship to ZZ, nnuz 9642 and nn0uz 9641 for its relationships to NN and NN0, and eluz1 9610 and eluz2 9612 for its membership relations. (Contributed by NM, 5-Sep-2005.)
|- ZZ>= = ( j e. ZZ |-> { k e. ZZ | j <_ k } )
Theorem	uzval 9608*	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 9609	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 9610	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 9611	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 9612	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 9613	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 9614	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 9615	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 9616	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 9617	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 9618	Implication of membership in an upper set of integers. (Contributed by NM, 6-Sep-2005.)
|- ( N e. ( ZZ>= ` M ) -> M <_ N )
Theorem	eluz 9619	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 9620	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 9621	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 9622	The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.)
|- ( M e. ran ZZ>= -> M =/= (/) )
Theorem	uztrn 9623	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 9624	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 9625	Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.)
|- ( N e. ( ZZ>= ` M ) -> -u M e. ( ZZ>= ` -u N ) )
Theorem	uzssz 9626	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 9627	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 9628	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 9629	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 9630	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 9631	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 9632	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 9633	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 9634	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 9635	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 9636	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 9637	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 9638	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 9639	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 9640	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 9641	Nonnegative integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
|- NN0 = ( ZZ>= ` 0 )
Theorem	nnuz 9642	Positive integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.)
|- NN = ( ZZ>= ` 1 )
Theorem	elnnuz 9643	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 9644	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 9645	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 9646	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 9647	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 9648	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 9649	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 9650	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 9651	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 9652	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 9653	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 9654	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 9655	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 9656	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 9657*	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 9658*	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 9659*	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 ) ) )
Theorem	rexuz2 9660*	Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
|- ( E. n e. ( ZZ>= ` M ) ph <-> ( M e. ZZ /\ E. n e. ZZ ( M <_ n /\ ph ) ) )
Theorem	2rexuz 9661*	Double existential quantification in an upper set of integers. (Contributed by NM, 3-Nov-2005.)
|- ( E. m E. n e. ( ZZ>= ` m ) ph <-> E. m e. ZZ E. n e. ZZ ( m <_ n /\ ph ) )
Theorem	peano2uz 9662	Second Peano postulate for an upper set of integers. (Contributed by NM, 7-Sep-2005.)
|- ( N e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( ZZ>= ` M ) )
Theorem	peano2uzs 9663	Second Peano postulate for an upper set of integers. (Contributed by Mario Carneiro, 26-Dec-2013.)
|- Z = ( ZZ>= ` M )   =>    |- ( N e. Z -> ( N + 1 ) e. Z )
Theorem	peano2uzr 9664	Reversed second Peano axiom for upper integers. (Contributed by NM, 2-Jan-2006.)
|- ( ( M e. ZZ /\ N e. ( ZZ>= ` ( M + 1 ) ) ) -> N e. ( ZZ>= ` M ) )
Theorem	uzaddcl 9665	Addition closure law for an upper set of integers. (Contributed by NM, 4-Jun-2006.)
|- ( ( N e. ( ZZ>= ` M ) /\ K e. NN0 ) -> ( N + K ) e. ( ZZ>= ` M ) )
Theorem	nn0pzuz 9666	The sum of a nonnegative integer and an integer is an integer greater than or equal to that integer. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
|- ( ( N e. NN0 /\ Z e. ZZ ) -> ( N + Z ) e. ( ZZ>= ` Z ) )
Theorem	uzind4 9667*	Induction on the upper set of integers that starts at an integer M. The first four hypotheses give us the substitution instances we need, and the last two are the basis and the induction step. (Contributed by NM, 7-Sep-2005.)
|- ( j = M -> ( ph <-> ps ) )   &    |- ( j = k -> ( ph <-> ch ) )   &    |- ( j = ( k + 1 ) -> ( ph <-> th ) )   &    |- ( j = N -> ( ph <-> ta ) )   &    |- ( M e. ZZ -> ps )   &    |- ( k e. ( ZZ>= ` M ) -> ( ch -> th ) )   =>    |- ( N e. ( ZZ>= ` M ) -> ta )
Theorem	uzind4ALT 9668*	Induction on the upper set of integers that starts at an integer M. The last four hypotheses give us the substitution instances we need; the first two are the basis and the induction step. Either uzind4 9667 or uzind4ALT 9668 may be used; see comment for nnind 9011. (Contributed by NM, 7-Sep-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ( M e. ZZ -> ps )   &    |- ( k e. ( ZZ>= ` M ) -> ( ch -> th ) )   &    |- ( j = M -> ( ph <-> ps ) )   &    |- ( j = k -> ( ph <-> ch ) )   &    |- ( j = ( k + 1 ) -> ( ph <-> th ) )   &    |- ( j = N -> ( ph <-> ta ) )   =>    |- ( N e. ( ZZ>= ` M ) -> ta )
Theorem	uzind4s 9669*	Induction on the upper set of integers that starts at an integer M, using explicit substitution. The hypotheses are the basis and the induction step. (Contributed by NM, 4-Nov-2005.)
|- ( M e. ZZ -> [. M / k ]. ph )   &    |- ( k e. ( ZZ>= ` M ) -> ( ph -> [. ( k + 1 ) / k ]. ph ) )   =>    |- ( N e. ( ZZ>= ` M ) -> [. N / k ]. ph )
Theorem	uzind4s2 9670*	Induction on the upper set of integers that starts at an integer M, using explicit substitution. The hypotheses are the basis and the induction step. Use this instead of uzind4s 9669 when j and k must be distinct in [. ( k + 1 ) / j ]. ph. (Contributed by NM, 16-Nov-2005.)
|- ( M e. ZZ -> [. M / j ]. ph )   &    |- ( k e. ( ZZ>= ` M ) -> ( [. k / j ]. ph -> [. ( k + 1 ) / j ]. ph ) )   =>    |- ( N e. ( ZZ>= ` M ) -> [. N / j ]. ph )
Theorem	uzind4i 9671*	Induction on the upper integers that start at M. The first four give us the substitution instances we need, and the last two are the basis and the induction step. This is a stronger version of uzind4 9667 assuming that ps holds unconditionally. Notice that N e. ( ZZ>= ` M ) implies that the lower bound M is an integer ( M e. ZZ, see eluzel2 9611). (Contributed by NM, 4-Sep-2005.) (Revised by AV, 13-Jul-2022.)
|- ( j = M -> ( ph <-> ps ) )   &    |- ( j = k -> ( ph <-> ch ) )   &    |- ( j = ( k + 1 ) -> ( ph <-> th ) )   &    |- ( j = N -> ( ph <-> ta ) )   &    |- ps   &    |- ( k e. ( ZZ>= ` M ) -> ( ch -> th ) )   =>    |- ( N e. ( ZZ>= ` M ) -> ta )
Theorem	indstr 9672*	Strong Mathematical Induction for positive integers (inference schema). (Contributed by NM, 17-Aug-2001.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ( x e. NN -> ( A. y e. NN ( y < x -> ps ) -> ph ) )   =>    |- ( x e. NN -> ph )
Theorem	infrenegsupex 9673*	The infimum of a set of reals A is the negative of the supremum of the negatives of its elements. (Contributed by Jim Kingdon, 14-Jan-2022.)
|- ( ph -> E. x e. RR ( A. y e. A -. y < x /\ A. y e. RR ( x < y -> E. z e. A z < y ) ) )   &    |- ( ph -> A C_ RR )   =>    |- ( ph -> inf ( A , RR , < ) = -u sup ( { z e. RR | -u z e. A } , RR , < ) )
Theorem	supinfneg 9674*	If a set of real numbers has a least upper bound, the set of the negation of those numbers has a greatest lower bound. For a theorem which is similar but only for the boundedness part, see ublbneg 9692. (Contributed by Jim Kingdon, 15-Jan-2022.)
|- ( ph -> E. x e. RR ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. z e. A y < z ) ) )   &    |- ( ph -> A C_ RR )   =>    |- ( ph -> E. x e. RR ( A. y e. { w e. RR | -u w e. A } -. y < x /\ A. y e. RR ( x < y -> E. z e. { w e. RR | -u w e. A } z < y ) ) )
Theorem	infsupneg 9675*	If a set of real numbers has a greatest lower bound, the set of the negation of those numbers has a least upper bound. To go in the other direction see supinfneg 9674. (Contributed by Jim Kingdon, 15-Jan-2022.)
|- ( ph -> E. x e. RR ( A. y e. A -. y < x /\ A. y e. RR ( x < y -> E. z e. A z < y ) ) )   &    |- ( ph -> A C_ RR )   =>    |- ( ph -> E. x e. RR ( A. y e. { w e. RR | -u w e. A } -. x < y /\ A. y e. RR ( y < x -> E. z e. { w e. RR | -u w e. A } y < z ) ) )
Theorem	supminfex 9676*	A supremum is the negation of the infimum of that set's image under negation. (Contributed by Jim Kingdon, 14-Jan-2022.)
|- ( ph -> E. x e. RR ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. z e. A y < z ) ) )   &    |- ( ph -> A C_ RR )   =>    |- ( ph -> sup ( A , RR , < ) = -uinf ( { w e. RR | -u w e. A } , RR , < ) )
Theorem	infregelbex 9677*	Any lower bound of a set of real numbers with an infimum is less than or equal to the infimum. (Contributed by Jim Kingdon, 27-Sep-2024.)
|- ( ph -> E. x e. RR ( A. y e. A -. y < x /\ A. y e. RR ( x < y -> E. z e. A z < y ) ) )   &    |- ( ph -> A C_ RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( B <_ inf ( A , RR , < ) <-> A. z e. A B <_ z ) )
Theorem	eluznn0 9678	Membership in a nonnegative upper set of integers implies membership in NN0. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ( N e. NN0 /\ M e. ( ZZ>= ` N ) ) -> M e. NN0 )
Theorem	eluznn 9679	Membership in a positive upper set of integers implies membership in NN. (Contributed by JJ, 1-Oct-2018.)
|- ( ( N e. NN /\ M e. ( ZZ>= ` N ) ) -> M e. NN )
Theorem	eluz2b1 9680	Two ways to say "an integer greater than or equal to 2". (Contributed by Paul Chapman, 23-Nov-2012.)
|- ( N e. ( ZZ>= ` 2 ) <-> ( N e. ZZ /\ 1 < N ) )
Theorem	eluz2gt1 9681	An integer greater than or equal to 2 is greater than 1. (Contributed by AV, 24-May-2020.)
|- ( N e. ( ZZ>= ` 2 ) -> 1 < N )
Theorem	eluz2b2 9682	Two ways to say "an integer greater than or equal to 2". (Contributed by Paul Chapman, 23-Nov-2012.)
|- ( N e. ( ZZ>= ` 2 ) <-> ( N e. NN /\ 1 < N ) )
Theorem	eluz2b3 9683	Two ways to say "an integer greater than or equal to 2". (Contributed by Paul Chapman, 23-Nov-2012.)
|- ( N e. ( ZZ>= ` 2 ) <-> ( N e. NN /\ N =/= 1 ) )
Theorem	uz2m1nn 9684	One less than an integer greater than or equal to 2 is a positive integer. (Contributed by Paul Chapman, 17-Nov-2012.)
|- ( N e. ( ZZ>= ` 2 ) -> ( N - 1 ) e. NN )
Theorem	1nuz2 9685	1 is not in ( ZZ>= ` 2 ). (Contributed by Paul Chapman, 21-Nov-2012.)
|- -. 1 e. ( ZZ>= ` 2 )
Theorem	elnn1uz2 9686	A positive integer is either 1 or greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.)
|- ( N e. NN <-> ( N = 1 \/ N e. ( ZZ>= ` 2 ) ) )
Theorem	uz2mulcl 9687	Closure of multiplication of integers greater than or equal to 2. (Contributed by Paul Chapman, 26-Oct-2012.)
|- ( ( M e. ( ZZ>= ` 2 ) /\ N e. ( ZZ>= ` 2 ) ) -> ( M x. N ) e. ( ZZ>= ` 2 ) )
Theorem	indstr2 9688*	Strong Mathematical Induction for positive integers (inference schema). The first two hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by Paul Chapman, 21-Nov-2012.)
|- ( x = 1 -> ( ph <-> ch ) )   &    |- ( x = y -> ( ph <-> ps ) )   &    |- ch   &    |- ( x e. ( ZZ>= ` 2 ) -> ( A. y e. NN ( y < x -> ps ) -> ph ) )   =>    |- ( x e. NN -> ph )
Theorem	eluzdc 9689	Membership of an integer in an upper set of integers is decidable. (Contributed by Jim Kingdon, 18-Apr-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> DECID N e. ( ZZ>= ` M ) )
Theorem	elnn0dc 9690	Membership of an integer in NN0 is decidable. (Contributed by Jim Kingdon, 8-Oct-2024.)
|- ( N e. ZZ -> DECID N e. NN0 )
Theorem	elnndc 9691	Membership of an integer in NN is decidable. (Contributed by Jim Kingdon, 17-Oct-2024.)
|- ( N e. ZZ -> DECID N e. NN )
Theorem	ublbneg 9692*	The image under negation of a bounded-above set of reals is bounded below. For a theorem which is similar but also adds that the bounds need to be the tightest possible, see supinfneg 9674. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( E. x e. RR A. y e. A y <_ x -> E. x e. RR A. y e. { z e. RR | -u z e. A } x <_ y )
Theorem	eqreznegel 9693*	Two ways to express the image under negation of a set of integers. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( A C_ ZZ -> { z e. RR | -u z e. A } = { z e. ZZ | -u z e. A } )
Theorem	negm 9694*	The image under negation of an inhabited set of reals is inhabited. (Contributed by Jim Kingdon, 10-Apr-2020.)
|- ( ( A C_ RR /\ E. x x e. A ) -> E. y y e. { z e. RR | -u z e. A } )
Theorem	lbzbi 9695*	If a set of reals is bounded below, it is bounded below by an integer. (Contributed by Paul Chapman, 21-Mar-2011.)
|- ( A C_ RR -> ( E. x e. RR A. y e. A x <_ y <-> E. x e. ZZ A. y e. A x <_ y ) )
Theorem	nn01to3 9696	A (nonnegative) integer between 1 and 3 must be 1, 2 or 3. (Contributed by Alexander van der Vekens, 13-Sep-2018.)
|- ( ( N e. NN0 /\ 1 <_ N /\ N <_ 3 ) -> ( N = 1 \/ N = 2 \/ N = 3 ) )
Theorem	nn0ge2m1nnALT 9697	Alternate proof of nn0ge2m1nn 9314: If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is a positive integer. This version is proved using eluz2 9612, a theorem for upper sets of integers, which are defined later than the positive and nonnegative integers. This proof is, however, much shorter than the proof of nn0ge2m1nn 9314. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ( ( N e. NN0 /\ 2 <_ N ) -> ( N - 1 ) e. NN )
4.4.12  Rational numbers (as a subset of complex numbers)
Syntax	cq 9698	Extend class notation to include the class of rationals.
class QQ
Definition	df-q 9699	Define the set of rational numbers. Based on definition of rationals in [Apostol] p. 22. See elq 9701 for the relation "is rational". (Contributed by NM, 8-Jan-2002.)
|- QQ = ( / " ( ZZ X. NN ) )
Theorem	divfnzn 9700	Division restricted to ZZ X. NN is a function. Given excluded middle, it would be easy to prove this for CC X. ( CC \ { 0 } ). The key difference is that an element of NN is apart from zero, whereas being an element of CC \ { 0 } implies being not equal to zero. (Contributed by Jim Kingdon, 19-Mar-2020.)
|- ( / |` ( ZZ X. NN ) ) Fn ( ZZ X. NN )