Most Recent Proofs - Intuitionistic Logic Explorer

Most recent proofs    These are the 100 (Unicode, GIF) or 1000 (Unicode, GIF) most recent proofs in the iset.mm database for the Intuitionistic Logic Explorer. The iset.mm database is maintained on GitHub with master (stable) and develop (development) versions. This page was created from the commit given on the MPE Most Recent Proofs page. The database from that commit is also available here: iset.mm.

See the MPE Most Recent Proofs page for news and some useful links.

Last updated on 25-Oct-2021 at 2:44 PM ET.

Recent Additions to the Intuitionistic Logic Explorer

Date	Label	Description
Theorem
24-Oct-2021	ax-ddkcomp 10114	Axiom of Dedekind completeness for Dedekind real numbers: every nonempty upper-bounded located set of reals has a real upper bound. Ideally, this axiom should be "proved" as "axddkcomp" for the real numbers constructed from IZF, and then the axiom ax-ddkcomp 10114 should be used in place of construction specific results. In particular, axcaucvg 6974 should be proved from it. (Contributed by BJ, 24-Oct-2021.)
|- ( ( ( A C_ RR /\ A =/= (/) ) /\ E. x e. RR A. y e. A y < x /\ A. x e. RR A. y e. RR ( x < y -> ( E. z e. A x < z \/ A. z e. A z < y ) ) ) -> E. x e. RR ( A. y e. A y <_ x /\ ( ( B e. R /\ A. y e. A y <_ B ) -> x <_ B ) ) )
20-Oct-2021	onunsnss 6355	Adding a singleton to create an ordinal. (Contributed by Jim Kingdon, 20-Oct-2021.)
|- ( ( B e. V /\ ( A u. { B } ) e. On ) -> B C_ A )
19-Oct-2021	snon0 6356	An ordinal which is a singleton is { (/) }. (Contributed by Jim Kingdon, 19-Oct-2021.)
|- ( ( A e. V /\ { A } e. On ) -> A = (/) )
18-Oct-2021	qdencn 10124	The set of complex numbers whose real and imaginary parts are rational is dense in the complex plane. This is a two dimensional analogue to qdenre 9798 (and also would hold for RR X. RR with the usual metric; this is not about complex numbers in particular). (Contributed by Jim Kingdon, 18-Oct-2021.)
|- Q = { z e. CC | ( ( Re ` z ) e. QQ /\ ( Im ` z ) e. QQ ) }   =>    |- ( ( A e. CC /\ B e. RR+ ) -> E. x e. Q ( abs ` ( x - A ) ) < B )
18-Oct-2021	modqmulnn 9184	Move a positive integer in and out of a floor in the first argument of a modulo operation. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( ( N e. NN /\ A e. QQ /\ M e. NN ) -> ( ( N x. ( |_ ` A ) ) mod ( N x. M ) ) <_ ( ( |_ ` ( N x. A ) ) mod ( N x. M ) ) )
18-Oct-2021	intqfrac 9181	Break a number into its integer part and its fractional part. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( A e. QQ -> A = ( ( |_ ` A ) + ( A mod 1 ) ) )
18-Oct-2021	flqmod 9180	The floor function expressed in terms of the modulo operation. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( A e. QQ -> ( |_ ` A ) = ( A - ( A mod 1 ) ) )
18-Oct-2021	modqfrac 9179	The fractional part of a number is the number modulo 1. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( A e. QQ -> ( A mod 1 ) = ( A - ( |_ ` A ) ) )
18-Oct-2021	modqdifz 9178	The modulo operation differs from A by an integer multiple of B. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A - ( A mod B ) ) / B ) e. ZZ )
18-Oct-2021	modqdiffl 9177	The modulo operation differs from A by an integer multiple of B. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A - ( A mod B ) ) / B ) = ( |_ ` ( A / B ) ) )
18-Oct-2021	modqelico 9176	Modular reduction produces a half-open interval. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) e. ( 0 [,) B ) )
18-Oct-2021	modqlt 9175	The modulo operation is less than its second argument. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) < B )
18-Oct-2021	modqge0 9174	The modulo operation is nonnegative. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> 0 <_ ( A mod B ) )
18-Oct-2021	negqmod0 9173	A is divisible by B iff its negative is. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A mod B ) = 0 <-> ( -u A mod B ) = 0 ) )
18-Oct-2021	mulqmod0 9172	The product of an integer and a positive rational number is 0 modulo the positive real number. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) -> ( ( A x. M ) mod M ) = 0 )
18-Oct-2021	flqdiv 9163	Cancellation of the embedded floor of a real divided by an integer. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- ( ( A e. QQ /\ N e. NN ) -> ( |_ ` ( ( |_ ` A ) / N ) ) = ( |_ ` ( A / N ) ) )
18-Oct-2021	intqfrac2 9161	Decompose a real into integer and fractional parts. (Contributed by Jim Kingdon, 18-Oct-2021.)
|- Z = ( |_ ` A )   &    |- F = ( A - Z )   =>    |- ( A e. QQ -> ( 0 <_ F /\ F < 1 /\ A = ( Z + F ) ) )
17-Oct-2021	modq0 9171	A mod B is zero iff A is evenly divisible by B. (Contributed by Jim Kingdon, 17-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( A mod B ) = 0 <-> ( A / B ) e. ZZ ) )
16-Oct-2021	modqcld 9170	Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.)
|- ( ph -> A e. QQ )   &    |- ( ph -> B e. QQ )   &    |- ( ph -> 0 < B )   =>    |- ( ph -> ( A mod B ) e. QQ )
16-Oct-2021	flqpmodeq 9169	Partition of a division into its integer part and the remainder. (Contributed by Jim Kingdon, 16-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( ( ( |_ ` ( A / B ) ) x. B ) + ( A mod B ) ) = A )
16-Oct-2021	modqcl 9168	Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) e. QQ )
16-Oct-2021	modqvalr 9167	The value of the modulo operation (multiplication in reversed order). (Contributed by Jim Kingdon, 16-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) = ( A - ( ( |_ ` ( A / B ) ) x. B ) ) )
16-Oct-2021	modqval 9166	The value of the modulo operation. The modulo congruence notation of number theory, J == K (modulo N), can be expressed in our notation as ( J mod N ) = ( K mod N ). Definition 1 in Knuth, The Art of Computer Programming, Vol. I (1972), p. 38. Knuth uses "mod" for the operation and "modulo" for the congruence. Unlike Knuth, we restrict the second argument to positive numbers to simplify certain theorems. (This also gives us future flexibility to extend it to any one of several different conventions for a zero or negative second argument, should there be an advantage in doing so.) As with flqcl 9117 we only prove this for rationals although other particular kinds of real numbers may be possible. (Contributed by Jim Kingdon, 16-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) )
15-Oct-2021	qdenre 9798	The rational numbers are dense in RR: any real number can be approximated with arbitrary precision by a rational number. For order theoretic density, see qbtwnre 9111. (Contributed by BJ, 15-Oct-2021.)
|- ( ( A e. RR /\ B e. RR+ ) -> E. x e. QQ ( abs ` ( x - A ) ) < B )
14-Oct-2021	qbtwnrelemcalc 9110	Lemma for qbtwnre 9111. Calculations involved in showing the constructed rational number is less than B. (Contributed by Jim Kingdon, 14-Oct-2021.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> N e. NN )   &    |- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> M < ( A x. ( 2 x. N ) ) )   &    |- ( ph -> ( 1 / N ) < ( B - A ) )   =>    |- ( ph -> ( ( M + 2 ) / ( 2 x. N ) ) < B )
13-Oct-2021	rebtwn2z 9109	A real number can be bounded by integers above and below which are two apart. The proof starts by finding two integers which are less than and greater than the given real number. Then this range can be shrunk by choosing an integer in between the endpoints of the range and then deciding which half of the range to keep based on weak linearity, and iterating until the range consists of integers which are two apart. (Contributed by Jim Kingdon, 13-Oct-2021.)
|- ( A e. RR -> E. x e. ZZ ( x < A /\ A < ( x + 2 ) ) )
13-Oct-2021	rebtwn2zlemshrink 9108	Lemma for rebtwn2z 9109. Shrinking the range around the given real number. (Contributed by Jim Kingdon, 13-Oct-2021.)
|- ( ( A e. RR /\ J e. ( ZZ>= ` 2 ) /\ E. m e. ZZ ( m < A /\ A < ( m + J ) ) ) -> E. x e. ZZ ( x < A /\ A < ( x + 2 ) ) )
13-Oct-2021	rebtwn2zlemstep 9107	Lemma for rebtwn2z 9109. Induction step. (Contributed by Jim Kingdon, 13-Oct-2021.)
|- ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR /\ E. m e. ZZ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> E. m e. ZZ ( m < A /\ A < ( m + K ) ) )
11-Oct-2021	flqeqceilz 9160	A rational number is an integer iff its floor equals its ceiling. (Contributed by Jim Kingdon, 11-Oct-2021.)
|- ( A e. QQ -> ( A e. ZZ <-> ( |_ ` A ) = (⌈ ` A ) ) )
11-Oct-2021	flqleceil 9159	The floor of a rational number is less than or equal to its ceiling. (Contributed by Jim Kingdon, 11-Oct-2021.)
|- ( A e. QQ -> ( |_ ` A ) <_ (⌈ ` A ) )
11-Oct-2021	ceilqidz 9158	A rational number equals its ceiling iff it is an integer. (Contributed by Jim Kingdon, 11-Oct-2021.)
|- ( A e. QQ -> ( A e. ZZ <-> (⌈ ` A ) = A ) )
11-Oct-2021	ceilqle 9156	The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by Jim Kingdon, 11-Oct-2021.)
|- ( ( A e. QQ /\ B e. ZZ /\ A <_ B ) -> (⌈ ` A ) <_ B )
11-Oct-2021	ceiqle 9155	The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by Jim Kingdon, 11-Oct-2021.)
|- ( ( A e. QQ /\ B e. ZZ /\ A <_ B ) -> -u ( |_ ` -u A ) <_ B )
11-Oct-2021	ceilqm1lt 9154	One less than the ceiling of a real number is strictly less than that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
|- ( A e. QQ -> ( (⌈ ` A ) - 1 ) < A )
11-Oct-2021	ceiqm1l 9153	One less than the ceiling of a real number is strictly less than that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
|- ( A e. QQ -> ( -u ( |_ ` -u A ) - 1 ) < A )
11-Oct-2021	ceilqge 9152	The ceiling of a real number is greater than or equal to that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
|- ( A e. QQ -> A <_ (⌈ ` A ) )
11-Oct-2021	ceiqge 9151	The ceiling of a real number is greater than or equal to that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
|- ( A e. QQ -> A <_ -u ( |_ ` -u A ) )
11-Oct-2021	ceilqcl 9150	Closure of the ceiling function. (Contributed by Jim Kingdon, 11-Oct-2021.)
|- ( A e. QQ -> (⌈ ` A ) e. ZZ )
11-Oct-2021	ceiqcl 9149	The ceiling function returns an integer (closure law). (Contributed by Jim Kingdon, 11-Oct-2021.)
|- ( A e. QQ -> -u ( |_ ` -u A ) e. ZZ )
11-Oct-2021	qdceq 9102	Equality of rationals is decidable. (Contributed by Jim Kingdon, 11-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ ) -> DECID A = B )
11-Oct-2021	qltlen 8575	Rational 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 7621 which is a similar result for real numbers. (Contributed by Jim Kingdon, 11-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ ) -> ( A < B <-> ( A <_ B /\ B =/= A ) ) )
10-Oct-2021	ceilqval 9148	The value of the ceiling function. (Contributed by Jim Kingdon, 10-Oct-2021.)
|- ( A e. QQ -> (⌈ ` A ) = -u ( |_ ` -u A ) )
10-Oct-2021	flqmulnn0 9141	Move a nonnegative integer in and out of a floor. (Contributed by Jim Kingdon, 10-Oct-2021.)
|- ( ( N e. NN0 /\ A e. QQ ) -> ( N x. ( |_ ` A ) ) <_ ( |_ ` ( N x. A ) ) )
10-Oct-2021	flqzadd 9140	An integer can be moved in and out of the floor of a sum. (Contributed by Jim Kingdon, 10-Oct-2021.)
|- ( ( N e. ZZ /\ A e. QQ ) -> ( |_ ` ( N + A ) ) = ( N + ( |_ ` A ) ) )
10-Oct-2021	flqaddz 9139	An integer can be moved in and out of the floor of a sum. (Contributed by Jim Kingdon, 10-Oct-2021.)
|- ( ( A e. QQ /\ N e. ZZ ) -> ( |_ ` ( A + N ) ) = ( ( |_ ` A ) + N ) )
10-Oct-2021	flqge1nn 9136	The floor of a number greater than or equal to 1 is a positive integer. (Contributed by Jim Kingdon, 10-Oct-2021.)
|- ( ( A e. QQ /\ 1 <_ A ) -> ( |_ ` A ) e. NN )
10-Oct-2021	flqge0nn0 9135	The floor of a number greater than or equal to 0 is a nonnegative integer. (Contributed by Jim Kingdon, 10-Oct-2021.)
|- ( ( A e. QQ /\ 0 <_ A ) -> ( |_ ` A ) e. NN0 )
10-Oct-2021	4ap0 8015	The number 4 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
|- 4 # 0
10-Oct-2021	3ap0 8012	The number 3 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
|- 3 # 0
9-Oct-2021	flqbi2 9133	A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.)
|- ( ( N e. ZZ /\ F e. QQ ) -> ( ( |_ ` ( N + F ) ) = N <-> ( 0 <_ F /\ F < 1 ) ) )
9-Oct-2021	flqbi 9132	A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.)
|- ( ( A e. QQ /\ B e. ZZ ) -> ( ( |_ ` A ) = B <-> ( B <_ A /\ A < ( B + 1 ) ) ) )
9-Oct-2021	flqword2 9131	Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ A <_ B ) -> ( |_ ` B ) e. ( ZZ>= ` ( |_ ` A ) ) )
9-Oct-2021	flqwordi 9130	Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ /\ A <_ B ) -> ( |_ ` A ) <_ ( |_ ` B ) )
9-Oct-2021	flqltnz 9129	If A is not an integer, then the floor of A is less than A. (Contributed by Jim Kingdon, 9-Oct-2021.)
|- ( ( A e. QQ /\ -. A e. ZZ ) -> ( |_ ` A ) < A )
9-Oct-2021	flqidz 9128	A rational number equals its floor iff it is an integer. (Contributed by Jim Kingdon, 9-Oct-2021.)
|- ( A e. QQ -> ( ( |_ ` A ) = A <-> A e. ZZ ) )
8-Oct-2021	flqidm 9127	The floor function is idempotent. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> ( |_ ` ( |_ ` A ) ) = ( |_ ` A ) )
8-Oct-2021	flqlt 9125	The floor function value is less than the next integer. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( ( A e. QQ /\ B e. ZZ ) -> ( A < B <-> ( |_ ` A ) < B ) )
8-Oct-2021	flqge 9124	The floor function value is the greatest integer less than or equal to its argument. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( ( A e. QQ /\ B e. ZZ ) -> ( B <_ A <-> B <_ ( |_ ` A ) ) )
8-Oct-2021	qfracge0 9123	The fractional part of a rational number is nonnegative. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> 0 <_ ( A - ( |_ ` A ) ) )
8-Oct-2021	qfraclt1 9122	The fractional part of a rational number is less than one. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> ( A - ( |_ ` A ) ) < 1 )
8-Oct-2021	flqltp1 9121	A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> A < ( ( |_ ` A ) + 1 ) )
8-Oct-2021	flqle 9120	A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> ( |_ ` A ) <_ A )
8-Oct-2021	flqcld 9119	The floor (greatest integer) function is an integer (closure law). (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( ph -> A e. QQ )   =>    |- ( ph -> ( |_ ` A ) e. ZZ )
8-Oct-2021	flqlelt 9118	A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> ( ( |_ ` A ) <_ A /\ A < ( ( |_ ` A ) + 1 ) ) )
8-Oct-2021	flqcl 9117	The floor (greatest integer) function yields an integer when applied to a rational (closure law). It would presumably be possible to prove a similar result for some real numbers (for example, those apart from any integer), but not real numbers in general. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> ( |_ ` A ) e. ZZ )
8-Oct-2021	qbtwnz 9106	There is a unique greatest integer less than or equal to a rational number. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> E! x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) )
8-Oct-2021	qbtwnzlemex 9105	Lemma for qbtwnz 9106. Existence of the integer. The proof starts by finding two integers which are less than and greater than the given rational number. Then this range can be shrunk by choosing an integer in between the endpoints of the range and then deciding which half of the range to keep based on rational number trichotomy, and iterating until the range consists of two consecutive integers. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( A e. QQ -> E. x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) )
8-Oct-2021	qbtwnzlemshrink 9104	Lemma for qbtwnz 9106. Shrinking the range around the given rational number. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( ( A e. QQ /\ J e. NN /\ E. m e. ZZ ( m <_ A /\ A < ( m + J ) ) ) -> E. x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) )
8-Oct-2021	qbtwnzlemstep 9103	Lemma for qbtwnz 9106. Induction step. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( ( K e. NN /\ A e. QQ /\ E. m e. ZZ ( m <_ A /\ A < ( m + ( K + 1 ) ) ) ) -> E. m e. ZZ ( m <_ A /\ A < ( m + K ) ) )
8-Oct-2021	qltnle 9101	'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 8-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ ) -> ( A < B <-> -. B <_ A ) )
7-Oct-2021	qlelttric 9100	Rational trichotomy. (Contributed by Jim Kingdon, 7-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ ) -> ( A <_ B \/ B < A ) )
6-Oct-2021	qletric 9099	Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)
|- ( ( A e. QQ /\ B e. QQ ) -> ( A <_ B \/ B <_ A ) )
6-Oct-2021	qtri3or 9098	Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)
|- ( ( M e. QQ /\ N e. QQ ) -> ( M < N \/ M = N \/ N < M ) )
3-Oct-2021	reg3exmid 4304	If any inhabited set satisfying df-wetr 4071 for _E has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Oct-2021.)
|- ( ( _E We z /\ E. w w e. z ) -> E. x e. z A. y e. z x C_ y )   =>    |- ( ph \/ -. ph )
3-Oct-2021	reg3exmidlemwe 4303	Lemma for reg3exmid 4304. Our counterexample A satisfies We. (Contributed by Jim Kingdon, 3-Oct-2021.)
|- A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) }   =>    |- _E We A
2-Oct-2021	reg2exmid 4261	If any inhabited set has a minimal element (when expressed by C_), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.)
|- A. z ( E. w w e. z -> E. x e. z A. y e. z x C_ y )   =>    |- ( ph \/ -. ph )
2-Oct-2021	reg2exmidlema 4259	Lemma for reg2exmid 4261. If A has a minimal element (expressed by C_), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.)
|- A = { x e. { (/) , { (/) } } | ( x = { (/) } \/ ( x = (/) /\ ph ) ) }   =>    |- ( E. u e. A A. v e. A u C_ v -> ( ph \/ -. ph ) )
30-Sep-2021	fin0or 6343	A finite set is either empty or inhabited. (Contributed by Jim Kingdon, 30-Sep-2021.)
|- ( A e. Fin -> ( A = (/) \/ E. x x e. A ) )
30-Sep-2021	wessep 4302	A subset of a set well-ordered by set membership is well-ordered by set membership. (Contributed by Jim Kingdon, 30-Sep-2021.)
|- ( ( _E We A /\ B C_ A ) -> _E We B )
28-Sep-2021	frind 4089	Induction over a well-founded set. (Contributed by Jim Kingdon, 28-Sep-2021.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ( ( ch /\ x e. A ) -> ( A. y e. A ( y R x -> ps ) -> ph ) )   &    |- ( ch -> R Fr A )   &    |- ( ch -> A e. V )   =>    |- ( ( ch /\ x e. A ) -> ph )
26-Sep-2021	wetriext 4301	A trichotomous well-order is extensional. (Contributed by Jim Kingdon, 26-Sep-2021.)
|- ( ph -> R We A )   &    |- ( ph -> A e. V )   &    |- ( ph -> A. a e. A A. b e. A ( a R b \/ a = b \/ b R a ) )   &    |- ( ph -> B e. A )   &    |- ( ph -> C e. A )   &    |- ( ph -> A. z e. A ( z R B <-> z R C ) )   =>    |- ( ph -> B = C )
25-Sep-2021	nnwetri 6354	A natural number is well-ordered by _E. More specifically, this order both satisfies We and is trichotomous. (Contributed by Jim Kingdon, 25-Sep-2021.)
|- ( A e. om -> ( _E We A /\ A. x e. A A. y e. A ( x _E y \/ x = y \/ y _E x ) ) )
23-Sep-2021	wepo 4096	A well-ordering is a partial ordering. (Contributed by Jim Kingdon, 23-Sep-2021.)
|- ( ( R We A /\ A e. V ) -> R Po A )
23-Sep-2021	df-wetr 4071	Define the well-ordering predicate. It is unusual to define "well-ordering" in the absence of excluded middle, but we mean an ordering which is like the ordering which we have for ordinals (for example, it does not entail trichotomy because ordinals don't have that as seen at ordtriexmid 4247). Given excluded middle, well-ordering is usually defined to require trichotomy (and the defintion of Fr is typically also different). (Contributed by Mario Carneiro and Jim Kingdon, 23-Sep-2021.)
|- ( R We A <-> ( R Fr A /\ A. x e. A A. y e. A A. z e. A ( ( x R y /\ y R z ) -> x R z ) ) )
22-Sep-2021	frforeq3 4084	Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
|- ( S = T -> (FrFor R A S <-> FrFor R A T ) )
22-Sep-2021	frforeq2 4082	Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
|- ( A = B -> (FrFor R A T <-> FrFor R B T ) )
22-Sep-2021	frforeq1 4080	Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
|- ( R = S -> (FrFor R A T <-> FrFor S A T ) )
22-Sep-2021	df-frfor 4068	Define the well-founded relation predicate where A might be a proper class. By passing in S we allow it potentially to be a proper class rather than a set. (Contributed by Jim Kingdon and Mario Carneiro, 22-Sep-2021.)
|- (FrFor R A S <-> ( A. x e. A ( A. y e. A ( y R x -> y e. S ) -> x e. S ) -> A C_ S ) )
21-Sep-2021	frirrg 4087	A well-founded relation is irreflexive. This is the case where A exists. (Contributed by Jim Kingdon, 21-Sep-2021.)
|- ( ( R Fr A /\ A e. V /\ B e. A ) -> -. B R B )
21-Sep-2021	df-frind 4069	Define the well-founded relation predicate. In the presence of excluded middle, there are a variety of equivalent ways to define this. In our case, this definition, in terms of an inductive principle, works better than one along the lines of "there is an element which is minimal when A is ordered by R". Because s is constrained to be a set (not a proper class) here, sometimes it may be necessary to use FrFor directly rather than via Fr. (Contributed by Jim Kingdon and Mario Carneiro, 21-Sep-2021.)
|- ( R Fr A <-> A. sFrFor R A s )
17-Sep-2021	ontr2exmid 4250	An ordinal transitivity law which implies excluded middle. (Contributed by Jim Kingdon, 17-Sep-2021.)
|- A. x e. On A. y A. z e. On ( ( x C_ y /\ y e. z ) -> x e. z )   =>    |- ( ph \/ -. ph )
16-Sep-2021	nnsseleq 6079	For natural numbers, inclusion is equivalent to membership or equality. (Contributed by Jim Kingdon, 16-Sep-2021.)
|- ( ( A e. om /\ B e. om ) -> ( A C_ B <-> ( A e. B \/ A = B ) ) )
15-Sep-2021	fientri3 6353	Trichotomy of dominance for finite sets. (Contributed by Jim Kingdon, 15-Sep-2021.)
|- ( ( A e. Fin /\ B e. Fin ) -> ( A ~<_ B \/ B ~<_ A ) )
15-Sep-2021	nntri2or2 6076	A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 15-Sep-2021.)
|- ( ( A e. om /\ B e. om ) -> ( A C_ B \/ B C_ A ) )
14-Sep-2021	findcard2sd 6349	Deduction form of finite set induction . (Contributed by Jim Kingdon, 14-Sep-2021.)
|- ( x = (/) -> ( ps <-> ch ) )   &    |- ( x = y -> ( ps <-> th ) )   &    |- ( x = ( y u. { z } ) -> ( ps <-> ta ) )   &    |- ( x = A -> ( ps <-> et ) )   &    |- ( ph -> ch )   &    |- ( ( ( ph /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> ( th -> ta ) )   &    |- ( ph -> A e. Fin )   =>    |- ( ph -> et )
13-Sep-2021	php5fin 6339	A finite set is not equinumerous to a set which adds one element. (Contributed by Jim Kingdon, 13-Sep-2021.)
|- ( ( A e. Fin /\ B e. ( _V \ A ) ) -> -. A ~~ ( A u. { B } ) )
13-Sep-2021	fiunsnnn 6338	Adding one element to a finite set which is equinumerous to a natural number. (Contributed by Jim Kingdon, 13-Sep-2021.)
|- ( ( ( A e. Fin /\ B e. ( _V \ A ) ) /\ ( N e. om /\ A ~~ N ) ) -> ( A u. { B } ) ~~ suc N )
12-Sep-2021	fisbth 6340	Schroeder-Bernstein Theorem for finite sets. (Contributed by Jim Kingdon, 12-Sep-2021.)
|- ( ( ( A e. Fin /\ B e. Fin ) /\ ( A ~<_ B /\ B ~<_ A ) ) -> A ~~ B )
11-Sep-2021	diffisn 6350	Subtracting a singleton from a finite set produces a finite set. (Contributed by Jim Kingdon, 11-Sep-2021.)
|- ( ( A e. Fin /\ B e. A ) -> ( A \ { B } ) e. Fin )
10-Sep-2021	fin0 6342	A nonempty finite set has at least one element. (Contributed by Jim Kingdon, 10-Sep-2021.)
|- ( A e. Fin -> ( A =/= (/) <-> E. x x e. A ) )
9-Sep-2021	fidifsnid 6332	If we remove a single element from a finite set then put it back in, we end up with the original finite set. This strengthens difsnss 3510 from subset to equality when the set is finite. (Contributed by Jim Kingdon, 9-Sep-2021.)
|- ( ( A e. Fin /\ B e. A ) -> ( ( A \ { B } ) u. { B } ) = A )
9-Sep-2021	fidifsnen 6331	All decrements of a finite set are equinumerous. (Contributed by Jim Kingdon, 9-Sep-2021.)
|- ( ( X e. Fin /\ A e. X /\ B e. X ) -> ( X \ { A } ) ~~ ( X \ { B } ) )
8-Sep-2021	diffifi 6351	Subtracting one finite set from another produces a finite set. (Contributed by Jim Kingdon, 8-Sep-2021.)
|- ( ( A e. Fin /\ B e. Fin /\ B C_ A ) -> ( A \ B ) e. Fin )
8-Sep-2021	diffitest 6344	If subtracting any set from a finite set gives a finite set, any proposition of the form -. ph is decidable. This is not a proof of full excluded middle, but it is close enough to show we won't be able to prove A e. Fin -> ( A \ B ) e. Fin. (Contributed by Jim Kingdon, 8-Sep-2021.)
|- A. a e. Fin A. b ( a \ b ) e. Fin   =>    |- ( -. ph \/ -. -. ph )
6-Sep-2021	phpelm 6328	Pigeonhole Principle. A natural number is not equinumerous to an element of itself. (Contributed by Jim Kingdon, 6-Sep-2021.)
|- ( ( A e. om /\ B e. A ) -> -. A ~~ B )
5-Sep-2021	fidceq 6330	Equality of members of a finite set is decidable. This may be counterintuitive: cannot any two sets be elements of a finite set? Well, to show, for example, that { B , C } is finite would require showing it is equinumerous to 1o or to 2o but to show that you'd need to know B = C or -. B = C, respectively. (Contributed by Jim Kingdon, 5-Sep-2021.)
|- ( ( A e. Fin /\ B e. A /\ C e. A ) -> DECID B = C )
5-Sep-2021	phplem4on 6329	Equinumerosity of successors of an ordinal and a natural number implies equinumerosity of the originals. (Contributed by Jim Kingdon, 5-Sep-2021.)
|- ( ( A e. On /\ B e. om ) -> ( suc A ~~ suc B -> A ~~ B ) )
4-Sep-2021	ex-ceil 9896	Example for df-ceil 9115. (Contributed by AV, 4-Sep-2021.)
|- ( (⌈ ` ( 3 / 2 ) ) = 2 /\ (⌈ ` -u ( 3 / 2 ) ) = -u 1 )
3-Sep-2021	0elsucexmid 4289	If the successor of any ordinal class contains the empty set, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2021.)
|- A. x e. On (/) e. suc x   =>    |- ( ph \/ -. ph )
1-Sep-2021	php5dom 6325	A natural number does not dominate its successor. (Contributed by Jim Kingdon, 1-Sep-2021.)
|- ( A e. om -> -. suc A ~<_ A )
1-Sep-2021	phplem4dom 6324	Dominance of successors implies dominance of the original natural numbers. (Contributed by Jim Kingdon, 1-Sep-2021.)
|- ( ( A e. om /\ B e. om ) -> ( suc A ~<_ suc B -> A ~<_ B ) )
1-Sep-2021	snnen2oprc 6323	A singleton { A } is never equinumerous with the ordinal number 2. If A is a set, see snnen2og 6322. (Contributed by Jim Kingdon, 1-Sep-2021.)
|- ( -. A e. _V -> -. { A } ~~ 2o )
1-Sep-2021	snnen2og 6322	A singleton { A } is never equinumerous with the ordinal number 2. If A is a proper class, see snnen2oprc 6323. (Contributed by Jim Kingdon, 1-Sep-2021.)
|- ( A e. V -> -. { A } ~~ 2o )
1-Sep-2021	phplem3g 6319	A natural number is equinumerous to its successor minus any element of the successor. Version of phplem3 6317 with unnecessary hypotheses removed. (Contributed by Jim Kingdon, 1-Sep-2021.)
|- ( ( A e. om /\ B e. suc A ) -> A ~~ ( suc A \ { B } ) )
31-Aug-2021	nndifsnid 6080	If we remove a single element from a natural number then put it back in, we end up with the original natural number. This strengthens difsnss 3510 from subset to equality but the proof relies on equality being decidable. (Contributed by Jim Kingdon, 31-Aug-2021.)
|- ( ( A e. om /\ B e. A ) -> ( ( A \ { B } ) u. { B } ) = A )
30-Aug-2021	carden2bex 6369	If two numerable sets are equinumerous, then they have equal cardinalities. (Contributed by Jim Kingdon, 30-Aug-2021.)
|- ( ( A ~~ B /\ E. x e. On x ~~ A ) -> ( card ` A ) = ( card ` B ) )
30-Aug-2021	cardval3ex 6365	The value of ( card ` A ). (Contributed by Jim Kingdon, 30-Aug-2021.)
|- ( E. x e. On x ~~ A -> ( card ` A ) = |^| { y e. On | y ~~ A } )
30-Aug-2021	cardcl 6361	The cardinality of a well-orderable set is an ordinal. (Contributed by Jim Kingdon, 30-Aug-2021.)
|- ( E. y e. On y ~~ A -> ( card ` A ) e. On )
30-Aug-2021	onintrab2im 4244	An existence condition which implies an intersection is an ordinal number. (Contributed by Jim Kingdon, 30-Aug-2021.)
|- ( E. x e. On ph -> |^| { x e. On | ph } e. On )
30-Aug-2021	onintonm 4243	The intersection of an inhabited collection of ordinal numbers is an ordinal number. Compare Exercise 6 of [TakeutiZaring] p. 44. (Contributed by Mario Carneiro and Jim Kingdon, 30-Aug-2021.)
|- ( ( A C_ On /\ E. x x e. A ) -> |^| A e. On )
29-Aug-2021	onintexmid 4297	If the intersection (infimum) of an inhabited class of ordinal numbers belongs to the class, excluded middle follows. The hypothesis would be provable given excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Aug-2021.)
|- ( ( y C_ On /\ E. x x e. y ) -> |^| y e. y )   =>    |- ( ph \/ -. ph )
29-Aug-2021	ordtri2or2exmid 4296	Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 29-Aug-2021.)
|- A. x e. On A. y e. On ( x C_ y \/ y C_ x )   =>    |- ( ph \/ -. ph )
29-Aug-2021	ordtri2or2exmidlem 4251	A set which is 2o if ph or (/) if -. ph is an ordinal. (Contributed by Jim Kingdon, 29-Aug-2021.)
|- { x e. { (/) , { (/) } } | ph } e. On
29-Aug-2021	2ordpr 4249	Version of 2on 6009 with the definition of 2o expanded and expressed in terms of Ord. (Contributed by Jim Kingdon, 29-Aug-2021.)
|- Ord { (/) , { (/) } }
25-Aug-2021	nnap0d 7959	A positive integer is apart from zero. (Contributed by Jim Kingdon, 25-Aug-2021.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A # 0 )
24-Aug-2021	climcaucn 9870	A converging sequence of complex numbers is a Cauchy sequence. This is like climcau 9866 but adds the part that ( F ` k ) is complex. (Contributed by Jim Kingdon, 24-Aug-2021.)
|- Z = ( ZZ>= ` M )   =>    |- ( ( M e. ZZ /\ F e. dom ~~> ) -> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - ( F ` j ) ) ) < x ) )
24-Aug-2021	climcvg1nlem 9868	Lemma for climcvg1n 9869. We construct sequences of the real and imaginary parts of each term of F, show those converge, and use that to show that F converges. (Contributed by Jim Kingdon, 24-Aug-2021.)
|- ( ph -> F : NN --> CC )   &    |- ( ph -> C e. RR+ )   &    |- ( ph -> A. n e. NN A. k e. ( ZZ>= ` n ) ( abs ` ( ( F ` k ) - ( F ` n ) ) ) < ( C / n ) )   &    |- G = ( x e. NN |-> ( Re ` ( F ` x ) ) )   &    |- H = ( x e. NN |-> ( Im ` ( F ` x ) ) )   &    |- J = ( x e. NN |-> ( _i x. ( H ` x ) ) )   =>    |- ( ph -> F e. dom ~~> )
23-Aug-2021	climcvg1n 9869	A Cauchy sequence of complex numbers converges, existence version. The rate of convergence is fixed: all terms after the nth term must be within C / n of the nth term, where C is a constant multiplier. (Contributed by Jim Kingdon, 23-Aug-2021.)
|- ( ph -> F : NN --> CC )   &    |- ( ph -> C e. RR+ )   &    |- ( ph -> A. n e. NN A. k e. ( ZZ>= ` n ) ( abs ` ( ( F ` k ) - ( F ` n ) ) ) < ( C / n ) )   =>    |- ( ph -> F e. dom ~~> )
23-Aug-2021	climrecvg1n 9867	A Cauchy sequence of real numbers converges, existence version. The rate of convergence is fixed: all terms after the nth term must be within C / n of the nth term, where C is a constant multiplier. (Contributed by Jim Kingdon, 23-Aug-2021.)
|- ( ph -> F : NN --> RR )   &    |- ( ph -> C e. RR+ )   &    |- ( ph -> A. n e. NN A. k e. ( ZZ>= ` n ) ( abs ` ( ( F ` k ) - ( F ` n ) ) ) < ( C / n ) )   =>    |- ( ph -> F e. dom ~~> )
22-Aug-2021	climserile 9865	The partial sums of a converging infinite series with nonnegative terms are bounded by its limit. (Contributed by Jim Kingdon, 22-Aug-2021.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> N e. Z )   &    |- ( ph -> seq M ( + , F , CC ) ~~> A )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR )   &    |- ( ( ph /\ k e. Z ) -> 0 <_ ( F ` k ) )   =>    |- ( ph -> ( seq M ( + , F , CC ) ` N ) <_ A )
22-Aug-2021	iserige0 9863	The limit of an infinite series of nonnegative reals is nonnegative. (Contributed by Jim Kingdon, 22-Aug-2021.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> seq M ( + , F , CC ) ~~> A )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR )   &    |- ( ( ph /\ k e. Z ) -> 0 <_ ( F ` k ) )   =>    |- ( ph -> 0 <_ A )
22-Aug-2021	iserile 9862	Comparison of the limits of two infinite series. (Contributed by Jim Kingdon, 22-Aug-2021.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> seq M ( + , F , CC ) ~~> A )   &    |- ( ph -> seq M ( + , G , CC ) ~~> B )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR )   &    |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. RR )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) <_ ( G ` k ) )   =>    |- ( ph -> A <_ B )
22-Aug-2021	serile 9253	Comparison of partial sums of two infinite series of reals. (Contributed by Jim Kingdon, 22-Aug-2021.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. RR )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. RR )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) <_ ( G ` k ) )   =>    |- ( ph -> ( seq M ( + , F , CC ) ` N ) <_ ( seq M ( + , G , CC ) ` N ) )
22-Aug-2021	serige0 9252	A finite sum of nonnegative terms is nonnegative. (Contributed by Jim Kingdon, 22-Aug-2021.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. RR )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> 0 <_ ( F ` k ) )   =>    |- ( ph -> 0 <_ ( seq M ( + , F , CC ) ` N ) )
21-Aug-2021	clim2iser2 9858	The limit of an infinite series with an initial segment added. (Contributed by Jim Kingdon, 21-Aug-2021.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> N e. Z )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   &    |- ( ph -> seq ( N + 1 ) ( + , F , CC ) ~~> A )   =>    |- ( ph -> seq M ( + , F , CC ) ~~> ( A + ( seq M ( + , F , CC ) ` N ) ) )
21-Aug-2021	iseqdistr 9249	The distributive property for series. (Contributed by Jim Kingdon, 21-Aug-2021.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( C T ( x .+ y ) ) = ( ( C T x ) .+ ( C T y ) ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) = ( C T ( G ` x ) ) )   &    |- ( ph -> S e. V )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x T y ) e. S )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ph -> C e. S )   =>    |- ( ph -> ( seq M ( .+ , F , S ) ` N ) = ( C T ( seq M ( .+ , G , S ) ` N ) ) )
21-Aug-2021	iseqhomo 9248	Apply a homomorphism to a sequence. (Contributed by Jim Kingdon, 21-Aug-2021.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ph -> S e. V )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( H ` ( x .+ y ) ) = ( ( H ` x ) Q ( H ` y ) ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( H ` ( F ` x ) ) = ( G ` x ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x Q y ) e. S )   =>    |- ( ph -> ( H ` ( seq M ( .+ , F , S ) ` N ) ) = ( seq M ( Q , G , S ) ` N ) )
20-Aug-2021	clim2iser 9857	The limit of an infinite series with an initial segment removed. (Contributed by Jim Kingdon, 20-Aug-2021.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> N e. Z )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   &    |- ( ph -> seq M ( + , F , CC ) ~~> A )   =>    |- ( ph -> seq ( N + 1 ) ( + , F , CC ) ~~> ( A - ( seq M ( + , F , CC ) ` N ) ) )
20-Aug-2021	iseqex 9213	Existence of the sequence builder operation. (Contributed by Jim Kingdon, 20-Aug-2021.)
|- seq M ( .+ , F , S ) e. _V
20-Aug-2021	frecex 5981	Finite recursion produces a set. (Contributed by Jim Kingdon, 20-Aug-2021.)
|- frec ( F , A ) e. _V
20-Aug-2021	tfrfun 5955	Transfinite recursion produces a function. (Contributed by Jim Kingdon, 20-Aug-2021.)
|- Fun recs ( F )
19-Aug-2021	iserclim0 9826	The zero series converges to zero. (Contributed by Jim Kingdon, 19-Aug-2021.)
|- ( M e. ZZ -> seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) , CC ) ~~> 0 )
19-Aug-2021	shftval4g 9438	Value of a sequence shifted by -u A. (Contributed by Jim Kingdon, 19-Aug-2021.)
|- ( ( F e. V /\ A e. CC /\ B e. CC ) -> ( ( F shift -u A ) ` B ) = ( F ` ( A + B ) ) )
19-Aug-2021	iser0f 9251	A zero-valued infinite series is equal to the constant zero function. (Contributed by Jim Kingdon, 19-Aug-2021.)
|- Z = ( ZZ>= ` M )   =>    |- ( M e. ZZ -> seq M ( + , ( Z X. { 0 } ) , CC ) = ( Z X. { 0 } ) )
19-Aug-2021	iser0 9250	The value of the partial sums in a zero-valued infinite series. (Contributed by Jim Kingdon, 19-Aug-2021.)
|- Z = ( ZZ>= ` M )   =>    |- ( N e. Z -> ( seq M ( + , ( Z X. { 0 } ) , CC ) ` N ) = 0 )
19-Aug-2021	ovexg 5539	Evaluating a set operation at two sets gives a set. (Contributed by Jim Kingdon, 19-Aug-2021.)
|- ( ( A e. V /\ F e. W /\ B e. X ) -> ( A F B ) e. _V )
18-Aug-2021	iseqid3s 9246	A sequence that consists of zeroes up to N sums to zero at N. In this case by "zero" we mean whatever the identity Z is for the operation .+). (Contributed by Jim Kingdon, 18-Aug-2021.)
|- ( ph -> ( Z .+ Z ) = Z )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ x e. ( M ... N ) ) -> ( F ` x ) = Z )   &    |- ( ph -> Z e. S )   &    |- ( ph -> S e. V )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> ( seq M ( .+ , F , S ) ` N ) = Z )
18-Aug-2021	iseqid3 9245	A sequence that consists entirely of zeroes (or whatever the identity Z is for operation .+) sums to zero. (Contributed by Jim Kingdon, 18-Aug-2021.)
|- ( ph -> ( Z .+ Z ) = Z )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) = Z )   &    |- ( ph -> Z e. V )   =>    |- ( ph -> ( seq M ( .+ , F , { Z } ) ` N ) = Z )
18-Aug-2021	iseqss 9226	Specifying a larger universe for seq. As long as F and .+ are closed over S, then any set which contains S can be used as the last argument to seq. This theorem does not allow T to be a proper class, however. It also currently requires that .+ be closed over T (as well as S). (Contributed by Jim Kingdon, 18-Aug-2021.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> T e. V )   &    |- ( ph -> S C_ T )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. T /\ y e. T ) ) -> ( x .+ y ) e. T )   =>    |- ( ph -> seq M ( .+ , F , S ) = seq M ( .+ , F , T ) )
17-Aug-2021	iseqcaopr 9242	The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Jim Kingdon, 17-Aug-2021.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) = ( y .+ x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( H ` k ) = ( ( F ` k ) .+ ( G ` k ) ) )   &    |- ( ph -> S e. V )   =>    |- ( ph -> ( seq M ( .+ , H , S ) ` N ) = ( ( seq M ( .+ , F , S ) ` N ) .+ ( seq M ( .+ , G , S ) ` N ) ) )
16-Aug-2021	iseqcaopr3 9240	Lemma for iseqcaopr2 . (Contributed by Jim Kingdon, 16-Aug-2021.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x Q y ) e. S )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( H ` k ) = ( ( F ` k ) Q ( G ` k ) ) )   &    |- ( ( ph /\ n e. ( M..^ N ) ) -> ( ( ( seq M ( .+ , F , S ) ` n ) Q ( seq M ( .+ , G , S ) ` n ) ) .+ ( ( F ` ( n + 1 ) ) Q ( G ` ( n + 1 ) ) ) ) = ( ( ( seq M ( .+ , F , S ) ` n ) .+ ( F ` ( n + 1 ) ) ) Q ( ( seq M ( .+ , G , S ) ` n ) .+ ( G ` ( n + 1 ) ) ) ) )   &    |- ( ph -> S e. V )   =>    |- ( ph -> ( seq M ( .+ , H , S ) ` N ) = ( ( seq M ( .+ , F , S ) ` N ) Q ( seq M ( .+ , G , S ) ` N ) ) )
16-Aug-2021	iseq1p 9239	Removing the first term from a sequence. (Contributed by Jim Kingdon, 16-Aug-2021.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> N e. ( ZZ>= ` ( M + 1 ) ) )   &    |- ( ph -> S e. V )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   =>    |- ( ph -> ( seq M ( .+ , F , S ) ` N ) = ( ( F ` M ) .+ ( seq ( M + 1 ) ( .+ , F , S ) ` N ) ) )
16-Aug-2021	iseqsplit 9238	Split a sequence into two sequences. (Contributed by Jim Kingdon, 16-Aug-2021.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> N e. ( ZZ>= ` ( M + 1 ) ) )   &    |- ( ph -> S e. V )   &    |- ( ph -> M e. ( ZZ>= ` K ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> ( F ` x ) e. S )   =>    |- ( ph -> ( seq K ( .+ , F , S ) ` N ) = ( ( seq K ( .+ , F , S ) ` M ) .+ ( seq ( M + 1 ) ( .+ , F , S ) ` N ) ) )
15-Aug-2021	shftfibg 9421	Value of a fiber of the relation F. (Contributed by Jim Kingdon, 15-Aug-2021.)
|- ( ( F e. V /\ A e. CC /\ B e. CC ) -> ( ( F shift A ) " { B } ) = ( F " { ( B - A ) } ) )
15-Aug-2021	ovshftex 9420	Existence of the result of applying shift. (Contributed by Jim Kingdon, 15-Aug-2021.)
|- ( ( F e. V /\ A e. CC ) -> ( F shift A ) e. _V )
15-Aug-2021	isermono 9237	The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by Jim Kingdon, 15-Aug-2021.)
|- ( ph -> K e. ( ZZ>= ` M ) )   &    |- ( ph -> N e. ( ZZ>= ` K ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. RR )   &    |- ( ( ph /\ x e. ( ( K + 1 ) ... N ) ) -> 0 <_ ( F ` x ) )   =>    |- ( ph -> ( seq M ( + , F , RR ) ` K ) <_ ( seq M ( + , F , RR ) ` N ) )
15-Aug-2021	iserf 9233	An infinite series of complex terms is a function from NN to CC. (Contributed by Jim Kingdon, 15-Aug-2021.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   =>    |- ( ph -> seq M ( + , F , CC ) : Z --> CC )
15-Aug-2021	iseqshft2 9232	Shifting the index set of a sequence. (Contributed by Jim Kingdon, 15-Aug-2021.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> K e. ZZ )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) = ( G ` ( k + K ) ) )   &    |- ( ph -> S e. V )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ x e. ( ZZ>= ` ( M + K ) ) ) -> ( G ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> ( seq M ( .+ , F , S ) ` N ) = ( seq ( M + K ) ( .+ , G , S ) ` ( N + K ) ) )
15-Aug-2021	iseqfeq 9231	Equality of sequences. (Contributed by Jim Kingdon, 15-Aug-2021.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> S e. V )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = ( G ` k ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> seq M ( .+ , F , S ) = seq M ( .+ , G , S ) )
13-Aug-2021	absdivapd 9791	Absolute value distributes over division. (Contributed by Jim Kingdon, 13-Aug-2021.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> B # 0 )   =>    |- ( ph -> ( abs ` ( A / B ) ) = ( ( abs ` A ) / ( abs ` B ) ) )
13-Aug-2021	absrpclapd 9784	The absolute value of a complex number apart from zero is a positive real. (Contributed by Jim Kingdon, 13-Aug-2021.)
|- ( ph -> A e. CC )   &    |- ( ph -> A # 0 )   =>    |- ( ph -> ( abs ` A ) e. RR+ )
13-Aug-2021	absdivapzi 9750	Absolute value distributes over division. (Contributed by Jim Kingdon, 13-Aug-2021.)
|- A e. CC   &    |- B e. CC   =>    |- ( B # 0 -> ( abs ` ( A / B ) ) = ( ( abs ` A ) / ( abs ` B ) ) )
13-Aug-2021	absgt0ap 9695	The absolute value of a number apart from zero is positive. (Contributed by Jim Kingdon, 13-Aug-2021.)
|- ( A e. CC -> ( A # 0 <-> 0 < ( abs ` A ) ) )
13-Aug-2021	recvalap 9693	Reciprocal expressed with a real denominator. (Contributed by Jim Kingdon, 13-Aug-2021.)
|- ( ( A e. CC /\ A # 0 ) -> ( 1 / A ) = ( ( * ` A ) / ( ( abs ` A ) ^ 2 ) ) )
13-Aug-2021	sqap0 9320	A number is apart from zero iff its square is apart from zero. (Contributed by Jim Kingdon, 13-Aug-2021.)
|- ( A e. CC -> ( ( A ^ 2 ) # 0 <-> A # 0 ) )
13-Aug-2021	leltapd 7628	'<_' implies 'less than' is 'apart'. (Contributed by Jim Kingdon, 13-Aug-2021.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A <_ B )   =>    |- ( ph -> ( A < B <-> B # A ) )
13-Aug-2021	leltap 7615	'<_' implies 'less than' is 'apart'. (Contributed by Jim Kingdon, 13-Aug-2021.)
|- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( A < B <-> B # A ) )
12-Aug-2021	abssubap0 9686	If the absolute value of a complex number is less than a real, its difference from the real is apart from zero. (Contributed by Jim Kingdon, 12-Aug-2021.)
|- ( ( A e. CC /\ B e. RR /\ ( abs ` A ) < B ) -> ( B - A ) # 0 )
12-Aug-2021	ltabs 9683	A number which is less than its absolute value is negative. (Contributed by Jim Kingdon, 12-Aug-2021.)
|- ( ( A e. RR /\ A < ( abs ` A ) ) -> A < 0 )
12-Aug-2021	absext 9661	Strong extensionality for absolute value. (Contributed by Jim Kingdon, 12-Aug-2021.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` A ) # ( abs ` B ) -> A # B ) )
12-Aug-2021	sq11ap 9414	Analogue to sq11 9326 but for apartness. (Contributed by Jim Kingdon, 12-Aug-2021.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( A ^ 2 ) # ( B ^ 2 ) <-> A # B ) )
12-Aug-2021	subap0d 7631	Two numbers apart from each other have difference apart from zero. (Contributed by Jim Kingdon, 12-Aug-2021.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> A # B )   =>    |- ( ph -> ( A - B ) # 0 )
12-Aug-2021	ltapd 7627	'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A < B )   =>    |- ( ph -> A # B )
12-Aug-2021	gtapd 7626	'Greater than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A < B )   =>    |- ( ph -> B # A )
12-Aug-2021	ltapi 7625	'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
|- A e. RR   &    |- B e. RR   =>    |- ( A < B -> B # A )
12-Aug-2021	ltapii 7624	'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
|- A e. RR   &    |- B e. RR   &    |- A < B   =>    |- A # B
12-Aug-2021	gtapii 7623	'Greater than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
|- A e. RR   &    |- B e. RR   &    |- A < B   =>    |- B # A
12-Aug-2021	ltap 7622	'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
|- ( ( A e. RR /\ B e. RR /\ A < B ) -> B # A )
11-Aug-2021	absexpzap 9676	Absolute value of integer exponentiation. (Contributed by Jim Kingdon, 11-Aug-2021.)
|- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) )
11-Aug-2021	absdivap 9668	Absolute value distributes over division. (Contributed by Jim Kingdon, 11-Aug-2021.)
|- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( abs ` ( A / B ) ) = ( ( abs ` A ) / ( abs ` B ) ) )
11-Aug-2021	abs00ap 9660	The absolute value of a number is apart from zero iff the number is apart from zero. (Contributed by Jim Kingdon, 11-Aug-2021.)
|- ( A e. CC -> ( ( abs ` A ) # 0 <-> A # 0 ) )
11-Aug-2021	absrpclap 9659	The absolute value of a number apart from zero is a positive real. (Contributed by Jim Kingdon, 11-Aug-2021.)
|- ( ( A e. CC /\ A # 0 ) -> ( abs ` A ) e. RR+ )
11-Aug-2021	sqrt11ap 9636	Analogue to sqrt11 9637 but for apartness. (Contributed by Jim Kingdon, 11-Aug-2021.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( sqr ` A ) # ( sqr ` B ) <-> A # B ) )
11-Aug-2021	cjap0d 9548	A number which is apart from zero has a complex conjugate which is apart from zero. (Contributed by Jim Kingdon, 11-Aug-2021.)
|- ( ph -> A e. CC )   &    |- ( ph -> A # 0 )   =>    |- ( ph -> ( * ` A ) # 0 )
11-Aug-2021	ap0gt0d 7630	A nonzero nonnegative number is positive. (Contributed by Jim Kingdon, 11-Aug-2021.)
|- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> A # 0 )   =>    |- ( ph -> 0 < A )
11-Aug-2021	ap0gt0 7629	A nonnegative number is apart from zero if and only if it is positive. (Contributed by Jim Kingdon, 11-Aug-2021.)
|- ( ( A e. RR /\ 0 <_ A ) -> ( A # 0 <-> 0 < A ) )
10-Aug-2021	rersqrtthlem 9628	Lemma for resqrtth 9629. (Contributed by Jim Kingdon, 10-Aug-2021.)
|- ( ( A e. RR /\ 0 <_ A ) -> ( ( ( sqr ` A ) ^ 2 ) = A /\ 0 <_ ( sqr ` A ) ) )
10-Aug-2021	rersqreu 9626	Existence and uniqueness for the real square root function. (Contributed by Jim Kingdon, 10-Aug-2021.)
|- ( ( A e. RR /\ 0 <_ A ) -> E! x e. RR ( ( x ^ 2 ) = A /\ 0 <_ x ) )
10-Aug-2021	rsqrmo 9625	Uniqueness for the square root function. (Contributed by Jim Kingdon, 10-Aug-2021.)
|- ( ( A e. RR /\ 0 <_ A ) -> E* x e. RR ( ( x ^ 2 ) = A /\ 0 <_ x ) )
9-Aug-2021	resqrexlemoverl 9619	Lemma for resqrex 9624. Every term in the sequence is an overestimate compared with the limit L. Although this theorem is stated in terms of a particular sequence the proof could be adapted for any decreasing convergent sequence. (Contributed by Jim Kingdon, 9-Aug-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) , RR+ )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> L e. RR )   &    |- ( ph -> A. e e. RR+ E. j e. NN A. i e. ( ZZ>= ` j ) ( ( F ` i ) < ( L + e ) /\ L < ( ( F ` i ) + e ) ) )   &    |- ( ph -> K e. NN )   =>    |- ( ph -> L <_ ( F ` K ) )
8-Aug-2021	resqrexlemga 9621	Lemma for resqrex 9624. The sequence formed by squaring each term of F converges to A. (Contributed by Mario Carneiro and Jim Kingdon, 8-Aug-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) , RR+ )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> L e. RR )   &    |- ( ph -> A. e e. RR+ E. j e. NN A. i e. ( ZZ>= ` j ) ( ( F ` i ) < ( L + e ) /\ L < ( ( F ` i ) + e ) ) )   &    |- G = ( x e. NN |-> ( ( F ` x ) ^ 2 ) )   =>    |- ( ph -> A. e e. RR+ E. j e. NN A. k e. ( ZZ>= ` j ) ( ( G ` k ) < ( A + e ) /\ A < ( ( G ` k ) + e ) ) )
8-Aug-2021	resqrexlemglsq 9620	Lemma for resqrex 9624. The sequence formed by squaring each term of F converges to ( L ^ 2 ). (Contributed by Mario Carneiro and Jim Kingdon, 8-Aug-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) , RR+ )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> L e. RR )   &    |- ( ph -> A. e e. RR+ E. j e. NN A. i e. ( ZZ>= ` j ) ( ( F ` i ) < ( L + e ) /\ L < ( ( F ` i ) + e ) ) )   &    |- G = ( x e. NN |-> ( ( F ` x ) ^ 2 ) )   =>    |- ( ph -> A. e e. RR+ E. j e. NN A. k e. ( ZZ>= ` j ) ( ( G ` k ) < ( ( L ^ 2 ) + e ) /\ ( L ^ 2 ) < ( ( G ` k ) + e ) ) )
8-Aug-2021	recvguniqlem 9592	Lemma for recvguniq 9593. Some of the rearrangements of the expressions. (Contributed by Jim Kingdon, 8-Aug-2021.)
|- ( ph -> F : NN --> RR )   &    |- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> K e. NN )   &    |- ( ph -> A < ( ( F ` K ) + ( ( A - B ) / 2 ) ) )   &    |- ( ph -> ( F ` K ) < ( B + ( ( A - B ) / 2 ) ) )   =>    |- ( ph -> F. )
8-Aug-2021	ifcldcd 3358	Membership (closure) of a conditional operator, deduction form. (Contributed by Jim Kingdon, 8-Aug-2021.)
|- ( ph -> A e. C )   &    |- ( ph -> B e. C )   &    |- ( ph -> DECID ps )   =>    |- ( ph -> if ( ps , A , B ) e. C )
8-Aug-2021	ifbothdc 3357	A wff th containing a conditional operator is true when both of its cases are true. (Contributed by Jim Kingdon, 8-Aug-2021.)
|- ( A = if ( ph , A , B ) -> ( ps <-> th ) )   &    |- ( B = if ( ph , A , B ) -> ( ch <-> th ) )   =>    |- ( ( ps /\ ch /\ DECID ph ) -> th )
7-Aug-2021	resqrexlemsqa 9622	Lemma for resqrex 9624. The square of a limit is A. (Contributed by Jim Kingdon, 7-Aug-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) , RR+ )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> L e. RR )   &    |- ( ph -> A. e e. RR+ E. j e. NN A. i e. ( ZZ>= ` j ) ( ( F ` i ) < ( L + e ) /\ L < ( ( F ` i ) + e ) ) )   =>    |- ( ph -> ( L ^ 2 ) = A )
7-Aug-2021	resqrexlemgt0 9618	Lemma for resqrex 9624. A limit is nonnegative. (Contributed by Jim Kingdon, 7-Aug-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) , RR+ )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> L e. RR )   &    |- ( ph -> A. e e. RR+ E. j e. NN A. i e. ( ZZ>= ` j ) ( ( F ` i ) < ( L + e ) /\ L < ( ( F ` i ) + e ) ) )   =>    |- ( ph -> 0 <_ L )
7-Aug-2021	recvguniq 9593	Limits are unique. (Contributed by Jim Kingdon, 7-Aug-2021.)
|- ( ph -> F : NN --> RR )   &    |- ( ph -> L e. RR )   &    |- ( ph -> A. x e. RR+ E. j e. NN A. k e. ( ZZ>= ` j ) ( ( F ` k ) < ( L + x ) /\ L < ( ( F ` k ) + x ) ) )   &    |- ( ph -> M e. RR )   &    |- ( ph -> A. x e. RR+ E. j e. NN A. k e. ( ZZ>= ` j ) ( ( F ` k ) < ( M + x ) /\ M < ( ( F ` k ) + x ) ) )   =>    |- ( ph -> L = M )
6-Aug-2021	resqrexlemcvg 9617	Lemma for resqrex 9624. The sequence has a limit. (Contributed by Jim Kingdon, 6-Aug-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) , RR+ )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   =>    |- ( ph -> E. r e. RR A. x e. RR+ E. j e. NN A. i e. ( ZZ>= ` j ) ( ( F ` i ) < ( r + x ) /\ r < ( ( F ` i ) + x ) ) )
6-Aug-2021	cvg1nlemcxze 9581	Lemma for cvg1n 9585. Rearranging an expression related to the rate of convergence. (Contributed by Jim Kingdon, 6-Aug-2021.)
|- ( ph -> C e. RR+ )   &    |- ( ph -> X e. RR+ )   &    |- ( ph -> Z e. NN )   &    |- ( ph -> E e. NN )   &    |- ( ph -> A e. NN )   &    |- ( ph -> ( ( ( ( C x. 2 ) / X ) / Z ) + A ) < E )   =>    |- ( ph -> ( C / ( E x. Z ) ) < ( X / 2 ) )
1-Aug-2021	cvg1n 9585	Convergence of real sequences. This is a version of caucvgre 9580 with a constant multiplier C on the rate of convergence. That is, all terms after the nth term must be within C / n of the nth term. (Contributed by Jim Kingdon, 1-Aug-2021.)
|- ( ph -> F : NN --> RR )   &    |- ( ph -> C e. RR+ )   &    |- ( ph -> A. n e. NN A. k e. ( ZZ>= ` n ) ( ( F ` n ) < ( ( F ` k ) + ( C / n ) ) /\ ( F ` k ) < ( ( F ` n ) + ( C / n ) ) ) )   =>    |- ( ph -> E. y e. RR A. x e. RR+ E. j e. NN A. i e. ( ZZ>= ` j ) ( ( F ` i ) < ( y + x ) /\ y < ( ( F ` i ) + x ) ) )
1-Aug-2021	cvg1nlemres 9584	Lemma for cvg1n 9585. The original sequence F has a limit (turns out it is the same as the limit of the modified sequence G). (Contributed by Jim Kingdon, 1-Aug-2021.)
|- ( ph -> F : NN --> RR )   &    |- ( ph -> C e. RR+ )   &    |- ( ph -> A. n e. NN A. k e. ( ZZ>= ` n ) ( ( F ` n ) < ( ( F ` k ) + ( C / n ) ) /\ ( F ` k ) < ( ( F ` n ) + ( C / n ) ) ) )   &    |- G = ( j e. NN |-> ( F ` ( j x. Z ) ) )   &    |- ( ph -> Z e. NN )   &    |- ( ph -> C < Z )   =>    |- ( ph -> E. y e. RR A. x e. RR+ E. j e. NN A. i e. ( ZZ>= ` j ) ( ( F ` i ) < ( y + x ) /\ y < ( ( F ` i ) + x ) ) )
1-Aug-2021	cvg1nlemcau 9583	Lemma for cvg1n 9585. By selecting spaced out terms for the modified sequence G, the terms are within 1 / n (without the constant C). (Contributed by Jim Kingdon, 1-Aug-2021.)
|- ( ph -> F : NN --> RR )   &    |- ( ph -> C e. RR+ )   &    |- ( ph -> A. n e. NN A. k e. ( ZZ>= ` n ) ( ( F ` n ) < ( ( F ` k ) + ( C / n ) ) /\ ( F ` k ) < ( ( F ` n ) + ( C / n ) ) ) )   &    |- G = ( j e. NN |-> ( F ` ( j x. Z ) ) )   &    |- ( ph -> Z e. NN )   &    |- ( ph -> C < Z )   =>    |- ( ph -> A. n e. NN A. k e. ( ZZ>= ` n ) ( ( G ` n ) < ( ( G ` k ) + ( 1 / n ) ) /\ ( G ` k ) < ( ( G ` n ) + ( 1 / n ) ) ) )
1-Aug-2021	cvg1nlemf 9582	Lemma for cvg1n 9585. The modified sequence G is a sequence. (Contributed by Jim Kingdon, 1-Aug-2021.)
|- ( ph -> F : NN --> RR )   &    |- ( ph -> C e. RR+ )   &    |- ( ph -> A. n e. NN A. k e. ( ZZ>= ` n ) ( ( F ` n ) < ( ( F ` k ) + ( C / n ) ) /\ ( F ` k ) < ( ( F ` n ) + ( C / n ) ) ) )   &    |- G = ( j e. NN |-> ( F ` ( j x. Z ) ) )   &    |- ( ph -> Z e. NN )   &    |- ( ph -> C < Z )   =>    |- ( ph -> G : NN --> RR )
31-Jul-2021	resqrexlemnm 9616	Lemma for resqrex 9624. The difference between two terms of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 31-Jul-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) , RR+ )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> N e. NN )   &    |- ( ph -> M e. NN )   &    |- ( ph -> N <_ M )   =>    |- ( ph -> ( ( F ` N ) - ( F ` M ) ) < ( ( ( ( F ` 1 ) ^ 2 ) x. 2 ) / ( 2 ^ ( N - 1 ) ) ) )
31-Jul-2021	resqrexlemdecn 9610	Lemma for resqrex 9624. The sequence is decreasing. (Contributed by Jim Kingdon, 31-Jul-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) , RR+ )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> N e. NN )   &    |- ( ph -> M e. NN )   &    |- ( ph -> N < M )   =>    |- ( ph -> ( F ` M ) < ( F ` N ) )
30-Jul-2021	resqrexlemnmsq 9615	Lemma for resqrex 9624. The difference between the squares of two terms of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 30-Jul-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) , RR+ )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> N e. NN )   &    |- ( ph -> M e. NN )   &    |- ( ph -> N <_ M )   =>    |- ( ph -> ( ( ( F ` N ) ^ 2 ) - ( ( F ` M ) ^ 2 ) ) < ( ( ( F ` 1 ) ^ 2 ) / ( 4 ^ ( N - 1 ) ) ) )
30-Jul-2021	divmuldivapd 7806	Multiplication of two ratios. (Contributed by Jim Kingdon, 30-Jul-2021.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> D e. CC )   &    |- ( ph -> B # 0 )   &    |- ( ph -> D # 0 )   =>    |- ( ph -> ( ( A / B ) x. ( C / D ) ) = ( ( A x. C ) / ( B x. D ) ) )
29-Jul-2021	resqrexlemcalc3 9614	Lemma for resqrex 9624. Some of the calculations involved in showing that the sequence converges. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) , RR+ )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   =>    |- ( ( ph /\ N e. NN ) -> ( ( ( F ` N ) ^ 2 ) - A ) <_ ( ( ( F ` 1 ) ^ 2 ) / ( 4 ^ ( N - 1 ) ) ) )
29-Jul-2021	resqrexlemcalc2 9613	Lemma for resqrex 9624. Some of the calculations involved in showing that the sequence converges. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) , RR+ )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   =>    |- ( ( ph /\ N e. NN ) -> ( ( ( F ` ( N + 1 ) ) ^ 2 ) - A ) <_ ( ( ( ( F ` N ) ^ 2 ) - A ) / 4 ) )
29-Jul-2021	resqrexlemcalc1 9612	Lemma for resqrex 9624. Some of the calculations involved in showing that the sequence converges. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) , RR+ )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   =>    |- ( ( ph /\ N e. NN ) -> ( ( ( F ` ( N + 1 ) ) ^ 2 ) - A ) = ( ( ( ( ( F ` N ) ^ 2 ) - A ) ^ 2 ) / ( 4 x. ( ( F ` N ) ^ 2 ) ) ) )
29-Jul-2021	resqrexlemlo 9611	Lemma for resqrex 9624. A (variable) lower bound for each term of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) , RR+ )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   =>    |- ( ( ph /\ N e. NN ) -> ( 1 / ( 2 ^ N ) ) < ( F ` N ) )
29-Jul-2021	resqrexlemdec 9609	Lemma for resqrex 9624. The sequence is decreasing. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) , RR+ )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   =>    |- ( ( ph /\ N e. NN ) -> ( F ` ( N + 1 ) ) < ( F ` N ) )
28-Jul-2021	resqrexlemp1rp 9604	Lemma for resqrex 9624. Applying the recursion rule yields a positive real (expressed in a way that will help apply iseqf 9224 and similar theorems). (Contributed by Jim Kingdon, 28-Jul-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) , RR+ )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   =>    |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> ( B ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) C ) e. RR+ )
28-Jul-2021	resqrexlem1arp 9603	Lemma for resqrex 9624. 1 + A is a positive real (expressed in a way that will help apply iseqf 9224 and similar theorems). (Contributed by Jim Kingdon, 28-Jul-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) , RR+ )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   =>    |- ( ( ph /\ N e. NN ) -> ( ( NN X. { ( 1 + A ) } ) ` N ) e. RR+ )
28-Jul-2021	rpap0d 8628	A positive real is apart from zero. (Contributed by Jim Kingdon, 28-Jul-2021.)
|- ( ph -> A e. RR+ )   =>    |- ( ph -> A # 0 )
27-Jul-2021	resqrexlemex 9623	Lemma for resqrex 9624. Existence of square root given a sequence which converges to the square root. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) , RR+ )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   =>    |- ( ph -> E. x e. RR ( 0 <_ x /\ ( x ^ 2 ) = A ) )
27-Jul-2021	resqrexlemover 9608	Lemma for resqrex 9624. Each element of the sequence is an overestimate. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) , RR+ )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   =>    |- ( ( ph /\ N e. NN ) -> A < ( ( F ` N ) ^ 2 ) )
27-Jul-2021	resqrexlemfp1 9607	Lemma for resqrex 9624. Recursion rule. This sequence is the ancient method for computing square roots, often known as the babylonian method, although known to many ancient cultures. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) , RR+ )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   =>    |- ( ( ph /\ N e. NN ) -> ( F ` ( N + 1 ) ) = ( ( ( F ` N ) + ( A / ( F ` N ) ) ) / 2 ) )
27-Jul-2021	resqrexlemf1 9606	Lemma for resqrex 9624. Initial value. Although this sequence converges to the square root with any positive initial value, this choice makes various steps in the proof of convergence easier. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) , RR+ )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   =>    |- ( ph -> ( F ` 1 ) = ( 1 + A ) )
27-Jul-2021	resqrexlemf 9605	Lemma for resqrex 9624. The sequence is a function. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)
|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) , RR+ )   &    |- ( ph -> A e. RR )   &    |- ( ph -> 0 <_ A )   =>    |- ( ph -> F : NN --> RR+ )
23-Jul-2021	iseqf 9224	Range of the recursive sequence builder. (Contributed by Jim Kingdon, 23-Jul-2021.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> S e. V )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ x e. Z ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> seq M ( .+ , F , S ) : Z --> S )
22-Jul-2021	ialgrlemconst 9882	Lemma for ialgr0 9883. Closure of a constant function, in a form suitable for theorems such as iseq1 9222 or iseqfn 9221. (Contributed by Jim Kingdon, 22-Jul-2021.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> A e. S )   =>    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( ( Z X. { A } ) ` x ) e. S )
22-Jul-2021	ialgrlem1st 9881	Lemma for ialgr0 9883. Expressing algrflemg 5851 in a form suitable for theorems such as iseq1 9222 or iseqfn 9221. (Contributed by Jim Kingdon, 22-Jul-2021.)
|- ( ph -> F : S --> S )   =>    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x ( F o. 1st ) y ) e. S )
22-Jul-2021	algrflemg 5851	Lemma for algrf and related theorems. (Contributed by Jim Kingdon, 22-Jul-2021.)
|- ( ( B e. V /\ C e. W ) -> ( B ( F o. 1st ) C ) = ( F ` B ) )
21-Jul-2021	zmod1congr 9183	Two arbitrary integers are congruent modulo 1, see example 4 in [ApostolNT] p. 107. (Contributed by AV, 21-Jul-2021.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( A mod 1 ) = ( B mod 1 ) )
20-Jul-2021	caucvgrelemcau 9579	Lemma for caucvgre 9580. Converting the Cauchy condition. (Contributed by Jim Kingdon, 20-Jul-2021.)
|- ( ph -> F : NN --> RR )   &    |- ( ph -> A. n e. NN A. k e. ( ZZ>= ` n ) ( ( F ` n ) < ( ( F ` k ) + ( 1 / n ) ) /\ ( F ` k ) < ( ( F ` n ) + ( 1 / n ) ) ) )   =>    |- ( ph -> A. n e. NN A. k e. NN ( n <RR k -> ( ( F ` n ) <RR ( ( F ` k ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) /\ ( F ` k ) <RR ( ( F ` n ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) ) ) )
20-Jul-2021	caucvgrelemrec 9578	Two ways to express a reciprocal. (Contributed by Jim Kingdon, 20-Jul-2021.)
|- ( ( A e. RR /\ A # 0 ) -> ( iota_ r e. RR ( A x. r ) = 1 ) = ( 1 / A ) )
19-Jul-2021	caucvgre 9580	Convergence of real sequences. A Cauchy sequence (as defined here, which has a rate of convergence built in) of real numbers converges to a real number. Specifically on rate of convergence, all terms after the nth term must be within 1 / n of the nth term. (Contributed by Jim Kingdon, 19-Jul-2021.)
|- ( ph -> F : NN --> RR )   &    |- ( ph -> A. n e. NN A. k e. ( ZZ>= ` n ) ( ( F ` n ) < ( ( F ` k ) + ( 1 / n ) ) /\ ( F ` k ) < ( ( F ` n ) + ( 1 / n ) ) ) )   =>    |- ( ph -> E. y e. RR A. x e. RR+ E. j e. NN A. i e. ( ZZ>= ` j ) ( ( F ` i ) < ( y + x ) /\ y < ( ( F ` i ) + x ) ) )
19-Jul-2021	ax-caucvg 7004	Completeness. Axiom for real and complex numbers, justified by theorem axcaucvg 6974. A Cauchy sequence (as defined here, which has a rate convergence built in) of real numbers converges to a real number. Specifically on rate of convergence, all terms after the nth term must be within 1 / n of the nth term. This axiom should not be used directly; instead use caucvgre 9580 (which is the same, but stated in terms of the NN and 1 / n notations). (Contributed by Jim Kingdon, 19-Jul-2021.) (New usage is discouraged.)
|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }   &    |- ( ph -> F : N --> RR )   &    |- ( ph -> A. n e. N A. k e. N ( n <RR k -> ( ( F ` n ) <RR ( ( F ` k ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) /\ ( F ` k ) <RR ( ( F ` n ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) ) ) )   =>    |- ( ph -> E. y e. RR A. x e. RR ( 0 <RR x -> E. j e. N A. k e. N ( j <RR k -> ( ( F ` k ) <RR ( y + x ) /\ y <RR ( ( F ` k ) + x ) ) ) ) )
18-Jul-2021	mulnqpr 6675	Multiplication of fractions embedded into positive reals. One can either multiply the fractions as fractions, or embed them into positive reals and multiply them as positive reals, and get the same result. (Contributed by Jim Kingdon, 18-Jul-2021.)
|- ( ( A e. Q. /\ B e. Q. ) -> <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. = ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) )
18-Jul-2021	mulnqprlemfu 6674	Lemma for mulnqpr 6675. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
|- ( ( A e. Q. /\ B e. Q. ) -> ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) C_ ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) )
18-Jul-2021	mulnqprlemfl 6673	Lemma for mulnqpr 6675. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
|- ( ( A e. Q. /\ B e. Q. ) -> ( 1st ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) C_ ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) )
18-Jul-2021	mulnqprlemru 6672	Lemma for mulnqpr 6675. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
|- ( ( A e. Q. /\ B e. Q. ) -> ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) C_ ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) )
18-Jul-2021	mulnqprlemrl 6671	Lemma for mulnqpr 6675. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
|- ( ( A e. Q. /\ B e. Q. ) -> ( 1st ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) C_ ( 1st ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) )
18-Jul-2021	lt2mulnq 6503	Ordering property of multiplication for positive fractions. (Contributed by Jim Kingdon, 18-Jul-2021.)
|- ( ( ( A e. Q. /\ B e. Q. ) /\ ( C e. Q. /\ D e. Q. ) ) -> ( ( A <Q B /\ C <Q D ) -> ( A .Q C ) <Q ( B .Q D ) ) )
17-Jul-2021	recidpirqlemcalc 6933	Lemma for recidpirq 6934. Rearranging some of the expressions. (Contributed by Jim Kingdon, 17-Jul-2021.)
|- ( ph -> A e. P. )   &    |- ( ph -> B e. P. )   &    |- ( ph -> ( A .P. B ) = 1P )   =>    |- ( ph -> ( ( ( ( A +P. 1P ) .P. ( B +P. 1P ) ) +P. ( 1P .P. 1P ) ) +P. 1P ) = ( ( ( ( A +P. 1P ) .P. 1P ) +P. ( 1P .P. ( B +P. 1P ) ) ) +P. ( 1P +P. 1P ) ) )
17-Jul-2021	recidpipr 6932	Another way of saying that a number times its reciprocal is one. (Contributed by Jim Kingdon, 17-Jul-2021.)
|- ( N e. N. -> ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. .P. <. { l | l <Q ( *Q ` [ <. N , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. N , 1o >. ] ~Q ) <Q u } >. ) = 1P )
15-Jul-2021	rereceu 6963	The reciprocal from axprecex 6954 is unique. (Contributed by Jim Kingdon, 15-Jul-2021.)
|- ( ( A e. RR /\ 0 <RR A ) -> E! x e. RR ( A x. x ) = 1 )
15-Jul-2021	recidpirq 6934	A real number times its reciprocal is one, where reciprocal is expressed with *Q. (Contributed by Jim Kingdon, 15-Jul-2021.)
|- ( N e. N. -> ( <. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. <. [ <. ( <. { l | l <Q ( *Q ` [ <. N , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. N , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = 1 )
15-Jul-2021	recnnre 6927	Embedding the reciprocal of a natural number into RR. (Contributed by Jim Kingdon, 15-Jul-2021.)
|- ( N e. N. -> <. [ <. ( <. { l | l <Q ( *Q ` [ <. N , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. N , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. RR )
15-Jul-2021	pitore 6926	Embedding from N. to RR. Similar to pitonn 6924 but separate in the sense that we have not proved nnssre 7918 yet. (Contributed by Jim Kingdon, 15-Jul-2021.)
|- ( N e. N. -> <. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. RR )
15-Jul-2021	pitoregt0 6925	Embedding from N. to RR yields a number greater than zero. (Contributed by Jim Kingdon, 15-Jul-2021.)
|- ( N e. N. -> 0 <RR <. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. )
14-Jul-2021	adddivflid 9134	The floor of a sum of an integer and a fraction is equal to the integer iff the denominator of the fraction is less than the numerator. (Contributed by AV, 14-Jul-2021.)
|- ( ( A e. ZZ /\ B e. NN0 /\ C e. NN ) -> ( B < C <-> ( |_ ` ( A + ( B / C ) ) ) = A ) )
14-Jul-2021	nnindnn 6967	Principle of Mathematical Induction (inference schema). This is a counterpart to nnind 7930 designed for real number axioms which involve natural numbers (notably, axcaucvg 6974). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)
|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }   &    |- ( z = 1 -> ( ph <-> ps ) )   &    |- ( z = k -> ( ph <-> ch ) )   &    |- ( z = ( k + 1 ) -> ( ph <-> th ) )   &    |- ( z = A -> ( ph <-> ta ) )   &    |- ps   &    |- ( k e. N -> ( ch -> th ) )   =>    |- ( A e. N -> ta )
14-Jul-2021	peano5nnnn 6966	Peano's inductive postulate. This is a counterpart to peano5nni 7917 designed for real number axioms which involve natural numbers (notably, axcaucvg 6974). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)
|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }   =>    |- ( ( 1 e. A /\ A. z e. A ( z + 1 ) e. A ) -> N C_ A )
14-Jul-2021	peano2nnnn 6929	A successor of a positive integer is a positive integer. This is a counterpart to peano2nn 7926 designed for real number axioms which involve to natural numbers (notably, axcaucvg 6974). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)
|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }   =>    |- ( A e. N -> ( A + 1 ) e. N )
14-Jul-2021	peano1nnnn 6928	One is an element of NN. This is a counterpart to 1nn 7925 designed for real number axioms which involve natural numbers (notably, axcaucvg 6974). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)
|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }   =>    |- 1 e. N
14-Jul-2021	addvalex 6920	Existence of a sum. This is dependent on how we define + so once we proceed to real number axioms we will replace it with theorems such as addcl 7006. (Contributed by Jim Kingdon, 14-Jul-2021.)
|- ( ( A e. V /\ B e. W ) -> ( A + B ) e. _V )
13-Jul-2021	nntopi 6968	Mapping from NN to N.. (Contributed by Jim Kingdon, 13-Jul-2021.)
|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }   =>    |- ( A e. N -> E. z e. N. <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = A )
13-Jul-2021	ltrennb 6930	Ordering of natural numbers with <N or <RR. (Contributed by Jim Kingdon, 13-Jul-2021.)
|- ( ( J e. N. /\ K e. N. ) -> ( J <N K <-> <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. <RR <. [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) )
12-Jul-2021	ltrenn 6931	Ordering of natural numbers with <N or <RR. (Contributed by Jim Kingdon, 12-Jul-2021.)
|- ( J <N K -> <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. <RR <. [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. )
11-Jul-2021	recriota 6964	Two ways to express the reciprocal of a natural number. (Contributed by Jim Kingdon, 11-Jul-2021.)
|- ( N e. N. -> ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) = <. [ <. ( <. { l | l <Q ( *Q ` [ <. N , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. N , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. )
11-Jul-2021	elrealeu 6906	The real number mapping in elreal 6905 is unique. (Contributed by Jim Kingdon, 11-Jul-2021.)
|- ( A e. RR <-> E! x e. R. <. x , 0R >. = A )
10-Jul-2021	axcaucvglemres 6973	Lemma for axcaucvg 6974. Mapping the limit from N. and R.. (Contributed by Jim Kingdon, 10-Jul-2021.)
|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }   &    |- ( ph -> F : N --> RR )   &    |- ( ph -> A. n e. N A. k e. N ( n <RR k -> ( ( F ` n ) <RR ( ( F ` k ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) /\ ( F ` k ) <RR ( ( F ` n ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) ) ) )   &    |- G = ( j e. N. |-> ( iota_ z e. R. ( F ` <. [ <. ( <. { l | l <Q [ <. j , 1o >. ] ~Q } , { u | [ <. j , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. ) )   =>    |- ( ph -> E. y e. RR A. x e. RR ( 0 <RR x -> E. j e. N A. k e. N ( j <RR k -> ( ( F ` k ) <RR ( y + x ) /\ y <RR ( ( F ` k ) + x ) ) ) ) )
10-Jul-2021	axcaucvglemval 6971	Lemma for axcaucvg 6974. Value of sequence when mapping to N. and R.. (Contributed by Jim Kingdon, 10-Jul-2021.)
|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }   &    |- ( ph -> F : N --> RR )   &    |- ( ph -> A. n e. N A. k e. N ( n <RR k -> ( ( F ` n ) <RR ( ( F ` k ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) /\ ( F ` k ) <RR ( ( F ` n ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) ) ) )   &    |- G = ( j e. N. |-> ( iota_ z e. R. ( F ` <. [ <. ( <. { l | l <Q [ <. j , 1o >. ] ~Q } , { u | [ <. j , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. ) )   =>    |- ( ( ph /\ J e. N. ) -> ( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. ( G ` J ) , 0R >. )
10-Jul-2021	axcaucvglemcl 6969	Lemma for axcaucvg 6974. Mapping to N. and R.. (Contributed by Jim Kingdon, 10-Jul-2021.)
|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }   &    |- ( ph -> F : N --> RR )   =>    |- ( ( ph /\ J e. N. ) -> ( iota_ z e. R. ( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. ) e. R. )
9-Jul-2021	axcaucvglemcau 6972	Lemma for axcaucvg 6974. The result of mapping to N. and R. satisfies the Cauchy condition. (Contributed by Jim Kingdon, 9-Jul-2021.)
|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }   &    |- ( ph -> F : N --> RR )   &    |- ( ph -> A. n e. N A. k e. N ( n <RR k -> ( ( F ` n ) <RR ( ( F ` k ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) /\ ( F ` k ) <RR ( ( F ` n ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) ) ) )   &    |- G = ( j e. N. |-> ( iota_ z e. R. ( F ` <. [ <. ( <. { l | l <Q [ <. j , 1o >. ] ~Q } , { u | [ <. j , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. ) )   =>    |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( G ` n ) <R ( ( G ` k ) +R [ <. ( <. { l | l <Q ( *Q