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**Statement List for Metamath Proof Explorer -** 7701-7800 - Page 78 of 175
Type	Label	Description
Statement

Theorem	fzshftral 7701	Shift the scanning order inside of a quantification over a finite set of sequential integers.
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Theorem	fsequb 7702	The values of a finite real sequence have an upper bound. Warning: The HTML proof page is 1/2 megabyte in size.
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Theorem	fsequb2 7703	The values of a finite real sequence have an upper bound.
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Theorem	fseqsupcl 7704	The values of a finite real sequence have a supremum.
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Theorem	fseqsupubi 7705	The values of a finite real sequence are bounded by their supremum.
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The infinite sequence builder "seq1"

Theorem	om2uz0i 7706	The mapping is a one-to-one mapping from onto upper integers that will be used to construct a recursive definition generator. Ordinal natural number 0 maps to complex number (normally 0 for the upper integers or 1 for the upper integers ), 1 maps to + 1, etc. This theorem shows the value of at ordinal natural number zero. (This series of theorems generalizes an earlier series for contributed by Raph Levien, 10-Apr-2004.)


Theorem	om2uzsuci 7707	The value of (see om2uz0i 7706) at a successor.


Theorem	om2uzuzi 7708	The value (see om2uz0i 7706) at an ordinal natural number is in the upper integers.


Theorem	om2uzlti 7709	Less-than relation for (see om2uz0i 7706).
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Theorem	om2uzlt2i 7710	The mapping (see om2uz0i 7706) preserves order.
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Theorem	om2uzrani 7711	Range of (see om2uz0i 7706).


Theorem	om2uzf1oi 7712	(see om2uz0i 7706) is a one-to-one onto mapping.


Theorem	om2uzisoi 7713	(see om2uz0i 7706) is an isomorphism from natural ordinals to upper integers.


Theorem	uzrdgvali 7714	A helper lemma for the value of a recursive definition generator on upper integers (typically either or ) with characteristic function and initial value . Normally is a function on the partition, and is a member of the partition. See also comment in om2uz0i 7706.


Theorem	uzrdginii 7715	Initial value of a recursive definition generator on upper integers. See comment in uzrdgvali 7714.


Theorem	uzrdgsuci 7716	Successor value of a recursive definition generator on upper integers. See comment in uzrdgvali 7714.


Theorem	uzrdginip1i 7717	A helper lemma for the value of an or -based recursive definition generator. Value at second index. See comment in uzrdgvali 7714.


Theorem	uzrdgfnuzi 7718	A helper lemma for the value of an or -based recursive definition generator. The generated function is a function on the upper integers starting at (usually 1 or 0). See comment in uzrdgvali 7714.


Theorem	cardfz 7719	The cardinality of a finite set of sequential integers. (See om2uz0i 7706 for a description of the antecedent.)


Syntax	cseq1 7720	Extend class notation with recursive sequence builder.


Definition	df-seq1 7721	Define a general-purpose operation that builds an recursive sequence (i.e. a function on the natural numbers ) whose value at an index is a function of its previous value and the value of an input sequence at that index. This definition is complicated, but fortunately it is not intended to be used directly. Instead, the only purpose of this definition is to provide us with an object that has the properties expressed by seq11 7730 and seq1p1 7731. Typically, those are the main theorems that would be used in practice. The first operand is the operation that is applied to the previous value and the value of the input sequence (second operand). For example, for the operation , an input sequence with values 1, 1/2, 1/4, 1/8,... would be transformed into the output sequence with values 1, 3/2, 7/4, 15/8,.., so that , 3/2, etc. In other words, transforms a sequence into an infinite series. means "the sum of F(n) from n = 1 to infinity is 2." Since limits are unique (climuni 8359), then by euabsn 3095 and unisn 3193 the "sum of F(n) from n = 1 to infinity" can be expressed as (provided the sequence converges) and evaluates to 2 in this example. Internally, we define a recursive function whose values are ordered pairs starting at . The first member of the ordered pair is a counter used to select the appropriate value of the input sequence . The first constructs this function on , and the converse of the second maps to . This definition has its roots in a series of theorems from om2uz0i 7706 through om2uzf1oi 7712, originally proved by Raph Levien for use with df-exp 7812 and later generalized for arbitrary recursive sequences. The related definitions df-seq0 7777 and df-seqz 7776 build recursive sequences on and general upper integer sets respectively. Definition df-sum 8240 extracts the summation values from partial (finite) and complete (infinite) series.
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Theorem	seq1lem1 7722	We prove by induction that the first member of the ordered pair value of the internal sequence of equals its index.


Theorem	seq1lem2 7723	Lemma for recursive sequence builder theorems.

Theorem	seq1rval 7724	Value of the characteristic function of the inner recursion in df-seq1 7721.


Theorem	seq1val 7725	Value of the recursive sequence builder operation.
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Theorem	seq1fnlem 7726	Lemma for seq1fn 7733.

Theorem	seq1val2 7727	Value of the value of the recursive sequence builder operation.


Theorem	seq11lem 7728	Lemma for seq11 7730.

Theorem	seq1suclem 7729	Lemma for seq1p1 7731.

Theorem	seq11 7730	Value of the recursive sequence builder at 1. See description in df-seq1 7721.


Theorem	seq1p1 7731	Value of the recursive sequence builder at a successor. See description in df-seq1 7721.


Theorem	seq1m1 7732	Value of the recursive sequence builder in terms of its previous value.
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Theorem	seq1fn 7733	Functionality and domain of the recursive sequence builder.


Theorem	seq1rn2 7734	Range of the recursive sequence builder.
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Theorem	seq1rn 7735	Range of the recursive sequence builder (special case of seq1rn2 7734).
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Theorem	seq1f 7736	Mapping of the 1-based recursive sequence builder.
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Theorem	seq1f2 7737	Mapping of the 1-based recursive sequence builder.
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Theorem	seq1cl 7738	Closure of the value of the recursive sequence builder.
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Theorem	seq1cl2 7739	Closure of the value of the recursive sequence builder.
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Theorem	seq1res 7740	Restricting its characteristic function to does not affect the function.


Theorem	ser1f 7741	An infinite series of complex terms is a function from to .


Theorem	ser1fi 7742	An infinite series is a function from to .


Theorem	ser1cl1i 7743	The partial sums in an infinite series of complex terms are complex.


Theorem	ser1recli 7744	The partial sums in an infinite series of real terms are real.


Theorem	ser1refi 7745	The partial sums of an infinite series of reals is an infinite real sequence.


Theorem	ser1cl2i 7746	Closure of the value of the th term of an infinite series.
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Theorem	ser1f2i 7747	An infinite series is a function from to .
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Theorem	ser11i 7748	The value of the first term in an infinite series.
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Theorem	ser1p1i 7749	The value of the next term in an infinite series.
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Theorem	ser1monoi 7750	The partial sums in an infinite series of positive terms form a monotonic sequence.


Theorem	ser1add2i 7751	The sum of two infinite series.
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Theorem	ser1addi 7752	The sum of two infinite series.
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The "shift" operation

Syntax	cshi 7753	Extend class notation with function shifter.


Definition	df-shft 7754	Define a function shifter. This operation offsets the value argument of a function (ordinarily on a subset of ) and produces a new function on . See shftval 7759 for its value.
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Theorem	shftfval 7755	The value of the sequence shifter operation is a function on . is ordinarily an integer.
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Theorem	shftfn 7756	Functionality and domain of a sequence shifted by .


Theorem	shftres 7757	Restriction of a shifted sequence.
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Theorem	shftresval 7758	Value of a restricted shifted sequence.


Theorem	shftval 7759	Value of a sequence shifted by .
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Theorem	shftval2 7760	Value of a sequence shifted by .
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Theorem	shftval3 7761	Value of a sequence shifted by .
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Theorem	shftval4 7762	Value of a sequence shifted by .
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Theorem	shftval5 7763	Value of a shifted sequence.
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Theorem	shftf 7764	Functionality of a restricted shifted sequence.
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Theorem	2shfti 7765	Composite shift operations.
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Theorem	shftcan2 7766	Cancellation law for the shift operation.
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Theorem	shftcan1 7767	Cancellation law for the shift operation.
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Theorem	shftidt 7768	Identity law for the shift operation.


Theorem	seq1shftid 7769	Identity law for the shift operation in a 1-based sequence builder.


Superior limit (lim sup)

Syntax	clsp 7770	Extend class notation to include the limsup function.


Definition	df-limsup 7771	Define the superior limit of an infinite sequence of extended real numbers. Definition 12-4.1 of [Gleason] p. 175. See limsupval 7772 for its value.


Theorem	limsupval 7772	The superior limit of an infinite sequence of extended real numbers, which is the infimum (indicated by ) of the set of suprema of all upper infinite subsequences of . Definition 12-4.1 of [Gleason] p. 175.


Theorem	limsupcl 7773	Closure of the superior limit.


Infinite sequence builders "seq" and "seq0"

Syntax	cseqz 7774	Extend class notation with arbitrarily-based recursive sequence builder.


Syntax	cseq0 7775	Extend class notation with 0-based recursive sequence builder.


Definition	df-seqz 7776	Define a recursive sequence builder operation that starts at an arbitrary integer index. See seqz1 7790 and seqzp1 7791 for its initial and successor values. Theorems seq0seqz 7785 and seq1seqz 7784 derive the 0-based and the 1-based as special cases.


Definition	df-seq0 7777	Define a recursive sequence builder operation that starts at index 0. This is a frequently-used variation of the operation (see df-seq1 7721), which starts at index 1. See seq00 7793 and seq0p1 7794 for its initial and successor values. See dfseq0 7806 for an alternate definition.


Theorem	seq0fval 7778	Value of the 0-based recursive sequence builder operation.


Theorem	seq0valt 7779	Value of the 0-based recursive sequence builder operation.


Theorem	seqzfval 7780	Value of the arbitrary-based recursive sequence builder operation.


Theorem	seqzfval2 7781	Value of the arbitrary-based recursive sequence builder operation.


Theorem	seqzfn 7782	Functionality and domain of a sequence generated by the arbitrary-based recursive sequence builder.


Theorem	seqzval 7783	Value of the arbitrary-based recursive sequence builder operation.
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Theorem	seq1seqz 7784	The 1-based recursive sequence in terms of the arbitrary-based one.


Theorem	seq0seqz 7785	The 0-based recursive sequence in terms of the arbitrary-based one.


Theorem	seq1seq02 7786	A relationship between the 1-based and 0-based recursive sequence builders.


Theorem	seq1seq01 7787	The 1-based recursive sequence builder operation in terms of the 0-based one.


Theorem	seq1seq0 7788	The 1-based recursive sequence builder operation in terms of the 0-based one.


Theorem	seq0fn 7789	Functionality and domain of the 0-based recursive sequence builder.


Theorem	seqz1 7790	Value of the arbitrary-based recursive sequence builder at its initial value.


Theorem	seqzp1 7791	Value of the arbitrary-based recursive sequence builder at a successor value.


Theorem	seqzm1 7792	Value of the recursive sequence builder in terms of its previous value.
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Theorem	seq00 7793	Value of the 0-based recursive sequence builder at 0.


Theorem	seq0p1 7794	Value of the 0-based recursive sequence builder at a successor.


Theorem	seq01 7795	Value of the 0-based recursive sequence builder at 1.


Theorem	seqzval2 7796	Value of the arbitrary-based recursive sequence builder operation.
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Theorem	seqzfveq 7797	Equality theorem for the recursive sequence builder.
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Theorem	seqzeq 7798	Equality theorem for the recursive sequence builder.
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Theorem	seqzrn2 7799	Range of a sequence generated by the arbitrary-based recursive sequence builder.
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Theorem	seqzrn 7800	Range of the arbitrary-based recursive sequence builder (special case of seqzrn2 7799).
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