mmtheorems67 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 6601-6700 - Page 67 of 123

Type	Label	Description
Statement
Theorem	elfzlem 6601	Lemma for elfzel1 6609 and others.
Theorem	elfz5 6602	Membership in a finite set of sequential integers.
|- ((K e. (ZZ>=` M) /\ N e. ZZ) -> (K e. (M...N) <-> K <_ N))
Theorem	elfz4 6603	Membership in a finite set of sequential integers.
|- (((M e. ZZ /\ N e. ZZ /\ K e. ZZ) /\ (M <_ K /\ K <_ N)) -> K e. (M...N))
Theorem	elfzuzb 6604	Membership in a finite set of sequential integers in terms of sets of upper integers.
|- (N e. A -> (K e. (M...N) <-> (K e. (ZZ>=` M) /\ N e. (ZZ>=` K))))
Theorem	eluzfz 6605	Membership in a finite set of sequential integers.
|- ((K e. (ZZ>=` M) /\ N e. (ZZ>=` K)) -> K e. (M...N))
Theorem	elfzuz3 6606	Membership in a finite set of sequential integers implies membership in a set of upper integers.
|- ((N e. A /\ K e. (M...N)) -> N e. (ZZ>=` K))
Theorem	elfzel2i 6607	Membership in a finite set of sequential integer implies the upper bound is an integer.
|- N e. V   =>   |- (K e. (M...N) -> N e. ZZ)
Theorem	elfzel2g 6608	Membership in a finite set of sequential integers implies the upper bound is an integer.
|- ((N e. A /\ K e. (M...N)) -> N e. ZZ)
Theorem	elfzel1 6609	Membership in a finite set of sequential integer implies the lower bound is an integer.
|- (K e. (M...N) -> M e. ZZ)
Theorem	elfzelz 6610	A member of a finite set of sequential integer is an integer.
|- (K e. (M...N) -> K e. ZZ)
Theorem	elfzle1 6611	A member of a finite set of sequential integer is greater than or equal to the lower bound.
|- (K e. (M...N) -> M <_ K)
Theorem	elfzle2 6612	A member of a finite set of sequential integer is less than or equal to the upper bound.
|- (K e. (M...N) -> K <_ N)
Theorem	elfzle3 6613	Membership in a finite set of sequential integer implies the bounds are comparable.
|- ((N e. A /\ K e. (M...N)) -> M <_ N)
Theorem	elfzuz2 6614	Implication of membership in a finite set of sequential integers.
|- ((N e. A /\ K e. (M...N)) -> N e. (ZZ>=` M))
Theorem	eluzfz1 6615	Membership in a finite set of sequential integers - special case.
|- (N e. (ZZ>=` M) -> M e. (M...N))
Theorem	elfzuz 6616	A member of a finite set of sequential integers belongs to a set of upper integers.
|- (K e. (M...N) -> K e. (ZZ>=` M))
Theorem	eluzfz2 6617	Membership in a finite set of sequential integers - special case.
|- (N e. (ZZ>=` M) -> N e. (M...N))
Theorem	eluzfz2b 6618	Membership in a finite set of sequential integers - special case.
|- (N e. (ZZ>=` M) <-> N e. (M...N))
Theorem	elfz3 6619	Membership in a finite set of sequential integers containing one integer.
|- (N e. ZZ -> N e. (N...N))
Theorem	elfz1eq 6620	Membership in a finite set of sequential integers containing one integer.
|- (K e. (N...N) -> K = N)
Theorem	fzn 6621	A finite set of sequential integers is empty if the bounds are reversed.
|- ((M e. ZZ /\ N e. ZZ) -> (N < M <-> (M...N) = (/)))
Theorem	fzen 6622	A shifted finite set of sequential integers is equinumerous to the original set. (Contributed by Paul Chapman, 11-Apr-2009.)
|- ((M e. ZZ /\ N e. ZZ /\ K e. ZZ) -> (M...N) ~~ ((M + K)...(N + K)))
Theorem	fz01en 6623	0-based and 1-based finite sets of sequential integers are equinumerous. (Contributed by Paul Chapman, 11-Apr-2009.)
|- (N e. ZZ -> (0...(N - 1)) ~~ (1...N))
Theorem	elfznn 6624	A member of a finite set of sequential integers starting at 1 is a natural number.
|- (K e. (1...N) -> K e. NN)
Theorem	elfz2nn0 6625	Membership in a finite set of sequential integers starting at 0.
|- (N e. A -> (K e. (0...N) <-> (K e. NN0 /\ N e. NN0 /\ K <_ N)))
Theorem	elfznn0 6626	A member of a finite set of sequential integers starting at 0 is a nonnegative integer.
|- (K e. (0...N) -> K e. NN0)
Theorem	elfz3nn0 6627	The upper bound of a nonempty finite set of sequential integers starting at 0 is a nonnegative integer.
|- ((N e. A /\ K e. (0...N)) -> N e. NN0)
Theorem	fznn0sub 6628	Subtraction closure for a member of a finite set of sequential integers.
|- ((N e. A /\ K e. (M...N)) -> (N - K) e. NN0)
Theorem	fznn0sub2 6629	Subtraction closure for a member of a finite set of sequential integers.
|- ((N e. A /\ K e. (0...N)) -> (N - K) e. (0...N))
Theorem	fzaddel 6630	Membership of a sum in a finite set of sequential integers.
|- (((M e. ZZ /\ N e. ZZ) /\ (J e. ZZ /\ K e. ZZ)) -> (J e. (M...N) <-> (J + K) e. ((M + K)...(N + K))))
Theorem	fzsubel 6631	Membership of a difference in a finite set of sequential integers.
|- (((M e. ZZ /\ N e. ZZ) /\ (J e. ZZ /\ K e. ZZ)) -> (J e. (M...N) <-> (J - K) e. ((M - K)...(N - K))))
Theorem	fzopth 6632	A finite set of sequential integers can represent an ordered pair.
|- ((N e. (ZZ>=` M) /\ K e. A) -> ((M...N) = (J...K) <-> (M = J /\ N = K)))
Theorem	fzss1 6633	Subset relationship for finite sets of sequential integers.
|- ((K e. (ZZ>=` M) /\ N e. ZZ) -> (K...N) (_ (M...N))
Theorem	fzss2 6634	Subset relationship for finite sets of sequential integers.
|- ((N e. (ZZ>=` K) /\ M e. ZZ) -> (M...K) (_ (M...N))
Theorem	fzssuz 6635	A finite set of sequential integers is a subset of a set of upper integers.
|- (M...N) (_ (ZZ>=` M)
Theorem	fzsuc 6636	Join a successor to the end of a finite set of sequential integers.
|- ((M e. ZZ /\ N e. ZZ /\ M <_ (N + 1)) -> (M...(N + 1)) = ((M...N) u. {(N + 1)}))
Theorem	fzssp1 6637	Subset relationship for finite sets of sequential integers.
|- ((M e. ZZ /\ N e. ZZ) -> (M...N) (_ (M...(N + 1)))
Theorem	fzp1ss 6638	Subset relationship for finite sets of sequential integers.
|- ((M e. ZZ /\ N e. ZZ) -> ((M + 1)...N) (_ (M...N))
Theorem	fzelp1 6639	Membership in a set of sequential integers with an appended element.
|- ((N e. A /\ K e. (M...N)) -> K e. (M...(N + 1)))
Theorem	fzelp1i 6640	Membership in a set of sequential integers with an appended element.
|- N e. V   =>   |- (K e. (M...N) -> K e. (M...(N + 1)))
Theorem	elfzp1 6641	Append an element to a finite set of sequential integers.
|- (N e. (ZZ>=` M) -> (K e. (M...(N + 1)) <-> (K e. (M...N) \/ K = (N + 1))))
Theorem	fzrev 6642	Reversal of start and end of a finite set of sequential integers.
|- (((M e. ZZ /\ N e. ZZ) /\ (J e. ZZ /\ K e. ZZ)) -> (K e. ((J - N)...(J - M)) <-> (J - K) e. (M...N)))
Theorem	fzrev2 6643	Reversal of start and end of a finite set of sequential integers.
|- (((M e. ZZ /\ N e. ZZ) /\ (J e. ZZ /\ K e. ZZ)) -> (K e. (M...N) <-> (J - K) e. ((J - N)...(J - M))))
Theorem	fzrev2i 6644	Reversal of start and end of a finite set of sequential integers.
|- ((N e. A /\ J e. ZZ /\ K e. (M...N)) -> (J - K) e. ((J - N)...(J - M)))
Theorem	fzrev3 6645	The "complement" of a member of a finite set of sequential integers.
|- ((N e. A /\ K e. ZZ) -> (K e. (M...N) <-> ((M + N) - K) e. (M...N)))
Theorem	fzrev3i 6646	The "complement" of a member of a finite set of sequential integers.
|- ((N e. A /\ K e. (M...N)) -> ((M + N) - K) e. (M...N))
Theorem	fznn0 6647	Finite set of sequential integers starting at 0.
|- (N e. NN0 -> (K e. (0...N) <-> (K e. NN0 /\ K <_ N)))
Theorem	fz1sbc 6648	Quantification over a one-member finite set of sequential integers in terms of substitution.
|- (N e. ZZ -> (A.k e. (N...N)ph <-> [N / k]ph))
Theorem	fzneuz 6649	No finite set of sequential integers equals a set of upper integers.
|- ((N e. (ZZ>=` M) /\ K e. ZZ) -> -. (M...N) = (ZZ>=` K))
Theorem	fzrevral 6650	Reversal of scanning order inside of a quantification over a finite set of sequential integers.
|- ((M e. ZZ /\ N e. ZZ /\ K e. ZZ) -> (A.j e. (M...N)ph <-> A.k e. ((K - N)...(K - M))[(K - k) / j]ph))
Theorem	fzrevral2 6651	Reversal of scanning order inside of a quantification over a finite set of sequential integers.
|- ((M e. ZZ /\ N e. ZZ /\ K e. ZZ) -> (A.j e. ((K - N)...(K - M))ph <-> A.k e. (M...N)[(K - k) / j]ph))
Theorem	fzrevral3 6652	Reversal of scanning order inside of a quantification over a finite set of sequential integers.
|- ((M e. ZZ /\ N e. ZZ) -> (A.j e. (M...N)ph <-> A.k e. (M...N)[((M + N) - k) / j]ph))
Theorem	fzshftral 6653	Shift the scanning order inside of a quantification over a finite set of sequential integers.
|- ((M e. ZZ /\ N e. ZZ /\ K e. ZZ) -> (A.j e. (M...N)ph <-> A.k e. ((M + K)...(N + K))[(k - K) / j]ph))
Theorem	fsequb 6654	The values of a finite real sequence have an upper bound. Warning: The HTML proof page is 1/2 megabyte in size.
|- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(F` k) e. RR) -> E.x e. RR A.k e. (M...N)(F` k) < x)
Theorem	fsequb2 6655	The values of a finite real sequence have an upper bound.
|- ((N e. (ZZ>=` M) /\ F:(M...N)-->RR) -> E.x e. RR A.y e. ran F y <_ x)
Theorem	fseqsupcl 6656	The values of a finite real sequence have a supremum.
|- ((N e. (ZZ>=` M) /\ F:(M...N)-->RR) -> sup(ran F, RR, < ) e. RR)
Theorem	fseqsupubi 6657	The values of a finite real sequence are bounded by their supremum.
|- N e. V   =>   |- ((K e. (M...N) /\ F:(M...N)-->RR) -> (F` K) <_ sup(ran F, RR, < ))
The infinite sequence builder "seq1"
Theorem	om2uz0i 6658	The mapping G is a one-to-one mapping from om onto upper integers that will be used to construct a recursive definition generator. Ordinal natural number 0 maps to complex number C (normally 0 for the upper integers NN0 or 1 for the upper integers NN), 1 maps to C + 1, etc. This theorem shows the value of G at ordinal natural number zero. (This series of theorems generalizes an earlier series for NN0 contributed by Raph Levien, 10-Apr-2004.)
|- C e. ZZ   &   |- G = (rec({<.x, y>. | y = (x + 1)}, C) |` om)   =>   |- (G` (/)) = C
Theorem	om2uzsuci 6659	The value of G (see om2uz0i 6658) at a successor.
|- C e. ZZ   &   |- G = (rec({<.x, y>. | y = (x + 1)}, C) |` om)   =>   |- (A e. om -> (G` suc A) = ((G` A) + 1))
Theorem	om2uzuzi 6660	The value G (see om2uz0i 6658) at an ordinal natural number is in the upper integers.
|- C e. ZZ   &   |- G = (rec({<.x, y>. | y = (x + 1)}, C) |` om)   =>   |- (A e. om -> (G` A) e. {z e. ZZ | C <_ z})
Theorem	om2uzlti 6661	Less-than relation for G (see om2uz0i 6658).
|- C e. ZZ   &   |- G = (rec({<.x, y>. | y = (x + 1)}, C) |` om)   =>   |- ((A e. om /\ B e. om) -> (A e. B -> (G` A) < (G` B)))
Theorem	om2uzlt2i 6662	The mapping G (see om2uz0i 6658) preserves order.
|- C e. ZZ   &   |- G = (rec({<.x, y>. | y = (x + 1)}, C) |` om)   =>   |- ((A e. om /\ B e. om) -> (A e. B <-> (G` A) < (G` B)))
Theorem	om2uzrani 6663	Range of G (see om2uz0i 6658).
|- C e. ZZ   &   |- G = (rec({<.x, y>. | y = (x + 1)}, C) |` om)   =>   |- ran G = {z e. ZZ | C <_ z}
Theorem	om2uzf1oi 6664	G (see om2uz0i 6658) is a one-to-one onto mapping.
|- C e. ZZ   &   |- G = (rec({<.x, y>. | y = (x + 1)}, C) |` om)   =>   |- G:om-1-1-onto->{z e. ZZ | C <_ z}
Theorem	om2uzisoi 6665	G (see om2uz0i 6658) is an isomorphism from natural ordinals to upper integers.
|- C e. ZZ   &   |- G = (rec({<.x, y>. | y = (x + 1)}, C) |` om)   =>   |- G Isom E, < (om, {z e. ZZ | C <_ z})
Theorem	uzrdgvali 6666	A helper lemma for the value of a recursive definition generator on upper integers (typically either NN or NN0) with characteristic function F and initial value A. Normally F is a function on the partition, and A is a member of the partition. See also comment in om2uz0i 6658.
|- C e. ZZ   &   |- G = (rec({<.x, y>. | y = (x + 1)}, C) |` om)   =>   |- (B e. {z e. ZZ | C <_ z} -> ((rec(F, A) o. `'G)` B) = (rec(F, A)` (`'G` B)))
Theorem	uzrdginii 6667	Initial value of a recursive definition generator on upper integers. See comment in uzrdgvali 6666.
|- C e. ZZ   &   |- G = (rec({<.x, y>. | y = (x + 1)}, C) |` om)   =>   |- (A e. B -> ((rec(F, A) o. `'G)` C) = A)
Theorem	uzrdgsuci 6668	Successor value of a recursive definition generator on upper integers. See comment in uzrdgvali 6666.
|- C e. ZZ   &   |- G = (rec({<.x, y>. | y = (x + 1)}, C) |` om)   =>   |- (B e. {z e. ZZ | C <_ z} -> ((rec(F, A) o. `'G)` (B + 1)) = (F` ((rec(F, A) o. `'G)` B)))
Theorem	uzrdginip1i 6669	A helper lemma for the value of an NN or NN0 -based recursive definition generator. Value at second index. See comment in uzrdgvali 6666.
|- C e. ZZ   &   |- G = (rec({<.x, y>. | y = (x + 1)}, C) |` om)   =>   |- (A e. B -> ((rec(F, A) o. `'G)` (C + 1)) = (F` A))
Theorem	uzrdgfnuzi 6670	A helper lemma for the value of an NN or NN0 -based recursive definition generator. The generated function is a function on the upper integers starting at C (usually 1 or 0). See comment in uzrdgvali 6666.
|- C e. ZZ   &   |- G = (rec({<.x, y>. | y = (x + 1)}, C) |` om)   =>   |- (rec(F, A) o. `'G) Fn {z e. ZZ | C <_ z}
Theorem	cardfz 6671	The cardinality of a finite set of sequential integers. (See om2uz0i 6658 for a description of the antecedent.)
|- G = (rec({<.j, k>. | k = (j + 1)}, 0) |` om)   =>   |- (N e. NN0 -> (card` (1...N)) = (`'G` N))
Syntax	cseq1 6672	Extend class notation with recursive sequence builder.
class seq1
Definition	df-seq1 6673	Define a general-purpose operation that builds an recursive sequence (i.e. a function on the natural numbers NN) 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 6682 and seq1p1 6683. 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 F with values 1, 1/2, 1/4, 1/8,... would be transformed into the output sequence ( + seq1 F) with values 1, 3/2, 7/4, 15/8,.., so that (( + seq1 F)` 1) = 1, (( + seq1 F)` 2) = 3/2, etc. In other words, ( + seq1 F) transforms a sequence F into an infinite series. ( + seq1 F) ~~> 2 means "the sum of F(n) from n = 1 to infinity is 2." Since limits are unique (climuni 7302), then by eusn 2507 and unisn 2583 the "sum of F(n) from n = 1 to infinity" can be expressed as U.{x | ( + seq1 F) ~~> x} (provided the sequence converges) and evaluates to 2 in this example. Internally, we define a recursive function whose values are ordered pairs starting at <.1, (g` 1)>.. The first member of the ordered pair is a counter used to select the appropriate value of the input sequence g. The first rec constructs this function on om, and the converse of the second rec maps NN to om. This definition has its roots in a series of theorems starting at om2uz0i 6658, originally proved by Raph Levien for use with df-exp 6764 and later generalized for arbitrary recursive sequences. The related definitions df-seq0 6729 and df-seqz 6728 build recursive sequences on NN0 and general upper integer sets respectively. Definition df-sum 7183 extracts the summation values from partial (finite) and complete (infinite) series.
|- seq1 = {<.<.f, g>., h>. | h = {<.x, y>. | (x e. NN /\ y = (2nd` ((rec({<.z, w>. | w = <.((1st` z) + 1), ((2nd` z)f(g` ((1st` z) + 1)))>.}, <.1, (g` 1)>.) o. `'(rec({<.z, w>. | w = (z + 1)}, 1) |` om))` x)))}}
Theorem	seq1lem1 6674	We prove by induction that the first member of the ordered pair value of the internal sequence of seq1 equals its index.
|- G = (rec({<.x, y>. | y = (x + 1)}, 1) |` om)   =>   |- (A e. NN -> (1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` A)) = A)
Theorem	seq1lem2 6675	Lemma for recursive sequence builder theorems.
Theorem	seq1rval 6676	Value of the characteristic function of the inner recursion in df-seq1 6673.
|- H = {<.z, w>. | w = <.((1st` z) + 1), ((2nd` z)S(F` ((1st` z) + 1)))>.}   &   |- A e. V   =>   |- (H` A) = <.((1st` A) + 1), ((2nd` A)S(F` ((1st` A) + 1)))>.
Theorem	seq1val 6677	Value of the recursive sequence builder operation.
|- S e. V   &   |- F e. V   &   |- G = (rec({<.z, w>. | w = (z + 1)}, 1) |` om)   &   |- H = {<.z, w>. | w = <.((1st` z) + 1), ((2nd` z)S(F` ((1st` z) + 1)))>.}   =>   |- (S seq1 F) = {<.x, y>. | (x e. NN /\ y = (2nd` ((rec(H, <.1, (F` 1)>.) o. `'G)` x)))}
Theorem	seq1fnlem 6678	Lemma for seq1fn 6685.
Theorem	seq1val2 6679	Value of the value of the recursive sequence builder operation.
|- S e. V   &   |- F e. V   &   |- G = (rec({<.z, w>. | w = (z + 1)}, 1) |` om)   &   |- H = {<.z, w>. | w = <.((1st` z) + 1), ((2nd` z)S(F` ((1st` z) + 1)))>.}   =>   |- (A e. NN -> ((S seq1 F)` A) = (2nd` ((rec(H, <.1, (F` 1)>.) o. `'G)` A)))
Theorem	seq11lem 6680	Lemma for seq11 6682.
Theorem	seq1suclem 6681	Lemma for seq1p1 6683.
Theorem	seq11 6682	Value of the recursive sequence builder at 1. See description in df-seq1 6673.
|- S e. V   &   |- F e. V   =>   |- ((S seq1 F)` 1) = (F` 1)
Theorem	seq1p1 6683	Value of the recursive sequence builder at a successor. See description in df-seq1 6673.
|- S e. V   &   |- F e. V   =>   |- (A e. NN -> ((S seq1 F)` (A + 1)) = (((S seq1 F)` A)S(F` (A + 1))))
Theorem	seq1m1 6684	Value of the recursive sequence builder in terms of its previous value.
|- S e. V   &   |- F e. V   =>   |- ((N e. NN /\ 1 < N) -> ((S seq1 F)` N) = (((S seq1 F)` (N - 1))S(F` N)))
Theorem	seq1fn 6685	Functionality and domain of the recursive sequence builder.
|- S e. V   &   |- F e. V   =>   |- (S seq1 F) Fn NN
Theorem	seq1rn2 6686	Range of the recursive sequence builder.
|- S e. V   &   |- F e. V   =>   |- (((F` 1) e. C /\ (F |` (NN \ {1})):(NN \ {1})-->D /\ S:(C X. D)-->C) -> ran ( S seq1 F) (_ C)
Theorem	seq1rn 6687	Range of the recursive sequence builder (special case of seq1rn2 6686).
|- S e. V   &   |- F e. V   =>   |- ((F:NN-->C /\ S:(C X. C)-->C) -> ran ( S seq1 F) (_ C)
Theorem	seq1f 6688	Mapping of the 1-based recursive sequence builder.
|- S e. V   &   |- F e. V   =>   |- ((F:NN-->C /\ S:(C X. C)-->C) -> (S seq1 F):NN-->C)
Theorem	seq1f2 6689	Mapping of the 1-based recursive sequence builder.
|- S e. V   &   |- F e. V   =>   |- (((F` 1) e. C /\ (F |` (NN \ {1})):(NN \ {1})-->D /\ S:(C X. D)-->C) -> (S seq1 F):NN-->C)
Theorem	seq1cl 6690	Closure of the value of the recursive sequence builder.
|- S e. V   &   |- F e. V   =>   |- ((A e. NN /\ F:NN-->C /\ S:(C X. C)-->C) -> ((S seq1 F)` A) e. C)
Theorem	seq1cl2 6691	Closure of the value of the recursive sequence builder.
|- S e. V   &   |- F e. V   =>   |- ((((F` 1) e. C /\ (F |` (NN \ {1})):(NN \ {1})-->D /\ S:(C X. D)-->C) /\ A e. NN) -> ((S seq1 F)` A) e. C)
Theorem	seq1res 6692	Restricting its characteristic function to NN does not affect the seq1 function.
|- S e. V   &   |- F e. V   =>   |- (S seq1 (F |` NN)) = (S seq1 F)
Theorem	ser1f 6693	An infinite series of complex terms is a function from NN to CC.
|- (F:NN-->CC -> ( + seq1 F):NN-->CC)
Theorem	ser1fi 6694	An infinite series is a function from NN to CC.
|- F:NN-->CC   =>   |- ( + seq1 F):NN-->CC
Theorem	ser1cl1i 6695	The partial sums in an infinite series of complex terms are complex.
|- F:NN-->CC   =>   |- (A e. NN -> (( + seq1 F)` A) e. CC)
Theorem	ser1recli 6696	The partial sums in an infinite series of real terms are real.
|- F:NN-->RR   =>   |- (A e. NN -> (( + seq1 F)` A) e. RR)
Theorem	ser1refi 6697	The partial sums of an infinite series of reals is an infinite real sequence.
|- F:NN-->RR   =>   |- ( + seq1 F):NN-->RR
Theorem	ser1cl2i 6698	Closure of the value of the B th term of an infinite series.
|- F = {<.x, y>. | (x e. NN /\ y = A)}   &   |- A.x e. NN A e. CC   =>   |- (B e. NN -> (( + seq1 F)` B) e. CC)
Theorem	ser1f2i 6699	An infinite series is a function from NN to CC.
|- F = {<.x, y>. | (x e. NN /\ y = A)}   &   |- A.x e. NN A e. CC   =>   |- ( + seq1 F):NN-->CC
Theorem	ser11i 6700	The value of the first term in an infinite series.
|- F = {<.x, y>. | (x e. NN /\ y = A)}   &   |- B e. V   &   |- (x = 1 -> A = B)   =>   |- (( + seq1 F)` 1) = B