mmtheorems71 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 7001-7100 - Page 71 of 107

Type	Label	Description
Statement
Theorem	fsumabs2mul 7001	The sum of absolute values of the product A(j) x. B(m) is less than or equal to the product of the two sums of absolute values.
|- (((K e. (ZZ>` J) /\ A.j e. (J...K)A e. CC) /\ (N e. (ZZ>` M) /\ A.m e. (M...N)B e. CC)) -> (abs` sum_j e. (J...K)sum_m e. (M...N)(A x. B)) <_ (sum_j e. (J...K)(abs` A) x. sum_m e. (M...N)(abs` B)))
Theorem	serzclt 7002	Closure of partial sums of an infinite series.
|- F e. V   =>   |- ((N e. (ZZ>` M) /\ A.k e. (M...N)(F` k) e. CC) -> ((<.M, + >. seq F)` N) e. CC)
Theorem	ser0clt 7003	Closure of partial sums of a 0-based infinite series.
|- F e. V   =>   |- ((N e. NN0 /\ A.k e. (0...N)(F` k) e. CC) -> (( + seq0 F)` N) e. CC)
Theorem	ser1clt 7004	Closure of partial sums of a 1-based infinite series.
|- F e. V   =>   |- ((N e. NN /\ A.k e. (1...N)(F` k) e. CC) -> (( + seq1 F)` N) e. CC)
Theorem	ser1ser0 7005	A 1-based infinite series in terms of a 0-based infinite series.
|- F e. V   &   |- (k e. NN0 -> (F` k) e. CC)   =>   |- (N e. NN -> (( + seq1 F)` N) = ((( + seq0 F)` N) - (F` 0)))
Theorem	serzcl2t 7006	Closure of partial sums of an infinite series.
|- F e. V   =>   |- ((N e. (ZZ>` M) /\ A.k e. (ZZ>` M)(F` k) e. CC) -> ((<.M, + >. seq F)` N) e. CC)
Theorem	serzreclt 7007	The partial sums in an infinite series of real terms are real.
|- F e. V   =>   |- ((N e. (ZZ>` M) /\ A.k e. (M...N)(F` k) e. RR) -> ((<.M, + >. seq F)` N) e. RR)
Theorem	serzref 7008	An infinite series of reals is an infinite real sequence.
|- M e. ZZ   &   |- F:(ZZ>` M)-->RR   =>   |- (<.M, + >. seq F):(ZZ>` M)-->RR
Theorem	serz1p 7009	Separate out the first term in an infinite series.
|- F e. V   =>   |- ((N e. (ZZ>` M) /\ M < N /\ A.k e. (M...N)(F` k) e. CC) -> ((<.M, + >. seq F)` N) = ((F` M) + ((<.(M + 1), + >. seq F)` N)))
Theorem	serz0 7010	The value of the partial sums in a zero-valued infinite series.
|- (N e. (ZZ>` M) -> ((<.M, + >. seq ((ZZ>` M) X. {0}))` N) = 0)
Theorem	serzcmp 7011	Comparison of partial sums of two infinite series of reals.
|- F e. V   &   |- G e. V   =>   |- ((N e. (ZZ>` M) /\ A.k e. (M...N)((F` k) e. RR /\ (G` k) e. RR /\ (F` k) <_ (G` k))) -> ((<.M, + >. seq F)` N) <_ ((<.M, + >. seq G)` N))
Theorem	serzcmp0 7012	A partial sum of an infinite series is nonnegative if each term is nonnegative.
|- F e. V   =>   |- ((N e. (ZZ>` M) /\ A.k e. (M...N)((F` k) e. RR /\ 0 <_ (F` k))) -> 0 <_ ((<.M, + >. seq F)` N))
Theorem	serzsplit 7013	Split off an initial piece of the partial sum of an infinite series.
|- F e. V   =>   |- ((N e. ZZ /\ K e. (M...(N - 1)) /\ A.k e. (M...N)(F` k) e. CC) -> ((<.M, + >. seq F)` N) = (((<.M, + >. seq F)` K) + ((<.(K + 1), + >. seq F)` N)))
Theorem	serzmulc1 7014	A constant C times a series.
|- F e. V   &   |- G e. V   =>   |- ((N e. (ZZ>` M) /\ C e. CC /\ A.k e. (M...N)((F` k) e. CC /\ (G` k) = (C x. (F` k)))) -> (C x. ((<.M, + >. seq F)` N)) = ((<.M, + >. seq G)` N))
Theorem	serzmulc 7015	A constant C times a series.
|- C e. CC   &   |- F:(ZZ>` M)-->CC   &   |- G e. V   &   |- (k e. (ZZ>` M) -> (G` k) = (C x. (F` k)))   =>   |- (N e. (ZZ>` M) -> ((<.M, + >. seq G)` N) = (C x. ((<.M, + >. seq F)` N)))
Theorem	ser0mulc 7016	A constant C times a 0-based series.
|- C e. CC   &   |- F:NN0-->CC   &   |- G e. V   &   |- (k e. NN0 -> (G` k) = (C x. (F` k)))   =>   |- (N e. NN0 -> (( + seq0 G)` N) = (C x. (( + seq0 F)` N)))
Theorem	ser1mulc 7017	A constant C times a 1-based series.
|- C e. CC   &   |- F:NN-->CC   &   |- G e. V   &   |- (k e. NN -> (G` k) = (C x. (F` k)))   =>   |- (N e. NN -> (( + seq1 G)` N) = (C x. (( + seq1 F)` N)))
Theorem	serzrelem 7018	Lemma for serzre 7019, serzim 7020 and serzcj 7021.
Theorem	serzre 7019	The real part of a series. (Contributed by Paul Chapman, 9-Nov-2007.)
|- F e. V   &   |- G e. V   &   |- (k e. (ZZ>` M) -> ((F` k) e. CC /\ (G` k) = (Re` (F` k))))   =>   |- (N e. (ZZ>` M) -> ((<.M, + >. seq G)` N) = (Re` ((<.M, + >. seq F)` N)))
Theorem	serzim 7020	The imaginary part of a series. (Contributed by Paul Chapman, 9-Nov-2007.)
|- F e. V   &   |- G e. V   &   |- (k e. (ZZ>` M) -> ((F` k) e. CC /\ (G` k) = (Im` (F` k))))   =>   |- (N e. (ZZ>` M) -> ((<.M, + >. seq G)` N) = (Im` ((<.M, + >. seq F)` N)))
Theorem	serzcj 7021	The complex conjugate of a series. (Contributed by Paul Chapman, 9-Nov-2007.)
|- F e. V   &   |- G e. V   &   |- (k e. (ZZ>` M) -> ((F` k) e. CC /\ (G` k) = (*` (F` k))))   =>   |- (N e. (ZZ>` M) -> ((<.M, + >. seq G)` N) = (*` ((<.M, + >. seq F)` N)))
Theorem	ser0cj 7022	The complex conjugate of a 0-based series.
|- F e. V   &   |- G e. V   &   |- (k e. NN0 -> ((F` k) e. CC /\ (G` k) = (*` (F` k))))   =>   |- (N e. NN0 -> (( + seq0 G)` N) = (*` (( + seq0 F)` N)))
The binomial theorem
Theorem	binomlem1 7023	Lemma for binom 7029 (binomial theorem). Break out and simplify the first term of the summation.
Theorem	binomlem2 7024	Lemma for binom 7029 (binomial theorem). Shift up the summation index with fsumshft 6988, then break out and simplify the last term of the summation.
Theorem	binomlem3 7025	Lemma for binom 7029 (binomial theorem). Break out the last term of the summation used by the induction hypothesis.
Theorem	binomlem4 7026	Lemma for binom 7029 (binomial theorem). Break out the first term of the summation used by the induction hypothesis.
Theorem	binomlem5 7027	Lemma for binom 7029 (binomial theorem). We use Pascal's rule bcpasct 6927 to combine the sum of the summations in binomlem1 7023 and binomlem2 7024 into a single summation.
Theorem	binomlem6 7028	Lemma for binom 7029 (binomial theorem). Do the final induction.
Theorem	binom 7029	The binomial theorem: (A + B)^N is the sum from k = 0 to N of (N C. k) x. ((A^k) x. (B^(N - k)). Theorem 15-2.8 of [Gleason] p. 296. This final piece of the proof combines the 0 < N case of binomlem6 7028 with the N = 0 case.
|- A e. CC   &   |- B e. CC   =>   |- (N e. NN0 -> ((A + B)^N) = sum_k e. (0...N)((N C. k) x. ((A^(N - k)) x. (B^k))))
Theorem	binom1p 7030	Special case of the binomial theorem for (1 + A)^N. (Contributed by Paul Chapman, 10-May-2007.)
|- A e. CC   =>   |- (N e. NN0 -> ((1 + A)^N) = sum_k e. (0...N)((N C. k) x. (A^k)))
Theorem	bcxmaslem1 7031	Lemma for bcxmas 7033.
Theorem	bcxmaslem2 7032	Lemma for bcxmas 7033.
Theorem	bcxmas 7033	Parallel summation (Christmas Stocking) theorem for Pascal's Triangle. (Contributed by Paul Chapman, 18-May-2007.)
|- ((N e. NN0 /\ M e. NN0) -> (((N + 1) + M) C. M) = sum_j e. (0...M)((N + j) C. j))
Limits (cont.)
Theorem	clm1 7034	Express the predicate: The limit of complex number sequence F is A, or F converges to A, with more general quantifier restrictions than clim 6934.
|- M e. ZZ   &   |- (ZZ>` M) (_ Z   &   |- Z (_ ZZ   &   |- N e. ZZ   &   |- (ZZ>` N) (_ W   &   |- W (_ ZZ   =>   |- ((F e. C /\ A e. D) -> (F