mmtheorems71 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 7001-7100 - Page 71 of 108

Type	Label	Description
Statement
Theorem	sumeq2i 7001	Equality inference for sum.
|- (k e. A -> B = C)   =>   |- sum_k e. A B = sum_k e. A C
Theorem	sumeq12i 7002	Equality inference for sum. (Contributed by FL, 10-Dec-2006.)
|- A = B   &   |- (k e. A -> C = D)   =>   |- sum_k e. A C = sum_k e. B D
Theorem	sumeq1d 7003	Equality deduction for sum.
|- (ph -> A = B)   =>   |- (ph -> sum_k e. A C = sum_k e. B C)
Theorem	sumeq2d 7004	Equality deduction for sum. Note that unlike sumeq2dv 7005, k may occur in ph.
|- (ph -> A.k e. A B = C)   =>   |- (ph -> sum_k e. A B = sum_k e. A C)
Theorem	sumeq2dv 7005	Equality deduction for sum.
|- ((ph /\ k e. A) -> B = C)   =>   |- (ph -> sum_k e. A B = sum_k e. A C)
Theorem	sumeq2sdv 7006	Equality deduction for sum.
|- (ph -> B = C)   =>   |- (ph -> sum_k e. A B = sum_k e. A C)
Theorem	2sumeq2dv 7007	Equality deduction for double sum.
|- ((ph /\ j e. A /\ k e. B) -> C = D)   =>   |- (ph -> sum_j e. A sum_k e. B C = sum_j e. A sum_k e. B D)
Theorem	sumeq12dv 7008	Equality deduction for sum.
|- (ph -> A = B)   &   |- ((ph /\ k e. A) -> C = D)   =>   |- (ph -> sum_k e. A C = sum_k e. B D)
Theorem	sumeq12rdv 7009	Equality deduction for sum.
|- (ph -> A = B)   &   |- ((ph /\ k e. B) -> C = D)   =>   |- (ph -> sum_k e. A C = sum_k e. B D)
Theorem	sumeqfv 7010	Convert a sum of function values to a sum of classes A(k).
|- A e. V   &   |- F = {<.k, y>. | (k e. B /\ y = A)}   =>   |- (C (_ B -> sum_k e. C (F` k) = sum_k e. C A)
Finite sums (cont.)
Theorem	dffsum 7011	Special case of series sum over a finite index set.
|- (N e. (ZZ>` M) -> sum_k e. (M...N)A = ((<.M, + >. seq ({<.k, y>. | y = A} |` ZZ))` N))
Theorem	fsumserz 7012	A finite sum expressed in terms of a partial sum of an infinite series. The recursive definition of follows as fsum1 7018 and fsump1 7019, which should make our notation clear and from which, along with closure fsumclt 7028, we will derive the basic properties of finite sums.
|- F e. V   =>   |- (N e. (ZZ>` M) -> sum_k e. (M...N)(F` k) = ((<.M, + >. seq F)` N))
Theorem	fsumserzf 7013	Version of fsumserz 7012 with a bound-variable hypothesis instead of a distinct variable condition.
|- F e. V   &   |- (x e. F -> A.k x e. F)   =>   |- (N e. (ZZ>` M) -> sum_k e. (M...N)(F` k) = ((<.M, + >. seq F)` N))
Theorem	fsumser0f 7014	A finite sum expressed in terms of a partial sum of an infinite 0-based series.
|- F e. V   &   |- (x e. F -> A.k x e. F)   =>   |- (N e. NN0 -> sum_k e. (0...N)(F` k) = (( + seq0 F)` N))
Theorem	fsumser1f 7015	A finite sum expressed in terms of a partial sum of an infinite 1-based series.
|- F e. V   &   |- (x e. F -> A.k x e. F)   =>   |- (N e. NN -> sum_k e. (1...N)(F` k) = (( + seq1 F)` N))
Theorem	fsumserz2 7016	A finite sum expressed in terms of a partial sum of an infinite series.
|- A e. V   &   |- F = {<.k, y>. | (k e. (ZZ>` M) /\ y = A)}   =>   |- (N e. (ZZ>` M) -> sum_k e. (M...N)A = ((<.M, + >. seq F)` N))
Theorem	serzfsum 7017	An infinite series in terms of finite partial sums of A(k).
|- A e. V   &   |- F = {<.k, y>. | (k e. (ZZ>` M) /\ y = A)}   &   |- G = {<.n, z>. | (n e. (ZZ>` M) /\ z = sum_k e. (M...n)A)}   =>   |- (M e. ZZ -> (<.M, + >. seq F) = G)
Theorem	fsum1 7018	The finite sum of A(k) from k = M to M (i.e. a sum with only one term) is B i.e. A(M).
|- (k = M -> A = B)   =>   |- ((B e. C /\ M e. ZZ) -> sum_k e. (M...M)A = B)
Theorem	fsump1 7019	The addition of the next term in a finite sum of A(k) is the current term plus B i.e. A(N + 1).
|- A e. V   &   |- B e. V   &   |- (k = (N + 1) -> A = B)   =>   |- (N e. (ZZ>` M) -> sum_k e. (M...(N + 1))A = (sum_k e. (M...N)A + B))
Theorem	fsum1f 7020	The finite sum of a term A(k) from M to M (i.e. a sum with only one term) is A(M) = B, where k is effectively not free in B.
|- (x e. B -> A.k x e. B)   &   |- (k = M -> A = B)   =>   |- ((B e. C /\ M e. ZZ) -> sum_k e. (M...M)A = B)
Theorem	fsum1slem 7021	Lemma for fsum1s 7022.
Theorem	fsum1s 7022	The finite sum of a sequence A(k) from M to M (i.e. a sum with only one term) is A(M).
|- ((M e. ZZ /\ A.k e. (M...M)A e. B) -> sum_k e. (M...M)A = [_M / k]_A)
Theorem	fsum1s2 7023	The finite sum of a sequence A(k) from M to M (i.e. a sum with only one term) is A(M).
|- ((M e. ZZ /\ [_M / k]_A e. B) -> sum_k e. (M...M)A = [_M / k]_A)
Theorem	fsump1f 7024	The addition of the next term in a finite sum of A(k) is the previous term plus A(N + 1) = B.
|- A e. V   &   |- B e. V   &   |- (x e. B -> A.k x e. B)   &   |- (k = (N + 1) -> A = B)   =>   |- (N e. (ZZ>` M) -> sum_k e. (M...(N + 1))A = (sum_k e. (M...N)A + B))
Theorem	fsump1slem 7025	Lemma for fsump1s 7026.
Theorem	fsump1s 7026	The addition of the next term in a finite sum of A(k) is the previous term plus A(N + 1).
|- ((N e. (ZZ>` M) /\ A.k e. (M...(N + 1))A e. B) -> sum_k e. (M...(N + 1))A = (sum_k e. (M...N)A + [_(N + 1) / k]_A))
Theorem	fsumcllem 7027	- Lemma for finite sum closures. (The "-" before "Lemma" forces the math content to be displayed in the Statement List - NM 11-Feb-2008.)
|- ((x e. C /\ y e. C) -> (x + y) e. C)   =>   |- ((N e. (ZZ>` M) /\ A.k e. (M...N)A e. C) -> sum_k e. (M...N)A e. C)
Theorem	fsumclt 7028	Closure of a finite sum of complex numbers A(k).
|- ((N e. (ZZ>` M) /\ A.k e. (M...N)A e. CC) -> sum_k e. (M...N)A e. CC)
Theorem	fsum0clt 7029	Closure of a finite sum of complex numbers A(k), starting at index zero.
|- ((N e. NN0 /\ A.k e. (0...N)A e. CC) -> sum_k e. (0...N)A e. CC)
Theorem	fsumreclt 7030	Closure of a finite sum of reals.
|- ((N e. (ZZ>` M) /\ A.k e. (M...N)A e. RR) -> sum_k e. (M...N)A e. RR)
Theorem	fsum1ps 7031	Separate out the first term in a finite sum.
|- ((N e. (ZZ>` M) /\ M < N /\ A.k e. (M...N)A e. CC) -> sum_k e. (M...N)A = ([_M / k]_A + sum_k e. ((M + 1)...N)A))
Theorem	fsum1p 7032	Separate out the first term in a finite sum.
|- (k = M -> A = B)   =>   |- ((N e. (ZZ>` M) /\ M < N /\ A.k e. (M...N)A e. CC) -> sum_k e. (M...N)A = (B + sum_k e. ((M + 1)...N)A))
Theorem	fsumsplit 7033	Split a finite sum into two parts. Warning: The HTML proof page is 0.6 megabyte in size.
|- ((N e. ZZ /\ K e. (M...(N - 1)) /\ A.k e. (M...N)A e. CC) -> sum_k e. (M...N)A = (sum_k e. (M...K)A + sum_k e. ((K + 1)...N)A))
Theorem	fsum0split 7034	Split a finite sum into two parts.
|- ((N e. ZZ /\ K e. (1...N) /\ A.k e. (0...N)A e. CC) -> sum_k e. (0...N)A = (sum_k e. (0...(N - K))A + sum_k e. (((N - K) + 1)...N)A))
Theorem	fsumadd 7035	The sum of two finite sums.
|- ((N e. (ZZ>` M) /\ A.k e. (M...N)(A e. CC /\ B e. CC)) -> sum_k e. (M...N)(A + B) = (sum_k e. (M...N)A + sum_k e. (M...N)B))
Theorem	fsum2 7036	The sum of two terms.
|- A e. V   =>   |- (M e. ZZ -> sum_k e. (M...(M + 1))A = ([_M / k