mmtheorems72 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 7101-7200 - Page 72 of 123

Type	Label	Description
Statement
Theorem	abstri 7101	Triangle inequality for absolute value. Proposition 10-3.7(h) of [Gleason] p. 133.
|- ((A e. CC /\ B e. CC) -> (abs` (A + B)) <_ ((abs` A) + (abs` B)))
Theorem	abs3dif 7102	Absolute value of differences around common element. (Contributed by FL, 9-Oct-2006.)
|- ((A e. CC /\ B e. CC /\ C e. CC) -> (abs` (A - B)) <_ ((abs` (A - C)) + (abs` (C - B))))
Theorem	abs3difi 7103	Absolute value of differences around common element.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   =>   |- (abs` (A - B)) <_ ((abs` (A - C)) + (abs` (C - B)))
Theorem	abs3lemi 7104	Lemma involving absolute value of differences.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   &   |- D e. RR   =>   |- (((abs` (A - C)) < (D / 2) /\ (abs` (C - B)) < (D / 2)) -> (abs` (A - B)) < D)
Theorem	abs2dif 7105	Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
|- ((A e. CC /\ B e. CC) -> ((abs` A) - (abs` B)) <_ (abs` (A - B)))
Theorem	abs2difabs 7106	Absolute value of difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
|- ((A e. CC /\ B e. CC) -> (abs` ((abs` A) - (abs` B))) <_ (abs` (A - B)))
Theorem	abs1mi 7107	For any complex number, there exists a unit-magnitude multiplier that produces its absolute value. Part of proof of Theorem 13-2.12 of [Gleason] p. 195.
|- A e. CC   =>   |- E.x e. CC ((abs` x) = 1 /\ (abs` A) = (x x. A))
Theorem	recan 7108	Cancellation law involving the real part of a complex number.
|- ((A e. CC /\ B e. CC) -> (A.x e. CC (Re` (x x. A)) = (Re` (x x. B)) <-> A = B))
Theorem	absf 7109	Mapping domain and codomain of the absolute value function.
|- abs:CC-->RR
Theorem	abs3lem 7110	Lemma involving absolute value of differences.
|- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. RR)) -> (((abs` (A - C)) < (D / 2) /\ (abs` (C - B)) < (D / 2)) -> (abs` (A - B)) < D))
Theorem	abslem2i 7111	Lemma involving absolute values.
|- A e. CC   &   |- A =/= 0   =>   |- (((*` (A / (abs` A))) x. A) + ((A / (abs` A)) x. (*` A))) = (2 x. (abs` A))
Theorem	abslem2 7112	Lemma involving absolute values.
|- A e. CC   =>   |- (A =/= 0 -> (((*` (A / (abs` A))) x. A) + ((A / (abs` A)) x. (*` A))) = (2 x. (abs` A)))
Theorem	seq1bndi 7113	An initial segment of an infinite sequence of complex numbers is bounded.
|- F:NN-->CC   =>   |- (A e. NN -> E.x e. RR A.y e. NN (y <_ A -> (abs` (F` y)) < x))
Theorem	seq1ublem 7114	Lemma for seq1ubi 7115.
Theorem	seq1ubi 7115	An upper bound for an initial segment of a sequence of reals.
|- F:NN-->RR   =>   |- ((A e. NN /\ B e. NN /\ A <_ B) -> (F` A) <_ sup(ran ( F |` {x e. NN | x <_ B}), RR, < ))
Theorem	cau2i 7116	Two ways to express that a sequence meets the Cauchy criterion. Remark in [Gleason] p. 181. R can be either < or <_.
|- F:NN-->CC   &   |- (x e. RR -> (0 < x -> 0Rx))   =>   |- (A.x e. RR (0 < x -> E.y e. NN A.z e. NN (y < z -> (abs` ((F` z) - (F` y)))Rx)) <-> A.x e. RR (0 < x -> E.y e. NN A.z e. NN (y <_ z -> (abs` ((F` z) - (F` y)))Rx)))
Theorem	cau3ii 7117	A relationship used to derive two ways to express a Cauchy sequence.
|- Z (_ ZZ   =>   |- (E.m e. Z A.j e. Z A.k e. W ((m <_ j /\ m <_ k) -> ph) -> E.j e. Z A.k e. W (j <_ k -> ph))
Theorem	cau3iri 7118	A relationship used to derive two ways to express a Cauchy sequence. Normally Z and W are subsets of ZZ, and R is <_ or <. ph is ph(j, k, x).
|- (k = m -> (ph <-> ps))   &   |- (j = k -> (ps <-> ch))   &   |- (x = (y / 2) -> ((ph /\ ps) <-> th))   &   |- (x = y -> (ch <-> ta))   &   |- (((et /\ y e. RR) /\ (j e. Z /\ k e. W /\ m e. W)) -> (th -> ta))   =>   |- (et -> (A.x e. RR (0 < x -> E.j e. Z A.k e. W (jRk -> ph)) -> A.x e. RR (0 < x -> E.m e. Z A.j e. W A.k e. W ((mRj /\ mRk) -> ph))))
Theorem	cau3i 7119	A relationship used to derive two ways to express a Cauchy sequence. ph is ph(j, k, x).
|- (k = m -> (ph <-> ps))   &   |- (j = k -> (ps <-> ch))   &   |- (x = (y / 2) -> ((ph /\ ps) <-> th))   &   |- (x = y -> (ch <-> ta))   &   |- (((et /\ y e. RR) /\ (j e. Z /\ k e. Z /\ m e. Z)) -> (th -> ta))   &   |- Z (_ ZZ   =>   |- (et -> (A.x e. RR (0 < x -> E.j e. Z A.k e. Z (j <_ k -> ph)) <-> A.x e. RR (0 < x -> E.m e. Z A.j e. Z A.k e. Z ((m <_ j /\ m <_ k) -> ph))))
Theorem	cau5ii 7120	A relationship used to derive two ways to express a Cauchy sequence. ph is ph(j, k).
|- M e. ZZ   &   |- Z = (ZZ>=` M)   &   |- W (_ ZZ   &   |- U (_ ZZ   =>   |- (E.m e. ZZ A.j e. ZZ A.k e. ZZ ((m <_ j /\ m <_ k) -> ph) -> E.m e. Z A.j e. W A.k e. U ((m <_ j /\ m <_ k) -> ph))
Theorem	cau4ii 7121	A relationship used to derive two ways to express a Cauchy sequence. ph is ph(j, k).
|- M e. ZZ   &   |- Z = (ZZ>=` M)   =>   |- (E.j e. Z A.k e. Z (j <_ k -> ph) -> E.j e. ZZ A.k e. ZZ (j <_ k -> ph))
Theorem	cau5i 7122	A relationship used to derive two ways to express a Cauchy sequence. ph is ph(j, k).
|- M e. ZZ   &   |- Z = (ZZ>=` M)   =>   |- (E.m e. ZZ A.j e. ZZ A.k e. ZZ ((m <_ j /\ m <_ k) -> ph) <-> E.m e. Z A.j e. Z A.k e. Z ((m <_ j /\ m <_ k) -> ph))
Theorem	cvg1i 7123	A relationship used to derive two ways to express that a sequence converges. Unlike cvg1 7124, j may be free in ph, so this can also be used for Cauchy sequences.
|- Z = (ZZ>=` M)   =>   |- (E.j e. Z A.k e. Z (j <_ k -> ph) -> E.j e. Z A.k e. Z (j < k -> ph))
Theorem	cvg1 7124	A relationship used to derive two ways to express that a sequence converges. ph is ph(k).
|- Z = (ZZ>=` M)   =>   |- (E.j e. Z A.k e. Z (j < k -> ph) <-> E.j e. Z A.k e. Z (j <_ k -> ph))
Theorem	cvg2i 7125	Two ways to express that a sequence converges or is Cauchy.
|- ((y e. F /\ z e. G) -> A e. RR)   =>   |- (A.x e. RR (0 < x -> E.y e. F A.z e. G (ph -> A < x)) <-> A.x e. RR (0 < x -> E.y e. F A.z e. G (ph -> A <_ x)))
Theorem	cvg3i 7126	A relationship used to derive two ways to express convergence. ph is ph(k).
|- M e. ZZ   &   |- (ZZ>=` M) (_ Z   &   |- Z (_ ZZ   &   |- N e. ZZ   &   |- (ZZ>=` N) (_ W   &   |- W (_ ZZ   =>   |- (E.j e. ZZ A.k e. ZZ (j <_ k -> ph) <-> E.j e. Z A.k e. W (j <_ k -> ph))
Theorem	cvganz 7127	Equivalence that lets us conjoin the properties of two independent converging sequences. k may be free in ph and ps. Compare r19.40 1808, where the implication holds in only one direction.
|- ((E.j e. ZZ A.k e. ZZ (j <_ k -> ph) /\ E.j e. ZZ A.k e. ZZ (j <_ k -> ps)) <-> E.j e. ZZ A.k e. ZZ (j <_ k -> (ph /\ ps)))
Theorem	cvganuzi 7128	Lemma that lets us combine the properties of two independent converging sequences. k may be free in ph and ps.
|- M e. ZZ   &   |- Z = (ZZ>=` M)   =>   |- ((E.j e. Z A.k e. Z (j <_ k -> ph) /\ E.j e. Z A.k e. Z (j <_ k -> ps)) <-> E.j e. Z A.k e. Z (j <_ k -> (ph /\ ps)))
Theorem	caubndi 7129	A Cauchy sequence of complex numbers is bounded.
|- F:NN-->CC   &   |- A.z e. RR (0 < z -> E.w e. NN A.y e. NN (w < y -> (abs` ((F` y) - (F` w))) < z))   =>   |- E.x e. RR A.y e. NN (abs` (F` y)) < x
Theorem	caurei 7130	The real part of a complex Caucy sequence is a Cauchy sequence.
|- F:NN-->CC   &   |- A.z e. RR (0 < z -> E.w e. NN A.y e. NN (w < y -> (abs` ((F` y) - (F` w))) < z))   &   |- G Fn NN   &   |- (x e. NN -> (G` x) = (Re` (F` x)))   =>   |- A.z e. RR (0 < z -> E.w e. NN A.y e. NN (w < y -> (abs` ((G` y) - (G` w))) < z))
Theorem	cauimi 7131	The imaginary part of a complex Caucy sequence is a Cauchy sequence.
|- F:NN-->CC   &   |- A.z e. RR (0 < z -> E.w e. NN A.y e. NN (w < y -> (abs` ((F` y) - (F` w))) < z))   &   |- G Fn NN   &   |- (x e. NN -> (G` x) = (Im` (F` x)))   =>   |- A.z e. RR (0 < z -> E.w e. NN A.y e. NN (w < y -> (abs` ((G` y) - (G` w))) < z))
Theorem	ser1absdiflem 7132	Lemma for ser1absdifi 7133. Warning: The HTML proof page is 1/2 megabyte in size.
Theorem	ser1absdifi 7133	Generalized triangle inequality: absolute value of a partial sum is less than or equal to sum of absolute values.
|- F:NN-->CC   &   |- G Fn NN   &   |- (x e. NN -> (G` x) = (abs` (F` x)))   =>   |- ((A e. NN /\ B e. NN /\ A < B) -> (abs` ((( + seq1 F)` B) - (( + seq1 F)` A))) <_ ((( + seq1 G)` B) - (( + seq1 G)` A)))
Factorial function
Syntax	cfa 7134	Extend class notation to include the factorial of nonnegative integers.
class !
Definition	df-fac 7135	Define the factorial function on nonnegative integers. In the literature, the factorial function is written as a postscript exclamation point.
|- ! = ({<.0, 1>.} u. ( x. seq1 (I |` NN)))
Theorem	facnn 7136	Value of the factorial function for positive integers.
|- (N e. NN -> (!` N) = (( x. seq1 (I |` NN))` N))
Theorem	fac0 7137	The factorial of 0.
|- (!` 0) = 1
Theorem	fac1 7138	The factorial of 1.
|- (!` 1) = 1
Theorem	facp1 7139	The factorial of a successor.
|- (N e. NN0 -> (!` (N + 1)) = ((!` N) x. (N + 1)))
Theorem	fac2 7140	The factorial of 2.
|- (!` 2) = 2
Theorem	fac3 7141	The factorial of 3.
|- (!` 3) = 6
Theorem	facnn2 7142	Value of the factorial function expressed recursively.
|- (N e. NN -> (!` N) = ((!` (N - 1)) x. N))
Theorem	faccl 7143	Closure of the factorial function.
|- (N e. NN0 -> (!` N) e. NN)
Theorem	facne0 7144	The factorial function is nonzero.
|- (N e. NN0 -> (!` N) =/= 0)
Theorem	facdiv 7145	A natural number divides the factorial of an equal or larger number.
|- ((M e. NN0 /\ N e. NN /\ N <_ M) -> ((!` M) / N) e. NN)
Theorem	facndiv 7146	No natural number (greater than one) divides the factorial plus one of an equal or larger number.
|- (((M e. NN0 /\ N e. NN) /\ (1 < N /\ N <_ M)) -> -. (((!` M) + 1) / N) e. ZZ)
Theorem	facwordi 7147	Ordering property of factorial.
|- ((M e. NN0 /\ N e. NN0 /\ M <_ N) -> (!` M) <_ (!` N))
Theorem	faclbnd 7148	A lower bound for the factorial function.
|- ((M e. NN0 /\ N e. NN0) -> (M^(N + 1)) <_ ((M^M) x. (!` N)))
Theorem	faclbnd2 7149	A lower bound for the factorial function.
|- (N e. NN0 -> ((2^N) / 2) <_ (!` N))
Theorem	faclbnd3 7150	A lower bound for the factorial function.
|- ((M e. NN0 /\ N e. NN0) -> (M^N) <_ ((M^M) x. (!` N)))
Theorem	faclbnd4lem1 7151	Lemma for faclbnd4 7155. Prepare the induction step. Warning: The HTML proof page is 0.6 megabyte in size.
Theorem	faclbnd4lem2 7152	Lemma for faclbnd4 7155. Use the weak deduction theorem to convert the hypotheses of faclbnd4lem1 7151 to antecedents.
Theorem	faclbnd4lem3 7153	Lemma for faclbnd4 7155. The N = 0 case.
Theorem	faclbnd4lem4 7154	Lemma for faclbnd4 7155. Prove the 0 < N case by induction on K.
Theorem	faclbnd4 7155	Variant of faclbnd5 7156 providing a non-strict lower bound.
|- ((N e. NN0 /\ K e. NN0 /\ M e. NN0) -> ((N^K) x. (M^N)) <_ (((2^(K^2)) x. (M^(M + K))) x. (!` N)))
Theorem	faclbnd5 7156	The factorial function grows faster than powers and exponentiations. If we consider K and M to be constants, the right-hand side of the inequality is a constant times N -factorial.
|- ((N e. NN0 /\ K e. NN0 /\ M e. NN) -> ((N^K) x. (M^N)) < ((2 x. ((2^(K^2)) x. (M^(M + K)))) x. (!` N)))
Theorem	faclbnd6 7157	Geometric lower bound for the factorial function, where N is usually held constant. (Contributed by Paul Chapman, 28-Dec-2007.)
|- ((N e. NN0 /\ M e. NN0) -> ((!` N) x. ((N + 1)^M)) <_ (!` (N + M)))
Theorem	facavg 7158	The product of two factorials is greater than or equal to the factorial of (the floor of) their average.
|- ((M e. NN0 /\ N e. NN0) -> (!` (|_` ((M + N) / 2))) <_ ((!` M) x. (!` N)))
The binomial coefficient operation
Syntax	cbc 7159	Extend class notation to include the binomial coefficient operation (combinatorial choose operation).
class C.
Definition	df-bc 7160	Define the binomial coefficient operation. In the literature, this function is often written as a column vector of the two arguments, or with the arguments as subscripts before and after the letter "C". (N C. K) is read "N choose K." Definition of binomial coefficient in [Gleason] p. 295. As suggested by Gleason, we define it to be 0 when 0 <_ k <_ n does not hold.
|- C. = {<.<.n, k>., m>. | ((n e. NN0 /\ k e. ZZ) /\ m = if((0 <_ k /\ k <_ n), ((!` n) / ((!` (n - k)) x. (!` k))), 0))}
Theorem	bcval 7161	Value of the binomial coefficient, N choose K. Definition of binomial coefficient in [Gleason] p. 295. As suggested by Gleason, we define it to be 0 when 0 <_ K <_ N does not hold. See bcval2 7162 for the value in the standard domain.
|- ((N e. NN0 /\ K e. ZZ) -> (N C. K) = if((0 <_ K /\ K <_ N), ((!` N) / ((!` (N - K)) x. (!` K))), 0))
Theorem	bcval2 7162	Value of the binomial coefficient, N choose K, in its standard domain.
|- ((N e. NN0 /\ K e. NN0 /\ K <_ N) -> (N C. K) = ((!` N) / ((!` (N - K)) x. (!` K))))
Theorem	bcval3 7163	Value of the binomial coefficient, N choose K, in its standard domain.
|- ((N e. NN0 /\ K e. (0...N)) -> (N C. K) = ((!` N) / ((!` (N - K)) x. (!` K))))
Theorem	bcval4 7164	Value of the binomial coefficient, N choose K, outside of its standard domain. Remark in [Gleason] p. 295.
|- ((N e. NN0 /\ K e. ZZ /\ (K < 0 \/ N < K)) -> (N C. K) = 0)
Theorem	bccmpl 7165	"Complementing" its second argument doesn't change a binary coefficient.
|- ((N e. NN0 /\ K e. NN0 /\ K <_ N) -> (N C. K) = (N C. (N - K)))
Theorem	bcn0 7166	N choose 0 is 1. Remark in [Gleason] p. 296.
|- (N e. NN0 -> (N C. 0) = 1)
Theorem	bcnn 7167	N choose N is 1. Remark in [Gleason] p. 296.
|- (N e. NN0 -> (N C. N) = 1)
Theorem	bcnp11 7168	Binomial coefficient: N + 1 choose 1.
|- (N e. NN0 -> ((N + 1) C. 1) = (N + 1))
Theorem	bcnp1n 7169	Binomial coefficient: N + 1 choose N.
|- (N e. NN0 -> ((N + 1) C. N) = (N + 1))
Theorem	bcpasc2i 7170	Pascal's rule for the binomial coefficient. Equation 2 of [Gleason] p. 295.
|- N e. NN   &   |- K e. NN   &   |- K <_ N   =>   |- ((N C. K) + (N C. (K - 1))) = ((N + 1) C. K)
Theorem	bcpasc2 7171	Pascal's rule for the binomial coefficient. Equation 2 of [Gleason] p. 295.
|- ((N e. NN /\ K e. NN /\ K <_ N) -> ((N C. K) + (N C. (K - 1))) = ((N + 1) C. K))
Theorem	bcpasci 7172	Pascal's rule for the binomial coefficient, generalized to all integers K. Equation 2 of [Gleason] p. 295.
|- N e. NN0   &   |- K e. ZZ   =>   |- ((N C. K) + (N C. (K - 1))) = ((N + 1) C. K)
Theorem	bcpasc 7173	Pascal's rule for the binomial coefficient, generalized to all integers K. Equation 2 of [Gleason] p. 295.
|- ((N e. NN0 /\ K e. ZZ) -> ((N C. K) + (N C. (K - 1))) = ((N + 1) C. K))
Theorem	bccl2 7174	A binomial coefficient, in its standard domain, is a natural number.
|- ((N e. NN0 /\ K e. (0...N)) -> (N C. K) e. NN)
Theorem	bccl 7175	A binomial coefficient, in its extended domain, is a nonnegative integer.
|- ((N e. NN0 /\ K e. ZZ) -> (N C. K) e. NN0)
Theorem	permnn 7176	The number of permutations of N - R objects from a collection of N objects is a natural number. (Contributed by Jason Orendorff, 24-Jan-2007.)
|- ((N e. NN0 /\ R e. (0...N)) -> ((!` N) / (!` R)) e. NN)
Limits
Syntax	cli 7177	Extend class notation with convergence relation for limits.
class ~~>
Definition	df-clim 7178	Define the limit relation for complex number sequences. See clim 7180 for its relational expression.
|- ~~> = {<.f, y>. | (y e. CC /\ A.x e. RR (0 < x -> E.j e. ZZ A.k e. ZZ (j <_ k -> ((f` k) e. CC /\ (abs` ((f` k) - y)) < x))))}
Theorem	climrel 7179	The limit relation is a relation.
|- Rel ~~>
Theorem	clim 7180	Express the predicate: The limit of complex number sequence F is A, or F converges to A. This means that for any real x, no matter how small, there always exists an integer j such that the absolute difference of any later complex number in the sequence and the limit is less than x.
|- ((F e. C /\ A e. D) -> (F ~~> A <-> (A e. CC /\ A.x e. RR (0 < x -> E.j e. ZZ A.k e. ZZ (j <_ k -> ((F` k) e. CC /\ (abs` ((F` k) - A)) < x))))))
Theorem	climcl 7181	Closure of the limit of a sequence of complex numbers.
|- ((A e. C /\ F ~~> A) -> A e. CC)
Finite and infinite sums
Syntax	csu 7182	Extend class notation to include finite summations. (An underscore was added to the ASCII token in order to facilitate set.mm text searches, since "sum" is is a commonly used word in comments.)
class sum_k e. A B
Definition	df-sum 7183	Define the sum of a series with an index set of integers A. k is normally a free variable in B, i.e. B can be thought of as B(k). The definition is meaningful when A is a finite set of sequential integers (representing a finite sum over them) or a set of upper integers (representing an infinite sum, when the sum converges). The left-hand side of the union expresses the finite sum case, and the right-hand side expresses the infinite sum case. In either case, the other side of the union equals the empty set. Examples: sum_k e. (2...4) k means 2 + 3 + 4 = 9, and sum_k e. NN (1 / (2^k)) means 1/2 + 1/4 + 1/8 + ... = 1. Note 1: The restrictions to ZZ force the class abstractions to be sets. Note 2: This definition is complex because it is "overloaded" with both finite and infinite sums as disjoint sets. An advantage is that equality theorems, for example, need to be proved only once for both cases.
|- sum_k e. A B = ({x | E.mE.n e. (ZZ>=` m)(A = (m...n) /\ x e. ((<.m, + >. seq ({<.k, y>. | y = B} |` ZZ))` n))} u. U.{x | E.m e. ZZ (A = (ZZ>=` m) /\ (<.m, + >. seq ({<.k, y>. | y = B} |` ZZ)) ~~> x)})
Theorem	sumex 7184	A sum is a set.
|- sum_k e. A B e. V
Theorem	sumeq1 7185	Equality theorem for a sum.
|- (A = B -> sum_k e. A C = sum_k e. B C)
Theorem	hbsum1 7186	Bound-variable hypothesis builder for sum.
|- (x e. A -> A.k x e. A)   =>   |- (x e. sum_k e. A B -> A.k x e. sum_k e. A B)
Theorem	hbsum 7187	Bound-variable hypothesis builder for sum: if x is (effectively) not free in A and B, it is not free in sum_k e. AB.
|- (y e. A -> A.x y e. A)   &   |- (y e. B -> A.x y e. B)   =>   |- (y e. sum_k e. A B -> A.x y e. sum_k e. A B)
Theorem	sumeq2 7188	Equality theorem for sum.
|- (A.k e. A B = C -> sum_k e. A B = sum_k e. A C)
Theorem	cbvsumi 7189	Change bound variable in a sum.
|- (x e. B -> A.k x e. B)   &   |- (x e. C -> A.j x e. C)   &   |- (j = k -> B = C)   =>   |- sum_j e. A B = sum_k e. A C
Theorem	sumeq1i 7190	Equality inference for sum.
|- A = B   =>   |- sum_k e. A C = sum_k e. B C
Theorem	sumeq2i 7191	Equality inference for sum.
|- (k e. A -> B = C)   =>   |- sum_k e. A B = sum_k e. A C
Theorem	sumeq12i 7192	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 7193	Equality deduction for sum.
|- (ph -> A = B)   =>   |- (ph -> sum_k e. A C = sum_k e. B C)
Theorem	sumeq2d 7194	Equality deduction for sum. Note that unlike sumeq2dv 7195, 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 7195	Equality deduction for sum.
|- ((ph /\ k e. A) -> B = C)   =>   |- (ph -> sum_k e. A B = sum_k e. A C)
Theorem	sumeq2sdv 7196	Equality deduction for sum.
|- (ph -> B = C)   =>   |- (ph -> sum_k e. A B = sum_k e. A C)
Theorem	2sumeq2dv 7197	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 7198	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 7199	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 7200	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)