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Theorem List for Metamath Proof Explorer - 30801-30900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhvpncan 30801 Addition/subtraction cancellation law for vectors in Hilbert space. (Contributed by NM, 7-Jun-2004.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((𝐴 +β„Ž 𝐡) βˆ’β„Ž 𝐡) = 𝐴)
 
Theoremhvpncan2 30802 Addition/subtraction cancellation law for vectors in Hilbert space. (Contributed by NM, 7-Jun-2004.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((𝐴 +β„Ž 𝐡) βˆ’β„Ž 𝐴) = 𝐡)
 
Theoremhvaddsubass 30803 Associativity of sum and difference of Hilbert space vectors. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 +β„Ž 𝐡) βˆ’β„Ž 𝐢) = (𝐴 +β„Ž (𝐡 βˆ’β„Ž 𝐢)))
 
Theoremhvpncan3 30804 Subtraction and addition of equal Hilbert space vectors. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (𝐴 +β„Ž (𝐡 βˆ’β„Ž 𝐴)) = 𝐡)
 
Theoremhvmulcom 30805 Scalar multiplication commutative law. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‹) β†’ (𝐴 Β·β„Ž (𝐡 Β·β„Ž 𝐢)) = (𝐡 Β·β„Ž (𝐴 Β·β„Ž 𝐢)))
 
Theoremhvsubass 30806 Hilbert vector space associative law for subtraction. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 βˆ’β„Ž 𝐡) βˆ’β„Ž 𝐢) = (𝐴 βˆ’β„Ž (𝐡 +β„Ž 𝐢)))
 
Theoremhvsub32 30807 Hilbert vector space commutative/associative law. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 βˆ’β„Ž 𝐡) βˆ’β„Ž 𝐢) = ((𝐴 βˆ’β„Ž 𝐢) βˆ’β„Ž 𝐡))
 
Theoremhvmulassi 30808 Scalar multiplication associative law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‚    &   π΅ ∈ β„‚    &   πΆ ∈ β„‹    β‡’   ((𝐴 Β· 𝐡) Β·β„Ž 𝐢) = (𝐴 Β·β„Ž (𝐡 Β·β„Ž 𝐢))
 
Theoremhvmulcomi 30809 Scalar multiplication commutative law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‚    &   π΅ ∈ β„‚    &   πΆ ∈ β„‹    β‡’   (𝐴 Β·β„Ž (𝐡 Β·β„Ž 𝐢)) = (𝐡 Β·β„Ž (𝐴 Β·β„Ž 𝐢))
 
Theoremhvmul2negi 30810 Double negative in scalar multiplication. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‚    &   π΅ ∈ β„‚    &   πΆ ∈ β„‹    β‡’   (-𝐴 Β·β„Ž (-𝐡 Β·β„Ž 𝐢)) = (𝐴 Β·β„Ž (𝐡 Β·β„Ž 𝐢))
 
Theoremhvsubdistr1 30811 Scalar multiplication distributive law for subtraction. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (𝐴 Β·β„Ž (𝐡 βˆ’β„Ž 𝐢)) = ((𝐴 Β·β„Ž 𝐡) βˆ’β„Ž (𝐴 Β·β„Ž 𝐢)))
 
Theoremhvsubdistr2 30812 Scalar multiplication distributive law for subtraction. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 βˆ’ 𝐡) Β·β„Ž 𝐢) = ((𝐴 Β·β„Ž 𝐢) βˆ’β„Ž (𝐡 Β·β„Ž 𝐢)))
 
Theoremhvdistr1i 30813 Scalar multiplication distributive law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‚    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    β‡’   (𝐴 Β·β„Ž (𝐡 +β„Ž 𝐢)) = ((𝐴 Β·β„Ž 𝐡) +β„Ž (𝐴 Β·β„Ž 𝐢))
 
Theoremhvsubdistr1i 30814 Scalar multiplication distributive law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‚    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    β‡’   (𝐴 Β·β„Ž (𝐡 βˆ’β„Ž 𝐢)) = ((𝐴 Β·β„Ž 𝐡) βˆ’β„Ž (𝐴 Β·β„Ž 𝐢))
 
Theoremhvassi 30815 Hilbert vector space associative law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    β‡’   ((𝐴 +β„Ž 𝐡) +β„Ž 𝐢) = (𝐴 +β„Ž (𝐡 +β„Ž 𝐢))
 
Theoremhvadd32i 30816 Hilbert vector space commutative/associative law. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    β‡’   ((𝐴 +β„Ž 𝐡) +β„Ž 𝐢) = ((𝐴 +β„Ž 𝐢) +β„Ž 𝐡)
 
Theoremhvsubassi 30817 Hilbert vector space associative law for subtraction. (Contributed by NM, 7-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    β‡’   ((𝐴 βˆ’β„Ž 𝐡) βˆ’β„Ž 𝐢) = (𝐴 βˆ’β„Ž (𝐡 +β„Ž 𝐢))
 
Theoremhvsub32i 30818 Hilbert vector space commutative/associative law. (Contributed by NM, 7-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    β‡’   ((𝐴 βˆ’β„Ž 𝐡) βˆ’β„Ž 𝐢) = ((𝐴 βˆ’β„Ž 𝐢) βˆ’β„Ž 𝐡)
 
Theoremhvadd12i 30819 Hilbert vector space commutative/associative law. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    β‡’   (𝐴 +β„Ž (𝐡 +β„Ž 𝐢)) = (𝐡 +β„Ž (𝐴 +β„Ž 𝐢))
 
Theoremhvadd4i 30820 Hilbert vector space addition law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    &   π· ∈ β„‹    β‡’   ((𝐴 +β„Ž 𝐡) +β„Ž (𝐢 +β„Ž 𝐷)) = ((𝐴 +β„Ž 𝐢) +β„Ž (𝐡 +β„Ž 𝐷))
 
Theoremhvsubsub4i 30821 Hilbert vector space addition law. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    &   π· ∈ β„‹    β‡’   ((𝐴 βˆ’β„Ž 𝐡) βˆ’β„Ž (𝐢 βˆ’β„Ž 𝐷)) = ((𝐴 βˆ’β„Ž 𝐢) βˆ’β„Ž (𝐡 βˆ’β„Ž 𝐷))
 
Theoremhvsubsub4 30822 Hilbert vector space addition/subtraction law. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
(((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) ∧ (𝐢 ∈ β„‹ ∧ 𝐷 ∈ β„‹)) β†’ ((𝐴 βˆ’β„Ž 𝐡) βˆ’β„Ž (𝐢 βˆ’β„Ž 𝐷)) = ((𝐴 βˆ’β„Ž 𝐢) βˆ’β„Ž (𝐡 βˆ’β„Ž 𝐷)))
 
Theoremhv2times 30823 Two times a vector. (Contributed by NM, 22-Jun-2006.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (2 Β·β„Ž 𝐴) = (𝐴 +β„Ž 𝐴))
 
Theoremhvnegdii 30824 Distribution of negative over subtraction. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    β‡’   (-1 Β·β„Ž (𝐴 βˆ’β„Ž 𝐡)) = (𝐡 βˆ’β„Ž 𝐴)
 
Theoremhvsubeq0i 30825 If the difference between two vectors is zero, they are equal. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    β‡’   ((𝐴 βˆ’β„Ž 𝐡) = 0β„Ž ↔ 𝐴 = 𝐡)
 
Theoremhvsubcan2i 30826 Vector cancellation law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    β‡’   ((𝐴 +β„Ž 𝐡) +β„Ž (𝐴 βˆ’β„Ž 𝐡)) = (2 Β·β„Ž 𝐴)
 
Theoremhvaddcani 30827 Cancellation law for vector addition. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    β‡’   ((𝐴 +β„Ž 𝐡) = (𝐴 +β„Ž 𝐢) ↔ 𝐡 = 𝐢)
 
Theoremhvsubaddi 30828 Relationship between vector subtraction and addition. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    β‡’   ((𝐴 βˆ’β„Ž 𝐡) = 𝐢 ↔ (𝐡 +β„Ž 𝐢) = 𝐴)
 
Theoremhvnegdi 30829 Distribution of negative over subtraction. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (-1 Β·β„Ž (𝐴 βˆ’β„Ž 𝐡)) = (𝐡 βˆ’β„Ž 𝐴))
 
Theoremhvsubeq0 30830 If the difference between two vectors is zero, they are equal. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((𝐴 βˆ’β„Ž 𝐡) = 0β„Ž ↔ 𝐴 = 𝐡))
 
Theoremhvaddeq0 30831 If the sum of two vectors is zero, one is the negative of the other. (Contributed by NM, 10-Jun-2006.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((𝐴 +β„Ž 𝐡) = 0β„Ž ↔ 𝐴 = (-1 Β·β„Ž 𝐡)))
 
Theoremhvaddcan 30832 Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 +β„Ž 𝐡) = (𝐴 +β„Ž 𝐢) ↔ 𝐡 = 𝐢))
 
Theoremhvaddcan2 30833 Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 +β„Ž 𝐢) = (𝐡 +β„Ž 𝐢) ↔ 𝐴 = 𝐡))
 
Theoremhvmulcan 30834 Cancellation law for scalar multiplication. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
(((𝐴 ∈ β„‚ ∧ 𝐴 β‰  0) ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 Β·β„Ž 𝐡) = (𝐴 Β·β„Ž 𝐢) ↔ 𝐡 = 𝐢))
 
Theoremhvmulcan2 30835 Cancellation law for scalar multiplication. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ (𝐢 ∈ β„‹ ∧ 𝐢 β‰  0β„Ž)) β†’ ((𝐴 Β·β„Ž 𝐢) = (𝐡 Β·β„Ž 𝐢) ↔ 𝐴 = 𝐡))
 
Theoremhvsubcan 30836 Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 βˆ’β„Ž 𝐡) = (𝐴 βˆ’β„Ž 𝐢) ↔ 𝐡 = 𝐢))
 
Theoremhvsubcan2 30837 Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 βˆ’β„Ž 𝐢) = (𝐡 βˆ’β„Ž 𝐢) ↔ 𝐴 = 𝐡))
 
Theoremhvsub0 30838 Subtraction of a zero vector. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (𝐴 βˆ’β„Ž 0β„Ž) = 𝐴)
 
Theoremhvsubadd 30839 Relationship between vector subtraction and addition. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 βˆ’β„Ž 𝐡) = 𝐢 ↔ (𝐡 +β„Ž 𝐢) = 𝐴))
 
Theoremhvaddsub4 30840 Hilbert vector space addition/subtraction law. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
(((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) ∧ (𝐢 ∈ β„‹ ∧ 𝐷 ∈ β„‹)) β†’ ((𝐴 +β„Ž 𝐡) = (𝐢 +β„Ž 𝐷) ↔ (𝐴 βˆ’β„Ž 𝐢) = (𝐷 βˆ’β„Ž 𝐡)))
 
20.1.6  Inner product postulates for a Hilbert space
 
Axiomax-hfi 30841 Inner product maps pairs from β„‹ to β„‚. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
Β·ih :( β„‹ Γ— β„‹)βŸΆβ„‚
 
Theoremhicl 30842 Closure of inner product. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (𝐴 Β·ih 𝐡) ∈ β„‚)
 
Theoremhicli 30843 Closure inference for inner product. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    β‡’   (𝐴 Β·ih 𝐡) ∈ β„‚
 
Axiomax-his1 30844 Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. Note that βˆ—β€˜π‘₯ is the complex conjugate cjval 15055 of π‘₯. In the literature, the inner product of 𝐴 and 𝐡 is usually written ⟨𝐴, 𝐡⟩, but our operation notation co 7405 allows to use existing theorems about operations and also avoids a clash with the definition of an ordered pair df-op 4630. Physicists use ⟨𝐡 ∣ 𝐴⟩, called Dirac bra-ket notation, to represent this operation; see comments in df-bra 31612. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (𝐴 Β·ih 𝐡) = (βˆ—β€˜(𝐡 Β·ih 𝐴)))
 
Axiomax-his2 30845 Distributive law for inner product. Postulate (S2) of [Beran] p. 95. (Contributed by NM, 31-Jul-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 +β„Ž 𝐡) Β·ih 𝐢) = ((𝐴 Β·ih 𝐢) + (𝐡 Β·ih 𝐢)))
 
Axiomax-his3 30846 Associative law for inner product. Postulate (S3) of [Beran] p. 95. Warning: Mathematics textbooks usually use our version of the axiom. Physics textbooks, on the other hand, usually replace the left-hand side with (𝐡 Β·ih (𝐴 Β·β„Ž 𝐢)) (e.g., Equation 1.21b of [Hughes] p. 44; Definition (iii) of [ReedSimon] p. 36). See the comments in df-bra 31612 for why the physics definition is swapped. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 Β·β„Ž 𝐡) Β·ih 𝐢) = (𝐴 Β· (𝐡 Β·ih 𝐢)))
 
Axiomax-his4 30847 Identity law for inner product. Postulate (S4) of [Beran] p. 95. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ 0 < (𝐴 Β·ih 𝐴))
 
20.2  Inner product and norms
 
20.2.1  Inner product
 
Theoremhis5 30848 Associative law for inner product. Lemma 3.1(S5) of [Beran] p. 95. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (𝐡 Β·ih (𝐴 Β·β„Ž 𝐢)) = ((βˆ—β€˜π΄) Β· (𝐡 Β·ih 𝐢)))
 
Theoremhis52 30849 Associative law for inner product. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (𝐡 Β·ih ((βˆ—β€˜π΄) Β·β„Ž 𝐢)) = (𝐴 Β· (𝐡 Β·ih 𝐢)))
 
Theoremhis35 30850 Move scalar multiplication to outside of inner product. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
(((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝐢 ∈ β„‹ ∧ 𝐷 ∈ β„‹)) β†’ ((𝐴 Β·β„Ž 𝐢) Β·ih (𝐡 Β·β„Ž 𝐷)) = ((𝐴 Β· (βˆ—β€˜π΅)) Β· (𝐢 Β·ih 𝐷)))
 
Theoremhis35i 30851 Move scalar multiplication to outside of inner product. (Contributed by NM, 1-Jul-2005.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐴 ∈ β„‚    &   π΅ ∈ β„‚    &   πΆ ∈ β„‹    &   π· ∈ β„‹    β‡’   ((𝐴 Β·β„Ž 𝐢) Β·ih (𝐡 Β·β„Ž 𝐷)) = ((𝐴 Β· (βˆ—β€˜π΅)) Β· (𝐢 Β·ih 𝐷))
 
Theoremhis7 30852 Distributive law for inner product. Lemma 3.1(S7) of [Beran] p. 95. (Contributed by NM, 31-Jul-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (𝐴 Β·ih (𝐡 +β„Ž 𝐢)) = ((𝐴 Β·ih 𝐡) + (𝐴 Β·ih 𝐢)))
 
Theoremhiassdi 30853 Distributive/associative law for inner product, useful for linearity proofs. (Contributed by NM, 10-May-2005.) (New usage is discouraged.)
(((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹) ∧ (𝐢 ∈ β„‹ ∧ 𝐷 ∈ β„‹)) β†’ (((𝐴 Β·β„Ž 𝐡) +β„Ž 𝐢) Β·ih 𝐷) = ((𝐴 Β· (𝐡 Β·ih 𝐷)) + (𝐢 Β·ih 𝐷)))
 
Theoremhis2sub 30854 Distributive law for inner product of vector subtraction. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 βˆ’β„Ž 𝐡) Β·ih 𝐢) = ((𝐴 Β·ih 𝐢) βˆ’ (𝐡 Β·ih 𝐢)))
 
Theoremhis2sub2 30855 Distributive law for inner product of vector subtraction. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (𝐴 Β·ih (𝐡 βˆ’β„Ž 𝐢)) = ((𝐴 Β·ih 𝐡) βˆ’ (𝐴 Β·ih 𝐢)))
 
Theoremhire 30856 A necessary and sufficient condition for an inner product to be real. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((𝐴 Β·ih 𝐡) ∈ ℝ ↔ (𝐴 Β·ih 𝐡) = (𝐡 Β·ih 𝐴)))
 
Theoremhiidrcl 30857 Real closure of inner product with self. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (𝐴 Β·ih 𝐴) ∈ ℝ)
 
Theoremhi01 30858 Inner product with the 0 vector. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (0β„Ž Β·ih 𝐴) = 0)
 
Theoremhi02 30859 Inner product with the 0 vector. (Contributed by NM, 13-Oct-1999.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (𝐴 Β·ih 0β„Ž) = 0)
 
Theoremhiidge0 30860 Inner product with self is not negative. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ 0 ≀ (𝐴 Β·ih 𝐴))
 
Theoremhis6 30861 Zero inner product with self means vector is zero. Lemma 3.1(S6) of [Beran] p. 95. (Contributed by NM, 27-Jul-1999.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ ((𝐴 Β·ih 𝐴) = 0 ↔ 𝐴 = 0β„Ž))
 
Theoremhis1i 30862 Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. (Contributed by NM, 15-May-2005.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    β‡’   (𝐴 Β·ih 𝐡) = (βˆ—β€˜(𝐡 Β·ih 𝐴))
 
Theoremabshicom 30863 Commuted inner products have the same absolute values. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (absβ€˜(𝐴 Β·ih 𝐡)) = (absβ€˜(𝐡 Β·ih 𝐴)))
 
Theoremhial0 30864* A vector whose inner product is always zero is zero. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (βˆ€π‘₯ ∈ β„‹ (𝐴 Β·ih π‘₯) = 0 ↔ 𝐴 = 0β„Ž))
 
Theoremhial02 30865* A vector whose inner product is always zero is zero. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (βˆ€π‘₯ ∈ β„‹ (π‘₯ Β·ih 𝐴) = 0 ↔ 𝐴 = 0β„Ž))
 
Theoremhisubcomi 30866 Two vector subtractions simultaneously commute in an inner product. (Contributed by NM, 1-Jul-2005.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    &   π· ∈ β„‹    β‡’   ((𝐴 βˆ’β„Ž 𝐡) Β·ih (𝐢 βˆ’β„Ž 𝐷)) = ((𝐡 βˆ’β„Ž 𝐴) Β·ih (𝐷 βˆ’β„Ž 𝐢))
 
Theoremhi2eq 30867 Lemma used to prove equality of vectors. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((𝐴 Β·ih (𝐴 βˆ’β„Ž 𝐡)) = (𝐡 Β·ih (𝐴 βˆ’β„Ž 𝐡)) ↔ 𝐴 = 𝐡))
 
Theoremhial2eq 30868* Two vectors whose inner product is always equal are equal. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (βˆ€π‘₯ ∈ β„‹ (𝐴 Β·ih π‘₯) = (𝐡 Β·ih π‘₯) ↔ 𝐴 = 𝐡))
 
Theoremhial2eq2 30869* Two vectors whose inner product is always equal are equal. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (βˆ€π‘₯ ∈ β„‹ (π‘₯ Β·ih 𝐴) = (π‘₯ Β·ih 𝐡) ↔ 𝐴 = 𝐡))
 
Theoremorthcom 30870 Orthogonality commutes. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((𝐴 Β·ih 𝐡) = 0 ↔ (𝐡 Β·ih 𝐴) = 0))
 
Theoremnormlem0 30871 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 7-Oct-1999.) (New usage is discouraged.)
𝑆 ∈ β„‚    &   πΉ ∈ β„‹    &   πΊ ∈ β„‹    β‡’   ((𝐹 βˆ’β„Ž (𝑆 Β·β„Ž 𝐺)) Β·ih (𝐹 βˆ’β„Ž (𝑆 Β·β„Ž 𝐺))) = (((𝐹 Β·ih 𝐹) + (-(βˆ—β€˜π‘†) Β· (𝐹 Β·ih 𝐺))) + ((-𝑆 Β· (𝐺 Β·ih 𝐹)) + ((𝑆 Β· (βˆ—β€˜π‘†)) Β· (𝐺 Β·ih 𝐺))))
 
Theoremnormlem1 30872 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 22-Aug-1999.) (New usage is discouraged.)
𝑆 ∈ β„‚    &   πΉ ∈ β„‹    &   πΊ ∈ β„‹    &   π‘… ∈ ℝ    &   (absβ€˜π‘†) = 1    β‡’   ((𝐹 βˆ’β„Ž ((𝑆 Β· 𝑅) Β·β„Ž 𝐺)) Β·ih (𝐹 βˆ’β„Ž ((𝑆 Β· 𝑅) Β·β„Ž 𝐺))) = (((𝐹 Β·ih 𝐹) + (((βˆ—β€˜π‘†) Β· -𝑅) Β· (𝐹 Β·ih 𝐺))) + (((𝑆 Β· -𝑅) Β· (𝐺 Β·ih 𝐹)) + ((𝑅↑2) Β· (𝐺 Β·ih 𝐺))))
 
Theoremnormlem2 30873 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 27-Jul-1999.) (New usage is discouraged.)
𝑆 ∈ β„‚    &   πΉ ∈ β„‹    &   πΊ ∈ β„‹    &   π΅ = -(((βˆ—β€˜π‘†) Β· (𝐹 Β·ih 𝐺)) + (𝑆 Β· (𝐺 Β·ih 𝐹)))    β‡’   π΅ ∈ ℝ
 
Theoremnormlem3 30874 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 21-Aug-1999.) (New usage is discouraged.)
𝑆 ∈ β„‚    &   πΉ ∈ β„‹    &   πΊ ∈ β„‹    &   π΅ = -(((βˆ—β€˜π‘†) Β· (𝐹 Β·ih 𝐺)) + (𝑆 Β· (𝐺 Β·ih 𝐹)))    &   π΄ = (𝐺 Β·ih 𝐺)    &   πΆ = (𝐹 Β·ih 𝐹)    &   π‘… ∈ ℝ    β‡’   (((𝐴 Β· (𝑅↑2)) + (𝐡 Β· 𝑅)) + 𝐢) = (((𝐹 Β·ih 𝐹) + (((βˆ—β€˜π‘†) Β· -𝑅) Β· (𝐹 Β·ih 𝐺))) + (((𝑆 Β· -𝑅) Β· (𝐺 Β·ih 𝐹)) + ((𝑅↑2) Β· (𝐺 Β·ih 𝐺))))
 
Theoremnormlem4 30875 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
𝑆 ∈ β„‚    &   πΉ ∈ β„‹    &   πΊ ∈ β„‹    &   π΅ = -(((βˆ—β€˜π‘†) Β· (𝐹 Β·ih 𝐺)) + (𝑆 Β· (𝐺 Β·ih 𝐹)))    &   π΄ = (𝐺 Β·ih 𝐺)    &   πΆ = (𝐹 Β·ih 𝐹)    &   π‘… ∈ ℝ    &   (absβ€˜π‘†) = 1    β‡’   ((𝐹 βˆ’β„Ž ((𝑆 Β· 𝑅) Β·β„Ž 𝐺)) Β·ih (𝐹 βˆ’β„Ž ((𝑆 Β· 𝑅) Β·β„Ž 𝐺))) = (((𝐴 Β· (𝑅↑2)) + (𝐡 Β· 𝑅)) + 𝐢)
 
Theoremnormlem5 30876 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 10-Aug-1999.) (New usage is discouraged.)
𝑆 ∈ β„‚    &   πΉ ∈ β„‹    &   πΊ ∈ β„‹    &   π΅ = -(((βˆ—β€˜π‘†) Β· (𝐹 Β·ih 𝐺)) + (𝑆 Β· (𝐺 Β·ih 𝐹)))    &   π΄ = (𝐺 Β·ih 𝐺)    &   πΆ = (𝐹 Β·ih 𝐹)    &   π‘… ∈ ℝ    &   (absβ€˜π‘†) = 1    β‡’   0 ≀ (((𝐴 Β· (𝑅↑2)) + (𝐡 Β· 𝑅)) + 𝐢)
 
Theoremnormlem6 30877 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 2-Aug-1999.) (Revised by Mario Carneiro, 4-Jun-2014.) (New usage is discouraged.)
𝑆 ∈ β„‚    &   πΉ ∈ β„‹    &   πΊ ∈ β„‹    &   π΅ = -(((βˆ—β€˜π‘†) Β· (𝐹 Β·ih 𝐺)) + (𝑆 Β· (𝐺 Β·ih 𝐹)))    &   π΄ = (𝐺 Β·ih 𝐺)    &   πΆ = (𝐹 Β·ih 𝐹)    &   (absβ€˜π‘†) = 1    β‡’   (absβ€˜π΅) ≀ (2 Β· ((βˆšβ€˜π΄) Β· (βˆšβ€˜πΆ)))
 
Theoremnormlem7 30878 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 11-Aug-1999.) (New usage is discouraged.)
𝑆 ∈ β„‚    &   πΉ ∈ β„‹    &   πΊ ∈ β„‹    &   (absβ€˜π‘†) = 1    β‡’   (((βˆ—β€˜π‘†) Β· (𝐹 Β·ih 𝐺)) + (𝑆 Β· (𝐺 Β·ih 𝐹))) ≀ (2 Β· ((βˆšβ€˜(𝐺 Β·ih 𝐺)) Β· (βˆšβ€˜(𝐹 Β·ih 𝐹))))
 
Theoremnormlem8 30879 Lemma used to derive properties of norm. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    &   π· ∈ β„‹    β‡’   ((𝐴 +β„Ž 𝐡) Β·ih (𝐢 +β„Ž 𝐷)) = (((𝐴 Β·ih 𝐢) + (𝐡 Β·ih 𝐷)) + ((𝐴 Β·ih 𝐷) + (𝐡 Β·ih 𝐢)))
 
Theoremnormlem9 30880 Lemma used to derive properties of norm. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    &   π· ∈ β„‹    β‡’   ((𝐴 βˆ’β„Ž 𝐡) Β·ih (𝐢 βˆ’β„Ž 𝐷)) = (((𝐴 Β·ih 𝐢) + (𝐡 Β·ih 𝐷)) βˆ’ ((𝐴 Β·ih 𝐷) + (𝐡 Β·ih 𝐢)))
 
Theoremnormlem7tALT 30881 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    β‡’   ((𝑆 ∈ β„‚ ∧ (absβ€˜π‘†) = 1) β†’ (((βˆ—β€˜π‘†) Β· (𝐴 Β·ih 𝐡)) + (𝑆 Β· (𝐡 Β·ih 𝐴))) ≀ (2 Β· ((βˆšβ€˜(𝐡 Β·ih 𝐡)) Β· (βˆšβ€˜(𝐴 Β·ih 𝐴)))))
 
Theorembcseqi 30882 Equality case of Bunjakovaskij-Cauchy-Schwarz inequality. Specifically, in the equality case the two vectors are collinear. Compare bcsiHIL 30942. (Contributed by NM, 16-Jul-2001.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    β‡’   (((𝐴 Β·ih 𝐡) Β· (𝐡 Β·ih 𝐴)) = ((𝐴 Β·ih 𝐴) Β· (𝐡 Β·ih 𝐡)) ↔ ((𝐡 Β·ih 𝐡) Β·β„Ž 𝐴) = ((𝐴 Β·ih 𝐡) Β·β„Ž 𝐡))
 
Theoremnormlem9at 30883 Lemma used to derive properties of norm. Part of Remark 3.4(B) of [Beran] p. 98. (Contributed by NM, 10-May-2005.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((𝐴 βˆ’β„Ž 𝐡) Β·ih (𝐴 βˆ’β„Ž 𝐡)) = (((𝐴 Β·ih 𝐴) + (𝐡 Β·ih 𝐡)) βˆ’ ((𝐴 Β·ih 𝐡) + (𝐡 Β·ih 𝐴))))
 
20.2.2  Norms
 
Theoremdfhnorm2 30884 Alternate definition of the norm of a vector of Hilbert space. Definition of norm in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
normβ„Ž = (π‘₯ ∈ β„‹ ↦ (βˆšβ€˜(π‘₯ Β·ih π‘₯)))
 
Theoremnormf 30885 The norm function maps from Hilbert space to reals. (Contributed by NM, 6-Sep-2007.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
normβ„Ž: β„‹βŸΆβ„
 
Theoremnormval 30886 The value of the norm of a vector in Hilbert space. Definition of norm in [Beran] p. 96. In the literature, the norm of 𝐴 is usually written as "|| 𝐴 ||", but we use function value notation to take advantage of our existing theorems about functions. (Contributed by NM, 29-May-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (normβ„Žβ€˜π΄) = (βˆšβ€˜(𝐴 Β·ih 𝐴)))
 
Theoremnormcl 30887 Real closure of the norm of a vector. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (normβ„Žβ€˜π΄) ∈ ℝ)
 
Theoremnormge0 30888 The norm of a vector is nonnegative. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ 0 ≀ (normβ„Žβ€˜π΄))
 
Theoremnormgt0 30889 The norm of nonzero vector is positive. (Contributed by NM, 10-Apr-2006.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (𝐴 β‰  0β„Ž ↔ 0 < (normβ„Žβ€˜π΄)))
 
Theoremnorm0 30890 The norm of a zero vector. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
(normβ„Žβ€˜0β„Ž) = 0
 
Theoremnorm-i 30891 Theorem 3.3(i) of [Beran] p. 97. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ ((normβ„Žβ€˜π΄) = 0 ↔ 𝐴 = 0β„Ž))
 
Theoremnormne0 30892 A norm is nonzero iff its argument is a nonzero vector. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ ((normβ„Žβ€˜π΄) β‰  0 ↔ 𝐴 β‰  0β„Ž))
 
Theoremnormcli 30893 Real closure of the norm of a vector. (Contributed by NM, 30-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    β‡’   (normβ„Žβ€˜π΄) ∈ ℝ
 
Theoremnormsqi 30894 The square of a norm. (Contributed by NM, 21-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    β‡’   ((normβ„Žβ€˜π΄)↑2) = (𝐴 Β·ih 𝐴)
 
Theoremnorm-i-i 30895 Theorem 3.3(i) of [Beran] p. 97. (Contributed by NM, 5-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    β‡’   ((normβ„Žβ€˜π΄) = 0 ↔ 𝐴 = 0β„Ž)
 
Theoremnormsq 30896 The square of a norm. (Contributed by NM, 12-May-2005.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ ((normβ„Žβ€˜π΄)↑2) = (𝐴 Β·ih 𝐴))
 
Theoremnormsub0i 30897 Two vectors are equal iff the norm of their difference is zero. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    β‡’   ((normβ„Žβ€˜(𝐴 βˆ’β„Ž 𝐡)) = 0 ↔ 𝐴 = 𝐡)
 
Theoremnormsub0 30898 Two vectors are equal iff the norm of their difference is zero. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((normβ„Žβ€˜(𝐴 βˆ’β„Ž 𝐡)) = 0 ↔ 𝐴 = 𝐡))
 
Theoremnorm-ii-i 30899 Triangle inequality for norms. Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 11-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    β‡’   (normβ„Žβ€˜(𝐴 +β„Ž 𝐡)) ≀ ((normβ„Žβ€˜π΄) + (normβ„Žβ€˜π΅))
 
Theoremnorm-ii 30900 Triangle inequality for norms. Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (normβ„Žβ€˜(𝐴 +β„Ž 𝐡)) ≀ ((normβ„Žβ€˜π΄) + (normβ„Žβ€˜π΅)))
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47900 480 47901-48000 481 48001-48100 482 48101-48126
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