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Theorem List for Metamath Proof Explorer - 22501-22600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremaxhcompl-zf 22501* Derive axiom ax-hcompl 22704 from Hilbert space under ZF set theory. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (New usage is discouraged.)

18.1.4  Introduce the vector space axioms for a Hilbert space

Here we introduce the axioms a complex Hilbert space, which is the foundation for quantum mechanics and quantum field theory. The 18 axioms for a complex Hilbert space consist of ax-hilex 22502, ax-hfvadd 22503, ax-hvcom 22504, ax-hvass 22505, ax-hv0cl 22506, ax-hvaddid 22507, ax-hfvmul 22508, ax-hvmulid 22509, ax-hvmulass 22510, ax-hvdistr1 22511, ax-hvdistr2 22512, ax-hvmul0 22513, ax-hfi 22581, ax-his1 22584, ax-his2 22585, ax-his3 22586, ax-his4 22587, and ax-hcompl 22704.

The axioms specify the properties of 5 primitive symbols, , , , , and .

If we can prove in ZFC set theory that a class is a complex Hilbert space, i.e. that , then these axioms can be proved as theorems axhilex-zf 22484, axhfvadd-zf 22485, axhvcom-zf 22486, axhvass-zf 22487, axhv0cl-zf 22488, axhvaddid-zf 22489, axhfvmul-zf 22490, axhvmulid-zf 22491, axhvmulass-zf 22492, axhvdistr1-zf 22493, axhvdistr2-zf 22494, axhvmul0-zf 22495, axhfi-zf 22496, axhis1-zf 22497, axhis2-zf 22498, axhis3-zf 22499, axhis4-zf 22500, and axhcompl-zf 22501 respectively. In that case, the theorems of the Hilbert Space Explorer will become theorems of ZFC set theory. See also the comments in axhilex-zf 22484.

Axiomax-hilex 22502 This is our first axiom for a complex Hilbert space, which is the foundation for quantum mechanics and quantum field theory. We assume that there exists a primitive class, , which contains objects called vectors. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)

Axiomax-hfvadd 22503 Vector addition is an operation on . (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)

Axiomax-hvcom 22504 Vector addition is commutative. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)

Axiomax-hvass 22505 Vector addition is associative. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)

Axiomax-hv0cl 22506 The zero vector is in the vector space. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)

Axiomax-hvaddid 22507 Addition with the zero vector. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)

Axiomax-hfvmul 22508 Scalar multiplication is an operation on and . (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)

Axiomax-hvmulid 22509 Scalar multiplication by one. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)

Axiomax-hvmulass 22510 Scalar multiplication associative law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)

Axiomax-hvdistr1 22511 Scalar multiplication distributive law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)

Axiomax-hvdistr2 22512 Scalar multiplication distributive law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)

Axiomax-hvmul0 22513 Scalar multiplication by zero. We can derive the existence of the negative of a vector from this axiom (see hvsubid 22528 and hvsubval 22519). (Contributed by NM, 29-May-1999.) (New usage is discouraged.)

18.1.5  Vector operations

Theoremhvmulex 22514 The Hilbert space scalar product operation is a set. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)

Theoremhvaddcl 22515 Closure of vector addition. (Contributed by NM, 18-Apr-2007.) (New usage is discouraged.)

Theoremhvmulcl 22516 Closure of scalar multiplication. (Contributed by NM, 19-Apr-2007.) (New usage is discouraged.)

Theoremhvmulcli 22517 Closure inference for scalar multiplication. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)

Theoremhvsubf 22518 Mapping domain and codomain of vector subtraction. (Contributed by NM, 6-Sep-2007.) (New usage is discouraged.)

Theoremhvsubval 22519 Value of vector subtraction. (Contributed by NM, 5-Sep-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremhvsubcl 22520 Closure of vector subtraction. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.)

Theoremhvaddcli 22521 Closure of vector addition. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)

Theoremhvcomi 22522 Commutation of vector addition. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)

Theoremhvsubvali 22523 Value of vector subtraction definition. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)

Theoremhvsubcli 22524 Closure of vector subtraction. (Contributed by NM, 2-Aug-1999.) (New usage is discouraged.)

Theoremhvaddid2 22525 Addition with the zero vector. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.)

Theoremhvmul0 22526 Scalar multiplication with the zero vector. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)

Theoremhvmul0or 22527 If a scalar product is zero, one of its factors must be zero. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)

Theoremhvsubid 22528 Subtraction of a vector from itself. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)

Theoremhvnegid 22529 Addition of negative of a vector to itself. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)

Theoremhv2neg 22530 Two ways to express the negative of a vector. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)

Theoremhvaddid2i 22531 Addition with the zero vector. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)

Theoremhvnegidi 22532 Addition of negative of a vector to itself. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)

Theoremhv2negi 22533 Two ways to express the negative of a vector. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)

Theoremhvm1neg 22534 Convert minus one times a scalar product to the negative of the scalar. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)

Theoremhvaddsubval 22535 Value of vector addition in terms of vector subtraction. (Contributed by NM, 10-Jun-2006.) (New usage is discouraged.)

Theoremhvadd32 22536 Commutative/associative law. (Contributed by NM, 16-Oct-1999.) (New usage is discouraged.)

Theoremhvadd12 22537 Commutative/associative law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)

Theoremhvadd4 22538 Hilbert vector space addition law. (Contributed by NM, 16-Oct-1999.) (New usage is discouraged.)

Theoremhvsub4 22539 Hilbert vector space addition/subtraction law. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)

Theoremhvaddsub12 22540 Commutative/associative law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)

Theoremhvpncan 22541 Addition/subtraction cancellation law for vectors in Hilbert space. (Contributed by NM, 7-Jun-2004.) (New usage is discouraged.)

Theoremhvpncan2 22542 Addition/subtraction cancellation law for vectors in Hilbert space. (Contributed by NM, 7-Jun-2004.) (New usage is discouraged.)

Theoremhvaddsubass 22543 Associativity of sum and difference of Hilbert space vectors. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)

Theoremhvpncan3 22544 Subtraction and addition of equal Hilbert space vectors. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)

Theoremhvmulcom 22545 Scalar multiplication commutative law. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)

Theoremhvsubass 22546 Hilbert vector space associative law for subtraction. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theoremhvsub32 22547 Hilbert vector space commutative/associative law. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theoremhvmulassi 22548 Scalar multiplication associative law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)

Theoremhvmulcomi 22549 Scalar multiplication commutative law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)

Theoremhvmul2negi 22550 Double negative in scalar multiplication. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)

Theoremhvsubdistr1 22551 Scalar multiplication distributive law for subtraction. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)

Theoremhvsubdistr2 22552 Scalar multiplication distributive law for subtraction. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)

Theoremhvdistr1i 22553 Scalar multiplication distributive law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)

Theoremhvsubdistr1i 22554 Scalar multiplication distributive law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)

Theoremhvassi 22555 Hilbert vector space associative law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)

Theoremhvadd32i 22556 Hilbert vector space commutative/associative law. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)

Theoremhvsubassi 22557 Hilbert vector space associative law for subtraction. (Contributed by NM, 7-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theoremhvsub32i 22558 Hilbert vector space commutative/associative law. (Contributed by NM, 7-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theoremhvadd12i 22559 Hilbert vector space commutative/associative law. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.)

Theoremhvadd4i 22560 Hilbert vector space addition law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)

Theoremhvsubsub4i 22561 Hilbert vector space addition law. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)

Theoremhvsubsub4 22562 Hilbert vector space addition/subtraction law. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)

Theoremhv2times 22563 Two times a vector. (Contributed by NM, 22-Jun-2006.) (New usage is discouraged.)

Theoremhvnegdii 22564 Distribution of negative over subtraction. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)

Theoremhvsubeq0i 22565 If the difference between two vectors is zero, they are equal. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)

Theoremhvsubcan2i 22566 Vector cancellation law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)

Theoremhvaddcani 22567 Cancellation law for vector addition. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.)

Theoremhvsubaddi 22568 Relationship between vector subtraction and addition. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.)

Theoremhvnegdi 22569 Distribution of negative over subtraction. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)

Theoremhvsubeq0 22570 If the difference between two vectors is zero, they are equal. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)

Theoremhvaddeq0 22571 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.)

Theoremhvaddcan 22572 Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)

Theoremhvaddcan2 22573 Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)

Theoremhvmulcan 22574 Cancellation law for scalar multiplication. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)

Theoremhvmulcan2 22575 Cancellation law for scalar multiplication. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)

Theoremhvsubcan 22576 Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)

Theoremhvsubcan2 22577 Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)

Theoremhvsub0 22578 Subtraction of a zero vector. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)

Theoremhvsubadd 22579 Relationship between vector subtraction and addition. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)

Theoremhvaddsub4 22580 Hilbert vector space addition/subtraction law. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)

18.1.6  Inner product postulates for a Hilbert space

Axiomax-hfi 22581 Inner product maps pairs from to . (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)

Theoremhicl 22582 Closure of inner product. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)

Theoremhicli 22583 Closure inference for inner product. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)

Axiomax-his1 22584 Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. Note that is the complex conjugate cjval 11907 of . In the literature, the inner product of and is usually written , but our operation notation co 6081 allows us to use existing theorems about operations and also avoids a clash with the definition of an ordered pair df-op 3823. Physicists use , called Dirac bra-ket notation, to represent this operation; see comments in df-bra 23353. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)

Axiomax-his2 22585 Distributive law for inner product. Postulate (S2) of [Beran] p. 95. (Contributed by NM, 31-Jul-1999.) (New usage is discouraged.)

Axiomax-his3 22586 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 (e.g. Equation 1.21b of [Hughes] p. 44; Definition (iii) of [ReedSimon] p. 36). See the comments in df-bra 23353 for why the physics definition is swapped. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)

Axiomax-his4 22587 Identity law for inner product. Postulate (S4) of [Beran] p. 95. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)

18.2  Inner product and norms

18.2.1  Inner product

Theoremhis5 22588 Associative law for inner product. Lemma 3.1(S5) of [Beran] p. 95. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)

Theoremhis52 22589 Associative law for inner product. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)

Theoremhis35 22590 Move scalar multiplication to outside of inner product. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theoremhis35i 22591 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.)

Theoremhis7 22592 Distributive law for inner product. Lemma 3.1(S7) of [Beran] p. 95. (Contributed by NM, 31-Jul-1999.) (New usage is discouraged.)

Theoremhiassdi 22593 Distributive/associative law for inner product, useful for linearity proofs. (Contributed by NM, 10-May-2005.) (New usage is discouraged.)

Theoremhis2sub 22594 Distributive law for inner product of vector subtraction. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.)

Theoremhis2sub2 22595 Distributive law for inner product of vector subtraction. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)

Theoremhire 22596 A necessary and sufficient condition for an inner product to be real. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.)

Theoremhiidrcl 22597 Real closure of inner product with self. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)

Theoremhi01 22598 Inner product with the 0 vector. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)

Theoremhi02 22599 Inner product with the 0 vector. (Contributed by NM, 13-Oct-1999.) (New usage is discouraged.)

Theoremhiidge0 22600 Inner product with self is not negative. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)

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