P. 312 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 31101-31200   *Has distinct variable group(s)

Type	Label	Description
Statement
Axiom	ax-his1 31101	Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. Note that ∗‘𝑥 is the complex conjugate cjval 15141 of 𝑥. In the literature, the inner product of 𝐴 and 𝐵 is usually written ⟨𝐴, 𝐵⟩, but our operation notation co 7431 allows to use existing theorems about operations and also avoids a clash with the definition of an ordered pair df-op 4633. Physicists use ⟨𝐵 ∣ 𝐴⟩, called Dirac bra-ket notation, to represent this operation; see comments in df-bra 31869. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴)))
Axiom	ax-his2 31102	Distributive law for inner product. Postulate (S2) of [Beran] p. 95. (Contributed by NM, 31-Jul-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) ·ih 𝐶) = ((𝐴 ·ih 𝐶) + (𝐵 ·ih 𝐶)))
Axiom	ax-his3 31103	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 (𝐵 ·ih (𝐴 ·ℎ 𝐶)) (e.g., Equation 1.21b of [Hughes] p. 44; Definition (iii) of [ReedSimon] p. 36). See the comments in df-bra 31869 for why the physics definition is swapped. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·ℎ 𝐵) ·ih 𝐶) = (𝐴 · (𝐵 ·ih 𝐶)))
Axiom	ax-his4 31104	Identity law for inner product. Postulate (S4) of [Beran] p. 95. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (𝐴 ·ih 𝐴))
20.2  Inner product and norms
20.2.1  Inner product
Theorem	his5 31105	Associative law for inner product. Lemma 3.1(S5) of [Beran] p. 95. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·ih (𝐴 ·ℎ 𝐶)) = ((∗‘𝐴) · (𝐵 ·ih 𝐶)))
Theorem	his52 31106	Associative law for inner product. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·ih ((∗‘𝐴) ·ℎ 𝐶)) = (𝐴 · (𝐵 ·ih 𝐶)))
Theorem	his35 31107	Move scalar multiplication to outside of inner product. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 ·ℎ 𝐶) ·ih (𝐵 ·ℎ 𝐷)) = ((𝐴 · (∗‘𝐵)) · (𝐶 ·ih 𝐷)))
Theorem	his35i 31108	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 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℋ    &   ⊢ 𝐷 ∈ ℋ    ⇒   ⊢ ((𝐴 ·ℎ 𝐶) ·ih (𝐵 ·ℎ 𝐷)) = ((𝐴 · (∗‘𝐵)) · (𝐶 ·ih 𝐷))
Theorem	his7 31109	Distributive law for inner product. Lemma 3.1(S7) of [Beran] p. 95. (Contributed by NM, 31-Jul-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ih (𝐵 +ℎ 𝐶)) = ((𝐴 ·ih 𝐵) + (𝐴 ·ih 𝐶)))
Theorem	hiassdi 31110	Distributive/associative law for inner product, useful for linearity proofs. (Contributed by NM, 10-May-2005.) (New usage is discouraged.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((𝐴 ·ℎ 𝐵) +ℎ 𝐶) ·ih 𝐷) = ((𝐴 · (𝐵 ·ih 𝐷)) + (𝐶 ·ih 𝐷)))
Theorem	his2sub 31111	Distributive law for inner product of vector subtraction. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −ℎ 𝐵) ·ih 𝐶) = ((𝐴 ·ih 𝐶) − (𝐵 ·ih 𝐶)))
Theorem	his2sub2 31112	Distributive law for inner product of vector subtraction. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ih (𝐵 −ℎ 𝐶)) = ((𝐴 ·ih 𝐵) − (𝐴 ·ih 𝐶)))
Theorem	hire 31113	A necessary and sufficient condition for an inner product to be real. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ih 𝐵) ∈ ℝ ↔ (𝐴 ·ih 𝐵) = (𝐵 ·ih 𝐴)))
Theorem	hiidrcl 31114	Real closure of inner product with self. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (𝐴 ·ih 𝐴) ∈ ℝ)
Theorem	hi01 31115	Inner product with the 0 vector. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (0ℎ ·ih 𝐴) = 0)
Theorem	hi02 31116	Inner product with the 0 vector. (Contributed by NM, 13-Oct-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (𝐴 ·ih 0ℎ) = 0)
Theorem	hiidge0 31117	Inner product with self is not negative. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → 0 ≤ (𝐴 ·ih 𝐴))
Theorem	his6 31118	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 ∈ ℋ → ((𝐴 ·ih 𝐴) = 0 ↔ 𝐴 = 0ℎ))
Theorem	his1i 31119	Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. (Contributed by NM, 15-May-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    ⇒   ⊢ (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴))
Theorem	abshicom 31120	Commuted inner products have the same absolute values. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (abs‘(𝐴 ·ih 𝐵)) = (abs‘(𝐵 ·ih 𝐴)))
Theorem	hial0 31121*	A vector whose inner product is always zero is zero. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (∀𝑥 ∈ ℋ (𝐴 ·ih 𝑥) = 0 ↔ 𝐴 = 0ℎ))
Theorem	hial02 31122*	A vector whose inner product is always zero is zero. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (∀𝑥 ∈ ℋ (𝑥 ·ih 𝐴) = 0 ↔ 𝐴 = 0ℎ))
Theorem	hisubcomi 31123	Two vector subtractions simultaneously commute in an inner product. (Contributed by NM, 1-Jul-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    &   ⊢ 𝐶 ∈ ℋ    &   ⊢ 𝐷 ∈ ℋ    ⇒   ⊢ ((𝐴 −ℎ 𝐵) ·ih (𝐶 −ℎ 𝐷)) = ((𝐵 −ℎ 𝐴) ·ih (𝐷 −ℎ 𝐶))
Theorem	hi2eq 31124	Lemma used to prove equality of vectors. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ih (𝐴 −ℎ 𝐵)) = (𝐵 ·ih (𝐴 −ℎ 𝐵)) ↔ 𝐴 = 𝐵))
Theorem	hial2eq 31125*	Two vectors whose inner product is always equal are equal. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ (𝐴 ·ih 𝑥) = (𝐵 ·ih 𝑥) ↔ 𝐴 = 𝐵))
Theorem	hial2eq2 31126*	Two vectors whose inner product is always equal are equal. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ (𝑥 ·ih 𝐴) = (𝑥 ·ih 𝐵) ↔ 𝐴 = 𝐵))
Theorem	orthcom 31127	Orthogonality commutes. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ih 𝐵) = 0 ↔ (𝐵 ·ih 𝐴) = 0))
Theorem	normlem0 31128	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 ∈ ℂ    &   ⊢ 𝐹 ∈ ℋ    &   ⊢ 𝐺 ∈ ℋ    ⇒   ⊢ ((𝐹 −ℎ (𝑆 ·ℎ 𝐺)) ·ih (𝐹 −ℎ (𝑆 ·ℎ 𝐺))) = (((𝐹 ·ih 𝐹) + (-(∗‘𝑆) · (𝐹 ·ih 𝐺))) + ((-𝑆 · (𝐺 ·ih 𝐹)) + ((𝑆 · (∗‘𝑆)) · (𝐺 ·ih 𝐺))))
Theorem	normlem1 31129	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 ∈ ℂ    &   ⊢ 𝐹 ∈ ℋ    &   ⊢ 𝐺 ∈ ℋ    &   ⊢ 𝑅 ∈ ℝ    &   ⊢ (abs‘𝑆) = 1    ⇒   ⊢ ((𝐹 −ℎ ((𝑆 · 𝑅) ·ℎ 𝐺)) ·ih (𝐹 −ℎ ((𝑆 · 𝑅) ·ℎ 𝐺))) = (((𝐹 ·ih 𝐹) + (((∗‘𝑆) · -𝑅) · (𝐹 ·ih 𝐺))) + (((𝑆 · -𝑅) · (𝐺 ·ih 𝐹)) + ((𝑅↑2) · (𝐺 ·ih 𝐺))))
Theorem	normlem2 31130	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 ∈ ℂ    &   ⊢ 𝐹 ∈ ℋ    &   ⊢ 𝐺 ∈ ℋ    &   ⊢ 𝐵 = -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))    ⇒   ⊢ 𝐵 ∈ ℝ
Theorem	normlem3 31131	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 ∈ ℂ    &   ⊢ 𝐹 ∈ ℋ    &   ⊢ 𝐺 ∈ ℋ    &   ⊢ 𝐵 = -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))    &   ⊢ 𝐴 = (𝐺 ·ih 𝐺)    &   ⊢ 𝐶 = (𝐹 ·ih 𝐹)    &   ⊢ 𝑅 ∈ ℝ    ⇒   ⊢ (((𝐴 · (𝑅↑2)) + (𝐵 · 𝑅)) + 𝐶) = (((𝐹 ·ih 𝐹) + (((∗‘𝑆) · -𝑅) · (𝐹 ·ih 𝐺))) + (((𝑆 · -𝑅) · (𝐺 ·ih 𝐹)) + ((𝑅↑2) · (𝐺 ·ih 𝐺))))
Theorem	normlem4 31132	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 ∈ ℂ    &   ⊢ 𝐹 ∈ ℋ    &   ⊢ 𝐺 ∈ ℋ    &   ⊢ 𝐵 = -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))    &   ⊢ 𝐴 = (𝐺 ·ih 𝐺)    &   ⊢ 𝐶 = (𝐹 ·ih 𝐹)    &   ⊢ 𝑅 ∈ ℝ    &   ⊢ (abs‘𝑆) = 1    ⇒   ⊢ ((𝐹 −ℎ ((𝑆 · 𝑅) ·ℎ 𝐺)) ·ih (𝐹 −ℎ ((𝑆 · 𝑅) ·ℎ 𝐺))) = (((𝐴 · (𝑅↑2)) + (𝐵 · 𝑅)) + 𝐶)
Theorem	normlem5 31133	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 ∈ ℂ    &   ⊢ 𝐹 ∈ ℋ    &   ⊢ 𝐺 ∈ ℋ    &   ⊢ 𝐵 = -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))    &   ⊢ 𝐴 = (𝐺 ·ih 𝐺)    &   ⊢ 𝐶 = (𝐹 ·ih 𝐹)    &   ⊢ 𝑅 ∈ ℝ    &   ⊢ (abs‘𝑆) = 1    ⇒   ⊢ 0 ≤ (((𝐴 · (𝑅↑2)) + (𝐵 · 𝑅)) + 𝐶)
Theorem	normlem6 31134	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 ∈ ℂ    &   ⊢ 𝐹 ∈ ℋ    &   ⊢ 𝐺 ∈ ℋ    &   ⊢ 𝐵 = -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))    &   ⊢ 𝐴 = (𝐺 ·ih 𝐺)    &   ⊢ 𝐶 = (𝐹 ·ih 𝐹)    &   ⊢ (abs‘𝑆) = 1    ⇒   ⊢ (abs‘𝐵) ≤ (2 · ((√‘𝐴) · (√‘𝐶)))
Theorem	normlem7 31135	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 ∈ ℂ    &   ⊢ 𝐹 ∈ ℋ    &   ⊢ 𝐺 ∈ ℋ    &   ⊢ (abs‘𝑆) = 1    ⇒   ⊢ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ≤ (2 · ((√‘(𝐺 ·ih 𝐺)) · (√‘(𝐹 ·ih 𝐹))))
Theorem	normlem8 31136	Lemma used to derive properties of norm. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    &   ⊢ 𝐶 ∈ ℋ    &   ⊢ 𝐷 ∈ ℋ    ⇒   ⊢ ((𝐴 +ℎ 𝐵) ·ih (𝐶 +ℎ 𝐷)) = (((𝐴 ·ih 𝐶) + (𝐵 ·ih 𝐷)) + ((𝐴 ·ih 𝐷) + (𝐵 ·ih 𝐶)))
Theorem	normlem9 31137	Lemma used to derive properties of norm. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    &   ⊢ 𝐶 ∈ ℋ    &   ⊢ 𝐷 ∈ ℋ    ⇒   ⊢ ((𝐴 −ℎ 𝐵) ·ih (𝐶 −ℎ 𝐷)) = (((𝐴 ·ih 𝐶) + (𝐵 ·ih 𝐷)) − ((𝐴 ·ih 𝐷) + (𝐵 ·ih 𝐶)))
Theorem	normlem7tALT 31138	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 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    ⇒   ⊢ ((𝑆 ∈ ℂ ∧ (abs‘𝑆) = 1) → (((∗‘𝑆) · (𝐴 ·ih 𝐵)) + (𝑆 · (𝐵 ·ih 𝐴))) ≤ (2 · ((√‘(𝐵 ·ih 𝐵)) · (√‘(𝐴 ·ih 𝐴)))))
Theorem	bcseqi 31139	Equality case of Bunjakovaskij-Cauchy-Schwarz inequality. Specifically, in the equality case the two vectors are collinear. Compare bcsiHIL 31199. (Contributed by NM, 16-Jul-2001.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    ⇒   ⊢ (((𝐴 ·ih 𝐵) · (𝐵 ·ih 𝐴)) = ((𝐴 ·ih 𝐴) · (𝐵 ·ih 𝐵)) ↔ ((𝐵 ·ih 𝐵) ·ℎ 𝐴) = ((𝐴 ·ih 𝐵) ·ℎ 𝐵))
Theorem	normlem9at 31140	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 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 −ℎ 𝐵) ·ih (𝐴 −ℎ 𝐵)) = (((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) − ((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴))))
20.2.2  Norms
Theorem	dfhnorm2 31141	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.)
⊢ normℎ = (𝑥 ∈ ℋ ↦ (√‘(𝑥 ·ih 𝑥)))
Theorem	normf 31142	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.)
⊢ normℎ: ℋ⟶ℝ
Theorem	normval 31143	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 ∈ ℋ → (normℎ‘𝐴) = (√‘(𝐴 ·ih 𝐴)))
Theorem	normcl 31144	Real closure of the norm of a vector. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ)
Theorem	normge0 31145	The norm of a vector is nonnegative. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → 0 ≤ (normℎ‘𝐴))
Theorem	normgt0 31146	The norm of nonzero vector is positive. (Contributed by NM, 10-Apr-2006.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (𝐴 ≠ 0ℎ ↔ 0 < (normℎ‘𝐴)))
Theorem	norm0 31147	The norm of a zero vector. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
⊢ (normℎ‘0ℎ) = 0
Theorem	norm-i 31148	Theorem 3.3(i) of [Beran] p. 97. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) = 0 ↔ 𝐴 = 0ℎ))
Theorem	normne0 31149	A norm is nonzero iff its argument is a nonzero vector. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0ℎ))
Theorem	normcli 31150	Real closure of the norm of a vector. (Contributed by NM, 30-Sep-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    ⇒   ⊢ (normℎ‘𝐴) ∈ ℝ
Theorem	normsqi 31151	The square of a norm. (Contributed by NM, 21-Aug-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    ⇒   ⊢ ((normℎ‘𝐴)↑2) = (𝐴 ·ih 𝐴)
Theorem	norm-i-i 31152	Theorem 3.3(i) of [Beran] p. 97. (Contributed by NM, 5-Sep-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    ⇒   ⊢ ((normℎ‘𝐴) = 0 ↔ 𝐴 = 0ℎ)
Theorem	normsq 31153	The square of a norm. (Contributed by NM, 12-May-2005.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴)↑2) = (𝐴 ·ih 𝐴))
Theorem	normsub0i 31154	Two vectors are equal iff the norm of their difference is zero. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    ⇒   ⊢ ((normℎ‘(𝐴 −ℎ 𝐵)) = 0 ↔ 𝐴 = 𝐵)
Theorem	normsub0 31155	Two vectors are equal iff the norm of their difference is zero. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((normℎ‘(𝐴 −ℎ 𝐵)) = 0 ↔ 𝐴 = 𝐵))
Theorem	norm-ii-i 31156	Triangle inequality for norms. Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 11-Aug-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    ⇒   ⊢ (normℎ‘(𝐴 +ℎ 𝐵)) ≤ ((normℎ‘𝐴) + (normℎ‘𝐵))
Theorem	norm-ii 31157	Triangle inequality for norms. Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (normℎ‘(𝐴 +ℎ 𝐵)) ≤ ((normℎ‘𝐴) + (normℎ‘𝐵)))
Theorem	norm-iii-i 31158	Theorem 3.3(iii) of [Beran] p. 97. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℋ    ⇒   ⊢ (normℎ‘(𝐴 ·ℎ 𝐵)) = ((abs‘𝐴) · (normℎ‘𝐵))
Theorem	norm-iii 31159	Theorem 3.3(iii) of [Beran] p. 97. (Contributed by NM, 25-Oct-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (normℎ‘(𝐴 ·ℎ 𝐵)) = ((abs‘𝐴) · (normℎ‘𝐵)))
Theorem	normsubi 31160	Negative doesn't change the norm of a Hilbert space vector. (Contributed by NM, 11-Aug-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    ⇒   ⊢ (normℎ‘(𝐴 −ℎ 𝐵)) = (normℎ‘(𝐵 −ℎ 𝐴))
Theorem	normpythi 31161	Analogy to Pythagorean theorem for orthogonal vectors. Remark 3.4(C) of [Beran] p. 98. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    ⇒   ⊢ ((𝐴 ·ih 𝐵) = 0 → ((normℎ‘(𝐴 +ℎ 𝐵))↑2) = (((normℎ‘𝐴)↑2) + ((normℎ‘𝐵)↑2)))
Theorem	normsub 31162	Swapping order of subtraction doesn't change the norm of a vector. (Contributed by NM, 14-Aug-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (normℎ‘(𝐴 −ℎ 𝐵)) = (normℎ‘(𝐵 −ℎ 𝐴)))
Theorem	normneg 31163	The norm of a vector equals the norm of its negative. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℋ → (normℎ‘(-1 ·ℎ 𝐴)) = (normℎ‘𝐴))
Theorem	normpyth 31164	Analogy to Pythagorean theorem for orthogonal vectors. Remark 3.4(C) of [Beran] p. 98. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ih 𝐵) = 0 → ((normℎ‘(𝐴 +ℎ 𝐵))↑2) = (((normℎ‘𝐴)↑2) + ((normℎ‘𝐵)↑2))))
Theorem	normpyc 31165	Corollary to Pythagorean theorem for orthogonal vectors. Remark 3.4(C) of [Beran] p. 98. (Contributed by NM, 26-Oct-1999.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ih 𝐵) = 0 → (normℎ‘𝐴) ≤ (normℎ‘(𝐴 +ℎ 𝐵))))
Theorem	norm3difi 31166	Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    &   ⊢ 𝐶 ∈ ℋ    ⇒   ⊢ (normℎ‘(𝐴 −ℎ 𝐵)) ≤ ((normℎ‘(𝐴 −ℎ 𝐶)) + (normℎ‘(𝐶 −ℎ 𝐵)))
Theorem	norm3adifii 31167	Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 30-Sep-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    &   ⊢ 𝐶 ∈ ℋ    ⇒   ⊢ (abs‘((normℎ‘(𝐴 −ℎ 𝐶)) − (normℎ‘(𝐵 −ℎ 𝐶)))) ≤ (normℎ‘(𝐴 −ℎ 𝐵))
Theorem	norm3lem 31168	Lemma involving norm of differences in Hilbert space. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    &   ⊢ 𝐶 ∈ ℋ    &   ⊢ 𝐷 ∈ ℝ    ⇒   ⊢ (((normℎ‘(𝐴 −ℎ 𝐶)) < (𝐷 / 2) ∧ (normℎ‘(𝐶 −ℎ 𝐵)) < (𝐷 / 2)) → (normℎ‘(𝐴 −ℎ 𝐵)) < 𝐷)
Theorem	norm3dif 31169	Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 20-Apr-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (normℎ‘(𝐴 −ℎ 𝐵)) ≤ ((normℎ‘(𝐴 −ℎ 𝐶)) + (normℎ‘(𝐶 −ℎ 𝐵))))
Theorem	norm3dif2 31170	Norm of differences around common element. (Contributed by NM, 18-Apr-2007.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (normℎ‘(𝐴 −ℎ 𝐵)) ≤ ((normℎ‘(𝐶 −ℎ 𝐴)) + (normℎ‘(𝐶 −ℎ 𝐵))))
Theorem	norm3lemt 31171	Lemma involving norm of differences in Hilbert space. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℝ)) → (((normℎ‘(𝐴 −ℎ 𝐶)) < (𝐷 / 2) ∧ (normℎ‘(𝐶 −ℎ 𝐵)) < (𝐷 / 2)) → (normℎ‘(𝐴 −ℎ 𝐵)) < 𝐷))
Theorem	norm3adifi 31172	Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 3-Oct-1999.) (New usage is discouraged.)
⊢ 𝐶 ∈ ℋ    ⇒   ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (abs‘((normℎ‘(𝐴 −ℎ 𝐶)) − (normℎ‘(𝐵 −ℎ 𝐶)))) ≤ (normℎ‘(𝐴 −ℎ 𝐵)))
Theorem	normpari 31173	Parallelogram law for norms. Remark 3.4(B) of [Beran] p. 98. (Contributed by NM, 21-Aug-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    ⇒   ⊢ (((normℎ‘(𝐴 −ℎ 𝐵))↑2) + ((normℎ‘(𝐴 +ℎ 𝐵))↑2)) = ((2 · ((normℎ‘𝐴)↑2)) + (2 · ((normℎ‘𝐵)↑2)))
Theorem	normpar 31174	Parallelogram law for norms. Remark 3.4(B) of [Beran] p. 98. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (((normℎ‘(𝐴 −ℎ 𝐵))↑2) + ((normℎ‘(𝐴 +ℎ 𝐵))↑2)) = ((2 · ((normℎ‘𝐴)↑2)) + (2 · ((normℎ‘𝐵)↑2))))
Theorem	normpar2i 31175	Corollary of parallelogram law for norms. Part of Lemma 3.6 of [Beran] p. 100. (Contributed by NM, 5-Oct-1999.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    &   ⊢ 𝐶 ∈ ℋ    ⇒   ⊢ ((normℎ‘(𝐴 −ℎ 𝐵))↑2) = (((2 · ((normℎ‘(𝐴 −ℎ 𝐶))↑2)) + (2 · ((normℎ‘(𝐵 −ℎ 𝐶))↑2))) − ((normℎ‘((𝐴 +ℎ 𝐵) −ℎ (2 ·ℎ 𝐶)))↑2))
Theorem	polid2i 31176	Generalized polarization identity. Generalization of Exercise 4(a) of [ReedSimon] p. 63. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    &   ⊢ 𝐶 ∈ ℋ    &   ⊢ 𝐷 ∈ ℋ    ⇒   ⊢ (𝐴 ·ih 𝐵) = (((((𝐴 +ℎ 𝐶) ·ih (𝐷 +ℎ 𝐵)) − ((𝐴 −ℎ 𝐶) ·ih (𝐷 −ℎ 𝐵))) + (i · (((𝐴 +ℎ (i ·ℎ 𝐶)) ·ih (𝐷 +ℎ (i ·ℎ 𝐵))) − ((𝐴 −ℎ (i ·ℎ 𝐶)) ·ih (𝐷 −ℎ (i ·ℎ 𝐵)))))) / 4)
Theorem	polidi 31177	Polarization identity. Recovers inner product from norm. Exercise 4(a) of [ReedSimon] p. 63. The outermost operation is + instead of - due to our mathematicians' (rather than physicists') version of Axiom ax-his3 31103. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    ⇒   ⊢ (𝐴 ·ih 𝐵) = (((((normℎ‘(𝐴 +ℎ 𝐵))↑2) − ((normℎ‘(𝐴 −ℎ 𝐵))↑2)) + (i · (((normℎ‘(𝐴 +ℎ (i ·ℎ 𝐵)))↑2) − ((normℎ‘(𝐴 −ℎ (i ·ℎ 𝐵)))↑2)))) / 4)
Theorem	polid 31178	Polarization identity. Recovers inner product from norm. Exercise 4(a) of [ReedSimon] p. 63. The outermost operation is + instead of - due to our mathematicians' (rather than physicists') version of Axiom ax-his3 31103. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) = (((((normℎ‘(𝐴 +ℎ 𝐵))↑2) − ((normℎ‘(𝐴 −ℎ 𝐵))↑2)) + (i · (((normℎ‘(𝐴 +ℎ (i ·ℎ 𝐵)))↑2) − ((normℎ‘(𝐴 −ℎ (i ·ℎ 𝐵)))↑2)))) / 4))
20.2.3  Relate Hilbert space to normed complex vector spaces
Theorem	hilablo 31179	Hilbert space vector addition is an Abelian group operation. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
⊢ +ℎ ∈ AbelOp
Theorem	hilid 31180	The group identity element of Hilbert space vector addition is the zero vector. (Contributed by NM, 16-Apr-2007.) (New usage is discouraged.)
⊢ (GId‘ +ℎ ) = 0ℎ
Theorem	hilvc 31181	Hilbert space is a complex vector space. Vector addition is +ℎ, and scalar product is ·ℎ. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
⊢ ⟨ +ℎ , ·ℎ ⟩ ∈ CVecOLD
Theorem	hilnormi 31182	Hilbert space norm in terms of vector space norm. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
⊢ ℋ = (BaseSet‘𝑈)    &   ⊢ ·ih = (·𝑖OLD‘𝑈)    &   ⊢ 𝑈 ∈ NrmCVec    ⇒   ⊢ normℎ = (normCV‘𝑈)
Theorem	hilhhi 31183	Deduce the structure of Hilbert space from its components. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
⊢ ℋ = (BaseSet‘𝑈)    &   ⊢ +ℎ = ( +𝑣 ‘𝑈)    &   ⊢ ·ℎ = ( ·𝑠OLD ‘𝑈)    &   ⊢ ·ih = (·𝑖OLD‘𝑈)    &   ⊢ 𝑈 ∈ NrmCVec    ⇒   ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩
Theorem	hhnv 31184	Hilbert space is a normed complex vector space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    ⇒   ⊢ 𝑈 ∈ NrmCVec
Theorem	hhva 31185	The group (addition) operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    ⇒   ⊢ +ℎ = ( +𝑣 ‘𝑈)
Theorem	hhba 31186	The base set of Hilbert space. This theorem provides an independent proof of df-hba 30988 (see comments in that definition). (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    ⇒   ⊢ ℋ = (BaseSet‘𝑈)
Theorem	hh0v 31187	The zero vector of Hilbert space. (Contributed by NM, 17-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    ⇒   ⊢ 0ℎ = (0vec‘𝑈)
Theorem	hhsm 31188	The scalar product operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    ⇒   ⊢ ·ℎ = ( ·𝑠OLD ‘𝑈)
Theorem	hhvs 31189	The vector subtraction operation of Hilbert space. (Contributed by NM, 13-Dec-2007.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    ⇒   ⊢ −ℎ = ( −𝑣 ‘𝑈)
Theorem	hhnm 31190	The norm function of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    ⇒   ⊢ normℎ = (normCV‘𝑈)
Theorem	hhims 31191	The induced metric of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝐷 = (normℎ ∘ −ℎ )    ⇒   ⊢ 𝐷 = (IndMet‘𝑈)
Theorem	hhims2 31192	Hilbert space distance metric. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝐷 = (IndMet‘𝑈)    ⇒   ⊢ 𝐷 = (normℎ ∘ −ℎ )
Theorem	hhmet 31193	The induced metric of Hilbert space. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝐷 = (IndMet‘𝑈)    ⇒   ⊢ 𝐷 ∈ (Met‘ ℋ)
Theorem	hhxmet 31194	The induced metric of Hilbert space. (Contributed by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝐷 = (IndMet‘𝑈)    ⇒   ⊢ 𝐷 ∈ (∞Met‘ ℋ)
Theorem	hhmetdval 31195	Value of the distance function of the metric space of Hilbert space. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    &   ⊢ 𝐷 = (IndMet‘𝑈)    ⇒   ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴𝐷𝐵) = (normℎ‘(𝐴 −ℎ 𝐵)))
Theorem	hhip 31196	The inner product operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    ⇒   ⊢ ·ih = (·𝑖OLD‘𝑈)
Theorem	hhph 31197	The Hilbert space of the Hilbert Space Explorer is an inner product space. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)
⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩    ⇒   ⊢ 𝑈 ∈ CPreHilOLD
20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality
Theorem	bcsiALT 31198	Bunjakovaskij-Cauchy-Schwarz inequality. Remark 3.4 of [Beran] p. 98. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    ⇒   ⊢ (abs‘(𝐴 ·ih 𝐵)) ≤ ((normℎ‘𝐴) · (normℎ‘𝐵))
Theorem	bcsiHIL 31199	Bunjakovaskij-Cauchy-Schwarz inequality. Remark 3.4 of [Beran] p. 98. (Proved from ZFC version.) (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)
⊢ 𝐴 ∈ ℋ    &   ⊢ 𝐵 ∈ ℋ    ⇒   ⊢ (abs‘(𝐴 ·ih 𝐵)) ≤ ((normℎ‘𝐴) · (normℎ‘𝐵))
Theorem	bcs 31200	Bunjakovaskij-Cauchy-Schwarz inequality. Remark 3.4 of [Beran] p. 98. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (abs‘(𝐴 ·ih 𝐵)) ≤ ((normℎ‘𝐴) · (normℎ‘𝐵)))