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Theorem List for Metamath Proof Explorer - 27901-28000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem5oalem5 27901 Lemma for orthoarguesian law 5OA. (Contributed by NM, 2-May-2000.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶S    &   𝐷S    &   𝐹S    &   𝐺S    &   𝑅S    &   𝑆S       (((((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐶𝑤𝐷)) ∧ ((𝑓𝐹𝑔𝐺) ∧ (𝑣𝑅𝑢𝑆))) ∧ (((𝑥 + 𝑦) = (𝑣 + 𝑢) ∧ (𝑧 + 𝑤) = (𝑣 + 𝑢)) ∧ (𝑓 + 𝑔) = (𝑣 + 𝑢))) → (𝑥 𝑧) ∈ ((((𝐴 + 𝐶) ∩ (𝐵 + 𝐷)) ∩ (((𝐴 + 𝑅) ∩ (𝐵 + 𝑆)) + ((𝐶 + 𝑅) ∩ (𝐷 + 𝑆)))) ∩ ((((𝐴 + 𝐹) ∩ (𝐵 + 𝐺)) ∩ (((𝐴 + 𝑅) ∩ (𝐵 + 𝑆)) + ((𝐹 + 𝑅) ∩ (𝐺 + 𝑆)))) + (((𝐶 + 𝐹) ∩ (𝐷 + 𝐺)) ∩ (((𝐶 + 𝑅) ∩ (𝐷 + 𝑆)) + ((𝐹 + 𝑅) ∩ (𝐺 + 𝑆)))))))

Theorem5oalem6 27902 Lemma for orthoarguesian law 5OA. (Contributed by NM, 4-May-2000.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶S    &   𝐷S    &   𝐹S    &   𝐺S    &   𝑅S    &   𝑆S       (((((𝑥𝐴𝑦𝐵) ∧ = (𝑥 + 𝑦)) ∧ ((𝑧𝐶𝑤𝐷) ∧ = (𝑧 + 𝑤))) ∧ (((𝑓𝐹𝑔𝐺) ∧ = (𝑓 + 𝑔)) ∧ ((𝑣𝑅𝑢𝑆) ∧ = (𝑣 + 𝑢)))) → ∈ (𝐵 + (𝐴 ∩ (𝐶 + ((((𝐴 + 𝐶) ∩ (𝐵 + 𝐷)) ∩ (((𝐴 + 𝑅) ∩ (𝐵 + 𝑆)) + ((𝐶 + 𝑅) ∩ (𝐷 + 𝑆)))) ∩ ((((𝐴 + 𝐹) ∩ (𝐵 + 𝐺)) ∩ (((𝐴 + 𝑅) ∩ (𝐵 + 𝑆)) + ((𝐹 + 𝑅) ∩ (𝐺 + 𝑆)))) + (((𝐶 + 𝐹) ∩ (𝐷 + 𝐺)) ∩ (((𝐶 + 𝑅) ∩ (𝐷 + 𝑆)) + ((𝐹 + 𝑅) ∩ (𝐺 + 𝑆))))))))))

Theorem5oalem7 27903 Lemma for orthoarguesian law 5OA. (Contributed by NM, 4-May-2000.) (New usage is discouraged.)
𝐴S    &   𝐵S    &   𝐶S    &   𝐷S    &   𝐹S    &   𝐺S    &   𝑅S    &   𝑆S       (((𝐴 + 𝐵) ∩ (𝐶 + 𝐷)) ∩ ((𝐹 + 𝐺) ∩ (𝑅 + 𝑆))) ⊆ (𝐵 + (𝐴 ∩ (𝐶 + ((((𝐴 + 𝐶) ∩ (𝐵 + 𝐷)) ∩ (((𝐴 + 𝑅) ∩ (𝐵 + 𝑆)) + ((𝐶 + 𝑅) ∩ (𝐷 + 𝑆)))) ∩ ((((𝐴 + 𝐹) ∩ (𝐵 + 𝐺)) ∩ (((𝐴 + 𝑅) ∩ (𝐵 + 𝑆)) + ((𝐹 + 𝑅) ∩ (𝐺 + 𝑆)))) + (((𝐶 + 𝐹) ∩ (𝐷 + 𝐺)) ∩ (((𝐶 + 𝑅) ∩ (𝐷 + 𝑆)) + ((𝐹 + 𝑅) ∩ (𝐺 + 𝑆)))))))))

Theorem5oai 27904 Orthoarguesian law 5OA. This 8-variable inference is called 5OA because it can be converted to a 5-variable equation (see Quantum Logic Explorer). (Contributed by NM, 5-May-2000.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C    &   𝐹C    &   𝐺C    &   𝑅C    &   𝑆C    &   𝐴 ⊆ (⊥‘𝐵)    &   𝐶 ⊆ (⊥‘𝐷)    &   𝐹 ⊆ (⊥‘𝐺)    &   𝑅 ⊆ (⊥‘𝑆)       (((𝐴 𝐵) ∩ (𝐶 𝐷)) ∩ ((𝐹 𝐺) ∩ (𝑅 𝑆))) ⊆ (𝐵 (𝐴 ∩ (𝐶 ((((𝐴 𝐶) ∩ (𝐵 𝐷)) ∩ (((𝐴 𝑅) ∩ (𝐵 𝑆)) ∨ ((𝐶 𝑅) ∩ (𝐷 𝑆)))) ∩ ((((𝐴 𝐹) ∩ (𝐵 𝐺)) ∩ (((𝐴 𝑅) ∩ (𝐵 𝑆)) ∨ ((𝐹 𝑅) ∩ (𝐺 𝑆)))) ∨ (((𝐶 𝐹) ∩ (𝐷 𝐺)) ∩ (((𝐶 𝑅) ∩ (𝐷 𝑆)) ∨ ((𝐹 𝑅) ∩ (𝐺 𝑆)))))))))

Theorem3oalem1 27905* Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐵C    &   𝐶C    &   𝑅C    &   𝑆C       ((((𝑥𝐵𝑦𝑅) ∧ 𝑣 = (𝑥 + 𝑦)) ∧ ((𝑧𝐶𝑤𝑆) ∧ 𝑣 = (𝑧 + 𝑤))) → (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑣 ∈ ℋ) ∧ (𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ)))

Theorem3oalem2 27906* Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐵C    &   𝐶C    &   𝑅C    &   𝑆C       ((((𝑥𝐵𝑦𝑅) ∧ 𝑣 = (𝑥 + 𝑦)) ∧ ((𝑧𝐶𝑤𝑆) ∧ 𝑣 = (𝑧 + 𝑤))) → 𝑣 ∈ (𝐵 + (𝑅 ∩ (𝑆 + ((𝐵 + 𝐶) ∩ (𝑅 + 𝑆))))))

Theorem3oalem3 27907 Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐵C    &   𝐶C    &   𝑅C    &   𝑆C       ((𝐵 + 𝑅) ∩ (𝐶 + 𝑆)) ⊆ (𝐵 + (𝑅 ∩ (𝑆 + ((𝐵 + 𝐶) ∩ (𝑅 + 𝑆)))))

Theorem3oalem4 27908 Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝑅 = ((⊥‘𝐵) ∩ (𝐵 𝐴))       𝑅 ⊆ (⊥‘𝐵)

Theorem3oalem5 27909 Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝑅 = ((⊥‘𝐵) ∩ (𝐵 𝐴))    &   𝑆 = ((⊥‘𝐶) ∩ (𝐶 𝐴))       ((𝐵 + 𝑅) ∩ (𝐶 + 𝑆)) = ((𝐵 𝑅) ∩ (𝐶 𝑆))

Theorem3oalem6 27910 Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝑅 = ((⊥‘𝐵) ∩ (𝐵 𝐴))    &   𝑆 = ((⊥‘𝐶) ∩ (𝐶 𝐴))       (𝐵 + (𝑅 ∩ (𝑆 + ((𝐵 + 𝐶) ∩ (𝑅 + 𝑆))))) ⊆ (𝐵 (𝑅 ∩ (𝑆 ((𝐵 𝐶) ∩ (𝑅 𝑆)))))

Theorem3oai 27911 3OA (weak) orthoarguesian law. Equation IV of [GodowskiGreechie] p. 249. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝑅 = ((⊥‘𝐵) ∩ (𝐵 𝐴))    &   𝑆 = ((⊥‘𝐶) ∩ (𝐶 𝐴))       ((𝐵 𝑅) ∩ (𝐶 𝑆)) ⊆ (𝐵 (𝑅 ∩ (𝑆 ((𝐵 𝐶) ∩ (𝑅 𝑆)))))

20.5.10  Projectors (cont.)

Theorempjorthi 27912 Projection components on orthocomplemented subspaces are orthogonal. (Contributed by NM, 26-Oct-1999.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (𝐻C → (((proj𝐻)‘𝐴) ·ih ((proj‘(⊥‘𝐻))‘𝐵)) = 0)

Theorempjch1 27913 Property of identity projection. Remark in [Beran] p. 111. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)
(𝐴 ∈ ℋ → ((proj‘ ℋ)‘𝐴) = 𝐴)

Theorempjo 27914 The orthogonal projection. Lemma 4.4(i) of [Beran] p. 111. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ((proj‘(⊥‘𝐻))‘𝐴) = (((proj‘ ℋ)‘𝐴) − ((proj𝐻)‘𝐴)))

Theorempjcompi 27915 Component of a projection. (Contributed by NM, 31-Oct-1999.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
𝐻C       ((𝐴𝐻𝐵 ∈ (⊥‘𝐻)) → ((proj𝐻)‘(𝐴 + 𝐵)) = 𝐴)

Theorempjidmi 27916 A projection is idempotent. Property (ii) of [Beran] p. 109. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       ((proj𝐻)‘((proj𝐻)‘𝐴)) = ((proj𝐻)‘𝐴)

Theorempjadjii 27917 A projection is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       (((proj𝐻)‘𝐴) ·ih 𝐵) = (𝐴 ·ih ((proj𝐻)‘𝐵))

Theorempjaddii 27918 Projection of vector sum is sum of projections. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       ((proj𝐻)‘(𝐴 + 𝐵)) = (((proj𝐻)‘𝐴) + ((proj𝐻)‘𝐵))

Theorempjinormii 27919 The inner product of a projection and its argument is the square of the norm of the projection. Remark in [Halmos] p. 44. (Contributed by NM, 13-Aug-2000.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       (((proj𝐻)‘𝐴) ·ih 𝐴) = ((norm‘((proj𝐻)‘𝐴))↑2)

Theorempjmulii 27920 Projection of (scalar) product is product of projection. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ    &   𝐶 ∈ ℂ       ((proj𝐻)‘(𝐶 · 𝐴)) = (𝐶 · ((proj𝐻)‘𝐴))

Theorempjsubii 27921 Projection of vector difference is difference of projections. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ    &   𝐵 ∈ ℋ       ((proj𝐻)‘(𝐴 𝐵)) = (((proj𝐻)‘𝐴) − ((proj𝐻)‘𝐵))

Theorempjsslem 27922 Lemma for subset relationships of projections. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ    &   𝐺C       (((proj‘(⊥‘𝐻))‘𝐴) − ((proj‘(⊥‘𝐺))‘𝐴)) = (((proj𝐺)‘𝐴) − ((proj𝐻)‘𝐴))

Theorempjss2i 27923 Subset relationship for projections. Theorem 4.5(i)->(ii) of [Beran] p. 112. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ    &   𝐺C       (𝐻𝐺 → ((proj𝐻)‘((proj𝐺)‘𝐴)) = ((proj𝐻)‘𝐴))

Theorempjssmii 27924 Projection meet property. Remark in [Kalmbach] p. 66. Also Theorem 4.5(i)->(iv) of [Beran] p. 112. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ    &   𝐺C       (𝐻𝐺 → (((proj𝐺)‘𝐴) − ((proj𝐻)‘𝐴)) = ((proj‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴))

Theorempjssge0ii 27925 Theorem 4.5(iv)->(v) of [Beran] p. 112. (Contributed by NM, 13-Aug-2000.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ    &   𝐺C       ((((proj𝐺)‘𝐴) − ((proj𝐻)‘𝐴)) = ((proj‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴) → 0 ≤ ((((proj𝐺)‘𝐴) − ((proj𝐻)‘𝐴)) ·ih 𝐴))

Theorempjdifnormii 27926 Theorem 4.5(v)<->(vi) of [Beran] p. 112. (Contributed by NM, 13-Aug-2000.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ    &   𝐺C       (0 ≤ ((((proj𝐺)‘𝐴) − ((proj𝐻)‘𝐴)) ·ih 𝐴) ↔ (norm‘((proj𝐻)‘𝐴)) ≤ (norm‘((proj𝐺)‘𝐴)))

Theorempjcji 27927 The projection on a subspace join is the sum of the projections. (Contributed by NM, 1-Nov-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ    &   𝐺C       (𝐻 ⊆ (⊥‘𝐺) → ((proj‘(𝐻 𝐺))‘𝐴) = (((proj𝐻)‘𝐴) + ((proj𝐺)‘𝐴)))

Theorempjadji 27928 A projection is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 6-Oct-2000.) (New usage is discouraged.)
𝐻C       ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (((proj𝐻)‘𝐴) ·ih 𝐵) = (𝐴 ·ih ((proj𝐻)‘𝐵)))

Theorempjaddi 27929 Projection of vector sum is sum of projections. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
𝐻C       ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((proj𝐻)‘(𝐴 + 𝐵)) = (((proj𝐻)‘𝐴) + ((proj𝐻)‘𝐵)))

Theorempjinormi 27930 The inner product of a projection and its argument is the square of the norm of the projection. Remark in [Halmos] p. 44. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
𝐻C       (𝐴 ∈ ℋ → (((proj𝐻)‘𝐴) ·ih 𝐴) = ((norm‘((proj𝐻)‘𝐴))↑2))

Theorempjsubi 27931 Projection of vector difference is difference of projections. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
𝐻C       ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((proj𝐻)‘(𝐴 𝐵)) = (((proj𝐻)‘𝐴) − ((proj𝐻)‘𝐵)))

Theorempjmuli 27932 Projection of scalar product is scalar product of projection. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
𝐻C       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → ((proj𝐻)‘(𝐴 · 𝐵)) = (𝐴 · ((proj𝐻)‘𝐵)))

Theorempjige0i 27933 The inner product of a projection and its argument is nonnegative. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
𝐻C       (𝐴 ∈ ℋ → 0 ≤ (((proj𝐻)‘𝐴) ·ih 𝐴))

Theorempjige0 27934 The inner product of a projection and its argument is nonnegative. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → 0 ≤ (((proj𝐻)‘𝐴) ·ih 𝐴))

Theorempjcjt2 27935 The projection on a subspace join is the sum of the projections. (Contributed by NM, 1-Nov-1999.) (New usage is discouraged.)
((𝐻C𝐺C𝐴 ∈ ℋ) → (𝐻 ⊆ (⊥‘𝐺) → ((proj‘(𝐻 𝐺))‘𝐴) = (((proj𝐻)‘𝐴) + ((proj𝐺)‘𝐴))))

Theorempj0i 27936 The projection of the zero vector. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐻C       ((proj𝐻)‘0) = 0

Theorempjch 27937 Projection of a vector in the projection subspace. Lemma 4.4(ii) of [Beran] p. 111. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → (𝐴𝐻 ↔ ((proj𝐻)‘𝐴) = 𝐴))

Theorempjid 27938 The projection of a vector in the projection subspace is itself. (Contributed by NM, 9-Apr-2006.) (New usage is discouraged.)
((𝐻C𝐴𝐻) → ((proj𝐻)‘𝐴) = 𝐴)

Theorempjvec 27939* The set of vectors belonging to the subspace of a projection. Part of Theorem 26.2 of [Halmos] p. 44. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)
(𝐻C𝐻 = {𝑥 ∈ ℋ ∣ ((proj𝐻)‘𝑥) = 𝑥})

Theorempjocvec 27940* The set of vectors belonging to the orthocomplemented subspace of a projection. Second part of Theorem 27.3 of [Halmos] p. 45. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
(𝐻C → (⊥‘𝐻) = {𝑥 ∈ ℋ ∣ ((proj𝐻)‘𝑥) = 0})

Theorempjocini 27941 Membership of projection in orthocomplement of intersection. (Contributed by NM, 21-Apr-2001.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐴 ∈ (⊥‘(𝐺𝐻)) → ((proj𝐺)‘𝐴) ∈ (⊥‘(𝐺𝐻)))

Theorempjini 27942 Membership of projection in an intersection. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐴 ∈ (𝐺𝐻) → ((proj𝐺)‘𝐴) ∈ (𝐺𝐻))

Theorempjjsi 27943* A sufficient condition for subspace join to be equal to subspace sum. (Contributed by NM, 29-May-2004.) (New usage is discouraged.)
𝐺C    &   𝐻S       (∀𝑥 ∈ (𝐺 𝐻)((proj‘(⊥‘𝐺))‘𝑥) ∈ 𝐻 → (𝐺 𝐻) = (𝐺 + 𝐻))

Theorempjfni 27944 Functionality of a projection. (Contributed by NM, 30-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
𝐻C       (proj𝐻) Fn ℋ

Theorempjrni 27945 The range of a projection. Part of Theorem 26.2 of [Halmos] p. 44. (Contributed by NM, 30-Oct-1999.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
𝐻C       ran (proj𝐻) = 𝐻

Theorempjfoi 27946 A projection maps onto its subspace. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
𝐻C       (proj𝐻): ℋ–onto𝐻

Theorempjfi 27947 The mapping of a projection. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.)
𝐻C       (proj𝐻): ℋ⟶ ℋ

Theorempjvi 27948 The value of a projection in terms of components. (Contributed by NM, 28-Nov-2000.) (New usage is discouraged.)
𝐻C       ((𝐴𝐻𝐵 ∈ (⊥‘𝐻)) → ((proj𝐻)‘(𝐴 + 𝐵)) = 𝐴)

Theorempjhfo 27949 A projection maps onto its subspace. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
(𝐻C → (proj𝐻): ℋ–onto𝐻)

Theorempjrn 27950 The range of a projection. Part of Theorem 26.2 of [Halmos] p. 44. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
(𝐻C → ran (proj𝐻) = 𝐻)

Theorempjhf 27951 The mapping of a projection. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
(𝐻C → (proj𝐻): ℋ⟶ ℋ)

Theorempjfn 27952 Functionality of a projection. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
(𝐻C → (proj𝐻) Fn ℋ)

Theorempjsumi 27953 The projection on a subspace sum is the sum of the projections. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐴 ∈ ℋ → (𝐺 ⊆ (⊥‘𝐻) → ((proj‘(𝐺 + 𝐻))‘𝐴) = (((proj𝐺)‘𝐴) + ((proj𝐻)‘𝐴))))

Theorempj11i 27954 One-to-one correspondence of projection and subspace. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       ((proj𝐺) = (proj𝐻) ↔ 𝐺 = 𝐻)

Theorempjdsi 27955 Vector decomposition into sum of projections on orthogonal subspaces. (Contributed by NM, 21-Jun-2006.) (New usage is discouraged.)
𝐺C    &   𝐻C       ((𝐴 ∈ (𝐺 𝐻) ∧ 𝐺 ⊆ (⊥‘𝐻)) → 𝐴 = (((proj𝐺)‘𝐴) + ((proj𝐻)‘𝐴)))

Theorempjds3i 27956 Vector decomposition into sum of projections on orthogonal subspaces. (Contributed by NM, 22-Jun-2006.) (New usage is discouraged.)
𝐹C    &   𝐺C    &   𝐻C       (((𝐴 ∈ ((𝐹 𝐺) ∨ 𝐻) ∧ 𝐹 ⊆ (⊥‘𝐺)) ∧ (𝐹 ⊆ (⊥‘𝐻) ∧ 𝐺 ⊆ (⊥‘𝐻))) → 𝐴 = ((((proj𝐹)‘𝐴) + ((proj𝐺)‘𝐴)) + ((proj𝐻)‘𝐴)))

Theorempj11 27957 One-to-one correspondence of projection and subspace. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
((𝐺C𝐻C ) → ((proj𝐺) = (proj𝐻) ↔ 𝐺 = 𝐻))

Theorempjmfn 27958 Functionality of the projection function. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
proj Fn C

Theorempjmf1 27959 The projector function maps one-to-one into the set of Hilbert space operators. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
proj: C1-1→( ℋ ↑𝑚 ℋ)

Theorempjoi0 27960 The inner product of projections on orthogonal subspaces vanishes. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
(((𝐺C𝐻C𝐴 ∈ ℋ) ∧ 𝐺 ⊆ (⊥‘𝐻)) → (((proj𝐺)‘𝐴) ·ih ((proj𝐻)‘𝐴)) = 0)

Theorempjoi0i 27961 The inner product of projections on orthogonal subspaces vanishes. (Contributed by NM, 1-Nov-1999.) (New usage is discouraged.)
𝐺C    &   𝐻C    &   𝐴 ∈ ℋ       (𝐺 ⊆ (⊥‘𝐻) → (((proj𝐺)‘𝐴) ·ih ((proj𝐻)‘𝐴)) = 0)

Theorempjopythi 27962 Pythagorean theorem for projections on orthogonal subspaces. (Contributed by NM, 1-Nov-1999.) (New usage is discouraged.)
𝐺C    &   𝐻C    &   𝐴 ∈ ℋ       (𝐺 ⊆ (⊥‘𝐻) → ((norm‘(((proj𝐺)‘𝐴) + ((proj𝐻)‘𝐴)))↑2) = (((norm‘((proj𝐺)‘𝐴))↑2) + ((norm‘((proj𝐻)‘𝐴))↑2)))

Theorempjopyth 27963 Pythagorean theorem for projections on orthogonal subspaces. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
((𝐻C𝐺C𝐴 ∈ ℋ) → (𝐻 ⊆ (⊥‘𝐺) → ((norm‘(((proj𝐻)‘𝐴) + ((proj𝐺)‘𝐴)))↑2) = (((norm‘((proj𝐻)‘𝐴))↑2) + ((norm‘((proj𝐺)‘𝐴))↑2))))

Theorempjnormi 27964 The norm of the projection is less than or equal to the norm. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       (norm‘((proj𝐻)‘𝐴)) ≤ (norm𝐴)

Theorempjpythi 27965 Pythagorean theorem for projections. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       ((norm𝐴)↑2) = (((norm‘((proj𝐻)‘𝐴))↑2) + ((norm‘((proj‘(⊥‘𝐻))‘𝐴))↑2))

Theorempjneli 27966 If a vector does not belong to subspace, the norm of its projection is less than its norm. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)
𝐻C    &   𝐴 ∈ ℋ       𝐴𝐻 ↔ (norm‘((proj𝐻)‘𝐴)) < (norm𝐴))

Theorempjnorm 27967 The norm of the projection is less than or equal to the norm. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → (norm‘((proj𝐻)‘𝐴)) ≤ (norm𝐴))

Theorempjpyth 27968 Pythagorean theorem for projectors. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → ((norm𝐴)↑2) = (((norm‘((proj𝐻)‘𝐴))↑2) + ((norm‘((proj‘(⊥‘𝐻))‘𝐴))↑2)))

Theorempjnel 27969 If a vector does not belong to subspace, the norm of its projection is less than its norm. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → (¬ 𝐴𝐻 ↔ (norm‘((proj𝐻)‘𝐴)) < (norm𝐴)))

Theorempjnorm2 27970 A vector belongs to the subspace of a projection iff the norm of its projection equals its norm. This and pjch 27937 yield Theorem 26.3 of [Halmos] p. 44. (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.)
((𝐻C𝐴 ∈ ℋ) → (𝐴𝐻 ↔ (norm‘((proj𝐻)‘𝐴)) = (norm𝐴)))

20.5.11  Mayet's equation E_3

Theoremmayete3i 27971 Mayet's equation E3. Part of Theorem 4.1 of [Mayet3] p. 1223. (Contributed by NM, 22-Jun-2006.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C    &   𝐹C    &   𝐺C    &   𝐴 ⊆ (⊥‘𝐶)    &   𝐴 ⊆ (⊥‘𝐹)    &   𝐶 ⊆ (⊥‘𝐹)    &   𝐴 ⊆ (⊥‘𝐵)    &   𝐶 ⊆ (⊥‘𝐷)    &   𝐹 ⊆ (⊥‘𝐺)    &   𝑋 = ((𝐴 𝐶) ∨ 𝐹)    &   𝑌 = (((𝐴 𝐵) ∩ (𝐶 𝐷)) ∩ (𝐹 𝐺))    &   𝑍 = ((𝐵 𝐷) ∨ 𝐺)       (𝑋𝑌) ⊆ 𝑍

Theoremmayetes3i 27972 Mayet's equation E^*3, derived from E3. Solution, for n = 3, to open problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. (Contributed by NM, 10-May-2009.) (New usage is discouraged.)
𝐴C    &   𝐵C    &   𝐶C    &   𝐷C    &   𝐹C    &   𝐺C    &   𝑅C    &   𝐴 ⊆ (⊥‘𝐶)    &   𝐴 ⊆ (⊥‘𝐹)    &   𝐶 ⊆ (⊥‘𝐹)    &   𝐴 ⊆ (⊥‘𝐵)    &   𝐶 ⊆ (⊥‘𝐷)    &   𝐹 ⊆ (⊥‘𝐺)    &   𝑅 ⊆ (⊥‘𝑋)    &   𝑋 = ((𝐴 𝐶) ∨ 𝐹)    &   𝑌 = (((𝐴 𝐵) ∩ (𝐶 𝐷)) ∩ (𝐹 𝐺))    &   𝑍 = ((𝐵 𝐷) ∨ 𝐺)       ((𝑋 𝑅) ∩ 𝑌) ⊆ (𝑍 𝑅)

20.6  Operators on Hilbert spaces

20.6.1  Operator sum, difference, and scalar multiplication

Note on operators. Unlike some authors, we use the term "operator" to mean any function from to . This is the definition of operator in [Hughes] p. 14, the definition of operator in [AkhiezerGlazman] p. 30, and the definition of operator in [Goldberg] p. 10. For Reed and Simon, an operator is linear (definition of operator in [ReedSimon] p. 2). For Halmos, an operator is bounded and linear (definition of operator in [Halmos] p. 35). For Kalmbach and Beran, an operator is continuous and linear (definition of operator in [Kalmbach] p. 353; definition of operator in [Beran] p. 99). Note that "bounded and linear" and "continuous and linear" are equivalent by lncnbd 28281.

Definitiondf-hosum 27973* Define the sum of two Hilbert space operators. Definition of [Beran] p. 111. (Contributed by NM, 9-Nov-2000.) (New usage is discouraged.)
+op = (𝑓 ∈ ( ℋ ↑𝑚 ℋ), 𝑔 ∈ ( ℋ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓𝑥) + (𝑔𝑥))))

Definitiondf-homul 27974* Define the scalar product with a Hilbert space operator. Definition of [Beran] p. 111. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
·op = (𝑓 ∈ ℂ, 𝑔 ∈ ( ℋ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔𝑥))))

Definitiondf-hodif 27975* Define the difference of two Hilbert space operators. Definition of [Beran] p. 111. (Contributed by NM, 9-Nov-2000.) (New usage is discouraged.)
op = (𝑓 ∈ ( ℋ ↑𝑚 ℋ), 𝑔 ∈ ( ℋ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓𝑥) − (𝑔𝑥))))

Definitiondf-hfsum 27976* Define the sum of two Hilbert space functionals. Definition of [Beran] p. 111. Note that unlike some authors, we define a functional as any function from to , not just linear (or bounded linear) ones. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
+fn = (𝑓 ∈ (ℂ ↑𝑚 ℋ), 𝑔 ∈ (ℂ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓𝑥) + (𝑔𝑥))))

Definitiondf-hfmul 27977* Define the scalar product with a Hilbert space functional. Definition of [Beran] p. 111. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
·fn = (𝑓 ∈ ℂ, 𝑔 ∈ (ℂ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔𝑥))))

Theoremhosmval 27978* Value of the sum of two Hilbert space operators. (Contributed by NM, 9-Nov-2000.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 +op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑇𝑥))))

Theoremhommval 27979* Value of the scalar product with a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))))

Theoremhodmval 27980* Value of the difference of two Hilbert space operators. (Contributed by NM, 9-Nov-2000.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) − (𝑇𝑥))))

Theoremhfsmval 27981* Value of the sum of two Hilbert space functionals. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
((𝑆: ℋ⟶ℂ ∧ 𝑇: ℋ⟶ℂ) → (𝑆 +fn 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑇𝑥))))

Theoremhfmmval 27982* Value of the scalar product with a Hilbert space functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ℂ) → (𝐴 ·fn 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇𝑥))))

Theoremhosval 27983 Value of the sum of two Hilbert space operators. (Contributed by NM, 10-Nov-2000.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝐴) = ((𝑆𝐴) + (𝑇𝐴)))

Theoremhomval 27984 Value of the scalar product with a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝐵) = (𝐴 · (𝑇𝐵)))

Theoremhodval 27985 Value of the difference of two Hilbert space operators. (Contributed by NM, 10-Nov-2000.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) → ((𝑆op 𝑇)‘𝐴) = ((𝑆𝐴) − (𝑇𝐴)))

Theoremhfsval 27986 Value of the sum of two Hilbert space functionals. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
((𝑆: ℋ⟶ℂ ∧ 𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → ((𝑆 +fn 𝑇)‘𝐴) = ((𝑆𝐴) + (𝑇𝐴)))

Theoremhfmval 27987 Value of the scalar product with a Hilbert space functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ℂ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·fn 𝑇)‘𝐵) = (𝐴 · (𝑇𝐵)))

Theoremhoscl 27988 Closure of the sum of two Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
(((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝐴 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝐴) ∈ ℋ)

Theoremhomcl 27989 Closure of the scalar product of a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝐵) ∈ ℋ)

Theoremhodcl 27990 Closure of the difference of two Hilbert space operators. (Contributed by NM, 15-Nov-2002.) (New usage is discouraged.)
(((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝐴 ∈ ℋ) → ((𝑆op 𝑇)‘𝐴) ∈ ℋ)

20.6.2  Zero and identity operators

Definitiondf-h0op 27991 Define the Hilbert space zero operator. See df0op2 27995 for alternate definition. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
0hop = (proj‘0)

Definitiondf-iop 27992 Define the Hilbert space identity operator. See dfiop2 27996 for alternate definition. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)
Iop = (proj‘ ℋ)

Theoremho0val 27993 Value of the zero Hilbert space operator (null projector). Remark in [Beran] p. 111. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
(𝐴 ∈ ℋ → ( 0hop𝐴) = 0)

Theoremho0f 27994 Functionality of the zero Hilbert space operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
0hop : ℋ⟶ ℋ

Theoremdf0op2 27995 Alternate definition of Hilbert space zero operator. (Contributed by NM, 7-Aug-2006.) (New usage is discouraged.)
0hop = ( ℋ × 0)

Theoremdfiop2 27996 Alternate definition of Hilbert space identity operator. (Contributed by NM, 7-Aug-2006.) (New usage is discouraged.)
Iop = ( I ↾ ℋ)

Theoremhoif 27997 Functionality of the Hilbert space identity operator. (Contributed by NM, 8-Aug-2006.) (New usage is discouraged.)
Iop : ℋ–1-1-onto→ ℋ

Theoremhoival 27998 The value of the Hilbert space identity operator. (Contributed by NM, 8-Aug-2006.) (New usage is discouraged.)
(𝐴 ∈ ℋ → ( Iop𝐴) = 𝐴)

Theoremhoico1 27999 Composition with the Hilbert space identity operator. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (𝑇 ∘ Iop ) = 𝑇)

Theoremhoico2 28000 Composition with the Hilbert space identity operator. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → ( Iop𝑇) = 𝑇)

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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 42301-42360
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