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Type | Label | Description |
---|---|---|
Statement | ||
Syntax | chg 38901 | Extend class notation with g-map. |
class HGMap | ||
Definition | df-hgmap 38902* | Define map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. (Contributed by NM, 25-Mar-2015.) |
⊢ HGMap = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑎 ∣ [((DVecH‘𝑘)‘𝑤) / 𝑢][(Base‘(Scalar‘𝑢)) / 𝑏][((HDMap‘𝑘)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠 ‘𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝑘)‘𝑤))(𝑚‘𝑣))))})) | ||
Theorem | hgmapffval 38903* | Map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. (Contributed by NM, 25-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) ⇒ ⊢ (𝐾 ∈ 𝑋 → (HGMap‘𝐾) = (𝑤 ∈ 𝐻 ↦ {𝑎 ∣ [((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘(Scalar‘𝑢)) / 𝑏][((HDMap‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠 ‘𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑤))(𝑚‘𝑣))))})) | ||
Theorem | hgmapfval 38904* | Map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. (Contributed by NM, 25-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ ∙ = ( ·𝑠 ‘𝐶) & ⊢ 𝑀 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝐼 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))) | ||
Theorem | hgmapval 38905* | Value of map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. Function sigma of scalar f in part 14 of [Baer] p. 50 line 4. TODO: variable names are inherited from older version. Maybe make more consistent with hdmap14lem15 38900. (Contributed by NM, 25-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ ∙ = ( ·𝑠 ‘𝐶) & ⊢ 𝑀 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝐼 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐼‘𝑋) = (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))) | ||
Theorem | hgmapfnN 38906 | Functionality of scalar sigma map. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → 𝐺 Fn 𝐵) | ||
Theorem | hgmapcl 38907 | Closure of scalar sigma map i.e. the map from the vector space scalar base to the dual space scalar base. (Contributed by NM, 6-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐺‘𝐹) ∈ 𝐵) | ||
Theorem | hgmapdcl 38908 | Closure of the vector space to dual space scalar map, in the scalar sigma map. (Contributed by NM, 6-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝑄 = (Scalar‘𝐶) & ⊢ 𝐴 = (Base‘𝑄) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐺‘𝐹) ∈ 𝐴) | ||
Theorem | hgmapvs 38909 | Part 15 of [Baer] p. 50 line 6. Also line 15 in [Holland95] p. 14. (Contributed by NM, 6-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ ∙ = ( ·𝑠 ‘𝐶) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) = ((𝐺‘𝐹) ∙ (𝑆‘𝑋))) | ||
Theorem | hgmapval0 38910 | Value of the scalar sigma map at zero. (Contributed by NM, 12-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → (𝐺‘ 0 ) = 0 ) | ||
Theorem | hgmapval1 38911 | Value of the scalar sigma map at one. (Contributed by NM, 12-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → (𝐺‘ 1 ) = 1 ) | ||
Theorem | hgmapadd 38912 | Part 15 of [Baer] p. 50 line 13. (Contributed by NM, 6-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐺‘(𝑋 + 𝑌)) = ((𝐺‘𝑋) + (𝐺‘𝑌))) | ||
Theorem | hgmapmul 38913 | Part 15 of [Baer] p. 50 line 16. The multiplication is reversed after converting to the dual space scalar to the vector space scalar. (Contributed by NM, 7-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐺‘(𝑋 · 𝑌)) = ((𝐺‘𝑌) · (𝐺‘𝑋))) | ||
Theorem | hgmaprnlem1N 38914 | Lemma for hgmaprnN 38919. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑃 = (Scalar‘𝐶) & ⊢ 𝐴 = (Base‘𝑃) & ⊢ ∙ = ( ·𝑠 ‘𝐶) & ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑧 ∈ 𝐴) & ⊢ (𝜑 → 𝑡 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑠 ∈ 𝑉) & ⊢ (𝜑 → (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))) & ⊢ (𝜑 → 𝑘 ∈ 𝐵) & ⊢ (𝜑 → 𝑠 = (𝑘 · 𝑡)) ⇒ ⊢ (𝜑 → 𝑧 ∈ ran 𝐺) | ||
Theorem | hgmaprnlem2N 38915 | Lemma for hgmaprnN 38919. Part 15 of [Baer] p. 50 line 20. We only require a subset relation, rather than equality, so that the case of zero 𝑧 is taken care of automatically. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑃 = (Scalar‘𝐶) & ⊢ 𝐴 = (Base‘𝑃) & ⊢ ∙ = ( ·𝑠 ‘𝐶) & ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑧 ∈ 𝐴) & ⊢ (𝜑 → 𝑡 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑠 ∈ 𝑉) & ⊢ (𝜑 → (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐿 = (LSpan‘𝐶) ⇒ ⊢ (𝜑 → (𝑁‘{𝑠}) ⊆ (𝑁‘{𝑡})) | ||
Theorem | hgmaprnlem3N 38916* | Lemma for hgmaprnN 38919. Eliminate 𝑘. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑃 = (Scalar‘𝐶) & ⊢ 𝐴 = (Base‘𝑃) & ⊢ ∙ = ( ·𝑠 ‘𝐶) & ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑧 ∈ 𝐴) & ⊢ (𝜑 → 𝑡 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑠 ∈ 𝑉) & ⊢ (𝜑 → (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐿 = (LSpan‘𝐶) ⇒ ⊢ (𝜑 → 𝑧 ∈ ran 𝐺) | ||
Theorem | hgmaprnlem4N 38917* | Lemma for hgmaprnN 38919. Eliminate 𝑠. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑃 = (Scalar‘𝐶) & ⊢ 𝐴 = (Base‘𝑃) & ⊢ ∙ = ( ·𝑠 ‘𝐶) & ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑧 ∈ 𝐴) & ⊢ (𝜑 → 𝑡 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → 𝑧 ∈ ran 𝐺) | ||
Theorem | hgmaprnlem5N 38918 | Lemma for hgmaprnN 38919. Eliminate 𝑡. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑃 = (Scalar‘𝐶) & ⊢ 𝐴 = (Base‘𝑃) & ⊢ ∙ = ( ·𝑠 ‘𝐶) & ⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑧 ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝑧 ∈ ran 𝐺) | ||
Theorem | hgmaprnN 38919 | Part of proof of part 16 in [Baer] p. 50 line 23, Fs=G, except that we use the original vector space scalars for the range. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → ran 𝐺 = 𝐵) | ||
Theorem | hgmap11 38920 | The scalar sigma map is one-to-one. (Contributed by NM, 7-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝐺‘𝑋) = (𝐺‘𝑌) ↔ 𝑋 = 𝑌)) | ||
Theorem | hgmapf1oN 38921 | The scalar sigma map is a one-to-one onto function. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐵) | ||
Theorem | hgmapeq0 38922 | The scalar sigma map is zero iff its argument is zero. (Contributed by NM, 12-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝐺‘𝑋) = 0 ↔ 𝑋 = 0 )) | ||
Theorem | hdmapipcl 38923 | The inner product (Hermitian form) (𝑋, 𝑌) will be defined as ((𝑆‘𝑌)‘𝑋). Show closure. (Contributed by NM, 7-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝑆‘𝑌)‘𝑋) ∈ 𝐵) | ||
Theorem | hdmapln1 38924 | Linearity property that will be used for inner product. TODO: try to combine hypotheses in hdmap*ln* series. (Contributed by NM, 7-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ + = (+g‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ ⨣ = (+g‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑆‘𝑍)‘((𝐴 · 𝑋) + 𝑌)) = ((𝐴 × ((𝑆‘𝑍)‘𝑋)) ⨣ ((𝑆‘𝑍)‘𝑌))) | ||
Theorem | hdmaplna1 38925 | Additive property of first (inner product) argument. (Contributed by NM, 11-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ + = (+g‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ ⨣ = (+g‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝑆‘𝑍)‘(𝑋 + 𝑌)) = (((𝑆‘𝑍)‘𝑋) ⨣ ((𝑆‘𝑍)‘𝑌))) | ||
Theorem | hdmaplns1 38926 | Subtraction property of first (inner product) argument. (Contributed by NM, 12-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝑁 = (-g‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝑆‘𝑍)‘(𝑋 − 𝑌)) = (((𝑆‘𝑍)‘𝑋)𝑁((𝑆‘𝑍)‘𝑌))) | ||
Theorem | hdmaplnm1 38927 | Multiplicative property of first (inner product) argument. (Contributed by NM, 11-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑆‘𝑌)‘(𝐴 · 𝑋)) = (𝐴 × ((𝑆‘𝑌)‘𝑋))) | ||
Theorem | hdmaplna2 38928 | Additive property of second (inner product) argument. (Contributed by NM, 10-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ + = (+g‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ ⨣ = (+g‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝑆‘(𝑌 + 𝑍))‘𝑋) = (((𝑆‘𝑌)‘𝑋) ⨣ ((𝑆‘𝑍)‘𝑋))) | ||
Theorem | hdmapglnm2 38929 | g-linear property of second (inner product) argument. Line 19 in [Holland95] p. 14. (Contributed by NM, 10-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑆‘(𝐴 · 𝑌))‘𝑋) = (((𝑆‘𝑌)‘𝑋) × (𝐺‘𝐴))) | ||
Theorem | hdmapgln2 38930 | g-linear property that will be used for inner product. (Contributed by NM, 14-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ + = (+g‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ ⨣ = (+g‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑆‘((𝐴 · 𝑌) + 𝑍))‘𝑋) = ((((𝑆‘𝑌)‘𝑋) × (𝐺‘𝐴)) ⨣ ((𝑆‘𝑍)‘𝑋))) | ||
Theorem | hdmaplkr 38931 | Kernel of the vector to dual map. Line 16 in [Holland95] p. 14. TODO: eliminate 𝐹 hypothesis. (Contributed by NM, 9-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝑌 = (LKer‘𝑈) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑌‘(𝑆‘𝑋)) = (𝑂‘{𝑋})) | ||
Theorem | hdmapellkr 38932 | Membership in the kernel (as shown by hdmaplkr 38931) of the vector to dual map. Line 17 in [Holland95] p. 14. (Contributed by NM, 16-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → (((𝑆‘𝑋)‘𝑌) = 0 ↔ 𝑌 ∈ (𝑂‘{𝑋}))) | ||
Theorem | hdmapip0 38933 | Zero property that will be used for inner product. (Contributed by NM, 9-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝑍 = (0g‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → (((𝑆‘𝑋)‘𝑋) = 𝑍 ↔ 𝑋 = 0 )) | ||
Theorem | hdmapip1 38934 | Construct a proportional vector 𝑌 whose inner product with the original 𝑋 equals one. (Contributed by NM, 13-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝑁 = (invr‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ 𝑌 = ((𝑁‘((𝑆‘𝑋)‘𝑋)) · 𝑋) ⇒ ⊢ (𝜑 → ((𝑆‘𝑋)‘𝑌) = 1 ) | ||
Theorem | hdmapip0com 38935 | Commutation property of Baer's sigma map (Holland's A map). Line 20 of [Holland95] p. 14. Also part of Lemma 1 of [Baer] p. 110 line 7. (Contributed by NM, 9-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → (((𝑆‘𝑋)‘𝑌) = 0 ↔ ((𝑆‘𝑌)‘𝑋) = 0 )) | ||
Theorem | hdmapinvlem1 38936 | Line 27 in [Baer] p. 110. We use 𝐶 for Baer's u. Our unit vector 𝐸 has the required properties for his w by hdmapevec2 38854. Our ((𝑆‘𝐸)‘𝐶) means the inner product 〈𝐶, 𝐸〉 i.e. his f(u,w) (note argument reversal). (Contributed by NM, 11-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸})) ⇒ ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐶) = 0 ) | ||
Theorem | hdmapinvlem2 38937 | Line 28 in [Baer] p. 110, 0 = f(w,u). (Contributed by NM, 11-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸})) ⇒ ⊢ (𝜑 → ((𝑆‘𝐶)‘𝐸) = 0 ) | ||
Theorem | hdmapinvlem3 38938 | Line 30 in [Baer] p. 110, f(sw + u, tw - v) = 0. (Contributed by NM, 12-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ + = (+g‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸})) & ⊢ (𝜑 → 𝐷 ∈ (𝑂‘{𝐸})) & ⊢ (𝜑 → 𝐼 ∈ 𝐵) & ⊢ (𝜑 → 𝐽 ∈ 𝐵) & ⊢ (𝜑 → (𝐼 × (𝐺‘𝐽)) = ((𝑆‘𝐷)‘𝐶)) ⇒ ⊢ (𝜑 → ((𝑆‘((𝐽 · 𝐸) − 𝐷))‘((𝐼 · 𝐸) + 𝐶)) = 0 ) | ||
Theorem | hdmapinvlem4 38939 | Part 1.1 of Proposition 1 of [Baer] p. 110. We use 𝐶, 𝐷, 𝐼, and 𝐽 for Baer's u, v, s, and t. Our unit vector 𝐸 has the required properties for his w by hdmapevec2 38854. Our ((𝑆‘𝐷)‘𝐶) means his f(u,v) (note argument reversal). (Contributed by NM, 12-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ + = (+g‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸})) & ⊢ (𝜑 → 𝐷 ∈ (𝑂‘{𝐸})) & ⊢ (𝜑 → 𝐼 ∈ 𝐵) & ⊢ (𝜑 → 𝐽 ∈ 𝐵) & ⊢ (𝜑 → (𝐼 × (𝐺‘𝐽)) = ((𝑆‘𝐷)‘𝐶)) ⇒ ⊢ (𝜑 → (𝐽 × (𝐺‘𝐼)) = ((𝑆‘𝐶)‘𝐷)) | ||
Theorem | hdmapglem5 38940 | Part 1.2 in [Baer] p. 110 line 34, f(u,v) alpha = f(v,u). (Contributed by NM, 12-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ + = (+g‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸})) & ⊢ (𝜑 → 𝐷 ∈ (𝑂‘{𝐸})) & ⊢ (𝜑 → 𝐼 ∈ 𝐵) & ⊢ (𝜑 → 𝐽 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐺‘((𝑆‘𝐷)‘𝐶)) = ((𝑆‘𝐶)‘𝐷)) | ||
Theorem | hgmapvvlem1 38941 | Involution property of scalar sigma map. Line 10 in [Baer] p. 111, t sigma squared = t. Our 𝐸, 𝐶, 𝐷, 𝑌, 𝑋 correspond to Baer's w, h, k, s, t. (Contributed by NM, 13-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝑁 = (invr‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ { 0 })) & ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸})) & ⊢ (𝜑 → 𝐷 ∈ (𝑂‘{𝐸})) & ⊢ (𝜑 → ((𝑆‘𝐷)‘𝐶) = 1 ) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ∖ { 0 })) & ⊢ (𝜑 → (𝑌 × (𝐺‘𝑋)) = 1 ) ⇒ ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) = 𝑋) | ||
Theorem | hgmapvvlem2 38942 | Lemma for hgmapvv 38944. Eliminate 𝑌 (Baer's s). (Contributed by NM, 13-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝑁 = (invr‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ { 0 })) & ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸})) & ⊢ (𝜑 → 𝐷 ∈ (𝑂‘{𝐸})) & ⊢ (𝜑 → ((𝑆‘𝐷)‘𝐶) = 1 ) ⇒ ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) = 𝑋) | ||
Theorem | hgmapvvlem3 38943 | Lemma for hgmapvv 38944. Eliminate ((𝑆‘𝐷)‘𝐶) = 1 (Baer's f(h,k)=1). (Contributed by NM, 13-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝑁 = (invr‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ { 0 })) ⇒ ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) = 𝑋) | ||
Theorem | hgmapvv 38944 | Value of a double involution. Part 1.2 of [Baer] p. 110 line 37. (Contributed by NM, 13-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) = 𝑋) | ||
Theorem | hdmapglem7a 38945* | Lemma for hdmapg 38948. (Contributed by NM, 14-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ + = (+g‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢)) | ||
Theorem | hdmapglem7b 38946 | Lemma for hdmapg 38948. (Contributed by NM, 14-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ + = (+g‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ × = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ ✚ = (+g‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → 𝑥 ∈ (𝑂‘{𝐸})) & ⊢ (𝜑 → 𝑦 ∈ (𝑂‘{𝐸})) & ⊢ (𝜑 → 𝑚 ∈ 𝐵) & ⊢ (𝜑 → 𝑛 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑆‘((𝑚 · 𝐸) + 𝑥))‘((𝑛 · 𝐸) + 𝑦)) = ((𝑛 × (𝐺‘𝑚)) ✚ ((𝑆‘𝑥)‘𝑦))) | ||
Theorem | hdmapglem7 38947 | Lemma for hdmapg 38948. Line 15 in [Baer] p. 111, f(x,y) alpha = f(y,x). In the proof, our 𝐸, (𝑂‘{𝐸}) 𝑋, 𝑌, 𝑘, 𝑢, 𝑙, 𝑣 correspond to Baer's w, H, x, y, x', x'', y' , y'', and our ((𝑆‘𝑌)‘𝑋) corresponds to Baer's f(x,y). (Contributed by NM, 14-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ + = (+g‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ × = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ ✚ = (+g‘𝑅) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐺‘((𝑆‘𝑌)‘𝑋)) = ((𝑆‘𝑋)‘𝑌)) | ||
Theorem | hdmapg 38948 | Apply the scalar sigma function (involution) 𝐺 to an inner product reverses the arguments. The inner product of 𝑋 and 𝑌 is represented by ((𝑆‘𝑌)‘𝑋). Line 15 in [Baer] p. 111, f(x,y) alpha = f(y,x). (Contributed by NM, 14-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐺‘((𝑆‘𝑌)‘𝑋)) = ((𝑆‘𝑋)‘𝑌)) | ||
Theorem | hdmapoc 38949* | Express our constructed orthocomplement (polarity) in terms of the Hilbert space definition of orthocomplement. Lines 24 and 25 in [Holland95] p. 14. (Contributed by NM, 17-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ⊆ 𝑉) ⇒ ⊢ (𝜑 → (𝑂‘𝑋) = {𝑦 ∈ 𝑉 ∣ ∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 }) | ||
Syntax | chlh 38950 | Extend class notation with the final constructed Hilbert space. |
class HLHil | ||
Definition | df-hlhil 38951* | Define our final Hilbert space constructed from a Hilbert lattice. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) |
⊢ HLHil = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ ⦋((DVecH‘𝑘)‘𝑤) / 𝑢⦌⦋(Base‘𝑢) / 𝑣⦌({〈(Base‘ndx), 𝑣〉, 〈(+g‘ndx), (+g‘𝑢)〉, 〈(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet 〈(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)〉)〉} ∪ {〈( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑢)〉, 〈(·𝑖‘ndx), (𝑥 ∈ 𝑣, 𝑦 ∈ 𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))〉}))) | ||
Theorem | hlhilset 38952* | The final Hilbert space constructed from a Hilbert lattice 𝐾 and an arbitrary hyperplane 𝑊 in 𝐾. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐿 = ((HLHil‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ + = (+g‘𝑈) & ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ 𝑅 = (𝐸 sSet 〈(*𝑟‘ndx), 𝐺〉) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ , = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ ((𝑆‘𝑦)‘𝑥)) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → 𝐿 = ({〈(Base‘ndx), 𝑉〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝑅〉} ∪ {〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), , 〉})) | ||
Theorem | hlhilsca 38953 | The scalar of the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ 𝑅 = (𝐸 sSet 〈(*𝑟‘ndx), 𝐺〉) ⇒ ⊢ (𝜑 → 𝑅 = (Scalar‘𝑈)) | ||
Theorem | hlhilbase 38954 | The base set of the final constructed Hilbert space. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑀 = (Base‘𝐿) ⇒ ⊢ (𝜑 → 𝑀 = (Base‘𝑈)) | ||
Theorem | hlhilplus 38955 | The vector addition for the final constructed Hilbert space. (Contributed by NM, 21-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) & ⊢ + = (+g‘𝐿) ⇒ ⊢ (𝜑 → + = (+g‘𝑈)) | ||
Theorem | hlhilslem 38956 | Lemma for hlhilsbase2 38960. (Contributed by Mario Carneiro, 28-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 𝐹 = Slot 𝑁 & ⊢ 𝑁 ∈ ℕ & ⊢ 𝑁 < 4 & ⊢ 𝐶 = (𝐹‘𝐸) ⇒ ⊢ (𝜑 → 𝐶 = (𝐹‘𝑅)) | ||
Theorem | hlhilsbase 38957 | The scalar base set of the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 𝐶 = (Base‘𝐸) ⇒ ⊢ (𝜑 → 𝐶 = (Base‘𝑅)) | ||
Theorem | hlhilsplus 38958 | Scalar addition for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ + = (+g‘𝐸) ⇒ ⊢ (𝜑 → + = (+g‘𝑅)) | ||
Theorem | hlhilsmul 38959 | Scalar multiplication for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ · = (.r‘𝐸) ⇒ ⊢ (𝜑 → · = (.r‘𝑅)) | ||
Theorem | hlhilsbase2 38960 | The scalar base set of the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑆 = (Scalar‘𝐿) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 𝐶 = (Base‘𝑆) ⇒ ⊢ (𝜑 → 𝐶 = (Base‘𝑅)) | ||
Theorem | hlhilsplus2 38961 | Scalar addition for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑆 = (Scalar‘𝐿) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ + = (+g‘𝑆) ⇒ ⊢ (𝜑 → + = (+g‘𝑅)) | ||
Theorem | hlhilsmul2 38962 | Scalar multiplication for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑆 = (Scalar‘𝐿) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ · = (.r‘𝑆) ⇒ ⊢ (𝜑 → · = (.r‘𝑅)) | ||
Theorem | hlhils0 38963 | The scalar ring zero for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑆 = (Scalar‘𝐿) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 0 = (0g‘𝑆) ⇒ ⊢ (𝜑 → 0 = (0g‘𝑅)) | ||
Theorem | hlhils1N 38964 | The scalar ring unity for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑆 = (Scalar‘𝐿) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 1 = (1r‘𝑆) ⇒ ⊢ (𝜑 → 1 = (1r‘𝑅)) | ||
Theorem | hlhilvsca 38965 | The scalar product for the final constructed Hilbert space. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) & ⊢ · = ( ·𝑠 ‘𝐿) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → · = ( ·𝑠 ‘𝑈)) | ||
Theorem | hlhilip 38966* | Inner product operation for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝐿) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ , = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ ((𝑆‘𝑦)‘𝑥)) ⇒ ⊢ (𝜑 → , = (·𝑖‘𝑈)) | ||
Theorem | hlhilipval 38967 | Value of inner product operation for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝐿) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ , = (·𝑖‘𝑈) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑋 , 𝑌) = ((𝑆‘𝑌)‘𝑋)) | ||
Theorem | hlhilnvl 38968 | The involution operation of the star division ring for the final constructed Hilbert space. (Contributed by NM, 20-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ ∗ = ((HGMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → ∗ = (*𝑟‘𝑅)) | ||
Theorem | hlhillvec 38969 | The final constructed Hilbert space is a vector space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → 𝑈 ∈ LVec) | ||
Theorem | hlhildrng 38970 | The star division ring for the final constructed Hilbert space is a division ring. (Contributed by NM, 20-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 𝑅 = (Scalar‘𝑈) ⇒ ⊢ (𝜑 → 𝑅 ∈ DivRing) | ||
Theorem | hlhilsrnglem 38971 | Lemma for hlhilsrng 38972. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑆 = (Scalar‘𝐿) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ + = (+g‘𝑆) & ⊢ · = (.r‘𝑆) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) ⇒ ⊢ (𝜑 → 𝑅 ∈ *-Ring) | ||
Theorem | hlhilsrng 38972 | The star division ring for the final constructed Hilbert space is a division ring. (Contributed by NM, 21-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 𝑅 = (Scalar‘𝑈) ⇒ ⊢ (𝜑 → 𝑅 ∈ *-Ring) | ||
Theorem | hlhil0 38973 | The zero vector for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 0 = (0g‘𝐿) ⇒ ⊢ (𝜑 → 0 = (0g‘𝑈)) | ||
Theorem | hlhillsm 38974 | The vector sum operation for the final constructed Hilbert space. (Contributed by NM, 23-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ ⊕ = (LSSum‘𝐿) ⇒ ⊢ (𝜑 → ⊕ = (LSSum‘𝑈)) | ||
Theorem | hlhilocv 38975 | The orthocomplement for the final constructed Hilbert space. (Contributed by NM, 23-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 𝑉 = (Base‘𝐿) & ⊢ 𝑁 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑂 = (ocv‘𝑈) & ⊢ (𝜑 → 𝑋 ⊆ 𝑉) ⇒ ⊢ (𝜑 → (𝑂‘𝑋) = (𝑁‘𝑋)) | ||
Theorem | hlhillcs 38976 | The closed subspaces of the final constructed Hilbert space. TODO: hlhilbase 38954 is applied over and over to conclusion rather than applied once to antecedent - would compressed proof be shorter if applied once to antecedent? (Contributed by NM, 23-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ 𝐶 = (ClSubSp‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → 𝐶 = ran 𝐼) | ||
Theorem | hlhilphllem 38977* | Lemma for hlhil 23975. (Contributed by NM, 23-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 𝐹 = (Scalar‘𝑈) & ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝐿) & ⊢ + = (+g‘𝐿) & ⊢ · = ( ·𝑠 ‘𝐿) & ⊢ 𝑅 = (Scalar‘𝐿) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ ⨣ = (+g‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ 𝑄 = (0g‘𝑅) & ⊢ 0 = (0g‘𝐿) & ⊢ , = (·𝑖‘𝑈) & ⊢ 𝐽 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ 𝐸 = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ ((𝐽‘𝑦)‘𝑥)) ⇒ ⊢ (𝜑 → 𝑈 ∈ PreHil) | ||
Theorem | hlhilhillem 38978* | Lemma for hlhil 23975. (Contributed by NM, 23-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 𝐹 = (Scalar‘𝑈) & ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝐿) & ⊢ + = (+g‘𝐿) & ⊢ · = ( ·𝑠 ‘𝐿) & ⊢ 𝑅 = (Scalar‘𝐿) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ ⨣ = (+g‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ 𝑄 = (0g‘𝑅) & ⊢ 0 = (0g‘𝐿) & ⊢ , = (·𝑖‘𝑈) & ⊢ 𝐽 = ((HDMap‘𝐾)‘𝑊) & ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) & ⊢ 𝐸 = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ ((𝐽‘𝑦)‘𝑥)) & ⊢ 𝑂 = (ocv‘𝑈) & ⊢ 𝐶 = (ClSubSp‘𝑈) ⇒ ⊢ (𝜑 → 𝑈 ∈ Hil) | ||
Theorem | hlathil 38979 |
Construction of a Hilbert space (df-hil 20778) 𝑈 from a Hilbert
lattice (df-hlat 36369) 𝐾, where 𝑊 is a fixed but arbitrary
hyperplane (co-atom) in 𝐾.
The Hilbert space 𝑈 is identical to the vector space ((DVecH‘𝐾)‘𝑊) (see dvhlvec 38127) except that it is extended with involution and inner product components. The construction of these two components is provided by Theorem 3.6 in [Holland95] p. 13, whose proof we follow loosely. An example of involution is the complex conjugate when the division ring is the field of complex numbers. The nature of the division ring we constructed is indeterminate, however, until we specialize the initial Hilbert lattice with additional conditions found by Maria Solèr in 1995 and refined by René Mayet in 1998 that result in a division ring isomorphic to ℂ. See additional discussion at https://us.metamath.org/qlegif/mmql.html#what 38127. 𝑊 corresponds to the w in the proof of Theorem 13.4 of [Crawley] p. 111. Such a 𝑊 always exists since HL has lattice rank of at least 4 by df-hil 20778. It can be eliminated if we just want to show the existence of a Hilbert space, as is done in the literature. (Contributed by NM, 23-Jun-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → 𝑈 ∈ Hil) | ||
Theorem | ioin9i8 38980 | Miscellaneous inference creating a biconditional from an implied converse implication. (Contributed by Steven Nguyen, 17-Jul-2022.) |
⊢ (𝜑 → (𝜓 ∨ 𝜒)) & ⊢ (𝜒 → ¬ 𝜃) & ⊢ (𝜓 → 𝜃) ⇒ ⊢ (𝜑 → (𝜓 ↔ 𝜃)) | ||
Theorem | jaodd 38981 | Double deduction form of jaoi 851. (Contributed by Steven Nguyen, 17-Jul-2022.) |
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) & ⊢ (𝜑 → (𝜓 → (𝜏 → 𝜃))) ⇒ ⊢ (𝜑 → (𝜓 → ((𝜒 ∨ 𝜏) → 𝜃))) | ||
Theorem | nsb 38982 | Generalization rule for negated wff. (Contributed by Steven Nguyen, 3-May-2023.) |
⊢ ¬ 𝜑 ⇒ ⊢ ¬ [𝑥 / 𝑦]𝜑 | ||
Theorem | sbn1 38983 | One direction of sbn 2279, using fewer axioms. Compare 19.2 1972. (Contributed by Steven Nguyen, 18-Aug-2023.) |
⊢ ([𝑡 / 𝑥] ¬ 𝜑 → ¬ [𝑡 / 𝑥]𝜑) | ||
Theorem | sbor2 38984 | One direction of sbor 2308, using fewer axioms. Compare 19.33 1876. (Contributed by Steven Nguyen, 18-Aug-2023.) |
⊢ (([𝑡 / 𝑥]𝜑 ∨ [𝑡 / 𝑥]𝜓) → [𝑡 / 𝑥](𝜑 ∨ 𝜓)) | ||
Theorem | 3rspcedvd 38985* | Triple application of rspcedvd 3625. (Contributed by Steven Nguyen, 27-Feb-2023.) |
⊢ (𝜑 → 𝐴 ∈ 𝐷) & ⊢ (𝜑 → 𝐵 ∈ 𝐷) & ⊢ (𝜑 → 𝐶 ∈ 𝐷) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) & ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → (𝜒 ↔ 𝜃)) & ⊢ ((𝜑 ∧ 𝑧 = 𝐶) → (𝜃 ↔ 𝜏)) & ⊢ (𝜑 → 𝜏) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐷 ∃𝑦 ∈ 𝐷 ∃𝑧 ∈ 𝐷 𝜓) | ||
Theorem | rabeqcda 38986* | When 𝜓 is always true in a context, a restricted class abstraction is equal to the restricting class. Deduction form of rabeqc 3677. (Contributed by Steven Nguyen, 7-Jun-2023.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = 𝐴) | ||
Theorem | rabdif 38987* | Move difference in and out of a restricted class abstraction. (Contributed by Steven Nguyen, 6-Jun-2023.) |
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∖ 𝐵) = {𝑥 ∈ (𝐴 ∖ 𝐵) ∣ 𝜑} | ||
Theorem | sn-axrep5v 38988* | A condensed form of axrep5 5188. (Contributed by SN, 21-Sep-2023.) |
⊢ (∀𝑤 ∈ 𝑥 ∃*𝑧𝜑 → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 𝜑)) | ||
Theorem | sn-axprlem3 38989* | axprlem3 5317 using only Tarski's FOL axiom schemes and ax-rep 5182. (Contributed by SN, 22-Sep-2023.) |
⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏)) | ||
Theorem | sn-el 38990* | A version of el 5262 with an inner existential quantifier on 𝑥, which avoids ax-7 2006 and ax-8 2107. (Contributed by SN, 18-Sep-2023.) |
⊢ ∃𝑦∃𝑥 𝑥 ∈ 𝑦 | ||
Theorem | sn-dtru 38991* | dtru 5263 without ax-8 2107 or ax-12 2167. (Contributed by SN, 21-Sep-2023.) |
⊢ ¬ ∀𝑥 𝑥 = 𝑦 | ||
Theorem | pssexg 38992 | The proper subset of a set is also a set. (Contributed by Steven Nguyen, 17-Jul-2022.) |
⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V) | ||
Theorem | pssn0 38993 | A proper superset is nonempty. (Contributed by Steven Nguyen, 17-Jul-2022.) |
⊢ (𝐴 ⊊ 𝐵 → 𝐵 ≠ ∅) | ||
Theorem | psspwb 38994 | Classes are proper subclasses if and only if their power classes are proper subclasses. (Contributed by Steven Nguyen, 17-Jul-2022.) |
⊢ (𝐴 ⊊ 𝐵 ↔ 𝒫 𝐴 ⊊ 𝒫 𝐵) | ||
Theorem | xppss12 38995 | Proper subset theorem for Cartesian product. (Contributed by Steven Nguyen, 17-Jul-2022.) |
⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐶 ⊊ 𝐷) → (𝐴 × 𝐶) ⊊ (𝐵 × 𝐷)) | ||
Theorem | elpwbi 38996 | Membership in a power set, biconditional. (Contributed by Steven Nguyen, 17-Jul-2022.) (Proof shortened by Steven Nguyen, 16-Sep-2022.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ 𝒫 𝐵) | ||
Theorem | opelxpii 38997 | Ordered pair membership in a Cartesian product (implication). (Contributed by Steven Nguyen, 17-Jul-2022.) |
⊢ 𝐴 ∈ 𝐶 & ⊢ 𝐵 ∈ 𝐷 ⇒ ⊢ 〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) | ||
Theorem | iunsn 38998* | Indexed union of a singleton. Compare dfiun2 4950 and rnmpt 5821. (Contributed by Steven Nguyen, 7-Jun-2023.) |
⊢ ∪ 𝑥 ∈ 𝐴 {𝐵} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} | ||
Theorem | imaopab 38999* | The image of a class of ordered pairs. (Contributed by Steven Nguyen, 6-Jun-2023.) |
⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} | ||
Theorem | fnsnbt 39000 | A function's domain is a singleton iff the function is a singleton. (Contributed by Steven Nguyen, 18-Aug-2023.) |
⊢ (𝐴 ∈ V → (𝐹 Fn {𝐴} ↔ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉})) |
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