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Type | Label | Description |
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Statement | ||
Theorem | hgmapvs 36201 | 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 36202 | 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 36203 | 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 36204 | 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 36205 | 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 36206 | Lemma for hgmaprnN 36211. (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 36207 | Lemma for hgmaprnN 36211. 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 36208* | Lemma for hgmaprnN 36211. 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 36209* | Lemma for hgmaprnN 36211. 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 36210 | Lemma for hgmaprnN 36211. 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 36211 | 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 36212 | 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 36213 | 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 36214 | 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 36215 | 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 36216 | 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 36217 | 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 36218 | 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 36219 | 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 36220 | 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 36221 | 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 36222 | 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 36223 | 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 36224 | Membership in the kernel (as shown by hdmaplkr 36223) 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 36225 | 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 36226 | 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 36227 | 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 36228 | 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 36146. 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 36229 | 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 36230 | 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 36231 | 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 36146. 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 36232 | 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 36233 | 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 36234 | Lemma for hgmapvv 36236. 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 36235 | Lemma for hgmapvv 36236. 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 36236 | 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 36237* | Lemma for hdmapg 36240. (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 36238 | Lemma for hdmapg 36240. (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 36239 | Lemma for hdmapg 36240. 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 36240 | 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 36241* | 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 36242 | Extend class notation with the final constructed Hilbert space. |
class HLHil | ||
Definition | df-hlhil 36243* | 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 36244* | 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 36245 | 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 36246 | 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 36247 | 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 36248 | Lemma for hlhilsbase2 36252. (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 36249 | 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 36250 | 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 36251 | 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 36252 | 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 36253 | 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 36254 | 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 36255 | 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 36256 | 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 36257 | 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 36258* | 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 36259 | 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 36260 | 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 36261 | 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 36262 | 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 36263 | Lemma for hlhilsrng 36264. (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 36264 | 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 36265 | 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 36266 | 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 36267 | 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 36268 | The closed subspaces of the final constructed Hilbert space. TODO: hlhilbase 36246 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 = (CSubSp‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → 𝐶 = ran 𝐼) | ||
Theorem | hlhilphllem 36269* | Lemma for hlhil 23022. (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 36270* | Lemma for hlhil 23022. (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 = (CSubSp‘𝑈) ⇒ ⊢ (𝜑 → 𝑈 ∈ Hil) | ||
Theorem | hlathil 36271 |
Construction of a Hilbert space (df-hil 19867) 𝑈 from a Hilbert
lattice (df-hlat 33656) 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 35416) 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 http://us.metamath.org/qlegif/mmql.html#what. 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 19867. 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 | rntrclfvOAI 36272 | The range of the transitive closure is equal to the range of the relation. (Contributed by OpenAI, 7-Jul-2020.) |
⊢ (𝑅 ∈ 𝑉 → ran (t+‘𝑅) = ran 𝑅) | ||
Theorem | moxfr 36273* | Transfer at-most-one between related expressions. (Contributed by Stefan O'Rear, 12-Feb-2015.) |
⊢ 𝐴 ∈ V & ⊢ ∃!𝑦 𝑥 = 𝐴 & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓) | ||
Theorem | imaiinfv 36274* | Indexed intersection of an image. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ∩ 𝑥 ∈ 𝐵 (𝐹‘𝑥) = ∩ (𝐹 “ 𝐵)) | ||
Theorem | elrfi 36275* | Elementhood in a set of relative finite intersections. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) ↔ ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 ∩ ∩ 𝑣))) | ||
Theorem | elrfirn 36276* | Elementhood in a set of relative finite intersections of an indexed family of sets. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran 𝐹)) ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 ∩ ∩ 𝑦 ∈ 𝑣 (𝐹‘𝑦)))) | ||
Theorem | elrfirn2 36277* | Elementhood in a set of relative finite intersections of an indexed family of sets (implicit). (Contributed by Stefan O'Rear, 22-Feb-2015.) |
⊢ ((𝐵 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐼 𝐶 ⊆ 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran (𝑦 ∈ 𝐼 ↦ 𝐶))) ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 ∩ ∩ 𝑦 ∈ 𝑣 𝐶))) | ||
Theorem | cmpfiiin 36278* | In a compact topology, a system of closed sets with nonempty finite intersections has a nonempty intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝑆 ∈ (Clsd‘𝐽)) & ⊢ ((𝜑 ∧ (𝑙 ⊆ 𝐼 ∧ 𝑙 ∈ Fin)) → (𝑋 ∩ ∩ 𝑘 ∈ 𝑙 𝑆) ≠ ∅) ⇒ ⊢ (𝜑 → (𝑋 ∩ ∩ 𝑘 ∈ 𝐼 𝑆) ≠ ∅) | ||
Theorem | ismrcd1 36279* | Any function from the subsets of a set to itself, which is extensive (satisfies mrcssid 16100), isotone (satisfies mrcss 16099), and idempotent (satisfies mrcidm 16102) has a collection of fixed points which is a Moore collection, and itself is the closure operator for that collection. This can be taken as an alternate definition for the closure operators. This is the first half, ismrcd2 36280 is the second. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝒫 𝐵⟶𝒫 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ (𝐹‘𝑥)) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)) ⇒ ⊢ (𝜑 → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵)) | ||
Theorem | ismrcd2 36280* | Second half of ismrcd1 36279. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝒫 𝐵⟶𝒫 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ (𝐹‘𝑥)) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)) ⇒ ⊢ (𝜑 → 𝐹 = (mrCls‘dom (𝐹 ∩ I ))) | ||
Theorem | istopclsd 36281* | A closure function which satisfies sscls 20670, clsidm 20681, cls0 20694, and clsun 31493 defines a (unique) topology which it is the closure function on. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝒫 𝐵⟶𝒫 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ (𝐹‘𝑥)) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥)) & ⊢ (𝜑 → (𝐹‘∅) = ∅) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝐵) → (𝐹‘(𝑥 ∪ 𝑦)) = ((𝐹‘𝑥) ∪ (𝐹‘𝑦))) & ⊢ 𝐽 = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐹‘(𝐵 ∖ 𝑧)) = (𝐵 ∖ 𝑧)} ⇒ ⊢ (𝜑 → (𝐽 ∈ (TopOn‘𝐵) ∧ (cls‘𝐽) = 𝐹)) | ||
Theorem | ismrc 36282* | A function is a Moore closure operator iff it satisfies mrcssid 16100, mrcss 16099, and mrcidm 16102. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ (𝐹 ∈ (mrCls “ (Moore‘𝐵)) ↔ (𝐵 ∈ V ∧ 𝐹:𝒫 𝐵⟶𝒫 𝐵 ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ⊆ (𝐹‘𝑥) ∧ (𝐹‘𝑦) ⊆ (𝐹‘𝑥) ∧ (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥))))) | ||
Syntax | cnacs 36283 | Class of Noetherian closure systems. |
class NoeACS | ||
Definition | df-nacs 36284* | Define a closure system of Noetherian type (not standard terminology) as an algebraic system where all closed sets are finitely generated. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
⊢ NoeACS = (𝑥 ∈ V ↦ {𝑐 ∈ (ACS‘𝑥) ∣ ∀𝑠 ∈ 𝑐 ∃𝑔 ∈ (𝒫 𝑥 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)}) | ||
Theorem | isnacs 36285* | Expand definition of Noetherian-type closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔))) | ||
Theorem | nacsfg 36286* | In a Noetherian-type closure system, all closed sets are finitely generated. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑆 ∈ 𝐶) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹‘𝑔)) | ||
Theorem | isnacs2 36287 | Express Noetherian-type closure system with fewer quantifiers. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ (𝐹 “ (𝒫 𝑋 ∩ Fin)) = 𝐶)) | ||
Theorem | mrefg2 36288* | Slight variation on finite genration for closure systems. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ (𝐶 ∈ (Moore‘𝑋) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 = (𝐹‘𝑔))) | ||
Theorem | mrefg3 36289* | Slight variation on finite genration for closure systems. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 ⊆ (𝐹‘𝑔))) | ||
Theorem | nacsacs 36290 | A closure system of Noetherian type is algebraic. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
⊢ (𝐶 ∈ (NoeACS‘𝑋) → 𝐶 ∈ (ACS‘𝑋)) | ||
Theorem | isnacs3 36291* | A choice-free order equivalent to the Noetherian condition on a closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
⊢ (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝑠))) | ||
Theorem | incssnn0 36292* | Transitivity induction of subsets, lemma for nacsfix 36293. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
⊢ ((∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1)) ∧ 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → (𝐹‘𝐴) ⊆ (𝐹‘𝐵)) | ||
Theorem | nacsfix 36293* | An increasing sequence of closed sets in a Noetherian-type closure system eventually fixates. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ∃𝑦 ∈ ℕ0 ∀𝑧 ∈ (ℤ≥‘𝑦)(𝐹‘𝑧) = (𝐹‘𝑦)) | ||
Theorem | constmap 36294 |
A constant (represented without dummy variables) is an element of a
function set.
_Note: In the following development, we will be quite often quantifying over functions and points in N-dimensional space (which are equivalent to functions from an "index set"). Many of the following theorems exist to transfer standard facts about functions to elements of function sets._ (Contributed by Stefan O'Rear, 30-Aug-2014.) (Revised by Stefan O'Rear, 5-May-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐶 ∈ V ⇒ ⊢ (𝐵 ∈ 𝐶 → (𝐴 × {𝐵}) ∈ (𝐶 ↑𝑚 𝐴)) | ||
Theorem | mapco2g 36295 | Renaming indexes in a tuple, with sethood as antecedents. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Mario Carneiro, 5-May-2015.) |
⊢ ((𝐸 ∈ V ∧ 𝐴 ∈ (𝐵 ↑𝑚 𝐶) ∧ 𝐷:𝐸⟶𝐶) → (𝐴 ∘ 𝐷) ∈ (𝐵 ↑𝑚 𝐸)) | ||
Theorem | mapco2 36296 | Post-composition (renaming indexes) of a mapping viewed as a point. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.) |
⊢ 𝐸 ∈ V ⇒ ⊢ ((𝐴 ∈ (𝐵 ↑𝑚 𝐶) ∧ 𝐷:𝐸⟶𝐶) → (𝐴 ∘ 𝐷) ∈ (𝐵 ↑𝑚 𝐸)) | ||
Theorem | mapfzcons 36297 | Extending a one-based mapping by adding a tuple at the end results in another mapping. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.) |
⊢ 𝑀 = (𝑁 + 1) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ (𝐵 ↑𝑚 (1...𝑁)) ∧ 𝐶 ∈ 𝐵) → (𝐴 ∪ {〈𝑀, 𝐶〉}) ∈ (𝐵 ↑𝑚 (1...𝑀))) | ||
Theorem | mapfzcons1 36298 | Recover prefix mapping from an extended mapping. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.) |
⊢ 𝑀 = (𝑁 + 1) ⇒ ⊢ (𝐴 ∈ (𝐵 ↑𝑚 (1...𝑁)) → ((𝐴 ∪ {〈𝑀, 𝐶〉}) ↾ (1...𝑁)) = 𝐴) | ||
Theorem | mapfzcons1cl 36299 | A nonempty mapping has a prefix. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.) |
⊢ 𝑀 = (𝑁 + 1) ⇒ ⊢ (𝐴 ∈ (𝐵 ↑𝑚 (1...𝑀)) → (𝐴 ↾ (1...𝑁)) ∈ (𝐵 ↑𝑚 (1...𝑁))) | ||
Theorem | mapfzcons2 36300 | Recover added element from an extended mapping. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.) |
⊢ 𝑀 = (𝑁 + 1) ⇒ ⊢ ((𝐴 ∈ (𝐵 ↑𝑚 (1...𝑁)) ∧ 𝐶 ∈ 𝐵) → ((𝐴 ∪ {〈𝑀, 𝐶〉})‘𝑀) = 𝐶) |
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