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Theorem List for Metamath Proof Explorer - 38901-39000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremhdmap14lem12 38901* Lemma for proof of part 14 in [Baer] p. 50. (Contributed by NM, 6-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    · = ( ·𝑠𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = ( ·𝑠𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐵)    &   𝑃 = (Scalar‘𝐶)    &   𝐴 = (Base‘𝑃)    &    0 = (0g𝑈)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐺𝐴)       (𝜑 → ((𝑆‘(𝐹 · 𝑋)) = (𝐺 (𝑆𝑋)) ↔ ∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 (𝑆𝑦))))

Theoremhdmap14lem13 38902* Lemma for proof of part 14 in [Baer] p. 50. (Contributed by NM, 6-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    · = ( ·𝑠𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = ( ·𝑠𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐵)    &   𝑃 = (Scalar‘𝐶)    &   𝐴 = (Base‘𝑃)    &    0 = (0g𝑈)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐺𝐴)       (𝜑 → ((𝑆‘(𝐹 · 𝑋)) = (𝐺 (𝑆𝑋)) ↔ ∀𝑦𝑉 (𝑆‘(𝐹 · 𝑦)) = (𝐺 (𝑆𝑦))))

Theoremhdmap14lem14 38903* Part of proof of part 14 in [Baer] p. 50 line 3. (Contributed by NM, 6-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    · = ( ·𝑠𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = ( ·𝑠𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐵)    &   𝑃 = (Scalar‘𝐶)    &   𝐴 = (Base‘𝑃)       (𝜑 → ∃!𝑔𝐴𝑥𝑉 (𝑆‘(𝐹 · 𝑥)) = (𝑔 (𝑆𝑥)))

Theoremhdmap14lem15 38904* Part of proof of part 14 in [Baer] p. 50 line 3. Convert scalar base of dual to scalar base of vector space. (Contributed by NM, 6-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    · = ( ·𝑠𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = ( ·𝑠𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐵)       (𝜑 → ∃!𝑔𝐵𝑥𝑉 (𝑆‘(𝐹 · 𝑥)) = (𝑔 (𝑆𝑥)))

Syntaxchg 38905 Extend class notation with g-map.
class HGMap

Definitiondf-hgmap 38906* 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 ∈ (𝑥𝑏 ↦ (𝑦𝑏𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝑘)‘𝑤))(𝑚𝑣))))}))

Theoremhgmapffval 38907* 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)       (𝐾𝑋 → (HGMap‘𝐾) = (𝑤𝐻 ↦ {𝑎[((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘(Scalar‘𝑢)) / 𝑏][((HDMap‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥𝑏 ↦ (𝑦𝑏𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑤))(𝑚𝑣))))}))

Theoremhgmapfval 38908* 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)    &    · = ( ·𝑠𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = ( ·𝑠𝐶)    &   𝑀 = ((HDMap‘𝐾)‘𝑊)    &   𝐼 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾𝑌𝑊𝐻))       (𝜑𝐼 = (𝑥𝐵 ↦ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣)))))

Theoremhgmapval 38909* 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 38904. (Contributed by NM, 25-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    · = ( ·𝑠𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = ( ·𝑠𝐶)    &   𝑀 = ((HDMap‘𝐾)‘𝑊)    &   𝐼 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾𝑌𝑊𝐻))    &   (𝜑𝑋𝐵)       (𝜑 → (𝐼𝑋) = (𝑦𝐵𝑣𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 (𝑀𝑣))))

TheoremhgmapfnN 38910 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 ∧ 𝑊𝐻))       (𝜑𝐺 Fn 𝐵)

Theoremhgmapcl 38911 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 ∧ 𝑊𝐻))    &   (𝜑𝐹𝐵)       (𝜑 → (𝐺𝐹) ∈ 𝐵)

Theoremhgmapdcl 38912 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 ∧ 𝑊𝐻))    &   (𝜑𝐹𝐵)       (𝜑 → (𝐺𝐹) ∈ 𝐴)

Theoremhgmapvs 38913 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)    &    · = ( ·𝑠𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = ( ·𝑠𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝐹𝐵)       (𝜑 → (𝑆‘(𝐹 · 𝑋)) = ((𝐺𝐹) (𝑆𝑋)))

Theoremhgmapval0 38914 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 = (0g𝑅)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 → (𝐺0 ) = 0 )

Theoremhgmapval1 38915 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 = (1r𝑅)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 → (𝐺1 ) = 1 )

Theoremhgmapadd 38916 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)    &    + = (+g𝑅)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝐺‘(𝑋 + 𝑌)) = ((𝐺𝑋) + (𝐺𝑌)))

Theoremhgmapmul 38917 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)    &    · = (.r𝑅)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝐺‘(𝑋 · 𝑌)) = ((𝐺𝑌) · (𝐺𝑋)))

Theoremhgmaprnlem1N 38918 Lemma for hgmaprnN 38923. (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)    &    · = ( ·𝑠𝑈)    &    0 = (0g𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑃 = (Scalar‘𝐶)    &   𝐴 = (Base‘𝑃)    &    = ( ·𝑠𝐶)    &   𝑄 = (0g𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑧𝐴)    &   (𝜑𝑡 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑠𝑉)    &   (𝜑 → (𝑆𝑠) = (𝑧 (𝑆𝑡)))    &   (𝜑𝑘𝐵)    &   (𝜑𝑠 = (𝑘 · 𝑡))       (𝜑𝑧 ∈ ran 𝐺)

Theoremhgmaprnlem2N 38919 Lemma for hgmaprnN 38923. 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)    &    · = ( ·𝑠𝑈)    &    0 = (0g𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑃 = (Scalar‘𝐶)    &   𝐴 = (Base‘𝑃)    &    = ( ·𝑠𝐶)    &   𝑄 = (0g𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑧𝐴)    &   (𝜑𝑡 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑠𝑉)    &   (𝜑 → (𝑆𝑠) = (𝑧 (𝑆𝑡)))    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑁 = (LSpan‘𝑈)    &   𝐿 = (LSpan‘𝐶)       (𝜑 → (𝑁‘{𝑠}) ⊆ (𝑁‘{𝑡}))

Theoremhgmaprnlem3N 38920* Lemma for hgmaprnN 38923. 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)    &    · = ( ·𝑠𝑈)    &    0 = (0g𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑃 = (Scalar‘𝐶)    &   𝐴 = (Base‘𝑃)    &    = ( ·𝑠𝐶)    &   𝑄 = (0g𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑧𝐴)    &   (𝜑𝑡 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑠𝑉)    &   (𝜑 → (𝑆𝑠) = (𝑧 (𝑆𝑡)))    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑁 = (LSpan‘𝑈)    &   𝐿 = (LSpan‘𝐶)       (𝜑𝑧 ∈ ran 𝐺)

Theoremhgmaprnlem4N 38921* Lemma for hgmaprnN 38923. 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)    &    · = ( ·𝑠𝑈)    &    0 = (0g𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑃 = (Scalar‘𝐶)    &   𝐴 = (Base‘𝑃)    &    = ( ·𝑠𝐶)    &   𝑄 = (0g𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑧𝐴)    &   (𝜑𝑡 ∈ (𝑉 ∖ { 0 }))       (𝜑𝑧 ∈ ran 𝐺)

Theoremhgmaprnlem5N 38922 Lemma for hgmaprnN 38923. 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)    &    · = ( ·𝑠𝑈)    &    0 = (0g𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑃 = (Scalar‘𝐶)    &   𝐴 = (Base‘𝑃)    &    = ( ·𝑠𝐶)    &   𝑄 = (0g𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑧𝐴)       (𝜑𝑧 ∈ ran 𝐺)

TheoremhgmaprnN 38923 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 ∧ 𝑊𝐻))       (𝜑 → ran 𝐺 = 𝐵)

Theoremhgmap11 38924 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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → ((𝐺𝑋) = (𝐺𝑌) ↔ 𝑋 = 𝑌))

Theoremhgmapf1oN 38925 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 ∧ 𝑊𝐻))       (𝜑𝐺:𝐵1-1-onto𝐵)

Theoremhgmapeq0 38926 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 = (0g𝑅)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝐵)       (𝜑 → ((𝐺𝑋) = 0𝑋 = 0 ))

Theoremhdmapipcl 38927 The inner product (Hermitian form) (𝑋, 𝑌) will be defined as ((𝑆𝑌)‘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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → ((𝑆𝑌)‘𝑋) ∈ 𝐵)

Theoremhdmapln1 38928 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)    &    + = (+g𝑈)    &    · = ( ·𝑠𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &    = (+g𝑅)    &    × = (.r𝑅)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑍𝑉)    &   (𝜑𝐴𝐵)       (𝜑 → ((𝑆𝑍)‘((𝐴 · 𝑋) + 𝑌)) = ((𝐴 × ((𝑆𝑍)‘𝑋)) ((𝑆𝑍)‘𝑌)))

Theoremhdmaplna1 38929 Additive property of first (inner product) argument. (Contributed by NM, 11-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &   𝑅 = (Scalar‘𝑈)    &    = (+g𝑅)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑍𝑉)       (𝜑 → ((𝑆𝑍)‘(𝑋 + 𝑌)) = (((𝑆𝑍)‘𝑋) ((𝑆𝑍)‘𝑌)))

Theoremhdmaplns1 38930 Subtraction property of first (inner product) argument. (Contributed by NM, 12-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝑁 = (-g𝑅)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑍𝑉)       (𝜑 → ((𝑆𝑍)‘(𝑋 𝑌)) = (((𝑆𝑍)‘𝑋)𝑁((𝑆𝑍)‘𝑌)))

Theoremhdmaplnm1 38931 Multiplicative property of first (inner product) argument. (Contributed by NM, 11-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    · = ( ·𝑠𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &    × = (.r𝑅)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝐴𝐵)       (𝜑 → ((𝑆𝑌)‘(𝐴 · 𝑋)) = (𝐴 × ((𝑆𝑌)‘𝑋)))

Theoremhdmaplna2 38932 Additive property of second (inner product) argument. (Contributed by NM, 10-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &   𝑅 = (Scalar‘𝑈)    &    = (+g𝑅)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑍𝑉)       (𝜑 → ((𝑆‘(𝑌 + 𝑍))‘𝑋) = (((𝑆𝑌)‘𝑋) ((𝑆𝑍)‘𝑋)))

Theoremhdmapglnm2 38933 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)    &    · = ( ·𝑠𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &    × = (.r𝑅)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝐴𝐵)       (𝜑 → ((𝑆‘(𝐴 · 𝑌))‘𝑋) = (((𝑆𝑌)‘𝑋) × (𝐺𝐴)))

Theoremhdmapgln2 38934 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)    &    + = (+g𝑈)    &    · = ( ·𝑠𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &    = (+g𝑅)    &    × = (.r𝑅)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑍𝑉)    &   (𝜑𝐴𝐵)       (𝜑 → ((𝑆‘((𝐴 · 𝑌) + 𝑍))‘𝑋) = ((((𝑆𝑌)‘𝑋) × (𝐺𝐴)) ((𝑆𝑍)‘𝑋)))

Theoremhdmaplkr 38935 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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)       (𝜑 → (𝑌‘(𝑆𝑋)) = (𝑂‘{𝑋}))

Theoremhdmapellkr 38936 Membership in the kernel (as shown by hdmaplkr 38935) 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 = (0g𝑅)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (((𝑆𝑋)‘𝑌) = 0𝑌 ∈ (𝑂‘{𝑋})))

Theoremhdmapip0 38937 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 = (0g𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝑍 = (0g𝑅)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)       (𝜑 → (((𝑆𝑋)‘𝑋) = 𝑍𝑋 = 0 ))

Theoremhdmapip1 38938 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)    &    · = ( ·𝑠𝑈)    &    0 = (0g𝑈)    &   𝑅 = (Scalar‘𝑈)    &    1 = (1r𝑅)    &   𝑁 = (invr𝑅)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   𝑌 = ((𝑁‘((𝑆𝑋)‘𝑋)) · 𝑋)       (𝜑 → ((𝑆𝑋)‘𝑌) = 1 )

Theoremhdmapip0com 38939 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 = (0g𝑅)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (((𝑆𝑋)‘𝑌) = 0 ↔ ((𝑆𝑌)‘𝑋) = 0 ))

Theoremhdmapinvlem1 38940 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 38858. Our ((𝑆𝐸)‘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)    &    · = (.r𝑅)    &    0 = (0g𝑅)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐶 ∈ (𝑂‘{𝐸}))       (𝜑 → ((𝑆𝐸)‘𝐶) = 0 )

Theoremhdmapinvlem2 38941 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)    &    · = (.r𝑅)    &    0 = (0g𝑅)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐶 ∈ (𝑂‘{𝐸}))       (𝜑 → ((𝑆𝐶)‘𝐸) = 0 )

Theoremhdmapinvlem3 38942 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)    &    + = (+g𝑈)    &    = (-g𝑈)    &    · = ( ·𝑠𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &    × = (.r𝑅)    &    0 = (0g𝑅)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐶 ∈ (𝑂‘{𝐸}))    &   (𝜑𝐷 ∈ (𝑂‘{𝐸}))    &   (𝜑𝐼𝐵)    &   (𝜑𝐽𝐵)    &   (𝜑 → (𝐼 × (𝐺𝐽)) = ((𝑆𝐷)‘𝐶))       (𝜑 → ((𝑆‘((𝐽 · 𝐸) 𝐷))‘((𝐼 · 𝐸) + 𝐶)) = 0 )

Theoremhdmapinvlem4 38943 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 38858. Our ((𝑆𝐷)‘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)    &    + = (+g𝑈)    &    = (-g𝑈)    &    · = ( ·𝑠𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &    × = (.r𝑅)    &    0 = (0g𝑅)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐶 ∈ (𝑂‘{𝐸}))    &   (𝜑𝐷 ∈ (𝑂‘{𝐸}))    &   (𝜑𝐼𝐵)    &   (𝜑𝐽𝐵)    &   (𝜑 → (𝐼 × (𝐺𝐽)) = ((𝑆𝐷)‘𝐶))       (𝜑 → (𝐽 × (𝐺𝐼)) = ((𝑆𝐶)‘𝐷))

Theoremhdmapglem5 38944 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)    &    + = (+g𝑈)    &    = (-g𝑈)    &    · = ( ·𝑠𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &    × = (.r𝑅)    &    0 = (0g𝑅)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐶 ∈ (𝑂‘{𝐸}))    &   (𝜑𝐷 ∈ (𝑂‘{𝐸}))    &   (𝜑𝐼𝐵)    &   (𝜑𝐽𝐵)       (𝜑 → (𝐺‘((𝑆𝐷)‘𝐶)) = ((𝑆𝐶)‘𝐷))

Theoremhgmapvvlem1 38945 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)    &    · = ( ·𝑠𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &    × = (.r𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝑁 = (invr𝑅)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝐵 ∖ { 0 }))    &   (𝜑𝐶 ∈ (𝑂‘{𝐸}))    &   (𝜑𝐷 ∈ (𝑂‘{𝐸}))    &   (𝜑 → ((𝑆𝐷)‘𝐶) = 1 )    &   (𝜑𝑌 ∈ (𝐵 ∖ { 0 }))    &   (𝜑 → (𝑌 × (𝐺𝑋)) = 1 )       (𝜑 → (𝐺‘(𝐺𝑋)) = 𝑋)

Theoremhgmapvvlem2 38946 Lemma for hgmapvv 38948. 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)    &    · = ( ·𝑠𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &    × = (.r𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝑁 = (invr𝑅)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝐵 ∖ { 0 }))    &   (𝜑𝐶 ∈ (𝑂‘{𝐸}))    &   (𝜑𝐷 ∈ (𝑂‘{𝐸}))    &   (𝜑 → ((𝑆𝐷)‘𝐶) = 1 )       (𝜑 → (𝐺‘(𝐺𝑋)) = 𝑋)

Theoremhgmapvvlem3 38947 Lemma for hgmapvv 38948. Eliminate ((𝑆𝐷)‘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)    &    · = ( ·𝑠𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &    × = (.r𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝑁 = (invr𝑅)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝐵 ∖ { 0 }))       (𝜑 → (𝐺‘(𝐺𝑋)) = 𝑋)

Theoremhgmapvv 38948 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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝐵)       (𝜑 → (𝐺‘(𝐺𝑋)) = 𝑋)

Theoremhdmapglem7a 38949* Lemma for hdmapg 38952. (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)    &    + = (+g𝑈)    &    · = ( ·𝑠𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &    = (LSSum‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)       (𝜑 → ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑘𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢))

Theoremhdmapglem7b 38950 Lemma for hdmapg 38952. (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)    &    + = (+g𝑈)    &    · = ( ·𝑠𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &    = (LSSum‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &    × = (.r𝑅)    &    0 = (0g𝑅)    &    = (+g𝑅)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑𝑥 ∈ (𝑂‘{𝐸}))    &   (𝜑𝑦 ∈ (𝑂‘{𝐸}))    &   (𝜑𝑚𝐵)    &   (𝜑𝑛𝐵)       (𝜑 → ((𝑆‘((𝑚 · 𝐸) + 𝑥))‘((𝑛 · 𝐸) + 𝑦)) = ((𝑛 × (𝐺𝑚)) ((𝑆𝑥)‘𝑦)))

Theoremhdmapglem7 38951 Lemma for hdmapg 38952. 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 ((𝑆𝑌)‘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)    &    + = (+g𝑈)    &    · = ( ·𝑠𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &    = (LSSum‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &    × = (.r𝑅)    &    0 = (0g𝑅)    &    = (+g𝑅)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑𝑌𝑉)       (𝜑 → (𝐺‘((𝑆𝑌)‘𝑋)) = ((𝑆𝑋)‘𝑌))

Theoremhdmapg 38952 Apply the scalar sigma function (involution) 𝐺 to an inner product reverses the arguments. The inner product of 𝑋 and 𝑌 is represented by ((𝑆𝑌)‘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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (𝐺‘((𝑆𝑌)‘𝑋)) = ((𝑆𝑋)‘𝑌))

Theoremhdmapoc 38953* 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 = (0g𝑅)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)       (𝜑 → (𝑂𝑋) = {𝑦𝑉 ∣ ∀𝑧𝑋 ((𝑆𝑧)‘𝑦) = 0 })

Syntaxchlh 38954 Extend class notation with the final constructed Hilbert space.
class HLHil

Definitiondf-hlhil 38955* 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), (+g𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·𝑠 ‘ndx), ( ·𝑠𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥𝑣, 𝑦𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩})))

Theoremhlhilset 38956* 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)    &    + = (+g𝑈)    &   𝐸 = ((EDRing‘𝐾)‘𝑊)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   𝑅 = (𝐸 sSet ⟨(*𝑟‘ndx), 𝐺⟩)    &    · = ( ·𝑠𝑈)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &    , = (𝑥𝑉, 𝑦𝑉 ↦ ((𝑆𝑦)‘𝑥))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝐿 = ({⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}))

Theoremhlhilsca 38957 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 ∧ 𝑊𝐻))    &   𝐸 = ((EDRing‘𝐾)‘𝑊)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   𝑅 = (𝐸 sSet ⟨(*𝑟‘ndx), 𝐺⟩)       (𝜑𝑅 = (Scalar‘𝑈))

Theoremhlhilbase 38958 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 ∧ 𝑊𝐻))    &   𝐿 = ((DVecH‘𝐾)‘𝑊)    &   𝑀 = (Base‘𝐿)       (𝜑𝑀 = (Base‘𝑈))

Theoremhlhilplus 38959 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 ∧ 𝑊𝐻))    &   𝐿 = ((DVecH‘𝐾)‘𝑊)    &    + = (+g𝐿)       (𝜑+ = (+g𝑈))

Theoremhlhilslem 38960 Lemma for hlhilsbase2 38964. (Contributed by Mario Carneiro, 28-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐸 = ((EDRing‘𝐾)‘𝑊)    &   𝑈 = ((HLHil‘𝐾)‘𝑊)    &   𝑅 = (Scalar‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   𝐹 = Slot 𝑁    &   𝑁 ∈ ℕ    &   𝑁 < 4    &   𝐶 = (𝐹𝐸)       (𝜑𝐶 = (𝐹𝑅))

Theoremhlhilsbase 38961 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 ∧ 𝑊𝐻))    &   𝐶 = (Base‘𝐸)       (𝜑𝐶 = (Base‘𝑅))

Theoremhlhilsplus 38962 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 ∧ 𝑊𝐻))    &    + = (+g𝐸)       (𝜑+ = (+g𝑅))

Theoremhlhilsmul 38963 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 ∧ 𝑊𝐻))    &    · = (.r𝐸)       (𝜑· = (.r𝑅))

Theoremhlhilsbase2 38964 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 ∧ 𝑊𝐻))    &   𝐶 = (Base‘𝑆)       (𝜑𝐶 = (Base‘𝑅))

Theoremhlhilsplus2 38965 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 ∧ 𝑊𝐻))    &    + = (+g𝑆)       (𝜑+ = (+g𝑅))

Theoremhlhilsmul2 38966 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 ∧ 𝑊𝐻))    &    · = (.r𝑆)       (𝜑· = (.r𝑅))

Theoremhlhils0 38967 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 ∧ 𝑊𝐻))    &    0 = (0g𝑆)       (𝜑0 = (0g𝑅))

Theoremhlhils1N 38968 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 ∧ 𝑊𝐻))    &    1 = (1r𝑆)       (𝜑1 = (1r𝑅))

Theoremhlhilvsca 38969 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)    &    · = ( ·𝑠𝐿)    &   𝑈 = ((HLHil‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑· = ( ·𝑠𝑈))

Theoremhlhilip 38970* 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 ∧ 𝑊𝐻))    &    , = (𝑥𝑉, 𝑦𝑉 ↦ ((𝑆𝑦)‘𝑥))       (𝜑, = (·𝑖𝑈))

Theoremhlhilipval 38971 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 ∧ 𝑊𝐻))    &    , = (·𝑖𝑈)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (𝑋 , 𝑌) = ((𝑆𝑌)‘𝑋))

Theoremhlhilnvl 38972 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 ∧ 𝑊𝐻))       (𝜑 = (*𝑟𝑅))

Theoremhlhillvec 38973 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 ∧ 𝑊𝐻))       (𝜑𝑈 ∈ LVec)

Theoremhlhildrng 38974 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 ∧ 𝑊𝐻))    &   𝑅 = (Scalar‘𝑈)       (𝜑𝑅 ∈ DivRing)

Theoremhlhilsrnglem 38975 Lemma for hlhilsrng 38976. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((HLHil‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   𝑅 = (Scalar‘𝑈)    &   𝐿 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (Scalar‘𝐿)    &   𝐵 = (Base‘𝑆)    &    + = (+g𝑆)    &    · = (.r𝑆)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)       (𝜑𝑅 ∈ *-Ring)

Theoremhlhilsrng 38976 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 ∧ 𝑊𝐻))    &   𝑅 = (Scalar‘𝑈)       (𝜑𝑅 ∈ *-Ring)

Theoremhlhil0 38977 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 ∧ 𝑊𝐻))    &    0 = (0g𝐿)       (𝜑0 = (0g𝑈))

Theoremhlhillsm 38978 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 ∧ 𝑊𝐻))    &    = (LSSum‘𝐿)       (𝜑 = (LSSum‘𝑈))

Theoremhlhilocv 38979 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 ∧ 𝑊𝐻))    &   𝑉 = (Base‘𝐿)    &   𝑁 = ((ocH‘𝐾)‘𝑊)    &   𝑂 = (ocv‘𝑈)    &   (𝜑𝑋𝑉)       (𝜑 → (𝑂𝑋) = (𝑁𝑋))

Theoremhlhillcs 38980 The closed subspaces of the final constructed Hilbert space. TODO: hlhilbase 38958 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 ∧ 𝑊𝐻))       (𝜑𝐶 = ran 𝐼)

Theoremhlhilphllem 38981* Lemma for hlhil 23980. (Contributed by NM, 23-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((HLHil‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   𝐹 = (Scalar‘𝑈)    &   𝐿 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝐿)    &    + = (+g𝐿)    &    · = ( ·𝑠𝐿)    &   𝑅 = (Scalar‘𝐿)    &   𝐵 = (Base‘𝑅)    &    = (+g𝑅)    &    × = (.r𝑅)    &   𝑄 = (0g𝑅)    &    0 = (0g𝐿)    &    , = (·𝑖𝑈)    &   𝐽 = ((HDMap‘𝐾)‘𝑊)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   𝐸 = (𝑥𝑉, 𝑦𝑉 ↦ ((𝐽𝑦)‘𝑥))       (𝜑𝑈 ∈ PreHil)

Theoremhlhilhillem 38982* Lemma for hlhil 23980. (Contributed by NM, 23-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((HLHil‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   𝐹 = (Scalar‘𝑈)    &   𝐿 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝐿)    &    + = (+g𝐿)    &    · = ( ·𝑠𝐿)    &   𝑅 = (Scalar‘𝐿)    &   𝐵 = (Base‘𝑅)    &    = (+g𝑅)    &    × = (.r𝑅)    &   𝑄 = (0g𝑅)    &    0 = (0g𝐿)    &    , = (·𝑖𝑈)    &   𝐽 = ((HDMap‘𝐾)‘𝑊)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   𝐸 = (𝑥𝑉, 𝑦𝑉 ↦ ((𝐽𝑦)‘𝑥))    &   𝑂 = (ocv‘𝑈)    &   𝐶 = (ClSubSp‘𝑈)       (𝜑𝑈 ∈ Hil)

Theoremhlathil 38983 Construction of a Hilbert space (df-hil 20783) 𝑈 from a Hilbert lattice (df-hlat 36373) 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 38131) 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 . See additional discussion at https://us.metamath.org/qlegif/mmql.html#what 38131.

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 20783. 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 ∧ 𝑊𝐻))       (𝜑𝑈 ∈ Hil)

20.25  Mathbox for Steven Nguyen

20.25.1  Utility theorems

Theoremioin9i8 38984 Miscellaneous inference creating a biconditional from an implied converse implication. (Contributed by Steven Nguyen, 17-Jul-2022.)
(𝜑 → (𝜓𝜒))    &   (𝜒 → ¬ 𝜃)    &   (𝜓𝜃)       (𝜑 → (𝜓𝜃))

Theoremjaodd 38985 Double deduction form of jaoi 853. (Contributed by Steven Nguyen, 17-Jul-2022.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑 → (𝜓 → (𝜏𝜃)))       (𝜑 → (𝜓 → ((𝜒𝜏) → 𝜃)))

Theoremnsb 38986 Generalization rule for negated wff. (Contributed by Steven Nguyen, 3-May-2023.)
¬ 𝜑        ¬ [𝑥 / 𝑦]𝜑

Theoremsbn1 38987 One direction of sbn 2281, using fewer axioms. Compare 19.2 1974. (Contributed by Steven Nguyen, 18-Aug-2023.)
([𝑡 / 𝑥] ¬ 𝜑 → ¬ [𝑡 / 𝑥]𝜑)

Theoremsbor2 38988 One direction of sbor 2310, using fewer axioms. Compare 19.33 1878. (Contributed by Steven Nguyen, 18-Aug-2023.)
(([𝑡 / 𝑥]𝜑 ∨ [𝑡 / 𝑥]𝜓) → [𝑡 / 𝑥](𝜑𝜓))

Theorem3rspcedvd 38989* Triple application of rspcedvd 3630. (Contributed by Steven Nguyen, 27-Feb-2023.)
(𝜑𝐴𝐷)    &   (𝜑𝐵𝐷)    &   (𝜑𝐶𝐷)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))    &   ((𝜑𝑦 = 𝐵) → (𝜒𝜃))    &   ((𝜑𝑧 = 𝐶) → (𝜃𝜏))    &   (𝜑𝜏)       (𝜑 → ∃𝑥𝐷𝑦𝐷𝑧𝐷 𝜓)

Theoremrabeqcda 38990* When 𝜓 is always true in a context, a restricted class abstraction is equal to the restricting class. Deduction form of rabeqc 3682. (Contributed by Steven Nguyen, 7-Jun-2023.)
((𝜑𝑥𝐴) → 𝜓)       (𝜑 → {𝑥𝐴𝜓} = 𝐴)

Theoremrabdif 38991* Move difference in and out of a restricted class abstraction. (Contributed by Steven Nguyen, 6-Jun-2023.)
({𝑥𝐴𝜑} ∖ 𝐵) = {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}

Theoremsn-axrep5v 38992* A condensed form of axrep5 5193. (Contributed by SN, 21-Sep-2023.)
(∀𝑤𝑥 ∃*𝑧𝜑 → ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤𝑥 𝜑))

Theoremsn-axprlem3 38993* axprlem3 5322 using only Tarski's FOL axiom schemes and ax-rep 5187. (Contributed by SN, 22-Sep-2023.)
𝑦𝑧(𝑧𝑦 ↔ ∃𝑤𝑥 if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏))

Theoremsn-el 38994* A version of el 5267 with an inner existential quantifier on 𝑥, which avoids ax-7 2008 and ax-8 2109. (Contributed by SN, 18-Sep-2023.)
𝑦𝑥 𝑥𝑦

Theoremsn-dtru 38995* dtru 5268 without ax-8 2109 or ax-12 2169. (Contributed by SN, 21-Sep-2023.)
¬ ∀𝑥 𝑥 = 𝑦

Theorempssexg 38996 The proper subset of a set is also a set. (Contributed by Steven Nguyen, 17-Jul-2022.)
((𝐴𝐵𝐵𝐶) → 𝐴 ∈ V)

Theorempssn0 38997 A proper superset is nonempty. (Contributed by Steven Nguyen, 17-Jul-2022.)
(𝐴𝐵𝐵 ≠ ∅)

Theorempsspwb 38998 Classes are proper subclasses if and only if their power classes are proper subclasses. (Contributed by Steven Nguyen, 17-Jul-2022.)
(𝐴𝐵 ↔ 𝒫 𝐴 ⊊ 𝒫 𝐵)

Theoremxppss12 38999 Proper subset theorem for Cartesian product. (Contributed by Steven Nguyen, 17-Jul-2022.)
((𝐴𝐵𝐶𝐷) → (𝐴 × 𝐶) ⊊ (𝐵 × 𝐷))

Theoremelpwbi 39000 Membership in a power set, biconditional. (Contributed by Steven Nguyen, 17-Jul-2022.) (Proof shortened by Steven Nguyen, 16-Sep-2022.)
𝐵 ∈ V       (𝐴𝐵𝐴 ∈ 𝒫 𝐵)

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