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Theorem List for Metamath Proof Explorer - 38901-39000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhgmapffval 38901* 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 38902* 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 38903* 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 38898. (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 38904 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 38905 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 38906 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 38907 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 38908 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 38909 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 38910 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 38911 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 38912 Lemma for hgmaprnN 38917. (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 38913 Lemma for hgmaprnN 38917. 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 38914* Lemma for hgmaprnN 38917. 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 38915* Lemma for hgmaprnN 38917. 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 38916 Lemma for hgmaprnN 38917. 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 38917 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 38918 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 38919 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 38920 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 38921 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 38922 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 38923 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 38924 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 38925 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 38926 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 38927 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 38928 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 38929 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 38930 Membership in the kernel (as shown by hdmaplkr 38929) 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 38931 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 38932 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 38933 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 38934 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 38852. 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 38935 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 38936 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 38937 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 38852. 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 38938 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 38939 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 38940 Lemma for hgmapvv 38942. 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 38941 Lemma for hgmapvv 38942. 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 38942 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 38943* Lemma for hdmapg 38946. (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 38944 Lemma for hdmapg 38946. (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 38945 Lemma for hdmapg 38946. 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 38946 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 38947* 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 38948 Extend class notation with the final constructed Hilbert space.
class HLHil
 
Definitiondf-hlhil 38949* 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 38950* 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 38951 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 38952 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 38953 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 38954 Lemma for hlhilsbase2 38958. (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 38955 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 38956 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 38957 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 38958 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 38959 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 38960 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 38961 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 38962 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 38963 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 38964* 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 38965 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 38966 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 38967 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 38968 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 38969 Lemma for hlhilsrng 38970. (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 38970 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 38971 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 38972 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 38973 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 38974 The closed subspaces of the final constructed Hilbert space. TODO: hlhilbase 38952 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 38975* Lemma for hlhil 23973. (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 38976* Lemma for hlhil 23973. (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 38977 Construction of a Hilbert space (df-hil 20776) 𝑈 from a Hilbert lattice (df-hlat 36367) 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 38125) 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 38125.

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 20776. 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 38978 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 38979 Double deduction form of jaoi 851. (Contributed by Steven Nguyen, 17-Jul-2022.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑 → (𝜓 → (𝜏𝜃)))       (𝜑 → (𝜓 → ((𝜒𝜏) → 𝜃)))
 
Theoremnsb 38980 Generalization rule for negated wff. (Contributed by Steven Nguyen, 3-May-2023.)
¬ 𝜑        ¬ [𝑥 / 𝑦]𝜑
 
Theoremsbn1 38981 One direction of sbn 2278, using fewer axioms. Compare 19.2 1972. (Contributed by Steven Nguyen, 18-Aug-2023.)
([𝑡 / 𝑥] ¬ 𝜑 → ¬ [𝑡 / 𝑥]𝜑)
 
Theoremsbor2 38982 One direction of sbor 2307, using fewer axioms. Compare 19.33 1876. (Contributed by Steven Nguyen, 18-Aug-2023.)
(([𝑡 / 𝑥]𝜑 ∨ [𝑡 / 𝑥]𝜓) → [𝑡 / 𝑥](𝜑𝜓))
 
Theorem3rspcedvd 38983* Triple application of rspcedvd 3623. (Contributed by Steven Nguyen, 27-Feb-2023.)
(𝜑𝐴𝐷)    &   (𝜑𝐵𝐷)    &   (𝜑𝐶𝐷)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))    &   ((𝜑𝑦 = 𝐵) → (𝜒𝜃))    &   ((𝜑𝑧 = 𝐶) → (𝜃𝜏))    &   (𝜑𝜏)       (𝜑 → ∃𝑥𝐷𝑦𝐷𝑧𝐷 𝜓)
 
Theoremrabeqcda 38984* When 𝜓 is always true in a context, a restricted class abstraction is equal to the restricting class. Deduction form of rabeqc 3675. (Contributed by Steven Nguyen, 7-Jun-2023.)
((𝜑𝑥𝐴) → 𝜓)       (𝜑 → {𝑥𝐴𝜓} = 𝐴)
 
Theoremrabdif 38985* Move difference in and out of a restricted class abstraction. (Contributed by Steven Nguyen, 6-Jun-2023.)
({𝑥𝐴𝜑} ∖ 𝐵) = {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}
 
Theoremsn-axrep5v 38986* A condensed form of axrep5 5187. (Contributed by SN, 21-Sep-2023.)
(∀𝑤𝑥 ∃*𝑧𝜑 → ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤𝑥 𝜑))
 
Theoremsn-axprlem3 38987* axprlem3 5316 using only Tarski's FOL axiom schemes and ax-rep 5181. (Contributed by SN, 22-Sep-2023.)
𝑦𝑧(𝑧𝑦 ↔ ∃𝑤𝑥 if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏))
 
Theoremsn-el 38988* A version of el 5261 with an inner existential quantifier on 𝑥, which avoids ax-7 2006 and ax-8 2107. (Contributed by SN, 18-Sep-2023.)
𝑦𝑥 𝑥𝑦
 
Theoremsn-dtru 38989* dtru 5262 without ax-8 2107 or ax-12 2167. (Contributed by SN, 21-Sep-2023.)
¬ ∀𝑥 𝑥 = 𝑦
 
Theorempssexg 38990 The proper subset of a set is also a set. (Contributed by Steven Nguyen, 17-Jul-2022.)
((𝐴𝐵𝐵𝐶) → 𝐴 ∈ V)
 
Theorempssn0 38991 A proper superset is nonempty. (Contributed by Steven Nguyen, 17-Jul-2022.)
(𝐴𝐵𝐵 ≠ ∅)
 
Theorempsspwb 38992 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 38993 Proper subset theorem for Cartesian product. (Contributed by Steven Nguyen, 17-Jul-2022.)
((𝐴𝐵𝐶𝐷) → (𝐴 × 𝐶) ⊊ (𝐵 × 𝐷))
 
Theoremelpwbi 38994 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)
 
Theoremopelxpii 38995 Ordered pair membership in a Cartesian product (implication). (Contributed by Steven Nguyen, 17-Jul-2022.)
𝐴𝐶    &   𝐵𝐷       𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷)
 
Theoremiunsn 38996* Indexed union of a singleton. Compare dfiun2 4949 and rnmpt 5820. (Contributed by Steven Nguyen, 7-Jun-2023.)
𝑥𝐴 {𝐵} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
 
Theoremimaopab 38997* The image of a class of ordered pairs. (Contributed by Steven Nguyen, 6-Jun-2023.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝜑}
 
Theoremfnsnbt 38998 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, (𝐹𝐴)⟩}))
 
Theoremfnimasnd 38999 The image of a function by a singleton whose element is in the domain of the function. (Contributed by Steven Nguyen, 7-Jun-2023.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝑆𝐴)       (𝜑 → (𝐹 “ {𝑆}) = {(𝐹𝑆)})
 
Theoremdfqs2 39000* Alternate definition of quotient set. (Contributed by Steven Nguyen, 7-Jun-2023.)
(𝐴 / 𝑅) = ran (𝑥𝐴 ↦ [𝑥]𝑅)
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