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Theorem List for Metamath Proof Explorer - 36201-36300   *Has distinct variable group(s)
TypeLabelDescription
Statement

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

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

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

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

Theoremhgmaprnlem1N 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)    &    · = ( ·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 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)    &    · = ( ·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 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)    &    · = ( ·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 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)    &    · = ( ·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 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)    &    · = ( ·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 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 ∧ 𝑊𝐻))       (𝜑 → ran 𝐺 = 𝐵)

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

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

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

Theoremhdmapipcl 36215 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 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)    &    + = (+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 36217 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 36218 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 36219 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 36220 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 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)    &    · = ( ·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 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)    &    + = (+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 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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)       (𝜑 → (𝑌‘(𝑆𝑋)) = (𝑂‘{𝑋}))

Theoremhdmapellkr 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 = (0g𝑅)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (((𝑆𝑋)‘𝑌) = 0𝑌 ∈ (𝑂‘{𝑋})))

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

Theoremhdmapip1 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)    &    · = ( ·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 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 = (0g𝑅)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (((𝑆𝑋)‘𝑌) = 0 ↔ ((𝑆𝑌)‘𝑋) = 0 ))

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

Theoremhdmapinvlem3 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)    &    + = (+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 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 ((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 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)    &    + = (+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 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)    &    · = ( ·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 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)    &    · = ( ·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 36235 Lemma for hgmapvv 36236. 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 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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝐵)       (𝜑 → (𝐺‘(𝐺𝑋)) = 𝑋)

Theoremhdmapglem7a 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)    &    + = (+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 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)    &    + = (+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 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 ((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 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 ((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 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 = (0g𝑅)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)       (𝜑 → (𝑂𝑋) = {𝑦𝑉 ∣ ∀𝑧𝑋 ((𝑆𝑧)‘𝑦) = 0 })

Syntaxchlh 36242 Extend class notation with the final constructed Hilbert space.
class HLHil

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

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

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

Theoremhlhilslem 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 ∧ 𝑊𝐻))    &   𝐹 = Slot 𝑁    &   𝑁 ∈ ℕ    &   𝑁 < 4    &   𝐶 = (𝐹𝐸)       (𝜑𝐶 = (𝐹𝑅))

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

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

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

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

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

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

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

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

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

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

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

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

Theoremhlhillvec 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 ∧ 𝑊𝐻))       (𝜑𝑈 ∈ LVec)

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

Theoremhlhilsrnglem 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 ∧ 𝑊𝐻))    &   𝑅 = (Scalar‘𝑈)    &   𝐿 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (Scalar‘𝐿)    &   𝐵 = (Base‘𝑆)    &    + = (+g𝑆)    &    · = (.r𝑆)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)       (𝜑𝑅 ∈ *-Ring)

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

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

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

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

Theoremhlhillcs 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 ∧ 𝑊𝐻))       (𝜑𝐶 = ran 𝐼)

Theoremhlhilphllem 36269* Lemma for hlhil 23022. (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 36270* Lemma for hlhil 23022. (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 = (CSubSp‘𝑈)       (𝜑𝑈 ∈ Hil)

Theoremhlathil 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 . 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 ∧ 𝑊𝐻))       (𝜑𝑈 ∈ Hil)

21.23  Mathbox for OpenAI

TheoremrntrclfvOAI 36272 The range of the transitive closure is equal to the range of the relation. (Contributed by OpenAI, 7-Jul-2020.)
(𝑅𝑉 → ran (t+‘𝑅) = ran 𝑅)

21.24  Mathbox for Stefan O'Rear

21.24.1  Additional elementary logic and set theory

Theoremmoxfr 36273* Transfer at-most-one between related expressions. (Contributed by Stefan O'Rear, 12-Feb-2015.)
𝐴 ∈ V    &   ∃!𝑦 𝑥 = 𝐴    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)

Theoremimaiinfv 36274* Indexed intersection of an image. (Contributed by Stefan O'Rear, 22-Feb-2015.)
((𝐹 Fn 𝐴𝐵𝐴) → 𝑥𝐵 (𝐹𝑥) = (𝐹𝐵))

Theoremelrfi 36275* Elementhood in a set of relative finite intersections. (Contributed by Stefan O'Rear, 22-Feb-2015.)
((𝐵𝑉𝐶 ⊆ 𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ 𝐶)) ↔ ∃𝑣 ∈ (𝒫 𝐶 ∩ Fin)𝐴 = (𝐵 𝑣)))

Theoremelrfirn 36276* Elementhood in a set of relative finite intersections of an indexed family of sets. (Contributed by Stefan O'Rear, 22-Feb-2015.)
((𝐵𝑉𝐹:𝐼⟶𝒫 𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran 𝐹)) ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 𝑦𝑣 (𝐹𝑦))))

Theoremelrfirn2 36277* Elementhood in a set of relative finite intersections of an indexed family of sets (implicit). (Contributed by Stefan O'Rear, 22-Feb-2015.)
((𝐵𝑉 ∧ ∀𝑦𝐼 𝐶𝐵) → (𝐴 ∈ (fi‘({𝐵} ∪ ran (𝑦𝐼𝐶))) ↔ ∃𝑣 ∈ (𝒫 𝐼 ∩ Fin)𝐴 = (𝐵 𝑦𝑣 𝐶)))

Theoremcmpfiiin 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    &   (𝜑𝐽 ∈ Comp)    &   ((𝜑𝑘𝐼) → 𝑆 ∈ (Clsd‘𝐽))    &   ((𝜑 ∧ (𝑙𝐼𝑙 ∈ Fin)) → (𝑋 𝑘𝑙 𝑆) ≠ ∅)       (𝜑 → (𝑋 𝑘𝐼 𝑆) ≠ ∅)

21.24.4  Characterization of closure operators. Kuratowski closure axioms

Theoremismrcd1 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.)
(𝜑𝐵𝑉)    &   (𝜑𝐹:𝒫 𝐵⟶𝒫 𝐵)    &   ((𝜑𝑥𝐵) → 𝑥 ⊆ (𝐹𝑥))    &   ((𝜑𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥))    &   ((𝜑𝑥𝐵) → (𝐹‘(𝐹𝑥)) = (𝐹𝑥))       (𝜑 → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))

Theoremismrcd2 36280* Second half of ismrcd1 36279. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝜑𝐵𝑉)    &   (𝜑𝐹:𝒫 𝐵⟶𝒫 𝐵)    &   ((𝜑𝑥𝐵) → 𝑥 ⊆ (𝐹𝑥))    &   ((𝜑𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥))    &   ((𝜑𝑥𝐵) → (𝐹‘(𝐹𝑥)) = (𝐹𝑥))       (𝜑𝐹 = (mrCls‘dom (𝐹 ∩ I )))

Theoremistopclsd 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.)
(𝜑𝐵𝑉)    &   (𝜑𝐹:𝒫 𝐵⟶𝒫 𝐵)    &   ((𝜑𝑥𝐵) → 𝑥 ⊆ (𝐹𝑥))    &   ((𝜑𝑥𝐵) → (𝐹‘(𝐹𝑥)) = (𝐹𝑥))    &   (𝜑 → (𝐹‘∅) = ∅)    &   ((𝜑𝑥𝐵𝑦𝐵) → (𝐹‘(𝑥𝑦)) = ((𝐹𝑥) ∪ (𝐹𝑦)))    &   𝐽 = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐹‘(𝐵𝑧)) = (𝐵𝑧)}       (𝜑 → (𝐽 ∈ (TopOn‘𝐵) ∧ (cls‘𝐽) = 𝐹))

Theoremismrc 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 ∧ ∀𝑥𝑦((𝑥𝐵𝑦𝑥) → (𝑥 ⊆ (𝐹𝑥) ∧ (𝐹𝑦) ⊆ (𝐹𝑥) ∧ (𝐹‘(𝐹𝑥)) = (𝐹𝑥)))))

21.24.5  Algebraic closure systems

Syntaxcnacs 36283 Class of Noetherian closure systems.
class NoeACS

Definitiondf-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) ∣ ∀𝑠𝑐𝑔 ∈ (𝒫 𝑥 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)})

Theoremisnacs 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) ∧ ∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔)))

Theoremnacsfg 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) ∧ 𝑆𝐶) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹𝑔))

Theoremisnacs2 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))

Theoremmrefg2 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 = (𝐹𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 = (𝐹𝑔)))

Theoremmrefg3 36289* Slight variation on finite genration for closure systems. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝐹 = (mrCls‘𝐶)       ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑆 ∩ Fin)𝑆 ⊆ (𝐹𝑔)))

Theoremnacsacs 36290 A closure system of Noetherian type is algebraic. (Contributed by Stefan O'Rear, 4-Apr-2015.)
(𝐶 ∈ (NoeACS‘𝑋) → 𝐶 ∈ (ACS‘𝑋))

Theoremisnacs3 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 → 𝑠𝑠)))

Theoremincssnn0 36292* Transitivity induction of subsets, lemma for nacsfix 36293. (Contributed by Stefan O'Rear, 4-Apr-2015.)
((∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1)) ∧ 𝐴 ∈ ℕ0𝐵 ∈ (ℤ𝐴)) → (𝐹𝐴) ⊆ (𝐹𝐵))

Theoremnacsfix 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:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ∃𝑦 ∈ ℕ0𝑧 ∈ (ℤ𝑦)(𝐹𝑧) = (𝐹𝑦))

21.24.6  Miscellanea 1. Map utilities

Theoremconstmap 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       (𝐵𝐶 → (𝐴 × {𝐵}) ∈ (𝐶𝑚 𝐴))

Theoremmapco2g 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 ∈ (𝐵𝑚 𝐶) ∧ 𝐷:𝐸𝐶) → (𝐴𝐷) ∈ (𝐵𝑚 𝐸))

Theoremmapco2 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 ∈ (𝐵𝑚 𝐶) ∧ 𝐷:𝐸𝐶) → (𝐴𝐷) ∈ (𝐵𝑚 𝐸))

Theoremmapfzcons 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 ∈ ℕ0𝐴 ∈ (𝐵𝑚 (1...𝑁)) ∧ 𝐶𝐵) → (𝐴 ∪ {⟨𝑀, 𝐶⟩}) ∈ (𝐵𝑚 (1...𝑀)))

Theoremmapfzcons1 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 ∈ (𝐵𝑚 (1...𝑁)) → ((𝐴 ∪ {⟨𝑀, 𝐶⟩}) ↾ (1...𝑁)) = 𝐴)

Theoremmapfzcons1cl 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 ∈ (𝐵𝑚 (1...𝑀)) → (𝐴 ↾ (1...𝑁)) ∈ (𝐵𝑚 (1...𝑁)))

Theoremmapfzcons2 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 ∈ (𝐵𝑚 (1...𝑁)) ∧ 𝐶𝐵) → ((𝐴 ∪ {⟨𝑀, 𝐶⟩})‘𝑀) = 𝐶)

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