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Theorem List for Metamath Proof Explorer - 35801-35900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlcdvsval 35801 Value of scalar product operation value for the closed kernel vector space dual. (Contributed by NM, 28-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝑆)    &    · = (.r𝑆)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐹 = (Base‘𝐶)    &    = ( ·𝑠𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑅)    &   (𝜑𝐺𝐹)    &   (𝜑𝐴𝑉)       (𝜑 → ((𝑋 𝐺)‘𝐴) = ((𝐺𝐴) · 𝑋))

Theoremlcdvscl 35802 The scalar product operation value is a functional. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝑆)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝐶)    &    · = ( ·𝑠𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑅)    &   (𝜑𝐺𝑉)       (𝜑 → (𝑋 · 𝐺) ∈ 𝑉)

Theoremlcdlssvscl 35803 Closure of scalar product in a closed kernel dual vector space. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝐹)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝐶)    &    · = ( ·𝑠𝐶)    &   𝑆 = (LSubSp‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐿𝑆)    &   (𝜑𝑋𝑅)    &   (𝜑𝑌𝐿)       (𝜑 → (𝑋 · 𝑌) ∈ 𝐿)

Theoremlcdvsass 35804 Associative law for scalar product in a closed kernel dual vector space. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑅 = (Scalar‘𝑈)    &   𝐿 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐹 = (Base‘𝐶)    &    = ( ·𝑠𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝐿)    &   (𝜑𝑌𝐿)    &   (𝜑𝐺𝐹)       (𝜑 → ((𝑌 · 𝑋) 𝐺) = (𝑋 (𝑌 𝐺)))

Theoremlcd0 35805 The zero scalar of the closed kernel dual of a vector space. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)    &    0 = (0g𝐹)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑆 = (Scalar‘𝐶)    &   𝑂 = (0g𝑆)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑂 = 0 )

Theoremlcd1 35806 The unit scalar of the closed kernel dual of a vector space. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)    &    1 = (1r𝐹)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑆 = (Scalar‘𝐶)    &   𝐼 = (1r𝑆)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝐼 = 1 )

Theoremlcdneg 35807 The unit scalar of the closed kernel dual of a vector space. (Contributed by NM, 11-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑅 = (Scalar‘𝑈)    &   𝑀 = (invg𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑆 = (Scalar‘𝐶)    &   𝑁 = (invg𝑆)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑁 = 𝑀)

Theoremlcd0v 35808 The zero functional in the set of functionals with closed kernels. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑅 = (Scalar‘𝑈)    &    0 = (0g𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑂 = (0g𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑂 = (𝑉 × { 0 }))

Theoremlcd0v2 35809 The zero functional in the set of functionals with closed kernels. (Contributed by NM, 27-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐷 = (LDual‘𝑈)    &    0 = (0g𝐷)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑂 = (0g𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑂 = 0 )

Theoremlcd0vvalN 35810 Value of the zero functional at any vector. (Contributed by NM, 28-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑆 = (Scalar‘𝑈)    &    0 = (0g𝑆)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑂 = (0g𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)       (𝜑 → (𝑂𝑋) = 0 )

Theoremlcd0vcl 35811 Closure of the zero functional in the set of functionals with closed kernels. (Contributed by NM, 15-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝐶)    &   𝑂 = (0g𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑂𝑉)

Theoremlcd0vs 35812 A scalar zero times a functional is the zero functional. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑅 = (Scalar‘𝑈)    &    0 = (0g𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝐶)    &    · = ( ·𝑠𝐶)    &   𝑂 = (0g𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝑉)       (𝜑 → ( 0 · 𝐺) = 𝑂)

Theoremlcdvs0N 35813 A scalar times the zero functional is the zero functional. (Contributed by NM, 20-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝑆)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    · = ( ·𝑠𝐶)    &    0 = (0g𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑅)       (𝜑 → (𝑋 · 0 ) = 0 )

Theoremlcdvsub 35814 The value of vector subtraction in the closed kernel dual space. (Contributed by NM, 22-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (Scalar‘𝑈)    &   𝑁 = (invg𝑆)    &    1 = (1r𝑆)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝐶)    &    + = (+g𝐶)    &    · = ( ·𝑠𝐶)    &    = (-g𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝑉)    &   (𝜑𝐺𝑉)       (𝜑 → (𝐹 𝐺) = (𝐹 + ((𝑁1 ) · 𝐺)))

Theoremlcdvsubval 35815 The value of the value of vector addition in the closed kernel vector space dual. (Contributed by NM, 11-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝑆 = (-g𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &    = (-g𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝐺𝐷)    &   (𝜑𝑋𝑉)       (𝜑 → ((𝐹 𝐺)‘𝑋) = ((𝐹𝑋)𝑆(𝐺𝑋)))

Theoremlcdlss 35816* Subspaces of a dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝐶)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑇 = (LSubSp‘𝐷)    &   𝐵 = {𝑓𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑆 = (𝑇 ∩ 𝒫 𝐵))

Theoremlcdlss2N 35817 Subspaces of a dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝐶)    &   𝑉 = (Base‘𝐶)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐷 = (LDual‘𝑈)    &   𝑇 = (LSubSp‘𝐷)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑆 = (𝑇 ∩ 𝒫 𝑉))

Theoremlcdlsp 35818 Span in the set of functionals with closed kernels. (Contributed by NM, 28-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐷 = (LDual‘𝑈)    &   𝑀 = (LSpan‘𝐷)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐹 = (Base‘𝐶)    &   𝑁 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)       (𝜑 → (𝑁𝐺) = (𝑀𝐺))

TheoremlcdlkreqN 35819 Colinear functionals have equal kernels. (Contributed by NM, 28-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐿 = (LKer‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    0 = (0g𝐶)    &   𝑁 = (LSpan‘𝐶)    &   𝑉 = (Base‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐼𝑉)    &   (𝜑𝐺 ∈ (𝑁‘{𝐼}))    &   (𝜑𝐺0 )       (𝜑 → (𝐿𝐺) = (𝐿𝐼))

Theoremlcdlkreq2N 35820 Colinear functionals have equal kernels. (Contributed by NM, 28-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝑆)    &    0 = (0g𝑆)    &   𝐿 = (LKer‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝐶)    &    · = ( ·𝑠𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐴 ∈ (𝑅 ∖ { 0 }))    &   (𝜑𝐼𝑉)    &   (𝜑𝐺 = (𝐴 · 𝐼))       (𝜑 → (𝐿𝐺) = (𝐿𝐼))

Syntaxcmpd 35821 Extend class notation with projectivity from subspaces of vector space H to subspaces of functionals with closed kernels.
class mapd

Definitiondf-mapd 35822* Extend class notation with a one-to-one onto (mapd1o 35845), order-preserving (mapdord 35835) map, called a projectivity (definition of projectivity in [Baer] p. 40), from subspaces of vector space H to those subspaces of the dual space having functionals with closed kernels. (Contributed by NM, 25-Jan-2015.)
mapd = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑠 ∈ (LSubSp‘((DVecH‘𝑘)‘𝑤)) ↦ {𝑓 ∈ (LFnl‘((DVecH‘𝑘)‘𝑤)) ∣ ((((ocH‘𝑘)‘𝑤)‘(((ocH‘𝑘)‘𝑤)‘((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓) ∧ (((ocH‘𝑘)‘𝑤)‘((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓)) ⊆ 𝑠)})))

Theoremmapdffval 35823* Projectivity from vector space H to dual space. (Contributed by NM, 25-Jan-2015.)
𝐻 = (LHyp‘𝐾)       (𝐾𝑋 → (mapd‘𝐾) = (𝑤𝐻 ↦ (𝑠 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤)) ↦ {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑤)) ∣ ((((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓) ∧ (((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)) ⊆ 𝑠)})))

Theoremmapdfval 35824* Projectivity from vector space H to dual space. (Contributed by NM, 25-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)       ((𝐾𝑋𝑊𝐻) → 𝑀 = (𝑠𝑆 ↦ {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑠)}))

Theoremmapdval 35825* Value of projectivity from vector space H to dual space. (Contributed by NM, 27-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   (𝜑 → (𝐾𝑋𝑊𝐻))    &   (𝜑𝑇𝑆)       (𝜑 → (𝑀𝑇) = {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑇)})

Theoremmapdvalc 35826* Value of projectivity from vector space H to dual space. (Contributed by NM, 27-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   (𝜑 → (𝐾𝑋𝑊𝐻))    &   (𝜑𝑇𝑆)    &   𝐶 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) = (𝐿𝑔)}       (𝜑 → (𝑀𝑇) = {𝑓𝐶 ∣ (𝑂‘(𝐿𝑓)) ⊆ 𝑇})

Theoremmapdval2N 35827* Value of projectivity from vector space H to dual space. (Contributed by NM, 31-Jan-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑇𝑆)    &   𝐶 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) = (𝐿𝑔)}       (𝜑 → (𝑀𝑇) = {𝑓𝐶 ∣ ∃𝑣𝑇 (𝑂‘(𝐿𝑓)) = (𝑁‘{𝑣})})

Theoremmapdval3N 35828* Value of projectivity from vector space H to dual space. (Contributed by NM, 31-Jan-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑇𝑆)    &   𝐶 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) = (𝐿𝑔)}       (𝜑 → (𝑀𝑇) = 𝑣𝑇 {𝑓𝐶 ∣ (𝑂‘(𝐿𝑓)) = (𝑁‘{𝑣})})

Theoremmapdval4N 35829* Value of projectivity from vector space H to dual space. TODO: 1. This is shorter than others - make it the official def? (but is not as obvious that it is 𝐶) 2. The unneeded direction of lcfl8a 35700 has awkward - add another thm with only one direction of it? 3. Swap 𝑂‘{𝑣} and 𝐿𝑓? (Contributed by NM, 31-Jan-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑇𝑆)       (𝜑 → (𝑀𝑇) = {𝑓𝐹 ∣ ∃𝑣𝑇 (𝑂‘{𝑣}) = (𝐿𝑓)})

Theoremmapdval5N 35830* Value of projectivity from vector space H to dual space. (Contributed by NM, 31-Jan-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑇𝑆)       (𝜑 → (𝑀𝑇) = 𝑣𝑇 {𝑓𝐹 ∣ (𝑂‘{𝑣}) = (𝐿𝑓)})

Theoremmapdordlem1a 35831* Lemma for mapdord 35835. (Contributed by NM, 27-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑌 = (LSHyp‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝑇 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) ∈ 𝑌}    &   𝐶 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) = (𝐿𝑔)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 → (𝐽𝑇 ↔ (𝐽𝐶 ∧ (𝑂‘(𝑂‘(𝐿𝐽))) ∈ 𝑌)))

Theoremmapdordlem1bN 35832* Lemma for mapdord 35835. (Contributed by NM, 27-Jan-2015.) (New usage is discouraged.)
𝐶 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) = (𝐿𝑔)}       (𝐽𝐶 ↔ (𝐽𝐹 ∧ (𝑂‘(𝑂‘(𝐿𝐽))) = (𝐿𝐽)))

Theoremmapdordlem1 35833* Lemma for mapdord 35835. (Contributed by NM, 27-Jan-2015.)
𝑇 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) ∈ 𝑌}       (𝐽𝑇 ↔ (𝐽𝐹 ∧ (𝑂‘(𝑂‘(𝐿𝐽))) ∈ 𝑌))

Theoremmapdordlem2 35834* Lemma for mapdord 35835. Ordering property of projectivity 𝑀. TODO: This was proved using some hacked-up older proofs. Maybe simplify; get rid of the 𝑇 hypothesis. (Contributed by NM, 27-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑆)    &   (𝜑𝑌𝑆)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝐴 = (LSAtoms‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐽 = (LSHyp‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝑇 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) ∈ 𝐽}    &   𝐶 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) = (𝐿𝑔)}       (𝜑 → ((𝑀𝑋) ⊆ (𝑀𝑌) ↔ 𝑋𝑌))

Theoremmapdord 35835 Ordering property of the map defined by df-mapd 35822. Property (b) of [Baer] p. 40. (Contributed by NM, 27-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑆)    &   (𝜑𝑌𝑆)       (𝜑 → ((𝑀𝑋) ⊆ (𝑀𝑌) ↔ 𝑋𝑌))

Theoremmapd11 35836 The map defined by df-mapd 35822 is one-to-one. Property (c) of [Baer] p. 40. (Contributed by NM, 12-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑆)    &   (𝜑𝑌𝑆)       (𝜑 → ((𝑀𝑋) = (𝑀𝑌) ↔ 𝑋 = 𝑌))

TheoremmapddlssN 35837 The mapping of a subspace of vector space H to the dual space is a subspace of the dual space. TODO: Make this obsolete, use mapdcl2 35853 instead. (Contributed by NM, 31-Jan-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑇 = (LSubSp‘𝐷)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑅𝑆)       (𝜑 → (𝑀𝑅) ∈ 𝑇)

Theoremmapdsn 35838* Value of the map defined by df-mapd 35822 at the span of a singleton. (Contributed by NM, 16-Feb-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)       (𝜑 → (𝑀‘(𝑁‘{𝑋})) = {𝑓𝐹 ∣ (𝑂‘{𝑋}) ⊆ (𝐿𝑓)})

Theoremmapdsn2 35839* Value of the map defined by df-mapd 35822 at the span of a singleton. (Contributed by NM, 16-Feb-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑 → (𝐿𝐺) = (𝑂‘{𝑋}))       (𝜑 → (𝑀‘(𝑁‘{𝑋})) = {𝑓𝐹 ∣ (𝐿𝐺) ⊆ (𝐿𝑓)})

Theoremmapdsn3 35840 Value of the map defined by df-mapd 35822 at the span of a singleton. (Contributed by NM, 17-Feb-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑃 = (LSpan‘𝐷)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝐿𝐺) = (𝑂‘{𝑋}))       (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝑃‘{𝐺}))

Theoremmapd1dim2lem1N 35841* Value of the map defined by df-mapd 35822 at an atom. (Contributed by NM, 10-Feb-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐴 = (LSAtoms‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑄𝐴)       (𝜑 → (𝑀𝑄) = {𝑓𝐹 ∣ ∃𝑣𝑄 (𝑂‘{𝑣}) = (𝐿𝑓)})

Theoremmapdrvallem2 35842* Lemma for mapdrval 35844. TODO: very long antecedents are dragged through proof in some places - see if it shortens proof to remove unused conjuncts. (Contributed by NM, 2-Feb-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑇 = (LSubSp‘𝐷)    &   𝐶 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) = (𝐿𝑔)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑅𝑇)    &   (𝜑𝑅𝐶)    &   𝑄 = 𝑅 (𝑂‘(𝐿))    &   𝑉 = (Base‘𝑈)    &   𝐴 = (LSAtoms‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &    0 = (0g𝑈)    &   𝑌 = (0g𝐷)       (𝜑 → {𝑓𝐶 ∣ (𝑂‘(𝐿𝑓)) ⊆ 𝑄} ⊆ 𝑅)

Theoremmapdrvallem3 35843* Lemma for mapdrval 35844. (Contributed by NM, 2-Feb-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑇 = (LSubSp‘𝐷)    &   𝐶 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) = (𝐿𝑔)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑅𝑇)    &   (𝜑𝑅𝐶)    &   𝑄 = 𝑅 (𝑂‘(𝐿))    &   𝑉 = (Base‘𝑈)    &   𝐴 = (LSAtoms‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &    0 = (0g𝑈)    &   𝑌 = (0g𝐷)       (𝜑 → {𝑓𝐶 ∣ (𝑂‘(𝐿𝑓)) ⊆ 𝑄} = 𝑅)

Theoremmapdrval 35844* Given a dual subspace 𝑅 (of functionals with closed kernels), reconstruct the subspace 𝑄 that maps to it. (Contributed by NM, 12-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑇 = (LSubSp‘𝐷)    &   𝐶 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) = (𝐿𝑔)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑅𝑇)    &   (𝜑𝑅𝐶)    &   𝑄 = 𝑅 (𝑂‘(𝐿))       (𝜑 → (𝑀𝑄) = 𝑅)

Theoremmapd1o 35845* The map defined by df-mapd 35822 is one-to-one and onto the set of dual subspaces of functionals with closed kernels. This shows 𝑀 satisfies part of the definition of projectivity of [Baer] p. 40. TODO: change theorems leading to this (lcfr 35782, mapdrval 35844, lclkrs 35736, lclkr 35730,...) to use 𝑇 ∩ 𝒫 𝐶? TODO: maybe get rid of \$d's for 𝑔 vs. 𝐾𝑈𝑊,. propagate to mapdrn 35846 and any others. (Contributed by NM, 12-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑇 = (LSubSp‘𝐷)    &   𝐶 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) = (𝐿𝑔)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑀:𝑆1-1-onto→(𝑇 ∩ 𝒫 𝐶))

Theoremmapdrn 35846* Range of the map defined by df-mapd 35822. (Contributed by NM, 12-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑇 = (LSubSp‘𝐷)    &   𝐶 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) = (𝐿𝑔)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 → ran 𝑀 = (𝑇 ∩ 𝒫 𝐶))

TheoremmapdunirnN 35847* Union of the range of the map defined by df-mapd 35822. (Contributed by NM, 13-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐶 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) = (𝐿𝑔)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 ran 𝑀 = 𝐶)

Theoremmapdrn2 35848 Range of the map defined by df-mapd 35822. TODO: this seems to be needed a lot in hdmaprnlem3eN 36058 etc. Would it be better to change df-mapd 35822 theorems to use LSubSp‘𝐶 instead of ran 𝑀? (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑇 = (LSubSp‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 → ran 𝑀 = 𝑇)

Theoremmapdcnvcl 35849 Closure of the converse of the map defined by df-mapd 35822. (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝑀)       (𝜑 → (𝑀𝑋) ∈ 𝑆)

Theoremmapdcl 35850 Closure the value of the map defined by df-mapd 35822. (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑆)       (𝜑 → (𝑀𝑋) ∈ ran 𝑀)

Theoremmapdcnvid1N 35851 Converse of the value of the map defined by df-mapd 35822. (Contributed by NM, 13-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑆)       (𝜑 → (𝑀‘(𝑀𝑋)) = 𝑋)

Theoremmapdsord 35852 Strong ordering property of themap defined by df-mapd 35822. (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑆)    &   (𝜑𝑌𝑆)       (𝜑 → ((𝑀𝑋) ⊊ (𝑀𝑌) ↔ 𝑋𝑌))

Theoremmapdcl2 35853 The mapping of a subspace of vector space H is a subspace in the dual space of functionals with closed kernels. (Contributed by NM, 31-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑇 = (LSubSp‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑅𝑆)       (𝜑 → (𝑀𝑅) ∈ 𝑇)

Theoremmapdcnvid2 35854 Value of the converse of the map defined by df-mapd 35822. (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝑀)       (𝜑 → (𝑀‘(𝑀𝑋)) = 𝑋)

TheoremmapdcnvordN 35855 Ordering property of the converse of the map defined by df-mapd 35822. (Contributed by NM, 13-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝑀)    &   (𝜑𝑌 ∈ ran 𝑀)       (𝜑 → ((𝑀𝑋) ⊆ (𝑀𝑌) ↔ 𝑋𝑌))

Theoremmapdcnv11N 35856 The converse of the map defined by df-mapd 35822 is one-to-one. (Contributed by NM, 13-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝑀)    &   (𝜑𝑌 ∈ ran 𝑀)       (𝜑 → ((𝑀𝑋) = (𝑀𝑌) ↔ 𝑋 = 𝑌))

Theoremmapdcv 35857 Covering property of the converse of the map defined by df-mapd 35822. (Contributed by NM, 14-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝐶 = ( ⋖L𝑈)    &   𝐷 = ((LCDual‘𝐾)‘𝑊)    &   𝐸 = ( ⋖L𝐷)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑆)    &   (𝜑𝑌𝑆)       (𝜑 → (𝑋𝐶𝑌 ↔ (𝑀𝑋)𝐸(𝑀𝑌)))

Theoremmapdincl 35858 Closure of dual subspace intersection for the map defined by df-mapd 35822. (Contributed by NM, 12-Apr-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝑀)    &   (𝜑𝑌 ∈ ran 𝑀)       (𝜑 → (𝑋𝑌) ∈ ran 𝑀)

Theoremmapdin 35859 Subspace intersection is preserved by the map defined by df-mapd 35822. Part of property (e) in [Baer] p. 40. (Contributed by NM, 12-Apr-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑆)    &   (𝜑𝑌𝑆)       (𝜑 → (𝑀‘(𝑋𝑌)) = ((𝑀𝑋) ∩ (𝑀𝑌)))

Theoremmapdlsmcl 35860 Closure of dual subspace sum for the map defined by df-mapd 35822. (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = (LSSum‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝑀)    &   (𝜑𝑌 ∈ ran 𝑀)       (𝜑 → (𝑋 𝑌) ∈ ran 𝑀)

Theoremmapdlsm 35861 Subspace sum is preserved by the map defined by df-mapd 35822. Part of property (e) in [Baer] p. 40. (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &    = (LSSum‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = (LSSum‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑆)    &   (𝜑𝑌𝑆)       (𝜑 → (𝑀‘(𝑋 𝑌)) = ((𝑀𝑋) (𝑀𝑌)))

Theoremmapd0 35862 Projectivity map of the zero subspace. Part of property (f) in [Baer] p. 40. TODO: does proof need to be this long for this simple fact? (Contributed by NM, 15-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑂 = (0g𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    0 = (0g𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 → (𝑀‘{𝑂}) = { 0 })

TheoremmapdcnvatN 35863 Atoms are preserved by the map defined by df-mapd 35822. (Contributed by NM, 29-May-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐴 = (LSAtoms‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐵 = (LSAtoms‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑄𝐵)       (𝜑 → (𝑀𝑄) ∈ 𝐴)

Theoremmapdat 35864 Atoms are preserved by the map defined by df-mapd 35822. Property (g) in [Baer] p. 41. (Contributed by NM, 14-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐴 = (LSAtoms‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐵 = (LSAtoms‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑄𝐴)       (𝜑 → (𝑀𝑄) ∈ 𝐵)

Theoremmapdspex 35865* The map of a span equals the dual span of some vector (functional). (Contributed by NM, 15-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐵 = (Base‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)       (𝜑 → ∃𝑔𝐵 (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝑔}))

Theoremmapdn0 35866 Transfer nonzero property from domain to range of projectivity mapd. (Contributed by NM, 12-Apr-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &    0 = (0g𝑈)    &   𝑍 = (0g𝐶)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))       (𝜑𝐹 ∈ (𝐷 ∖ {𝑍}))

Theoremmapdncol 35867 Transfer non-colinearity from domain to range of projectivity mapd. (Contributed by NM, 11-Apr-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝐺𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))       (𝜑 → (𝐽‘{𝐹}) ≠ (𝐽‘{𝐺}))

Theoremmapdindp 35868 Transfer (part of) vector independence condition from domain to range of projectivity mapd. (Contributed by NM, 11-Apr-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝐺𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}))    &   (𝜑𝑍𝑉)    &   (𝜑𝐸𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑍})) = (𝐽‘{𝐸}))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))       (𝜑 → ¬ 𝐹 ∈ (𝐽‘{𝐺, 𝐸}))

Theoremmapdpglem1 35869 Lemma for mapdpg 35903. Baer p. 44, last line: "(F(x-y))* =< (Fx)*+(Fy)*." (Contributed by NM, 15-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)       (𝜑 → (𝑀‘(𝑁‘{(𝑋 𝑌)})) ⊆ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌}))))

Theoremmapdpglem2 35870* Lemma for mapdpg 35903. Baer p. 45, lines 1 and 2: "we have (F(x-y))* = Gt where t necessarily belongs to (Fx)*+(Fy)*." (We scope \$d 𝑡𝜑 locally to avoid clashes with later substitutions into 𝜑.) (Contributed by NM, 15-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)    &   𝐽 = (LSpan‘𝐶)       (𝜑 → ∃𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌})))(𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{𝑡}))

Theoremmapdpglem2a 35871* Lemma for mapdpg 35903. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝐹 = (Base‘𝐶)    &   (𝜑𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌}))))       (𝜑𝑡𝐹)

Theoremmapdpglem3 35872* Lemma for mapdpg 35903. Baer p. 45, line 3: "infer...the existence of a number g in G and of an element z in (Fy)* such that t = gx'-z." (We scope \$d 𝑔𝑤𝑧𝜑 locally to avoid clashes with later substitutions into 𝜑.) (Contributed by NM, 18-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝐹 = (Base‘𝐶)    &   (𝜑𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑅 = (-g𝐶)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))       (𝜑 → ∃𝑔𝐵𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = ((𝑔 · 𝐺)𝑅𝑧))

Theoremmapdpglem4N 35873* Lemma for mapdpg 35903. (Contributed by NM, 20-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝐹 = (Base‘𝐶)    &   (𝜑𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑅 = (-g𝐶)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   𝑄 = (0g𝑈)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))       (𝜑 → (𝑋 𝑌) ≠ 𝑄)

Theoremmapdpglem5N 35874* Lemma for mapdpg 35903. (Contributed by NM, 20-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝐹 = (Base‘𝐶)    &   (𝜑𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑅 = (-g𝐶)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   𝑄 = (0g𝑈)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{𝑡}))       (𝜑𝑡 ≠ (0g𝐶))

Theoremmapdpglem6 35875* Lemma for mapdpg 35903. Baer p. 45, line 4: "If g were 0, then t would be in (Fy)*..." (Contributed by NM, 18-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝐹 = (Base‘𝐶)    &   (𝜑𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑅 = (-g𝐶)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   𝑄 = (0g𝑈)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{𝑡}))    &    0 = (0g𝐴)    &   (𝜑𝑔𝐵)    &   (𝜑𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))    &   (𝜑𝑡 = ((𝑔 · 𝐺)𝑅𝑧))    &   (𝜑𝑋𝑄)    &   (𝜑𝑔 = 0 )       (𝜑𝑡 ∈ (𝑀‘(𝑁‘{𝑌})))

Theoremmapdpglem8 35876* Lemma for mapdpg 35903. Baer p. 45, line 4: "...so that (F(x-y))* =< (Fy)*. This would imply that F(x-y) =< F(y)..." (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝐹 = (Base‘𝐶)    &   (𝜑𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑅 = (-g𝐶)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   𝑄 = (0g𝑈)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{𝑡}))    &    0 = (0g𝐴)    &   (𝜑𝑔𝐵)    &   (𝜑𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))    &   (𝜑𝑡 = ((𝑔 · 𝐺)𝑅𝑧))    &   (𝜑𝑋𝑄)    &   (𝜑𝑔 = 0 )       (𝜑 → (𝑁‘{(𝑋 𝑌)}) ⊆ (𝑁‘{𝑌}))

Theoremmapdpglem9 35877* Lemma for mapdpg 35903. Baer p. 45, line 4: "...so that x would consequently belong to Fy." (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝐹 = (Base‘𝐶)    &   (𝜑𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑅 = (-g𝐶)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   𝑄 = (0g𝑈)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{𝑡}))    &    0 = (0g𝐴)    &   (𝜑𝑔𝐵)    &   (𝜑𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))    &   (𝜑𝑡 = ((𝑔 · 𝐺)𝑅𝑧))    &   (𝜑𝑋𝑄)    &   (𝜑𝑔 = 0 )       (𝜑𝑋 ∈ (𝑁‘{𝑌}))

Theoremmapdpglem10 35878* Lemma for mapdpg 35903. Baer p. 45, line 6: "Hence Fx=Fy, an impossibility." (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝐹 = (Base‘𝐶)    &   (𝜑𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑅 = (-g𝐶)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   𝑄 = (0g𝑈)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{𝑡}))    &    0 = (0g𝐴)    &   (𝜑𝑔𝐵)    &   (𝜑𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))    &   (𝜑𝑡 = ((𝑔 · 𝐺)𝑅𝑧))    &   (𝜑𝑋𝑄)    &   (𝜑𝑔 = 0 )       (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))

Theoremmapdpglem11 35879* Lemma for mapdpg 35903. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝐹 = (Base‘𝐶)    &   (𝜑𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑅 = (-g𝐶)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   𝑄 = (0g𝑈)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{𝑡}))    &    0 = (0g𝐴)    &   (𝜑𝑔𝐵)    &   (𝜑𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))    &   (𝜑𝑡 = ((𝑔 · 𝐺)𝑅𝑧))    &   (𝜑𝑋𝑄)       (𝜑𝑔0 )

Theoremmapdpglem12 35880* Lemma for mapdpg 35903. TODO: Can some commonality with mapdpglem6 35875 through mapdpglem11 35879 be exploited? Also, some consolidation of small lemmas here could be done. (Contributed by NM, 18-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝐹 = (Base‘𝐶)    &   (𝜑𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑅 = (-g𝐶)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   𝑄 = (0g𝑈)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{𝑡}))    &    0 = (0g𝐴)    &   (𝜑𝑔𝐵)    &   (𝜑𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))    &   (𝜑𝑡 = ((𝑔 · 𝐺)𝑅𝑧))    &   (𝜑𝑋𝑄)    &   (𝜑𝑌𝑄)    &   (𝜑𝑧 = (0g𝐶))       (𝜑𝑡 ∈ (𝑀‘(𝑁‘{𝑋})))

Theoremmapdpglem13 35881* Lemma for mapdpg 35903. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝐹 = (Base‘𝐶)    &   (𝜑𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑅 = (-g𝐶)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   𝑄 = (0g𝑈)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{𝑡}))    &    0 = (0g𝐴)    &   (𝜑𝑔𝐵)    &   (𝜑𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))    &   (𝜑𝑡 = ((𝑔 · 𝐺)𝑅𝑧))    &   (𝜑𝑋𝑄)    &   (𝜑𝑌𝑄)    &   (𝜑𝑧 = (0g𝐶))       (𝜑 → (𝑁‘{(𝑋 𝑌)}) ⊆ (𝑁‘{𝑋}))

Theoremmapdpglem14 35882* Lemma for mapdpg 35903. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝐹 = (Base‘𝐶)    &   (𝜑𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑅 = (-g𝐶)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   𝑄 = (0g𝑈)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{𝑡}))    &    0 = (0g𝐴)    &   (𝜑𝑔𝐵)    &   (𝜑𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))    &   (𝜑𝑡 = ((𝑔 · 𝐺)𝑅𝑧))    &   (𝜑𝑋𝑄)    &   (𝜑𝑌𝑄)    &   (𝜑𝑧 = (0g𝐶))       (𝜑𝑌 ∈ (𝑁‘{𝑋}))

Theoremmapdpglem15 35883* Lemma for mapdpg 35903. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝐹 = (Base‘𝐶)    &   (𝜑𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑅 = (-g𝐶)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   𝑄 = (0g𝑈)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{𝑡}))    &    0 = (0g𝐴)    &   (𝜑𝑔𝐵)    &   (𝜑𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))    &   (𝜑𝑡 = ((𝑔 · 𝐺)𝑅𝑧))    &   (𝜑𝑋𝑄)    &   (𝜑𝑌𝑄)    &   (𝜑𝑧 = (0g𝐶))       (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))

Theoremmapdpglem16 35884* Lemma for mapdpg 35903. Baer p. 45, line 7: "Likewise we see that z =/= 0." (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝐹 = (Base‘𝐶)    &   (𝜑𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑅 = (-g𝐶)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   𝑄 = (0g𝑈)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{𝑡}))    &    0 = (0g𝐴)    &   (𝜑𝑔𝐵)    &   (𝜑𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))    &   (𝜑𝑡 = ((𝑔 · 𝐺)𝑅𝑧))    &   (𝜑𝑋𝑄)    &   (𝜑𝑌𝑄)       (𝜑𝑧 ≠ (0g𝐶))

Theoremmapdpglem17N 35885* Lemma for mapdpg 35903. Baer p. 45, line 7: "Hence we may form y' = g^-1 z." (Contributed by NM, 20-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝐹 = (Base‘𝐶)    &   (𝜑𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑅 = (-g𝐶)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   𝑄 = (0g𝑈)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{𝑡}))    &    0 = (0g𝐴)    &   (𝜑𝑔𝐵)    &   (𝜑𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))    &   (𝜑𝑡 = ((𝑔 · 𝐺)𝑅𝑧))    &   (𝜑𝑋𝑄)    &   (𝜑𝑌𝑄)    &   𝐸 = (((invr𝐴)‘𝑔) · 𝑧)       (𝜑𝐸𝐹)

Theoremmapdpglem18 35886* Lemma for mapdpg 35903. Baer p. 45, line 7: "Then y =/= 0..." (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝐹 = (Base‘𝐶)    &   (𝜑𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑅 = (-g𝐶)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   𝑄 = (0g𝑈)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{𝑡}))    &    0 = (0g𝐴)    &   (𝜑𝑔𝐵)    &   (𝜑𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))    &   (𝜑𝑡 = ((𝑔 · 𝐺)𝑅𝑧))    &   (𝜑𝑋𝑄)    &   (𝜑𝑌𝑄)    &   𝐸 = (((invr𝐴)‘𝑔) · 𝑧)       (𝜑𝐸 ≠ (0g𝐶))

Theoremmapdpglem19 35887* Lemma for mapdpg 35903. Baer p. 45, line 8: "...is in (Fy)*..." (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝐹 = (Base‘𝐶)    &   (𝜑𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑅 = (-g𝐶)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   𝑄 = (0g𝑈)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{𝑡}))    &    0 = (0g𝐴)    &   (𝜑𝑔𝐵)    &   (𝜑𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))    &   (𝜑𝑡 = ((𝑔 · 𝐺)𝑅𝑧))    &   (𝜑𝑋𝑄)    &   (𝜑𝑌𝑄)    &   𝐸 = (((invr𝐴)‘𝑔) · 𝑧)       (𝜑𝐸 ∈ (𝑀‘(𝑁‘{𝑌})))

Theoremmapdpglem20 35888* Lemma for mapdpg 35903. Baer p. 45, line 8: "...so that (Fy)*=Gy'." (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝐹 = (Base‘𝐶)    &   (𝜑𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑅 = (-g𝐶)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   𝑄 = (0g𝑈)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{𝑡}))    &    0 = (0g𝐴)    &   (𝜑𝑔𝐵)    &   (𝜑𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))    &   (𝜑𝑡 = ((𝑔 · 𝐺)𝑅𝑧))    &   (𝜑𝑋𝑄)    &   (𝜑𝑌𝑄)    &   𝐸 = (((invr𝐴)‘𝑔) · 𝑧)       (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐸}))

Theoremmapdpglem21 35889* Lemma for mapdpg 35903. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝐹 = (Base‘𝐶)    &   (𝜑𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑅 = (-g𝐶)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   𝑄 = (0g𝑈)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{𝑡}))    &    0 = (0g𝐴)    &   (𝜑𝑔𝐵)    &   (𝜑𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))    &   (𝜑𝑡 = ((𝑔 · 𝐺)𝑅𝑧))    &   (𝜑𝑋𝑄)    &   (𝜑𝑌𝑄)    &   𝐸 = (((invr𝐴)‘𝑔) · 𝑧)       (𝜑 → (((invr𝐴)‘𝑔) · 𝑡) = (𝐺𝑅𝐸))

Theoremmapdpglem22 35890* Lemma for mapdpg 35903. Baer p. 45, line 9: "(F(x-y))* = ... = G(x'-y')." (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝐹 = (Base‘𝐶)    &   (𝜑𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑅 = (-g𝐶)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   𝑄 = (0g𝑈)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{𝑡}))    &    0 = (0g𝐴)    &   (𝜑𝑔𝐵)    &   (𝜑𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))    &   (𝜑𝑡 = ((𝑔 · 𝐺)𝑅𝑧))    &   (𝜑𝑋𝑄)    &   (𝜑𝑌𝑄)    &   𝐸 = (((invr𝐴)‘𝑔) · 𝑧)       (𝜑 → (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐺𝑅𝐸)}))

Theoremmapdpglem23 35891* Lemma for mapdpg 35903. Baer p. 45, line 10: "and so y' meets all our requirements." Our is Baer's y'. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝐹 = (Base‘𝐶)    &   (𝜑𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑅 = (-g𝐶)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   𝑄 = (0g𝑈)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{𝑡}))    &    0 = (0g𝐴)    &   (𝜑𝑔𝐵)    &   (𝜑𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))    &   (𝜑𝑡 = ((𝑔 · 𝐺)𝑅𝑧))    &   (𝜑𝑋𝑄)    &   (𝜑𝑌𝑄)    &   𝐸 = (((invr𝐴)‘𝑔) · 𝑧)       (𝜑 → ∃𝐹 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐺𝑅)})))

Theoremmapdpglem30a 35892 Lemma for mapdpg 35903. (Contributed by NM, 22-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐹 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))       (𝜑𝐺 ≠ (0g𝐶))

Theoremmapdpglem30b 35893 Lemma for mapdpg 35903. (Contributed by NM, 22-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐹 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   (𝜑 → (𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐺𝑅)}))))    &   (𝜑 → (𝑖𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)}))))       (𝜑𝑖 ≠ (0g𝐶))

Theoremmapdpglem25 35894 Lemma for mapdpg 35903. Baer p. 45 line 12: "Then we have Gy' = Gy'' and G(x'-y') = G(x'-y'')." (Contributed by NM, 21-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐹 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   (𝜑 → (𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐺𝑅)}))))    &   (𝜑 → (𝑖𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)}))))       (𝜑 → ((𝐽‘{}) = (𝐽‘{𝑖}) ∧ (𝐽‘{(𝐺𝑅)}) = (𝐽‘{(𝐺𝑅𝑖)})))

Theoremmapdpglem26 35895* Lemma for mapdpg 35903. Baer p. 45 line 14: "Consequently there exist numbers u,v in G neither of which is 0 such that y = uy'' and..." (We scope \$d 𝑢𝜑 locally to avoid clashes with later substitutions into 𝜑.) (Contributed by NM, 22-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐹 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   (𝜑 → (𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐺𝑅)}))))    &   (𝜑 → (𝑖𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑂 = (0g𝐴)       (𝜑 → ∃𝑢 ∈ (𝐵 ∖ {𝑂}) = (𝑢 · 𝑖))

Theoremmapdpglem27 35896* Lemma for mapdpg 35903. Baer p. 45 line 16: "v(x'-y'') = x'-y'" (with equality swapped). (Contributed by NM, 22-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐹 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   (𝜑 → (𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐺𝑅)}))))    &   (𝜑 → (𝑖𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑂 = (0g𝐴)       (𝜑 → ∃𝑣 ∈ (𝐵 ∖ {𝑂})(𝐺𝑅) = (𝑣 · (𝐺𝑅𝑖)))

Theoremmapdpglem29 35897* Lemma for mapdpg 35903. Baer p. 45 line 16: "But Gx' and Gy'' are distinct points and so x' and y'' are independent elements in B. (Contributed by NM, 22-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐹 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   (𝜑 → (𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐺𝑅)}))))    &   (𝜑 → (𝑖𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑂 = (0g𝐴)    &   (𝜑𝑣𝐵)    &   (𝜑 = (𝑢 · 𝑖))    &   (𝜑 → (𝐺𝑅) = (𝑣 · (𝐺𝑅𝑖)))       (𝜑 → (𝐽‘{𝐺}) ≠ (𝐽‘{𝑖}))

Theoremmapdpglem28 35898* Lemma for mapdpg 35903. Baer p. 45 line 18: "vx'-vy'' = x'-uy''". (Contributed by NM, 22-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐹 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   (𝜑 → (𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐺𝑅)}))))    &   (𝜑 → (𝑖𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑂 = (0g𝐴)    &   (𝜑𝑣𝐵)    &   (𝜑 = (𝑢 · 𝑖))    &   (𝜑 → (𝐺𝑅) = (𝑣 · (𝐺𝑅𝑖)))       (𝜑 → ((𝑣 · 𝐺)𝑅(𝑣 · 𝑖)) = (𝐺𝑅(𝑢 · 𝑖)))

Theoremmapdpglem30 35899* Lemma for mapdpg 35903. Baer p. 45 line 18: "Hence we deduce (from mapdpglem28 35898, using lvecindp2 18864) that v = 1 and v = u...". TODO: would it be shorter to have only the 𝑣 = (1r𝐴) part and use mapdpglem28.u2 in mapdpglem31 35900? (Contributed by NM, 22-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐹 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   (𝜑 → (𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐺𝑅)}))))    &   (𝜑 → (𝑖𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑂 = (0g𝐴)    &   (𝜑𝑣𝐵)    &   (𝜑 = (𝑢 · 𝑖))    &   (𝜑 → (𝐺𝑅) = (𝑣 · (𝐺𝑅𝑖)))    &   (𝜑𝑢𝐵)       (𝜑 → (𝑣 = (1r𝐴) ∧ 𝑣 = 𝑢))

Theoremmapdpglem31 35900* Lemma for mapdpg 35903. Baer p. 45 line 19: "...and we have consequently that y' = y'', as we claimed." (Contributed by NM, 23-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐹 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   (𝜑 → (𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐺𝑅)}))))    &   (𝜑 → (𝑖𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑂 = (0g𝐴)    &   (𝜑𝑣𝐵)    &   (𝜑 = (𝑢 · 𝑖))    &   (𝜑 → (𝐺𝑅) = (𝑣 · (𝐺𝑅𝑖)))    &   (𝜑𝑢𝐵)       (𝜑 = 𝑖)

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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42426
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