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Theorem List for Metamath Proof Explorer - 39101-39200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmapdcnvid1N 39101 Converse of the value of the map defined by df-mapd 39072. (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 39102 Strong ordering property of themap defined by df-mapd 39072. (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 39103 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 39104 Value of the converse of the map defined by df-mapd 39072. (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 39105 Ordering property of the converse of the map defined by df-mapd 39072. (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 39106 The converse of the map defined by df-mapd 39072 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 39107 Covering property of the converse of the map defined by df-mapd 39072. (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 39108 Closure of dual subspace intersection for the map defined by df-mapd 39072. (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 39109 Subspace intersection is preserved by the map defined by df-mapd 39072. 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 39110 Closure of dual subspace sum for the map defined by df-mapd 39072. (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 39111 Subspace sum is preserved by the map defined by df-mapd 39072. 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 39112 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 39113 Atoms are preserved by the map defined by df-mapd 39072. (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 39114 Atoms are preserved by the map defined by df-mapd 39072. 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 39115* 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 39116 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 39117 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 39118 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 39119 Lemma for mapdpg 39153. 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 39120* Lemma for mapdpg 39153. 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 39121* Lemma for mapdpg 39153. (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 39122* Lemma for mapdpg 39153. 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 39123* Lemma for mapdpg 39153. (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 39124* Lemma for mapdpg 39153. (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 39125* Lemma for mapdpg 39153. 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 39126* Lemma for mapdpg 39153. 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 39127* Lemma for mapdpg 39153. 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 39128* Lemma for mapdpg 39153. 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 39129* Lemma for mapdpg 39153. (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 39130* Lemma for mapdpg 39153. TODO: Can some commonality with mapdpglem6 39125 through mapdpglem11 39129 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 39131* Lemma for mapdpg 39153. (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 39132* Lemma for mapdpg 39153. (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 39133* Lemma for mapdpg 39153. (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 39134* Lemma for mapdpg 39153. 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 39135* Lemma for mapdpg 39153. 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 39136* Lemma for mapdpg 39153. 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 39137* Lemma for mapdpg 39153. 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 39138* Lemma for mapdpg 39153. 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 39139* Lemma for mapdpg 39153. (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 39140* Lemma for mapdpg 39153. 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 39141* Lemma for mapdpg 39153. 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 39142 Lemma for mapdpg 39153. (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 39143 Lemma for mapdpg 39153. (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 39144 Lemma for mapdpg 39153. 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 39145* Lemma for mapdpg 39153. 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 39146* Lemma for mapdpg 39153. 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 39147* Lemma for mapdpg 39153. 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 39148* Lemma for mapdpg 39153. 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 39149* Lemma for mapdpg 39153. Baer p. 45 line 18: "Hence we deduce (from mapdpglem28 39148, using lvecindp2 19925) that v = 1 and v = u...". TODO: would it be shorter to have only the 𝑣 = (1r𝐴) part and use mapdpglem28.u2 in mapdpglem31 39150? (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 39150* Lemma for mapdpg 39153. 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)

Theoremmapdpglem24 39151* Lemma for mapdpg 39153. Existence part - consolidate hypotheses in mapdpglem23 39141. (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)})))

Theoremmapdpglem32 39152* Lemma for mapdpg 39153. Uniqueness part - consolidate hypotheses in mapdpglem31 39150. (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 ∧ (𝐹𝑖𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐺𝑅)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → = 𝑖)

Theoremmapdpg 39153* Part 1 of proof of the first fundamental theorem of projective geometry. Part (1) in [Baer] p. 44. Our notation corresponds to Baer's as follows: 𝑀 for *, 𝑁‘{} for F(), 𝐽‘{} for G(), 𝑋 for x, 𝐺 for x', 𝑌 for y, for y'. TODO: Rename variables per mapdhval 39171. (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)})))

Theorembaerlem3lem1 39154 Lemma for baerlem3 39160. (Contributed by NM, 9-Apr-2015.)
𝑉 = (Base‘𝑊)    &    = (-g𝑊)    &    0 = (0g𝑊)    &    = (LSSum‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋𝑉)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &    + = (+g𝑊)    &    · = ( ·𝑠𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝑅)    &    = (+g𝑅)    &   𝐿 = (-g𝑅)    &   𝑄 = (0g𝑅)    &   𝐼 = (invg𝑅)    &   (𝜑𝑎𝐵)    &   (𝜑𝑏𝐵)    &   (𝜑𝑑𝐵)    &   (𝜑𝑒𝐵)    &   (𝜑𝑗 = ((𝑎 · 𝑌) + (𝑏 · 𝑍)))    &   (𝜑𝑗 = ((𝑑 · (𝑋 𝑌)) + (𝑒 · (𝑋 𝑍))))       (𝜑𝑗 = (𝑎 · (𝑌 𝑍)))

Theorembaerlem5alem1 39155 Lemma for baerlem5a 39161. (Contributed by NM, 13-Apr-2015.)
𝑉 = (Base‘𝑊)    &    = (-g𝑊)    &    0 = (0g𝑊)    &    = (LSSum‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋𝑉)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &    + = (+g𝑊)    &    · = ( ·𝑠𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝑅)    &    = (+g𝑅)    &   𝐿 = (-g𝑅)    &   𝑄 = (0g𝑅)    &   𝐼 = (invg𝑅)    &   (𝜑𝑎𝐵)    &   (𝜑𝑏𝐵)    &   (𝜑𝑑𝐵)    &   (𝜑𝑒𝐵)    &   (𝜑𝑗 = ((𝑎 · (𝑋 𝑌)) + (𝑏 · 𝑍)))    &   (𝜑𝑗 = ((𝑑 · (𝑋 𝑍)) + (𝑒 · 𝑌)))       (𝜑𝑗 = (𝑎 · (𝑋 (𝑌 + 𝑍))))

Theorembaerlem5blem1 39156 Lemma for baerlem5b 39162. (Contributed by NM, 9-Apr-2015.)
𝑉 = (Base‘𝑊)    &    = (-g𝑊)    &    0 = (0g𝑊)    &    = (LSSum‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋𝑉)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &    + = (+g𝑊)    &    · = ( ·𝑠𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝑅)    &    = (+g𝑅)    &   𝐿 = (-g𝑅)    &   𝑄 = (0g𝑅)    &   𝐼 = (invg𝑅)    &   (𝜑𝑎𝐵)    &   (𝜑𝑏𝐵)    &   (𝜑𝑑𝐵)    &   (𝜑𝑒𝐵)    &   (𝜑𝑗 = ((𝑎 · 𝑌) + (𝑏 · 𝑍)))    &   (𝜑𝑗 = ((𝑑 · (𝑋 (𝑌 + 𝑍))) + (𝑒 · 𝑋)))       (𝜑𝑗 = ((𝐼𝑑) · (𝑌 + 𝑍)))

Theorembaerlem3lem2 39157 Lemma for baerlem3 39160. (Contributed by NM, 9-Apr-2015.)
𝑉 = (Base‘𝑊)    &    = (-g𝑊)    &    0 = (0g𝑊)    &    = (LSSum‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋𝑉)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &    + = (+g𝑊)    &    · = ( ·𝑠𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝑅)    &    = (+g𝑅)    &   𝐿 = (-g𝑅)    &   𝑄 = (0g𝑅)    &   𝐼 = (invg𝑅)       (𝜑 → (𝑁‘{(𝑌 𝑍)}) = (((𝑁‘{𝑌}) (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 𝑌)}) (𝑁‘{(𝑋 𝑍)}))))

Theorembaerlem5alem2 39158 Lemma for baerlem5a 39161. (Contributed by NM, 9-Apr-2015.)
𝑉 = (Base‘𝑊)    &    = (-g𝑊)    &    0 = (0g𝑊)    &    = (LSSum‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋𝑉)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &    + = (+g𝑊)    &    · = ( ·𝑠𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝑅)    &    = (+g𝑅)    &   𝐿 = (-g𝑅)    &   𝑄 = (0g𝑅)    &   𝐼 = (invg𝑅)       (𝜑 → (𝑁‘{(𝑋 (𝑌 + 𝑍))}) = (((𝑁‘{(𝑋 𝑌)}) (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 𝑍)}) (𝑁‘{𝑌}))))

Theorembaerlem5blem2 39159 Lemma for baerlem5b 39162. (Contributed by NM, 13-Apr-2015.)
𝑉 = (Base‘𝑊)    &    = (-g𝑊)    &    0 = (0g𝑊)    &    = (LSSum‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋𝑉)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &    + = (+g𝑊)    &    · = ( ·𝑠𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝑅)    &    = (+g𝑅)    &   𝐿 = (-g𝑅)    &   𝑄 = (0g𝑅)    &   𝐼 = (invg𝑅)       (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) = (((𝑁‘{𝑌}) (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 (𝑌 + 𝑍))}) (𝑁‘{𝑋}))))

Theorembaerlem3 39160 An equality that holds when 𝑋, 𝑌, 𝑍 are independent (non-colinear) vectors. Part (3) in [Baer] p. 45. TODO fix ref. (Contributed by NM, 9-Apr-2015.)
𝑉 = (Base‘𝑊)    &    = (-g𝑊)    &    0 = (0g𝑊)    &    = (LSSum‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋𝑉)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))       (𝜑 → (𝑁‘{(𝑌 𝑍)}) = (((𝑁‘{𝑌}) (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 𝑌)}) (𝑁‘{(𝑋 𝑍)}))))

Theorembaerlem5a 39161 An equality that holds when 𝑋, 𝑌, 𝑍 are independent (non-colinear) vectors. First equation of part (5) in [Baer] p. 46. (Contributed by NM, 10-Apr-2015.)
𝑉 = (Base‘𝑊)    &    = (-g𝑊)    &    0 = (0g𝑊)    &    = (LSSum‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋𝑉)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &    + = (+g𝑊)       (𝜑 → (𝑁‘{(𝑋 (𝑌 + 𝑍))}) = (((𝑁‘{(𝑋 𝑌)}) (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 𝑍)}) (𝑁‘{𝑌}))))

Theorembaerlem5b 39162 An equality that holds when 𝑋, 𝑌, 𝑍 are independent (non-colinear) vectors. Second equation of part (5) in [Baer] p. 46. (Contributed by NM, 13-Apr-2015.)
𝑉 = (Base‘𝑊)    &    = (-g𝑊)    &    0 = (0g𝑊)    &    = (LSSum‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋𝑉)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &    + = (+g𝑊)       (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) = (((𝑁‘{𝑌}) (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 (𝑌 + 𝑍))}) (𝑁‘{𝑋}))))

Theorembaerlem5amN 39163 An equality that holds when 𝑋, 𝑌, 𝑍 are independent (non-colinear) vectors. Subtraction version of first equation of part (5) in [Baer] p. 46. TODO: This is the subtraction version, may not be needed. TODO: delete if baerlem5abmN 39165 is used. (Contributed by NM, 24-May-2015.) (New usage is discouraged.)
𝑉 = (Base‘𝑊)    &    = (-g𝑊)    &    0 = (0g𝑊)    &    = (LSSum‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋𝑉)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &    + = (+g𝑊)       (𝜑 → (𝑁‘{(𝑋 (𝑌 𝑍))}) = (((𝑁‘{(𝑋 𝑌)}) (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 + 𝑍)}) (𝑁‘{𝑌}))))

Theorembaerlem5bmN 39164 An equality that holds when 𝑋, 𝑌, 𝑍 are independent (non-colinear) vectors. Subtraction version of second equation of part (5) in [Baer] p. 46. TODO: This is the subtraction version, may not be needed. TODO: delete if baerlem5abmN 39165 is used. (Contributed by NM, 24-May-2015.) (New usage is discouraged.)
𝑉 = (Base‘𝑊)    &    = (-g𝑊)    &    0 = (0g𝑊)    &    = (LSSum‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋𝑉)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &    + = (+g𝑊)       (𝜑 → (𝑁‘{(𝑌 𝑍)}) = (((𝑁‘{𝑌}) (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 (𝑌 𝑍))}) (𝑁‘{𝑋}))))

Theorembaerlem5abmN 39165 An equality that holds when 𝑋, 𝑌, 𝑍 are independent (non-colinear) vectors. Subtraction versions of first and second equations of part (5) in [Baer] p. 46, conjoined to share commonality in their proofs. TODO: Delete if not needed. (Contributed by NM, 24-May-2015.) (New usage is discouraged.)
𝑉 = (Base‘𝑊)    &    = (-g𝑊)    &    0 = (0g𝑊)    &    = (LSSum‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋𝑉)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &    + = (+g𝑊)       (𝜑 → ((𝑁‘{(𝑋 (𝑌 𝑍))}) = (((𝑁‘{(𝑋 𝑌)}) (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 + 𝑍)}) (𝑁‘{𝑌}))) ∧ (𝑁‘{(𝑌 𝑍)}) = (((𝑁‘{𝑌}) (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 (𝑌 𝑍))}) (𝑁‘{𝑋})))))

Theoremmapdindp0 39166 Vector independence lemma. (Contributed by NM, 29-Apr-2015.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑤 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))    &   (𝜑 → (𝑌 + 𝑍) ≠ 0 )       (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) = (𝑁‘{𝑌}))

Theoremmapdindp1 39167 Vector independence lemma. (Contributed by NM, 1-May-2015.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑤 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))       (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑌 + 𝑍)}))

Theoremmapdindp2 39168 Vector independence lemma. (Contributed by NM, 1-May-2015.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑤 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))       (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, (𝑌 + 𝑍)}))

Theoremmapdindp3 39169 Vector independence lemma. (Contributed by NM, 29-Apr-2015.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑤 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))       (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑤 + 𝑌)}))

Theoremmapdindp4 39170 Vector independence lemma. (Contributed by NM, 29-Apr-2015.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑤 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))       (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, (𝑤 + 𝑌)}))

Theoremmapdhval 39171* Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 6-May-2015.)
𝑄 = (0g𝐶)    &   𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))    &   (𝜑𝑋𝐴)    &   (𝜑𝐹𝐵)    &   (𝜑𝑌𝐸)       (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if(𝑌 = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐹𝑅)})))))

Theoremmapdhval0 39172* Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015.)
𝑄 = (0g𝐶)    &   𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))    &    0 = (0g𝑈)    &   (𝜑𝑋𝐴)    &   (𝜑𝐹𝐵)       (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = 𝑄)

Theoremmapdhval2 39173* Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015.)
𝑄 = (0g𝐶)    &   𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))    &   (𝜑𝑋𝐴)    &   (𝜑𝐹𝐵)    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))       (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐹𝑅)}))))

Theoremmapdhcl 39174* Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015.)
𝑄 = (0g𝐶)    &   𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))    &   𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌𝑉)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))       (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷)

Theoremmapdheq 39175* Lemmma for ~? mapdh . The defining equation for h(x,x',y)=y' in part (2) in [Baer] p. 45 line 24. (Contributed by NM, 4-Apr-2015.)
𝑄 = (0g𝐶)    &   𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))    &   𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐺𝐷)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))       (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺 ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)}))))

Theoremmapdheq2 39176* Lemmma for ~? mapdh . One direction of part (2) in [Baer] p. 45. (Contributed by NM, 4-Apr-2015.)
𝑄 = (0g𝐶)    &   𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))    &   𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐺𝐷)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))       (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺 → (𝐼‘⟨𝑌, 𝐺, 𝑋⟩) = 𝐹))

Theoremmapdheq2biN 39177* Lemmma for ~? mapdh . Part (2) in [Baer] p. 45. The bidirectional version of mapdheq2 39176 seems to require an additional hypothesis not mentioned in Baer. TODO fix ref. TODO: We probably don't need this; delete if never used. (Contributed by NM, 4-Apr-2015.) (New usage is discouraged.)
𝑄 = (0g𝐶)    &   𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))    &   𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐺𝐷)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}))       (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺 ↔ (𝐼‘⟨𝑌, 𝐺, 𝑋⟩) = 𝐹))

Theoremmapdheq4lem 39178* Lemma for mapdheq4 39179. Part (4) in [Baer] p. 46. (Contributed by NM, 12-Apr-2015.)
𝑄 = (0g𝐶)    &   𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))    &   𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))    &   (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)    &   (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)       (𝜑 → (𝑀‘(𝑁‘{(𝑌 𝑍)})) = (𝐽‘{(𝐺𝑅𝐸)}))

Theoremmapdheq4 39179* Lemma for ~? mapdh . Part (4) in [Baer] p. 46. (Contributed by NM, 12-Apr-2015.)
𝑄 = (0g𝐶)    &   𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))    &   𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))    &   (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)    &   (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)       (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑍⟩) = 𝐸)

Theoremmapdh6lem1N 39180* Lemma for mapdh6N 39194. Part (6) in [Baer] p. 47, lines 16-18. (Contributed by NM, 13-Apr-2015.) (New usage is discouraged.)
𝑄 = (0g𝐶)    &   𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))    &   𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &    + = (+g𝑈)    &    = (+g𝐶)    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))    &   (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)    &   (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)       (𝜑 → (𝑀‘(𝑁‘{(𝑋 (𝑌 + 𝑍))})) = (𝐽‘{(𝐹𝑅(𝐺 𝐸))}))

Theoremmapdh6lem2N 39181* Lemma for mapdh6N 39194. Part (6) in [Baer] p. 47, lines 20-22. (Contributed by NM, 13-Apr-2015.) (New usage is discouraged.)
𝑄 = (0g𝐶)    &   𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))    &   𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &    + = (+g𝑈)    &    = (+g𝐶)    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))    &   (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)    &   (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)       (𝜑 → (𝑀‘(𝑁‘{(𝑌 + 𝑍)})) = (𝐽‘{(𝐺 𝐸)}))

Theoremmapdh6aN 39182* Lemma for mapdh6N 39194. Part (6) in [Baer] p. 47, case 1. (Contributed by NM, 23-Apr-2015.) (New usage is discouraged.)
𝑄 = (0g𝐶)    &   𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))    &   𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &    + = (+g𝑈)    &    = (+g𝐶)    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))    &   (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)    &   (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)       (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

Theoremmapdh6b0N 39183* Lemmma for mapdh6N 39194. (Contributed by NM, 23-Apr-2015.) (New usage is discouraged.)
𝑄 = (0g𝐶)    &   𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))    &   𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &    + = (+g𝑈)    &    = (+g𝐶)    &   (𝜑𝑌𝑉)    &   (𝜑𝑍𝑉)    &   (𝜑 → ((𝑁‘{𝑋}) ∩ (𝑁‘{𝑌, 𝑍})) = { 0 })       (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))

Theoremmapdh6bN 39184* Lemmma for mapdh6N 39194. (Contributed by NM, 24-Apr-2015.) (New usage is discouraged.)
𝑄 = (0g𝐶)    &   𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))    &   𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &    + = (+g𝑈)    &    = (+g𝐶)    &   (𝜑𝑌 = 0 )    &   (𝜑𝑍𝑉)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))       (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

Theoremmapdh6cN 39185* Lemmma for mapdh6N 39194. (Contributed by NM, 24-Apr-2015.) (New usage is discouraged.)
𝑄 = (0g𝐶)    &   𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))    &   𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &    + = (+g𝑈)    &    = (+g𝐶)    &   (𝜑𝑌𝑉)    &   (𝜑𝑍 = 0 )    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))       (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

Theoremmapdh6dN 39186* Lemmma for mapdh6N 39194. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
𝑄 = (0g𝐶)    &   𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))    &   𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &    + = (+g𝑈)    &    = (+g𝐶)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑤 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))       (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑤 + (𝑌 + 𝑍))⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩)))

Theoremmapdh6eN 39187* Lemmma for mapdh6N 39194. Part (6) in [Baer] p. 47 line 38. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
𝑄 = (0g𝐶)    &   𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))    &   𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &    + = (+g𝑈)    &    = (+g𝐶)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑤 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))       (𝜑 → (𝐼‘⟨𝑋, 𝐹, ((𝑤 + 𝑌) + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, (𝑤 + 𝑌)⟩) (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

Theoremmapdh6fN 39188* Lemmma for mapdh6N 39194. Part (6) in [Baer] p. 47 line 38. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
𝑄 = (0g𝐶)    &   𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))    &   𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &    + = (+g𝑈)    &    = (+g𝐶)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑤 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))       (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑤 + 𝑌)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) (𝐼‘⟨𝑋, 𝐹, 𝑌⟩)))

Theoremmapdh6gN 39189* Lemmma for mapdh6N 39194. Part (6) of [Baer] p. 47 line 39. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
𝑄 = (0g𝐶)    &   𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))    &   𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &    + = (+g𝑈)    &    = (+g𝐶)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑤 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))       (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩)) = (((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) (𝐼‘⟨𝑋, 𝐹, 𝑌⟩)) (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

Theoremmapdh6hN 39190* Lemmma for mapdh6N 39194. Part (6) of [Baer] p. 48 line 2. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
𝑄 = (0g𝐶)    &   𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))    &   𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &    + = (+g𝑈)    &    = (+g𝐶)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑤 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))       (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

Theoremmapdh6iN 39191* Lemmma for mapdh6N 39194. Eliminate auxiliary vector 𝑤. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
𝑄 = (0g𝐶)    &   𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))    &   𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &    + = (+g𝑈)    &    = (+g𝐶)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))       (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

Theoremmapdh6jN 39192* Lemmma for mapdh6N 39194. Eliminate (𝑁‘{𝑌}) = (𝑁‘{𝑍}) hypothesis. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
𝑄 = (0g𝐶)    &   𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))    &   𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &    + = (+g𝑈)    &    = (+g𝐶)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))       (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

Theoremmapdh6kN 39193* Lemmma for mapdh6N 39194. Eliminate nonzero vector requirement. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
𝑄 = (0g𝐶)    &   𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))    &   𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &    + = (+g𝑈)    &    = (+g𝐶)    &   (𝜑𝑌𝑉)    &   (𝜑𝑍𝑉)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))       (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

Theoremmapdh6N 39194* Part (6) of [Baer] p. 47 line 6. Note that we use ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}) which is equivalent to Baer's "Fx (Fy + Fz)" by lspdisjb 19912. TODO: If disjoint variable conditions with 𝐼 and 𝜑 become a problem later, use cbv* theorems on 𝐼 variables here to get rid of them. Maybe reorder hypotheses in lemmas to the more consistent order of this theorem, so they can be shared with this theorem. TODO: may be deleted (with its lemmas), if not needed, in view of hdmap1l6 39268. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &    = (+g𝐶)    &   𝑅 = (-g𝐶)    &   𝑄 = (0g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌𝑉)    &   (𝜑𝑍𝑉)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))       (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

Theoremmapdh7eN 39195* Part (7) of [Baer] p. 48 line 10 (5 of 6 cases). (Note: 1 of 6 and 2 of 6 are hypotheses a and b.) (Contributed by NM, 2-May-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝑄 = (0g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑢})) = (𝐽‘{𝐹}))    &   (𝜑𝑢 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑣 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑤 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑢}) ≠ (𝑁‘{𝑣}))    &   (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑢, 𝑣}))    &   (𝜑 → (𝐼‘⟨𝑢, 𝐹, 𝑤⟩) = 𝐸)       (𝜑 → (𝐼‘⟨𝑤, 𝐸, 𝑢⟩) = 𝐹)

Theoremmapdh7cN 39196* Part (7) of [Baer] p. 48 line 10 (3 of 6 cases). (Contributed by NM, 2-May-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝑄 = (0g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑢})) = (𝐽‘{𝐹}))    &   (𝜑𝑢 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑣 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑤 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑢}) ≠ (𝑁‘{𝑣}))    &   (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑢, 𝑣}))    &   (𝜑 → (𝐼‘⟨𝑢, 𝐹, 𝑣⟩) = 𝐺)       (𝜑 → (𝐼‘⟨𝑣, 𝐺, 𝑢⟩) = 𝐹)

Theoremmapdh7dN 39197* Part (7) of [Baer] p. 48 line 10 (4 of 6 cases). (Contributed by NM, 2-May-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝑄 = (0g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑢})) = (𝐽‘{𝐹}))    &   (𝜑𝑢 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑣 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑤 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑢}) ≠ (𝑁‘{𝑣}))    &   (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑢, 𝑣}))    &   (𝜑 → (𝐼‘⟨𝑢, 𝐹, 𝑣⟩) = 𝐺)    &   (𝜑 → (𝐼‘⟨𝑢, 𝐹, 𝑤⟩) = 𝐸)       (𝜑 → (𝐼‘⟨𝑣, 𝐺, 𝑤⟩) = 𝐸)

Theoremmapdh7fN 39198* Part (7) of [Baer] p. 48 line 10 (6 of 6 cases). (Contributed by NM, 2-May-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝑄 = (0g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑢})) = (𝐽‘{𝐹}))    &   (𝜑𝑢 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑣 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑤 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑢}) ≠ (𝑁‘{𝑣}))    &   (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑢, 𝑣}))    &   (𝜑 → (𝐼‘⟨𝑢, 𝐹, 𝑣⟩) = 𝐺)    &   (𝜑 → (𝐼‘⟨𝑢, 𝐹, 𝑤⟩) = 𝐸)       (𝜑 → (𝐼‘⟨𝑤, 𝐸, 𝑣⟩) = 𝐺)

Theoremmapdh75e 39199* Part (7) of [Baer] p. 48 line 10 (5 of 6 cases). 𝑋, 𝑌, 𝑍 are Baer's u, v, w. (Note: Cases 1 of 6 and 2 of 6 are hypotheses mapdh75b here and mapdh75a in mapdh75cN 39200.) (Contributed by NM, 2-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝑄 = (0g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))       (𝜑 → (𝐼‘⟨𝑍, 𝐸, 𝑋⟩) = 𝐹)

Theoremmapdh75cN 39200* Part (7) of [Baer] p. 48 line 10 (3 of 6 cases). (Contributed by NM, 2-May-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝑄 = (0g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))       (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑋⟩) = 𝐹)

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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45499
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