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Theorem List for Metamath Proof Explorer - 38301-38400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmapdindp1 38301 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 38302 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 38303 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 38304 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 38305* 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 38306* 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 38307* 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 38308* 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 38309* 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 38310* 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 38311* Lemmma for ~? mapdh . Part (2) in [Baer] p. 45. The bidirectional version of mapdheq2 38310 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 38312* Lemma for mapdheq4 38313. 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 38313* 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 38314* Lemma for mapdh6N 38328. 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 38315* Lemma for mapdh6N 38328. 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 38316* Lemma for mapdh6N 38328. 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 38317* Lemmma for mapdh6N 38328. (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 38318* Lemmma for mapdh6N 38328. (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 38319* Lemmma for mapdh6N 38328. (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 38320* Lemmma for mapdh6N 38328. (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 38321* Lemmma for mapdh6N 38328. 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 38322* Lemmma for mapdh6N 38328. 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 38323* Lemmma for mapdh6N 38328. 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 38324* Lemmma for mapdh6N 38328. 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 38325* Lemmma for mapdh6N 38328. 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 38326* Lemmma for mapdh6N 38328. 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 38327* Lemmma for mapdh6N 38328. 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 38328* 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 19623. 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 38402. (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 38329* 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 38330* 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 38331* 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 38332* 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 38333* 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 38334.) (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 38334* 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)

Theoremmapdh75d 38335* Part (7) of [Baer] p. 48 line 10 (4 of 6 cases). (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, 𝑌⟩) = 𝐺)    &   (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))       (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑍⟩) = 𝐸)

Theoremmapdh75fN 38336* 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‘{𝑋})) = (𝐽‘{𝐹}))    &   (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)    &   (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))       (𝜑 → (𝐼‘⟨𝑍, 𝐸, 𝑌⟩) = 𝐺)

Syntaxchvm 38337 Extend class notation with vector to dual map.
class HVMap

Definitiondf-hvmap 38338* Extend class notation with a map from each nonzero vector 𝑥 to a unique nonzero functional in the closed kernel dual space. (We could extend it to include the zero vector, but that is unnecessary for our purposes.) TODO: This pattern is used several times earlier, e.g., lcf1o 38132, dochfl1 38057- should we update those to use this definition? (Contributed by NM, 23-Mar-2015.)
HVMap = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ ((Base‘((DVecH‘𝑘)‘𝑤)) ∖ {(0g‘((DVecH‘𝑘)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝑘)‘𝑤)))∃𝑡 ∈ (((ocH‘𝑘)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝑘)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝑘)‘𝑤))𝑥)))))))

Theoremhvmapffval 38339* Map from nonzero vectors to nonzero functionals in the closed kernel dual space. (Contributed by NM, 23-Mar-2015.)
𝐻 = (LHyp‘𝐾)       (𝐾𝑋 → (HVMap‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ ((Base‘((DVecH‘𝐾)‘𝑤)) ∖ {(0g‘((DVecH‘𝐾)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥)))))))

Theoremhvmapfval 38340* Map from nonzero vectors to nonzero functionals in the closed kernel dual space. (Contributed by NM, 23-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    · = ( ·𝑠𝑈)    &    0 = (0g𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝑆)    &   𝑀 = ((HVMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾𝐴𝑊𝐻))       (𝜑𝑀 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥))))))

Theoremhvmapval 38341* Value of map from nonzero vectors to nonzero functionals in the closed kernel dual space. (Contributed by NM, 23-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    · = ( ·𝑠𝑈)    &    0 = (0g𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝑆)    &   𝑀 = ((HVMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾𝐴𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))       (𝜑 → (𝑀𝑋) = (𝑣𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑣 = (𝑡 + (𝑗 · 𝑋)))))

TheoremhvmapvalvalN 38342* Value of value of map (i.e. functional value) from nonzero vectors to nonzero functionals in the closed kernel dual space. (Contributed by NM, 23-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    · = ( ·𝑠𝑈)    &    0 = (0g𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝑆)    &   𝑀 = ((HVMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾𝐴𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌𝑉)       (𝜑 → ((𝑀𝑋)‘𝑌) = (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))))

TheoremhvmapidN 38343 The value of the vector to functional map, at the vector, is one. (Contributed by NM, 23-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &   𝑆 = (Scalar‘𝑈)    &    1 = (1r𝑆)    &   𝑀 = ((HVMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))       (𝜑 → ((𝑀𝑋)‘𝑋) = 1 )

Theoremhvmap1o 38344* The vector to functional map provides a bijection from nonzero vectors 𝑉 to nonzero functionals with closed kernels 𝐶. (Contributed by NM, 27-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑄 = (0g𝐷)    &   𝐶 = {𝑓𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓)}    &   𝑀 = ((HVMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑀:(𝑉 ∖ { 0 })–1-1-onto→(𝐶 ∖ {𝑄}))

TheoremhvmapclN 38345* Closure of the vector to functional map. (Contributed by NM, 27-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑄 = (0g𝐷)    &   𝐶 = {𝑓𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓)}    &   𝑀 = ((HVMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))       (𝜑 → (𝑀𝑋) ∈ (𝐶 ∖ {𝑄}))

Theoremhvmap1o2 38346 The vector to functional map provides a bijection from nonzero vectors 𝑉 to nonzero functionals with closed kernels 𝐶. (Contributed by NM, 27-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐹 = (Base‘𝐶)    &   𝑂 = (0g𝐶)    &   𝑀 = ((HVMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑀:(𝑉 ∖ { 0 })–1-1-onto→(𝐹 ∖ {𝑂}))

Theoremhvmapcl2 38347 Closure of the vector to functional map. (Contributed by NM, 27-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐹 = (Base‘𝐶)    &   𝑂 = (0g𝐶)    &   𝑀 = ((HVMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))       (𝜑 → (𝑀𝑋) ∈ (𝐹 ∖ {𝑂}))

Theoremhvmaplfl 38348 The vector to functional map value is a functional. (Contributed by NM, 28-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝑀 = ((HVMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))       (𝜑 → (𝑀𝑋) ∈ 𝐹)

Theoremhvmaplkr 38349 Kernel of the vector to functional map. TODO: make this become lcfrlem11 38134. (Contributed by NM, 29-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝑀 = ((HVMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))       (𝜑 → (𝐿‘(𝑀𝑋)) = (𝑂‘{𝑋}))

Theoremmapdhvmap 38350 Relationship between mapd and HVMap, which can be used to satisfy the last hypothesis of mapdpg 38287. Equation 10 of [Baer] p. 48. (Contributed by NM, 29-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐽 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑃 = ((HVMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))       (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{(𝑃𝑋)}))

Theoremlspindp5 38351 Obtain an independent vector set 𝑈, 𝑋, 𝑌 from a vector 𝑈 dependent on 𝑋 and 𝑍 and another independent set 𝑍, 𝑋, 𝑌. (Here we don't show the (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) part of the independence, which passes straight through. We also don't show nonzero vector requirements that are redundant for this theorem. Different orderings can be obtained using lspexch 19626 and prcom 4543.) (Contributed by NM, 4-May-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑈𝑉)    &   (𝜑𝑍 ∈ (𝑁‘{𝑋, 𝑈}))    &   (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌}))       (𝜑 → ¬ 𝑈 ∈ (𝑁‘{𝑋, 𝑌}))

Theoremhdmaplem1 38352 Lemma to convert a frequently-used union condition. TODO: see if this can be applied to other hdmap* theorems. (Contributed by NM, 17-May-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑍𝑉)    &   (𝜑 → ¬ 𝑍 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌})))    &   (𝜑𝑋𝑉)       (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑋}))

Theoremhdmaplem2N 38353 Lemma to convert a frequently-used union condition. TODO: see if this can be applied to other hdmap* theorems. (Contributed by NM, 17-May-2015.) (New usage is discouraged.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑍𝑉)    &   (𝜑 → ¬ 𝑍 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌})))    &   (𝜑𝑌𝑉)       (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑌}))

Theoremhdmaplem3 38354 Lemma to convert a frequently-used union condition. TODO: see if this can be applied to other hdmap* theorems. (Contributed by NM, 17-May-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑍𝑉)    &   (𝜑 → ¬ 𝑍 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌})))    &   (𝜑𝑌𝑉)    &    0 = (0g𝑊)       (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))

Theoremhdmaplem4 38355 Lemma to convert a frequently-used union condition. TODO: see if this can be applied to other hdmap* theorems. (Contributed by NM, 17-May-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    0 = (0g𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑋}))    &   (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑌}))       (𝜑 → ¬ 𝑍 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌})))

Theoremmapdh8a 38356* Part of Part (8) in [Baer] p. 48. (Contributed by NM, 5-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, 𝑌⟩) = 𝐺)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇}))    &   (𝜑𝑇 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇}))       (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑇⟩))

Theoremmapdh8aa 38357* Part of Part (8) in [Baer] p. 48. (Contributed by NM, 12-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, 𝑌⟩) = 𝐺)    &   (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑇}))    &   (𝜑𝑇 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇}))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))       (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑍, 𝐸, 𝑇⟩))

Theoremmapdh8ab 38358* Part of Part (8) in [Baer] p. 48. (Contributed by NM, 13-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, 𝑌⟩) = 𝐺)    &   (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑇 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑇}))       (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑍, 𝐸, 𝑇⟩))

Theoremmapdh8ac 38359* Part of Part (8) in [Baer] p. 48. (Contributed by NM, 13-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, 𝑌⟩) = 𝐺)    &   (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑇 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑇}))    &   (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑤⟩) = 𝐵)    &   (𝜑𝑤 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤}))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤}))    &   (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑍}))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑤, 𝑍}))       (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑍, 𝐸, 𝑇⟩))

Theoremmapdh8ad 38360* Part of Part (8) in [Baer] p. 48. (Contributed by NM, 13-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, 𝑌⟩) = 𝐺)    &   (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑇 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑇}))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))       (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑍, 𝐸, 𝑇⟩))

Theoremmapdh8b 38361* Part of Part (8) in [Baer] p. 48. (Contributed by NM, 6-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))    &   (𝜑𝐺𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}))    &   (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑤⟩) = 𝐸)    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑤 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))    &   (𝜑𝑇 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤}))    &   (𝜑𝑋 ∈ (𝑁‘{𝑌, 𝑇}))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤}))       (𝜑 → (𝐼‘⟨𝑤, 𝐸, 𝑇⟩) = (𝐼‘⟨𝑌, 𝐺, 𝑇⟩))

Theoremmapdh8c 38362* Part of Part (8) in [Baer] p. 48. (Contributed by NM, 6-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, 𝑤⟩) = 𝐸)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑇 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇}))    &   (𝜑𝑤 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇}))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤}))    &   (𝜑𝑋 ∈ (𝑁‘{𝑌, 𝑇}))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤}))       (𝜑 → (𝐼‘⟨𝑤, 𝐸, 𝑇⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑇⟩))

Theoremmapdh8d0N 38363* Part of Part (8) in [Baer] p. 48. (Contributed by NM, 10-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)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑇 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇}))    &   (𝜑𝑤 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇}))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤}))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤}))    &   (𝜑𝑋 ∈ (𝑁‘{𝑌, 𝑇}))       (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑇⟩))

Theoremmapdh8d 38364* Part of Part (8) in [Baer] p. 48. (Contributed by NM, 6-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, 𝑌⟩) = 𝐺)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑇 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇}))    &   (𝜑𝑤 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇}))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤}))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤}))       (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑇⟩))

Theoremmapdh8e 38365* Part of Part (8) in [Baer] p. 48. Eliminate 𝑤. (Contributed by NM, 10-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, 𝑌⟩) = 𝐺)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑇 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇}))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇}))    &   (𝜑𝑋 ∈ (𝑁‘{𝑌, 𝑇}))       (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑇⟩))

Theoremmapdh8g 38366* Part of Part (8) in [Baer] p. 48. Eliminate 𝑋 ∈ (𝑁‘{𝑌, 𝑇}). TODO: break out 𝑇0 in mapdh8e 38365 so we can share hypotheses. Also, look at hypothesis sharing for earlier mapdh8* and mapdh75* stuff. (Contributed by NM, 10-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, 𝑌⟩) = 𝐺)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑇 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇}))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇}))       (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑇⟩))

Theoremmapdh8i 38367* Part of Part (8) in [Baer] p. 48. (Contributed by NM, 11-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}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇}))    &   (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑇}))    &   (𝜑𝑇 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇}))       (𝜑 → (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 𝑇⟩) = (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 𝑇⟩))

Theoremmapdh8j 38368* Part of Part (8) in [Baer] p. 48. (Contributed by NM, 13-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}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇}))    &   (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑇}))    &   (𝜑𝑇 ∈ (𝑉 ∖ { 0 }))       (𝜑 → (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 𝑇⟩) = (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 𝑇⟩))

Theoremmapdh8 38369* Part (8) in [Baer] p. 48. Given a reference vector 𝑋, the value of function 𝐼 at a vector 𝑇 is independent of the choice of auxiliary vectors 𝑌 and 𝑍. Unlike Baer's, our version does not require 𝑋, 𝑌, and 𝑍 to be independent, and also is defined for all 𝑌 and 𝑍 that are not colinear with 𝑋 or 𝑇. We do this to make the definition of Baer's sigma function more straightforward. (This part eliminates 𝑇0.) (Contributed by NM, 13-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}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇}))    &   (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑇}))    &   (𝜑𝑇𝑉)       (𝜑 → (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 𝑇⟩) = (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 𝑇⟩))

Theoremmapdh9a 38370* Lemma for part (9) in [Baer] p. 48. TODO: why is this 50% larger than mapdh9aOLDN 38371? (Contributed by NM, 14-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}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑇𝑉)       (𝜑 → ∃!𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩)))

Theoremmapdh9aOLDN 38371* Lemma for part (9) in [Baer] p. 48. (Contributed by NM, 14-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}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑇𝑉)       (𝜑 → ∃!𝑦𝐷𝑧𝑉𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩)))

Syntaxchdma1 38372 Extend class notation with preliminary map from vectors to functionals in the closed kernel dual space.
class HDMap1

Syntaxchdma 38373 Extend class notation with map from vectors to functionals in the closed kernel dual space.
class HDMap

Definitiondf-hdmap1 38374* Define preliminary map from vectors to functionals in the closed kernel dual space. See hdmap1fval 38377 description for more details. (Contributed by NM, 14-May-2015.)
HDMap1 = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑎[((DVecH‘𝑘)‘𝑤) / 𝑢][(Base‘𝑢) / 𝑣][(LSpan‘𝑢) / 𝑛][((LCDual‘𝑘)‘𝑤) / 𝑐][(Base‘𝑐) / 𝑑][(LSpan‘𝑐) / 𝑗][((mapd‘𝑘)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), (0g𝑐), (𝑑 ((𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑚‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))})))))}))

Definitiondf-hdmap 38375* Define map from vectors to functionals in the closed kernel dual space. See hdmapfval 38408 description for more details. (Contributed by NM, 15-May-2015.)
HDMap = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑎[⟨( I ↾ (Base‘𝑘)), ( I ↾ ((LTrn‘𝑘)‘𝑤))⟩ / 𝑒][((DVecH‘𝑘)‘𝑤) / 𝑢][(Base‘𝑢) / 𝑣][((HDMap1‘𝑘)‘𝑤) / 𝑖]𝑎 ∈ (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝑘)‘𝑤))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝑘)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩))))}))

Theoremhdmap1ffval 38376* Preliminary map from vectors to functionals in the closed kernel dual space. (Contributed by NM, 14-May-2015.)
𝐻 = (LHyp‘𝐾)       (𝐾𝑋 → (HDMap1‘𝐾) = (𝑤𝐻 ↦ {𝑎[((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘𝑢) / 𝑣][(LSpan‘𝑢) / 𝑛][((LCDual‘𝐾)‘𝑤) / 𝑐][(Base‘𝑐) / 𝑑][(LSpan‘𝑐) / 𝑗][((mapd‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2nd𝑥) = (0g𝑢), (0g𝑐), (𝑑 ((𝑚‘(𝑛‘{(2nd𝑥)})) = (𝑗‘{}) ∧ (𝑚‘(𝑛‘{((1st ‘(1st𝑥))(-g𝑢)(2nd𝑥))})) = (𝑗‘{((2nd ‘(1st𝑥))(-g𝑐))})))))}))

Theoremhdmap1fval 38377* Preliminary map from vectors to functionals in the closed kernel dual space. TODO: change span 𝐽 to the convention 𝐿 for this section. (Contributed by NM, 15-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 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾𝐴𝑊𝐻))       (𝜑𝐼 = (𝑥 ∈ ((𝑉 × 𝐷) × 𝑉) ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}))))))

Theoremhdmap1vallem 38378* Value of preliminary map from vectors to functionals in the closed kernel dual space. (Contributed by NM, 15-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 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾𝐴𝑊𝐻))    &   (𝜑𝑇 ∈ ((𝑉 × 𝐷) × 𝑉))       (𝜑 → (𝐼𝑇) = if((2nd𝑇) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑇)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑇)) (2nd𝑇))})) = (𝐽‘{((2nd ‘(1st𝑇))𝑅)})))))

Theoremhdmap1val 38379* Value of preliminary map from vectors to functionals in the closed kernel dual space. (Restatement of mapdhval 38305.) TODO: change 𝐼 = (𝑥 ∈ V ↦... to (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌 > ) =... in e.g. mapdh8 38369 to shorten proofs with no \$d on 𝑥. (Contributed by NM, 15-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 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾𝐴𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝐹𝐷)    &   (𝜑𝑌𝑉)       (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if(𝑌 = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐹𝑅)})))))

Theoremhdmap1val0 38380 Value of preliminary map from vectors to functionals at zero. (Restated mapdhval0 38306.) (Contributed by NM, 17-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑄 = (0g𝐶)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋𝑉)       (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = 𝑄)

Theoremhdmap1val2 38381* Value of preliminary map from vectors to functionals in the closed kernel dual space, for nonzero 𝑌. (Contributed by NM, 16-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)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝐹𝐷)    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))       (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐿‘{(𝐹𝑅)}))))

Theoremhdmap1eq 38382 The defining equation for h(x,x',y)=y' in part (2) in [Baer] p. 45 line 24. (Contributed by NM, 16-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)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐹𝐷)    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐺𝐷)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))       (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺 ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐿‘{(𝐹𝑅𝐺)}))))

Theoremhdmap1cbv 38383* Frequently used lemma to change bound variables in 𝐿 hypothesis. (Contributed by NM, 15-May-2015.)
𝐿 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))       𝐿 = (𝑦 ∈ V ↦ if((2nd𝑦) = 0 , 𝑄, (𝑖𝐷 ((𝑀‘(𝑁‘{(2nd𝑦)})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑦)) (2nd𝑦))})) = (𝐽‘{((2nd ‘(1st𝑦))𝑅𝑖)})))))

Theoremhdmap1valc 38384* Connect the value of the preliminary map from vectors to functionals 𝐼 to the hypothesis 𝐿 used by earlier theorems. Note: the 𝑋 ∈ (𝑉 ∖ { 0 }) hypothesis could be the more general 𝑋𝑉 but the former will be easier to use. TODO: use the 𝐼 function directly in those theorems, so this theorem becomes unnecessary? TODO: The hdmap1cbv 38383 is probably unnecessary, but it would mean different \$d's later on. (Contributed by NM, 15-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 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐹𝐷)    &   (𝜑𝑌𝑉)    &   𝐿 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))       (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = (𝐿‘⟨𝑋, 𝐹, 𝑌⟩))

Theoremhdmap1cl 38385 Convert closure theorem mapdhcl 38308 to use HDMap1 function. (Contributed by NM, 15-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌𝑉)       (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷)

Theoremhdmap1eq2 38386 Convert mapdheq2 38310 to use HDMap1 function. (Contributed by NM, 16-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)       (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑋⟩) = 𝐹)

Theoremhdmap1eq4N 38387 Convert mapdheq4 38313 to use HDMap1 function. (Contributed by NM, 17-May-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)    &   (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐵)       (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑍⟩) = 𝐵)

Theoremhdmap1l6lem1 38388 Lemma for hdmap1l6 38402. Part (6) in [Baer] p. 47, lines 16-18. (Contributed by NM, 13-Apr-2015.)
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)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (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))})) = (𝐿‘{(𝐹𝑅(𝐺 𝐸))}))

Theoremhdmap1l6lem2 38389 Lemma for hdmap1l6 38402. Part (6) in [Baer] p. 47, lines 20-22. (Contributed by NM, 13-Apr-2015.)
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)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (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)})) = (𝐿‘{(𝐺 𝐸)}))

Theoremhdmap1l6a 38390 Lemma for hdmap1l6 38402. Part (6) in [Baer] p. 47, case 1. (Contributed by NM, 23-Apr-2015.)
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)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (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⟩)))

Theoremhdmap1l6b0N 38391 Lemmma for hdmap1l6 38402. (Contributed by NM, 23-Apr-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)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (𝜑𝑌𝑉)    &   (𝜑𝑍𝑉)    &   (𝜑 → ((𝑁‘{𝑋}) ∩ (𝑁‘{𝑌, 𝑍})) = { 0 })       (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))

Theoremhdmap1l6b 38392 Lemmma for hdmap1l6 38402. (Contributed by NM, 24-Apr-2015.)
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)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (𝜑𝑌 = 0 )    &   (𝜑𝑍𝑉)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))       (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

Theoremhdmap1l6c 38393 Lemmma for hdmap1l6 38402. (Contributed by NM, 24-Apr-2015.)
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)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (𝜑𝑌𝑉)    &   (𝜑𝑍 = 0 )    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))       (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

Theoremhdmap1l6d 38394 Lemmma for hdmap1l6 38402. (Contributed by NM, 1-May-2015.)
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)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (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)⟩)))

Theoremhdmap1l6e 38395 Lemmma for hdmap1l6 38402. Part (6) in [Baer] p. 47 line 38. (Contributed by NM, 1-May-2015.)
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)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (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⟩)))

Theoremhdmap1l6f 38396 Lemmma for hdmap1l6 38402. Part (6) in [Baer] p. 47 line 38. (Contributed by NM, 1-May-2015.)
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)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (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⟩)))

Theoremhdmap1l6g 38397 Lemmma for hdmap1l6 38402. Part (6) of [Baer] p. 47 line 39. (Contributed by NM, 1-May-2015.)
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)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (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⟩)))

Theoremhdmap1l6h 38398 Lemmma for hdmap1l6 38402. Part (6) of [Baer] p. 48 line 2. (Contributed by NM, 1-May-2015.)
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)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (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⟩)))

Theoremhdmap1l6i 38399 Lemmma for hdmap1l6 38402. Eliminate auxiliary vector 𝑤. (Contributed by NM, 1-May-2015.)
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)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (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⟩)))

Theoremhdmap1l6j 38400 Lemmma for hdmap1l6 38402. Eliminate (𝑁 { Y } ) = ( N {𝑍}) hypothesis. (Contributed by NM, 1-May-2015.)
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)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (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⟩)))

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