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Theorem List for Metamath Proof Explorer - 37901-38000   *Has distinct variable group(s)
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Statement

Theoremmapdh6gN 37901* Lemmma for mapdh6N 37906. 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 37902* Lemmma for mapdh6N 37906. 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 37903* Lemmma for mapdh6N 37906. 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 37904* Lemmma for mapdh6N 37906. 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 37905* Lemmma for mapdh6N 37906. 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 37906* 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 19525. 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 37980. (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 37907* 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 37908* 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 37909* 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 37910* 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 37911* 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 37912.) (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 37912* 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 37913* 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 37914* 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 37915 Extend class notation with vector to dual map.
class HVMap

Definitiondf-hvmap 37916* 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 37710, dochfl1 37635- 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 37917* 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 37918* 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 37919* 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 37920* 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 37921 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 37922* 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 37923* 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 37924 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 37925 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 37926 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 37927 Kernel of the vector to functional map. TODO: make this become lcfrlem11 37712. (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 37928 Relationship between mapd and HVMap, which can be used to satisfy the last hypothesis of mapdpg 37865. 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 37929 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 19529 and prcom 4499.) (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 37930 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 37931 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 37932 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 37933 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 37934* 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 37935* 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 37936* 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 37937* 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 37938* 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 37939* 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 37940* 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 37941* 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 37942* 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 37943* 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 37944* Part of Part (8) in [Baer] p. 48. Eliminate 𝑋 ∈ (𝑁‘{𝑌, 𝑇}). TODO: break out 𝑇0 in mapdh8e 37943 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 37945* 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 37946* 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 37947* 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 37948* Lemma for part (9) in [Baer] p. 48. TODO: why is this 50% larger than mapdh9aOLDN 37949? (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 37949* 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 37950 Extend class notation with preliminary map from vectors to functionals in the closed kernel dual space.
class HDMap1

Syntaxchdma 37951 Extend class notation with map from vectors to functionals in the closed kernel dual space.
class HDMap

Definitiondf-hdmap1 37952* Define preliminary map from vectors to functionals in the closed kernel dual space. See hdmap1fval 37955 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 37953* Define map from vectors to functionals in the closed kernel dual space. See hdmapfval 37986 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 37954* 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 37955* 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 37956* 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 37957* Value of preliminary map from vectors to functionals in the closed kernel dual space. (Restatement of mapdhval 37883.) TODO: change 𝐼 = (𝑥 ∈ V ↦... to (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌 > ) =... in e.g. mapdh8 37947 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 37958 Value of preliminary map from vectors to functionals at zero. (Restated mapdhval0 37884.) (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 37959* 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 37960 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 37961* 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 37962* 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 37961 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 37963 Convert closure theorem mapdhcl 37886 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 37964 Convert mapdheq2 37888 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 37965 Convert mapdheq4 37891 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 37966 Lemma for hdmap1l6 37980. 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 37967 Lemma for hdmap1l6 37980. 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 37968 Lemma for hdmap1l6 37980. 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 37969 Lemmma for hdmap1l6 37980. (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 37970 Lemmma for hdmap1l6 37980. (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 37971 Lemmma for hdmap1l6 37980. (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 37972 Lemmma for hdmap1l6 37980. (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 37973 Lemmma for hdmap1l6 37980. 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 37974 Lemmma for hdmap1l6 37980. 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 37975 Lemmma for hdmap1l6 37980. 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 37976 Lemmma for hdmap1l6 37980. 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 37977 Lemmma for hdmap1l6 37980. 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 37978 Lemmma for hdmap1l6 37980. 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⟩)))

Theoremhdmap1l6k 37979 Lemmma for hdmap1l6 37980. Eliminate nonzero vector requirement. (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}))    &   (𝜑𝑌𝑉)    &   (𝜑𝑍𝑉)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))       (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

Theoremhdmap1l6 37980 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 19525. (Convert mapdh6N 37906 to use the function HDMap1.) (Contributed by NM, 17-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)    &    = (+g𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌𝑉)    &   (𝜑𝑍𝑉)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))       (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

Theoremhdmap1eulem 37981* Lemma for hdmap1eu 37983. TODO: combine with hdmap1eu 37983 or at least share some hypotheses. (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))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐹𝐷)    &   (𝜑𝑇𝑉)    &   𝐿 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))       (𝜑 → ∃!𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩)))

Theoremhdmap1eulemOLDN 37982* Lemma for hdmap1euOLDN 37984. TODO: combine with hdmap1euOLDN 37984 or at least share some hypotheses. (Contributed by NM, 15-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 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐹𝐷)    &   (𝜑𝑇𝑉)    &   𝐿 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))       (𝜑 → ∃!𝑦𝐷𝑧𝑉𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩)))

Theoremhdmap1eu 37983* Convert mapdh9a 37948 to use the HDMap1 notation. (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))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐹𝐷)    &   (𝜑𝑇𝑉)       (𝜑 → ∃!𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩)))

Theoremhdmap1euOLDN 37984* Convert mapdh9aOLDN 37949 to use the HDMap1 notation. (Contributed by NM, 15-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))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐹𝐷)    &   (𝜑𝑇𝑉)       (𝜑 → ∃!𝑦𝐷𝑧𝑉𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩)))

Theoremhdmapffval 37985* Map from vectors to functionals in the closed kernel dual space. (Contributed by NM, 15-May-2015.)
𝐻 = (LHyp‘𝐾)       (𝐾𝑋 → (HDMap‘𝐾) = (𝑤𝐻 ↦ {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⟩))))}))

Theoremhdmapfval 37986* Map from vectors to functionals in the closed kernel dual space. (Contributed by NM, 15-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝐽 = ((HVMap‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾𝐴𝑊𝐻))       (𝜑𝑆 = (𝑡𝑉 ↦ (𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩)))))

Theoremhdmapval 37987* Value of map from vectors to functionals in the closed kernel dual space. This is the function sigma on line 27 above part 9 in [Baer] p. 48. We select a convenient fixed reference vector 𝐸 to be ⟨0, 1⟩ (corresponding to vector u on p. 48 line 7) whose span is the lattice isomorphism map of the fiducial atom 𝑃 = ((oc‘𝐾)‘𝑊) (see dvheveccl 37271). (𝐽𝐸) is a fixed reference functional determined by this vector (corresponding to u' on line 8; mapdhvmap 37928 shows in Baer's notation (Fu)* = Gu'). Baer's independent vectors v and w on line 7 correspond to our 𝑧 that the 𝑧𝑉 ranges over. The middle term (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩) provides isolation to allow 𝐸 and 𝑇 to assume the same value without conflict. Closure is shown by hdmapcl 37989. If a separate auxiliary vector is known, hdmapval2 37991 provides a version without quantification. (Contributed by NM, 15-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝐽 = ((HVMap‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾𝐴𝑊𝐻))    &   (𝜑𝑇𝑉)       (𝜑 → (𝑆𝑇) = (𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑇⟩))))

TheoremhdmapfnN 37988 Functionality of map from vectors to functionals with closed kernels. (Contributed by NM, 30-May-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑆 Fn 𝑉)

Theoremhdmapcl 37989 Closure of map from vectors to functionals with closed kernels. (Contributed by NM, 15-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑇𝑉)       (𝜑 → (𝑆𝑇) ∈ 𝐷)

Theoremhdmapval2lem 37990* Lemma for hdmapval2 37991. (Contributed by NM, 15-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝐽 = ((HVMap‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑇𝑉)    &   (𝜑𝐹𝐷)       (𝜑 → ((𝑆𝑇) = 𝐹 ↔ ∀𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝐹 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑇⟩))))

Theoremhdmapval2 37991 Value of map from vectors to functionals with a specific auxiliary vector. TODO: Would shorter proofs result if the .ne hypothesis were changed to two hypothesis? Consider hdmaplem1 37930 through hdmaplem4 37933, which would become obsolete. (Contributed by NM, 15-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝐽 = ((HVMap‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑇𝑉)    &   (𝜑𝑋𝑉)    &   (𝜑 → ¬ 𝑋 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})))       (𝜑 → (𝑆𝑇) = (𝐼‘⟨𝑋, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑋⟩), 𝑇⟩))

Theoremhdmapval0 37992 Value of map from vectors to functionals at zero. Note: we use dvh3dim 37605 for convenience, even though 3 dimensions aren't necessary at this point. TODO: I think either this or hdmapeq0 38003 could be derived from the other to shorten proof. (Contributed by NM, 17-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    0 = (0g𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑄 = (0g𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 → (𝑆0 ) = 𝑄)

Theoremhdmapeveclem 37993 Lemma for hdmapevec 37994. TODO: combine with hdmapevec 37994 if it shortens overall. (Contributed by NM, 16-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   𝐽 = ((HVMap‘𝐾)‘𝑊)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑𝑋𝑉)    &   (𝜑 → ¬ 𝑋 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝐸})))       (𝜑 → (𝑆𝐸) = (𝐽𝐸))

Theoremhdmapevec 37994 Value of map from vectors to functionals at the reference vector 𝐸. (Contributed by NM, 16-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   𝐽 = ((HVMap‘𝐾)‘𝑊)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 → (𝑆𝐸) = (𝐽𝐸))

Theoremhdmapevec2 37995 The inner product of the reference vector 𝐸 with itself is nonzero. This shows the inner product condition in the proof of Theorem 3.6 of [Holland95] p. 14 line 32, [ e , e ] ≠ 0 is satisfied. TODO: remove redundant hypothesis hdmapevec.j. (Contributed by NM, 1-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   𝐽 = ((HVMap‘𝐾)‘𝑊)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑅 = (Scalar‘𝑈)    &    1 = (1r𝑅)       (𝜑 → ((𝑆𝐸)‘𝐸) = 1 )

Theoremhdmapval3lemN 37996 Value of map from vectors to functionals at arguments not colinear with the reference vector 𝐸. (Contributed by NM, 17-May-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝐽 = ((HVMap‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑 → (𝑁‘{𝑇}) ≠ (𝑁‘{𝐸}))    &   (𝜑𝑇 ∈ (𝑉 ∖ {(0g𝑈)}))    &   (𝜑𝑥𝑉)    &   (𝜑 → ¬ 𝑥 ∈ (𝑁‘{𝐸, 𝑇}))       (𝜑 → (𝑆𝑇) = (𝐼‘⟨𝐸, (𝐽𝐸), 𝑇⟩))

Theoremhdmapval3N 37997 Value of map from vectors to functionals at arguments not colinear with the reference vector 𝐸. (Contributed by NM, 17-May-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝐽 = ((HVMap‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑 → (𝑁‘{𝑇}) ≠ (𝑁‘{𝐸}))    &   (𝜑𝑇𝑉)       (𝜑 → (𝑆𝑇) = (𝐼‘⟨𝐸, (𝐽𝐸), 𝑇⟩))

Theoremhdmap10lem 37998 Lemma for hdmap10 37999. (Contributed by NM, 17-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &    0 = (0g𝑈)    &   𝐷 = (Base‘𝐶)    &   𝐽 = ((HVMap‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑𝑇 ∈ (𝑉 ∖ { 0 }))       (𝜑 → (𝑀‘(𝑁‘{𝑇})) = (𝐿‘{(𝑆𝑇)}))

Theoremhdmap10 37999 Part 10 in [Baer] p. 48 line 33, (Ft)* = G(tS) in their notation (S = sigma). (Contributed by NM, 17-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑇𝑉)       (𝜑 → (𝑀‘(𝑁‘{𝑇})) = (𝐿‘{(𝑆𝑇)}))

Theoremhdmap11lem1 38000 Lemma for hdmapadd 38002. (Contributed by NM, 26-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = (+g𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐷 = (Base‘𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐽 = ((HVMap‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑𝑧𝑉)    &   (𝜑 → ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}))    &   (𝜑 → (𝑁‘{𝑧}) ≠ (𝑁‘{𝐸}))       (𝜑 → (𝑆‘(𝑋 + 𝑌)) = ((𝑆𝑋) (𝑆𝑌)))

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