P. 422 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 42101-42200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mapdh6b0N 42101*	Lemmma for mapdh6N 42112. (Contributed by NM, 23-Apr-2015.) (New usage is discouraged.)
⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ + = (+g‘𝑈)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → ((𝑁‘{𝑋}) ∩ (𝑁‘{𝑌, 𝑍})) = { 0 })    ⇒   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
Theorem	mapdh6bN 42102*	Lemmma for mapdh6N 42112. (Contributed by NM, 24-Apr-2015.) (New usage is discouraged.)
⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ + = (+g‘𝑈)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ (𝜑 → 𝑌 = 0 )    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
Theorem	mapdh6cN 42103*	Lemmma for mapdh6N 42112. (Contributed by NM, 24-Apr-2015.) (New usage is discouraged.)
⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ + = (+g‘𝑈)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 = 0 )    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
Theorem	mapdh6dN 42104*	Lemmma for mapdh6N 42112. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ + = (+g‘𝑈)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑤 + (𝑌 + 𝑍))⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩)))
Theorem	mapdh6eN 42105*	Lemmma for mapdh6N 42112. Part (6) in [Baer] p. 47 line 38. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ + = (+g‘𝑈)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, ((𝑤 + 𝑌) + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, (𝑤 + 𝑌)⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
Theorem	mapdh6fN 42106*	Lemmma for mapdh6N 42112. Part (6) in [Baer] p. 47 line 38. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ + = (+g‘𝑈)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑤 + 𝑌)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑌⟩)))
Theorem	mapdh6gN 42107*	Lemmma for mapdh6N 42112. Part (6) of [Baer] p. 47 line 39. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ + = (+g‘𝑈)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑤 ∈ (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⟩)))
Theorem	mapdh6hN 42108*	Lemmma for mapdh6N 42112. Part (6) of [Baer] p. 48 line 2. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ + = (+g‘𝑈)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
Theorem	mapdh6iN 42109*	Lemmma for mapdh6N 42112. Eliminate auxiliary vector 𝑤. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ + = (+g‘𝑈)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
Theorem	mapdh6jN 42110*	Lemmma for mapdh6N 42112. Eliminate (𝑁‘{𝑌}) = (𝑁‘{𝑍}) hypothesis. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ + = (+g‘𝑈)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
Theorem	mapdh6kN 42111*	Lemmma for mapdh6N 42112. Eliminate nonzero vector requirement. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ + = (+g‘𝑈)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
Theorem	mapdh6N 42112*	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 21093. 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 42186. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (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⟩)))
Theorem	mapdh7eN 42113*	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)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑢})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → 𝑢 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑣 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑢}) ≠ (𝑁‘{𝑣}))    &   ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑢, 𝑣}))    &   ⊢ (𝜑 → (𝐼‘⟨𝑢, 𝐹, 𝑤⟩) = 𝐸)    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑤, 𝐸, 𝑢⟩) = 𝐹)
Theorem	mapdh7cN 42114*	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)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑢})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → 𝑢 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑣 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑢}) ≠ (𝑁‘{𝑣}))    &   ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑢, 𝑣}))    &   ⊢ (𝜑 → (𝐼‘⟨𝑢, 𝐹, 𝑣⟩) = 𝐺)    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑣, 𝐺, 𝑢⟩) = 𝐹)
Theorem	mapdh7dN 42115*	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)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑢})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → 𝑢 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑣 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑤 ∈ (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)
Theorem	mapdh7fN 42116*	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)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑢})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → 𝑢 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑣 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑤 ∈ (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)
Theorem	mapdh75e 42117*	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 42118.) (Contributed by NM, 2-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑍, 𝐸, 𝑋⟩) = 𝐹)
Theorem	mapdh75cN 42118*	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)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑋⟩) = 𝐹)
Theorem	mapdh75d 42119*	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)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (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}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑍⟩) = 𝐸)
Theorem	mapdh75fN 42120*	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)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (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}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑍, 𝐸, 𝑌⟩) = 𝐺)
Syntax	chvm 42121	Extend class notation with vector to dual map.
class HVMap
Definition	df-hvmap 42122*	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 41916, dochfl1 41841- 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)))))))
Theorem	hvmapffval 42123*	Map from nonzero vectors to nonzero functionals in the closed kernel dual space. (Contributed by NM, 23-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝑋 → (HVMap‘𝐾) = (𝑤 ∈ 𝐻 ↦ (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)))))))
Theorem	hvmapfval 42124*	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)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 𝑀 = ((HVMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝑀 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥))))))
Theorem	hvmapval 42125*	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)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 𝑀 = ((HVMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝑀‘𝑋) = (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑣 = (𝑡 + (𝑗 · 𝑋)))))
Theorem	hvmapvalvalN 42126*	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)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 𝑀 = ((HVMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝑀‘𝑋)‘𝑌) = (℩𝑗 ∈ 𝑅 ∃𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))))
Theorem	hvmapidN 42127	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 = (0g‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 1 = (1r‘𝑆)    &   ⊢ 𝑀 = ((HVMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → ((𝑀‘𝑋)‘𝑋) = 1 )
Theorem	hvmap1o 42128*	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 = (0g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ 𝑄 = (0g‘𝐷)    &   ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ 𝑀 = ((HVMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝑀:(𝑉 ∖ { 0 })–1-1-onto→(𝐶 ∖ {𝑄}))
Theorem	hvmapclN 42129*	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 = (0g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ 𝑄 = (0g‘𝐷)    &   ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ 𝑀 = ((HVMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝑀‘𝑋) ∈ (𝐶 ∖ {𝑄}))
Theorem	hvmap1o2 42130	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 = (0g‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐹 = (Base‘𝐶)    &   ⊢ 𝑂 = (0g‘𝐶)    &   ⊢ 𝑀 = ((HVMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝑀:(𝑉 ∖ { 0 })–1-1-onto→(𝐹 ∖ {𝑂}))
Theorem	hvmapcl2 42131	Closure of the vector to functional map. (Contributed by NM, 27-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐹 = (Base‘𝐶)    &   ⊢ 𝑂 = (0g‘𝐶)    &   ⊢ 𝑀 = ((HVMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝑀‘𝑋) ∈ (𝐹 ∖ {𝑂}))
Theorem	hvmaplfl 42132	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 = (0g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝑀 = ((HVMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝑀‘𝑋) ∈ 𝐹)
Theorem	hvmaplkr 42133	Kernel of the vector to functional map. TODO: make this become lcfrlem11 41918. (Contributed by NM, 29-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝑀 = ((HVMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝐿‘(𝑀‘𝑋)) = (𝑂‘{𝑋}))
Theorem	mapdhvmap 42134	Relationship between mapd and HVMap, which can be used to satisfy the last hypothesis of mapdpg 42071. Equation 10 of [Baer] p. 48. (Contributed by NM, 29-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝑃 = ((HVMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{(𝑃‘𝑋)}))
Theorem	lspindp5 42135	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 21096 and prcom 4691.) (Contributed by NM, 4-May-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ (𝑁‘{𝑋, 𝑈}))    &   ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌}))    ⇒   ⊢ (𝜑 → ¬ 𝑈 ∈ (𝑁‘{𝑋, 𝑌}))
Theorem	hdmaplem1 42136	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)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑍 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌})))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑋}))
Theorem	hdmaplem2N 42137	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)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑍 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌})))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑌}))
Theorem	hdmaplem3 42138	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)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑍 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌})))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ 0 = (0g‘𝑊)    ⇒   ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
Theorem	hdmaplem4 42139	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 = (0g‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑋}))    &   ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑌}))    ⇒   ⊢ (𝜑 → ¬ 𝑍 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌})))
Theorem	mapdh8a 42140*	Part of Part (8) in [Baer] p. 48. (Contributed by NM, 5-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇}))    &   ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑇}))    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑇⟩))
Theorem	mapdh8aa 42141*	Part of Part (8) in [Baer] p. 48. (Contributed by NM, 12-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)    &   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑇}))    &   ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇}))    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑍, 𝐸, 𝑇⟩))
Theorem	mapdh8ab 42142*	Part of Part (8) in [Baer] p. 48. (Contributed by NM, 13-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)    &   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑇 ∈ (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⟩))
Theorem	mapdh8ac 42143*	Part of Part (8) in [Baer] p. 48. (Contributed by NM, 13-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)    &   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑇}))    &   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑤⟩) = 𝐵)    &   ⊢ (𝜑 → 𝑤 ∈ (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⟩))
Theorem	mapdh8ad 42144*	Part of Part (8) in [Baer] p. 48. (Contributed by NM, 13-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)    &   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑇 ∈ (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⟩))
Theorem	mapdh8b 42145*	Part of Part (8) in [Baer] p. 48. (Contributed by NM, 6-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}))    &   ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑤⟩) = 𝐸)    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))    &   ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌, 𝑇}))    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤}))    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑤, 𝐸, 𝑇⟩) = (𝐼‘⟨𝑌, 𝐺, 𝑇⟩))
Theorem	mapdh8c 42146*	Part of Part (8) in [Baer] p. 48. (Contributed by NM, 6-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑤⟩) = 𝐸)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇}))    &   ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇}))    &   ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌, 𝑇}))    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤}))    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑤, 𝐸, 𝑇⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑇⟩))
Theorem	mapdh8d0N 42147*	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)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇}))    &   ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇}))    &   ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑤}))    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑤}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌, 𝑇}))    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑇⟩))
Theorem	mapdh8d 42148*	Part of Part (8) in [Baer] p. 48. (Contributed by NM, 6-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇}))    &   ⊢ (𝜑 → 𝑤 ∈ (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⟩))
Theorem	mapdh8e 42149*	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)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇}))    &   ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌, 𝑇}))    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑇⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑇⟩))
Theorem	mapdh8g 42150*	Part of Part (8) in [Baer] p. 48. Eliminate 𝑋 ∈ (𝑁‘{𝑌, 𝑇}). TODO: break out 𝑇 ≠ 0 in mapdh8e 42149 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)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑇 ∈ (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⟩))
Theorem	mapdh8i 42151*	Part of Part (8) in [Baer] p. 48. (Contributed by NM, 11-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))    &   ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇}))    &   ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑇}))    &   ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑇}))    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 𝑇⟩) = (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 𝑇⟩))
Theorem	mapdh8j 42152*	Part of Part (8) in [Baer] p. 48. (Contributed by NM, 13-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))    &   ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇}))    &   ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑇}))    &   ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 𝑇⟩) = (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 𝑇⟩))
Theorem	mapdh8 42153*	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 𝑇 ≠ 0.) (Contributed by NM, 13-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))    &   ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑇}))    &   ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑇}))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑌, (𝐼‘⟨𝑋, 𝐹, 𝑌⟩), 𝑇⟩) = (𝐼‘⟨𝑍, (𝐼‘⟨𝑋, 𝐹, 𝑍⟩), 𝑇⟩))
Theorem	mapdh9a 42154*	Lemma for part (9) in [Baer] p. 48. TODO: why is this 50% larger than mapdh9aOLDN 42155? (Contributed by NM, 14-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∃!𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩)))
Theorem	mapdh9aOLDN 42155*	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)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∃!𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩)))
Syntax	chdma1 42156	Extend class notation with preliminary map from vectors to functionals in the closed kernel dual space.
class HDMap1
Syntax	chdma 42157	Extend class notation with map from vectors to functionals in the closed kernel dual space.
class HDMap
Definition	df-hdmap1 42158*	Define preliminary map from vectors to functionals in the closed kernel dual space. See hdmap1fval 42161 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((2nd ‘𝑥) = (0g‘𝑢), (0g‘𝑐), (℩ℎ ∈ 𝑑 ((𝑚‘(𝑛‘{(2nd ‘𝑥)})) = (𝑗‘{ℎ}) ∧ (𝑚‘(𝑛‘{((1st ‘(1st ‘𝑥))(-g‘𝑢)(2nd ‘𝑥))})) = (𝑗‘{((2nd ‘(1st ‘𝑥))(-g‘𝑐)ℎ)})))))}))
Definition	df-hdmap 42159*	Define map from vectors to functionals in the closed kernel dual space. See hdmapfval 42192 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 ∈ (𝑡 ∈ 𝑣 ↦ (℩𝑦 ∈ (Base‘((LCDual‘𝑘)‘𝑤))∀𝑧 ∈ 𝑣 (¬ 𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝑘)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩))))}))
Theorem	hdmap1ffval 42160*	Preliminary map from vectors to functionals in the closed kernel dual space. (Contributed by NM, 14-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝑋 → (HDMap1‘𝐾) = (𝑤 ∈ 𝐻 ↦ {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((2nd ‘𝑥) = (0g‘𝑢), (0g‘𝑐), (℩ℎ ∈ 𝑑 ((𝑚‘(𝑛‘{(2nd ‘𝑥)})) = (𝑗‘{ℎ}) ∧ (𝑚‘(𝑛‘{((1st ‘(1st ‘𝑥))(-g‘𝑢)(2nd ‘𝑥))})) = (𝑗‘{((2nd ‘(1st ‘𝑥))(-g‘𝑐)ℎ)})))))}))
Theorem	hdmap1fval 42161*	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)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝐼 = (𝑥 ∈ ((𝑉 × 𝐷) × 𝑉) ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}))))))
Theorem	hdmap1vallem 42162*	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)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑇 ∈ ((𝑉 × 𝐷) × 𝑉))    ⇒   ⊢ (𝜑 → (𝐼‘𝑇) = if((2nd ‘𝑇) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑇)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑇)) − (2nd ‘𝑇))})) = (𝐽‘{((2nd ‘(1st ‘𝑇))𝑅ℎ)})))))
Theorem	hdmap1val 42163*	Value of preliminary map from vectors to functionals in the closed kernel dual space. (Restatement of mapdhval 42089.) TODO: change 𝐼 = (𝑥 ∈ V ↦... to (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌 > ) =... in e.g. mapdh8 42153 to shorten proofs with no $d on 𝑥. (Contributed by NM, 15-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if(𝑌 = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)})))))
Theorem	hdmap1val0 42164	Value of preliminary map from vectors to functionals at zero. (Restated mapdhval0 42090.) (Contributed by NM, 17-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = 𝑄)
Theorem	hdmap1val2 42165*	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)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐿‘{(𝐹𝑅ℎ)}))))
Theorem	hdmap1eq 42166	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)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐷)    &   ⊢ (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)}))))
Theorem	hdmap1cbv 42167*	Frequently used lemma to change bound variables in 𝐿 hypothesis. (Contributed by NM, 15-May-2015.)
⊢ 𝐿 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    ⇒   ⊢ 𝐿 = (𝑦 ∈ V ↦ if((2nd ‘𝑦) = 0 , 𝑄, (℩𝑖 ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑦)})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑦)) − (2nd ‘𝑦))})) = (𝐽‘{((2nd ‘(1st ‘𝑦))𝑅𝑖)})))))
Theorem	hdmap1valc 42168*	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 𝑋 ∈ 𝑉 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 42167 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)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ 𝐿 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = (𝐿‘⟨𝑋, 𝐹, 𝑌⟩))
Theorem	hdmap1cl 42169	Convert closure theorem mapdhcl 42092 to use HDMap1 function. (Contributed by NM, 15-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷)
Theorem	hdmap1eq2 42170	Convert mapdheq2 42094 to use HDMap1 function. (Contributed by NM, 16-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑋⟩) = 𝐹)
Theorem	hdmap1eq4N 42171	Convert mapdheq4 42097 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 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑍 ∈ (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)
Theorem	hdmap1l6lem1 42172	Lemma for hdmap1l6 42186. 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)    &   ⊢ + = (+g‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑍 ∈ (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))}))
Theorem	hdmap1l6lem2 42173	Lemma for hdmap1l6 42186. 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)    &   ⊢ + = (+g‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑍 ∈ (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)}))
Theorem	hdmap1l6a 42174	Lemma for hdmap1l6 42186. Part (6) in [Baer] p. 47, case 1. (Contributed by NM, 23-Apr-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑍 ∈ (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⟩)))
Theorem	hdmap1l6b0N 42175	Lemmma for hdmap1l6 42186. (Contributed by NM, 23-Apr-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → ((𝑁‘{𝑋}) ∩ (𝑁‘{𝑌, 𝑍})) = { 0 })    ⇒   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
Theorem	hdmap1l6b 42176	Lemmma for hdmap1l6 42186. (Contributed by NM, 24-Apr-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   ⊢ (𝜑 → 𝑌 = 0 )    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
Theorem	hdmap1l6c 42177	Lemmma for hdmap1l6 42186. (Contributed by NM, 24-Apr-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 = 0 )    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
Theorem	hdmap1l6d 42178	Lemmma for hdmap1l6 42186. (Contributed by NM, 1-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑤 + (𝑌 + 𝑍))⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩)))
Theorem	hdmap1l6e 42179	Lemmma for hdmap1l6 42186. Part (6) in [Baer] p. 47 line 38. (Contributed by NM, 1-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, ((𝑤 + 𝑌) + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, (𝑤 + 𝑌)⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
Theorem	hdmap1l6f 42180	Lemmma for hdmap1l6 42186. Part (6) in [Baer] p. 47 line 38. (Contributed by NM, 1-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑤 + 𝑌)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑌⟩)))
Theorem	hdmap1l6g 42181	Lemmma for hdmap1l6 42186. Part (6) of [Baer] p. 47 line 39. (Contributed by NM, 1-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑤 ∈ (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⟩)))
Theorem	hdmap1l6h 42182	Lemmma for hdmap1l6 42186. Part (6) of [Baer] p. 48 line 2. (Contributed by NM, 1-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
Theorem	hdmap1l6i 42183	Lemmma for hdmap1l6 42186. Eliminate auxiliary vector 𝑤. (Contributed by NM, 1-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
Theorem	hdmap1l6j 42184	Lemmma for hdmap1l6 42186. Eliminate (𝑁 { Y } ) = ( N {𝑍}) hypothesis. (Contributed by NM, 1-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
Theorem	hdmap1l6k 42185	Lemmma for hdmap1l6 42186. Eliminate nonzero vector requirement. (Contributed by NM, 1-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    ⇒   ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ✚ (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
Theorem	hdmap1l6 42186	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 21093. (Convert mapdh6N 42112 to use the function HDMap1.) (Contributed by NM, 17-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (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⟩)))
Theorem	hdmap1eulem 42187*	Lemma for hdmap1eu 42189. TODO: combine with hdmap1eu 42189 or at least share some hypotheses. (Contributed by NM, 15-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑉)    &   ⊢ 𝐿 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    ⇒   ⊢ (𝜑 → ∃!𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩)))
Theorem	hdmap1eulemOLDN 42188*	Lemma for hdmap1euOLDN 42190. TODO: combine with hdmap1euOLDN 42190 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)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑅 = (-g‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐽 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑉)    &   ⊢ 𝐿 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)})))))    ⇒   ⊢ (𝜑 → ∃!𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩)))
Theorem	hdmap1eu 42189*	Convert mapdh9a 42154 to use the HDMap1 notation. (Contributed by NM, 15-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∃!𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩)))
Theorem	hdmap1euOLDN 42190*	Convert mapdh9aOLDN 42155 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 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∃!𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩)))
Theorem	hdmapffval 42191*	Map from vectors to functionals in the closed kernel dual space. (Contributed by NM, 15-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝑋 → (HDMap‘𝐾) = (𝑤 ∈ 𝐻 ↦ {𝑎 ∣ [⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑤))⟩ / 𝑒][((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘𝑢) / 𝑣][((HDMap1‘𝐾)‘𝑤) / 𝑖]𝑎 ∈ (𝑡 ∈ 𝑣 ↦ (℩𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑤))∀𝑧 ∈ 𝑣 (¬ 𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩))))}))
Theorem	hdmapfval 42192*	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 → (𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝑆 = (𝑡 ∈ 𝑉 ↦ (℩𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑡⟩)))))
Theorem	hdmapval 42193*	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 41477). (𝐽‘𝐸) is a fixed reference functional determined by this vector (corresponding to u' on line 8; mapdhvmap 42134 shows in Baer's notation (Fu)* = Gu'). Baer's independent vectors v and w on line 7 correspond to our 𝑧 that the ∀𝑧 ∈ 𝑉 ranges over. The middle term (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩) provides isolation to allow 𝐸 and 𝑇 to assume the same value without conflict. Closure is shown by hdmapcl 42195. If a separate auxiliary vector is known, hdmapval2 42197 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 → (𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑆‘𝑇) = (℩𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩))))
Theorem	hdmapfnN 42194	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 ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝑆 Fn 𝑉)
Theorem	hdmapcl 42195	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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑆‘𝑇) ∈ 𝐷)
Theorem	hdmapval2lem 42196*	Lemma for hdmapval2 42197. (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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    ⇒   ⊢ (𝜑 → ((𝑆‘𝑇) = 𝐹 ↔ ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝐹 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩))))
Theorem	hdmapval2 42197	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 42136 through hdmaplem4 42139, 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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})))    ⇒   ⊢ (𝜑 → (𝑆‘𝑇) = (𝐼‘⟨𝑋, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑋⟩), 𝑇⟩))
Theorem	hdmapval0 42198	Value of map from vectors to functionals at zero. Note: we use dvh3dim 41811 for convenience, even though 3 dimensions aren't necessary at this point. TODO: I think either this or hdmapeq0 42209 could be derived from the other to shorten proof. (Contributed by NM, 17-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → (𝑆‘ 0 ) = 𝑄)
Theorem	hdmapeveclem 42199	Lemma for hdmapevec 42200. TODO: combine with hdmapevec 42200 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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝐸})))    ⇒   ⊢ (𝜑 → (𝑆‘𝐸) = (𝐽‘𝐸))
Theorem	hdmapevec 42200	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 ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → (𝑆‘𝐸) = (𝐽‘𝐸))