P. 430 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 42901-43000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mapdh7dN 42901*	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 42902*	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 42903*	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 42904.) (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 42904*	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 42905*	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 42906*	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 42907	Extend class notation with vector to dual map.
class HVMap
Definition	df-hvmap 42908*	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 42702, dochfl1 42627- 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 42909*	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 42910*	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 42911*	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 42912*	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 42913	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 42914*	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 42915*	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 42916	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 42917	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 42918	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 42919	Kernel of the vector to functional map. TODO: make this become lcfrlem11 42704. (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 42920	Relationship between mapd and HVMap, which can be used to satisfy the last hypothesis of mapdpg 42857. 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 42921	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 21413 and prcom 4689.) (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 42922	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 42923	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 42924	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 42925	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 42926*	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 42927*	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 42928*	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 42929*	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 42930*	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 42931*	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 42932*	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 42933*	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 42934*	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 42935*	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 42936*	Part of Part (8) in [Baer] p. 48. Eliminate 𝑋 ∈ (𝑁‘{𝑌, 𝑇}). TODO: break out 𝑇 ≠ 0 in mapdh8e 42935 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 42937*	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 42938*	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 42939*	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 42940*	Lemma for part (9) in [Baer] p. 48. TODO: why is this 50% larger than mapdh9aOLDN 42941? (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 42941*	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 42942	Extend class notation with preliminary map from vectors to functionals in the closed kernel dual space.
class HDMap1
Syntax	chdma 42943	Extend class notation with map from vectors to functionals in the closed kernel dual space.
class HDMap
Definition	df-hdmap1 42944*	Define preliminary map from vectors to functionals in the closed kernel dual space. See hdmap1fval 42947 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 42945*	Define map from vectors to functionals in the closed kernel dual space. See hdmapfval 42978 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 42946*	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 42947*	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 42948*	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 42949*	Value of preliminary map from vectors to functionals in the closed kernel dual space. (Restatement of mapdhval 42875.) TODO: change 𝐼 = (𝑥 ∈ V ↦... to (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌 > ) =... in e.g. mapdh8 42939 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 42950	Value of preliminary map from vectors to functionals at zero. (Restated mapdhval0 42876.) (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 42951*	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 42952	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 42953*	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 42954*	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 42953 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 42955	Convert closure theorem mapdhcl 42878 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 42956	Convert mapdheq2 42880 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 42957	Convert mapdheq4 42883 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 42958	Lemma for hdmap1l6 42972. 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 42959	Lemma for hdmap1l6 42972. 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 42960	Lemma for hdmap1l6 42972. 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 42961	Lemmma for hdmap1l6 42972. (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 42962	Lemmma for hdmap1l6 42972. (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 42963	Lemmma for hdmap1l6 42972. (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 42964	Lemmma for hdmap1l6 42972. (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 42965	Lemmma for hdmap1l6 42972. 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 42966	Lemmma for hdmap1l6 42972. 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 42967	Lemmma for hdmap1l6 42972. 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 42968	Lemmma for hdmap1l6 42972. 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 42969	Lemmma for hdmap1l6 42972. 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 42970	Lemmma for hdmap1l6 42972. 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 42971	Lemmma for hdmap1l6 42972. 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 42972	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 21410. (Convert mapdh6N 42898 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 42973*	Lemma for hdmap1eu 42975. TODO: combine with hdmap1eu 42975 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 42974*	Lemma for hdmap1euOLDN 42976. TODO: combine with hdmap1euOLDN 42976 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 42975*	Convert mapdh9a 42940 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 42976*	Convert mapdh9aOLDN 42941 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 42977*	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 42978*	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 42979*	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 42263). (𝐽‘𝐸) is a fixed reference functional determined by this vector (corresponding to u' on line 8; mapdhvmap 42920 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 42981. If a separate auxiliary vector is known, hdmapval2 42983 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 42980	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 42981	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 42982*	Lemma for hdmapval2 42983. (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 42983	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 42922 through hdmaplem4 42925, 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 42984	Value of map from vectors to functionals at zero. Note: we use dvh3dim 42597 for convenience, even though 3 dimensions aren't necessary at this point. TODO: I think either this or hdmapeq0 42995 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 42985	Lemma for hdmapevec 42986. TODO: combine with hdmapevec 42986 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 42986	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))
Theorem	hdmapevec2 42987	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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐸) = 1 )
Theorem	hdmapval3lemN 42988	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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝑁‘{𝑇}) ≠ (𝑁‘{𝐸}))    &   ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ {(0g‘𝑈)}))    &   ⊢ (𝜑 → 𝑥 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑥 ∈ (𝑁‘{𝐸, 𝑇}))    ⇒   ⊢ (𝜑 → (𝑆‘𝑇) = (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑇⟩))
Theorem	hdmapval3N 42989	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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝑁‘{𝑇}) ≠ (𝑁‘{𝐸}))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑆‘𝑇) = (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑇⟩))
Theorem	hdmap10lem 42990	Lemma for hdmap10 42991. (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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑇})) = (𝐿‘{(𝑆‘𝑇)}))
Theorem	hdmap10 42991	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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑇})) = (𝐿‘{(𝑆‘𝑇)}))
Theorem	hdmap11lem1 42992	Lemma for hdmapadd 42994. (Contributed by NM, 26-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ + = (+g‘𝐶)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   ⊢ (𝜑 → 𝑧 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}))    &   ⊢ (𝜑 → (𝑁‘{𝑧}) ≠ (𝑁‘{𝐸}))    ⇒   ⊢ (𝜑 → (𝑆‘(𝑋 + 𝑌)) = ((𝑆‘𝑋) + (𝑆‘𝑌)))
Theorem	hdmap11lem2 42993	Lemma for hdmapadd 42994. (Contributed by NM, 26-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ + = (+g‘𝐶)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)    ⇒   ⊢ (𝜑 → (𝑆‘(𝑋 + 𝑌)) = ((𝑆‘𝑋) + (𝑆‘𝑌)))
Theorem	hdmapadd 42994	Part 11 in [Baer] p. 48 line 35, (a+b)S = aS+bS in their notation (S = sigma). (Contributed by NM, 22-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ + = (+g‘𝐶)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑆‘(𝑋 + 𝑌)) = ((𝑆‘𝑋) + (𝑆‘𝑌)))
Theorem	hdmapeq0 42995	Part of proof of part 12 in [Baer] p. 49 line 3. (Contributed by NM, 22-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝑆‘𝑇) = 𝑄 ↔ 𝑇 = 0 ))
Theorem	hdmapnzcl 42996	Nonzero vector closure of map from vectors to functionals with closed kernels. (Contributed by NM, 27-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝑆‘𝑇) ∈ (𝐷 ∖ {𝑄}))
Theorem	hdmapneg 42997	Part of proof of part 12 in [Baer] p. 49 line 4. The sigma map of a negative is the negative of the sigma map. (Contributed by NM, 24-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑀 = (invg‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐼 = (invg‘𝐶)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑆‘(𝑀‘𝑇)) = (𝐼‘(𝑆‘𝑇)))
Theorem	hdmapsub 42998	Part of proof of part 12 in [Baer] p. 49 line 5, (a-b)S = aS-bS in their notation (S = sigma). (Contributed by NM, 26-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝑁 = (-g‘𝐶)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑆‘(𝑋 − 𝑌)) = ((𝑆‘𝑋)𝑁(𝑆‘𝑌)))
Theorem	hdmap11 42999	Part of proof of part 12 in [Baer] p. 49 line 4, aS=bS iff a=b in their notation (S = sigma). The sigma map is one-to-one. (Contributed by NM, 26-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝑆‘𝑋) = (𝑆‘𝑌) ↔ 𝑋 = 𝑌))
Theorem	hdmaprnlem1N 43000	Part of proof of part 12 in [Baer] p. 49 line 10, Gu' ≠ Gs. Our (𝑁‘{𝑣}) is Baer's T. (Contributed by NM, 26-May-2015.) (New usage is discouraged.)
⊢ 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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄}))    &   ⊢ (𝜑 → 𝑣 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))    &   ⊢ (𝜑 → 𝑢 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))    ⇒   ⊢ (𝜑 → (𝐿‘{(𝑆‘𝑢)}) ≠ (𝐿‘{𝑠}))