P. 403 of Theorem List - Metamath Proof Explorer

Home	Metamath Proof Explorer Theorem List (p. 403 of 473)	< Previous Next >
		Bad symbols? Try the GIF version.
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List

Color key:

Metamath Proof Explorer
(1-29860)

Hilbert Space Explorer
(29861-31383)

Users' Mathboxes
(31384-47242)

**Theorem List for Metamath Proof Explorer -** 40201-40300 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	mapdh6cN 40201*	Lemmma for mapdh6N 40210. (Contributed by NM, 24-Apr-2015.) (New usage is discouraged.)
⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-_g‘𝑈) & ⊢ 0 = (0_g‘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 40202*	Lemmma for mapdh6N 40210. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-_g‘𝑈) & ⊢ 0 = (0_g‘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 40203*	Lemmma for mapdh6N 40210. Part (6) in [Baer] p. 47 line 38. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-_g‘𝑈) & ⊢ 0 = (0_g‘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 40204*	Lemmma for mapdh6N 40210. Part (6) in [Baer] p. 47 line 38. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-_g‘𝑈) & ⊢ 0 = (0_g‘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 40205*	Lemmma for mapdh6N 40210. Part (6) of [Baer] p. 47 line 39. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-_g‘𝑈) & ⊢ 0 = (0_g‘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 40206*	Lemmma for mapdh6N 40210. Part (6) of [Baer] p. 48 line 2. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-_g‘𝑈) & ⊢ 0 = (0_g‘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 40207*	Lemmma for mapdh6N 40210. Eliminate auxiliary vector 𝑤. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-_g‘𝑈) & ⊢ 0 = (0_g‘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 40208*	Lemmma for mapdh6N 40210. Eliminate (𝑁‘{𝑌}) = (𝑁‘{𝑍}) hypothesis. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-_g‘𝑈) & ⊢ 0 = (0_g‘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 40209*	Lemmma for mapdh6N 40210. Eliminate nonzero vector requirement. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-_g‘𝑈) & ⊢ 0 = (0_g‘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 40210*	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 20587. 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 40284. (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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ ✚ = (+_g‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘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 40211*	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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘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 40212*	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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘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 40213*	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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘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 40214*	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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘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 40215*	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 40216.) (Contributed by NM, 2-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-_g‘𝑈) & ⊢ 0 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘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 40216*	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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘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 40217*	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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘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 40218*	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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘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 40219	Extend class notation with vector to dual map.
class HVMap

Definition	df-hvmap 40220*	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 40014, dochfl1 39939- 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)) ∖ {(0_g‘((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 40221*	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)) ∖ {(0_g‘((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 40222*	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 = (0_g‘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 40223*	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 = (0_g‘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 40224*	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 = (0_g‘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 40225	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 = (0_g‘𝑈) & ⊢ 𝑆 = (Scalar‘𝑈) & ⊢ 1 = (1_r‘𝑆) & ⊢ 𝑀 = ((HVMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → ((𝑀‘𝑋)‘𝑋) = 1 )

Theorem	hvmap1o 40226*	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 = (0_g‘𝑈) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝐿 = (LKer‘𝑈) & ⊢ 𝐷 = (LDual‘𝑈) & ⊢ 𝑄 = (0_g‘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 40227*	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 = (0_g‘𝑈) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝐿 = (LKer‘𝑈) & ⊢ 𝐷 = (LDual‘𝑈) & ⊢ 𝑄 = (0_g‘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 40228	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 = (0_g‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ 𝑂 = (0_g‘𝐶) & ⊢ 𝑀 = ((HVMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → 𝑀:(𝑉 ∖ { 0 })–1-1-onto→(𝐹 ∖ {𝑂}))

Theorem	hvmapcl2 40229	Closure of the vector to functional map. (Contributed by NM, 27-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0_g‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ 𝑂 = (0_g‘𝐶) & ⊢ 𝑀 = ((HVMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → (𝑀‘𝑋) ∈ (𝐹 ∖ {𝑂}))

Theorem	hvmaplfl 40230	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 = (0_g‘𝑈) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝑀 = ((HVMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → (𝑀‘𝑋) ∈ 𝐹)

Theorem	hvmaplkr 40231	Kernel of the vector to functional map. TODO: make this become lcfrlem11 40016. (Contributed by NM, 29-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0_g‘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 40232	Relationship between mapd and HVMap, which can be used to satisfy the last hypothesis of mapdpg 40169. Equation 10 of [Baer] p. 48. (Contributed by NM, 29-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0_g‘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 40233	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 20590 and prcom 4693.) (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 40234	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 40235	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 40236	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 = (0_g‘𝑊) ⇒ ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))

Theorem	hdmaplem4 40237	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 = (0_g‘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 40238*	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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘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 40239*	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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘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 40240*	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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘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 40241*	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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘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 40242*	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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘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 40243*	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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘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 40244*	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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘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 40245*	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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘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 40246*	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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘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 40247*	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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘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 40248*	Part of Part (8) in [Baer] p. 48. Eliminate 𝑋 ∈ (𝑁‘{𝑌, 𝑇}). TODO: break out 𝑇 ≠ 0 in mapdh8e 40247 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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘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 40249*	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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘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 40250*	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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘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 40251*	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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘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 40252*	Lemma for part (9) in [Baer] p. 48. TODO: why is this 50% larger than mapdh9aOLDN 40253? (Contributed by NM, 14-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-_g‘𝑈) & ⊢ 0 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘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 40253*	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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘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 40254	Extend class notation with preliminary map from vectors to functionals in the closed kernel dual space.
class HDMap1

Syntax	chdma 40255	Extend class notation with map from vectors to functionals in the closed kernel dual space.
class HDMap

Definition	df-hdmap1 40256*	Define preliminary map from vectors to functionals in the closed kernel dual space. See hdmap1fval 40259 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((2^nd ‘𝑥) = (0_g‘𝑢), (0_g‘𝑐), (℩ℎ ∈ 𝑑 ((𝑚‘(𝑛‘{(2^nd ‘𝑥)})) = (𝑗‘{ℎ}) ∧ (𝑚‘(𝑛‘{((1^st ‘(1^st ‘𝑥))(-_g‘𝑢)(2^nd ‘𝑥))})) = (𝑗‘{((2^nd ‘(1^st ‘𝑥))(-_g‘𝑐)ℎ)})))))}))

Definition	df-hdmap 40257*	Define map from vectors to functionals in the closed kernel dual space. See hdmapfval 40290 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 40258*	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((2^nd ‘𝑥) = (0_g‘𝑢), (0_g‘𝑐), (℩ℎ ∈ 𝑑 ((𝑚‘(𝑛‘{(2^nd ‘𝑥)})) = (𝑗‘{ℎ}) ∧ (𝑚‘(𝑛‘{((1^st ‘(1^st ‘𝑥))(-_g‘𝑢)(2^nd ‘𝑥))})) = (𝑗‘{((2^nd ‘(1^st ‘𝑥))(-_g‘𝑐)ℎ)})))))}))

Theorem	hdmap1fval 40259*	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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → 𝐼 = (𝑥 ∈ ((𝑉 × 𝐷) × 𝑉) ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)}))))))

Theorem	hdmap1vallem 40260*	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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_g‘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((2^nd ‘𝑇) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑇)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑇)) − (2^nd ‘𝑇))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑇))𝑅ℎ)})))))

Theorem	hdmap1val 40261*	Value of preliminary map from vectors to functionals in the closed kernel dual space. (Restatement of mapdhval 40187.) TODO: change 𝐼 = (𝑥 ∈ V ↦... to (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌 > ) =... in e.g. mapdh8 40251 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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_g‘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 40262	Value of preliminary map from vectors to functionals at zero. (Restated mapdhval0 40188.) (Contributed by NM, 17-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0_g‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = 𝑄)

Theorem	hdmap1val2 40263*	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 = (0_g‘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 40264	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 = (0_g‘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 40265*	Frequently used lemma to change bound variables in 𝐿 hypothesis. (Contributed by NM, 15-May-2015.)
⊢ 𝐿 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)}))))) ⇒ ⊢ 𝐿 = (𝑦 ∈ V ↦ if((2^nd ‘𝑦) = 0 , 𝑄, (℩𝑖 ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑦)})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑦)) − (2^nd ‘𝑦))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑦))𝑅𝑖)})))))

Theorem	hdmap1valc 40266*	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 40265 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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_g‘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((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)}))))) ⇒ ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = (𝐿‘⟨𝑋, 𝐹, 𝑌⟩))

Theorem	hdmap1cl 40267	Convert closure theorem mapdhcl 40190 to use HDMap1 function. (Contributed by NM, 15-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0_g‘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 40268	Convert mapdheq2 40192 to use HDMap1 function. (Contributed by NM, 16-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0_g‘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 40269	Convert mapdheq4 40195 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 = (0_g‘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 40270	Lemma for hdmap1l6 40284. 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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ ✚ = (+_g‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_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 → (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 40271	Lemma for hdmap1l6 40284. 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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ ✚ = (+_g‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_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 → (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 40272	Lemma for hdmap1l6 40284. 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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ ✚ = (+_g‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_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 → (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 40273	Lemmma for hdmap1l6 40284. (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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ ✚ = (+_g‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_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 → (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 40274	Lemmma for hdmap1l6 40284. (Contributed by NM, 24-Apr-2015.)
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ + = (+_g‘𝑈) & ⊢ − = (-_g‘𝑈) & ⊢ 0 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ ✚ = (+_g‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_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 → (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 40275	Lemmma for hdmap1l6 40284. (Contributed by NM, 24-Apr-2015.)
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ + = (+_g‘𝑈) & ⊢ − = (-_g‘𝑈) & ⊢ 0 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ ✚ = (+_g‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_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 → (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 40276	Lemmma for hdmap1l6 40284. (Contributed by NM, 1-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ + = (+_g‘𝑈) & ⊢ − = (-_g‘𝑈) & ⊢ 0 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ ✚ = (+_g‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_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 → (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 40277	Lemmma for hdmap1l6 40284. 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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ ✚ = (+_g‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_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 → (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 40278	Lemmma for hdmap1l6 40284. 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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ ✚ = (+_g‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_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 → (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 40279	Lemmma for hdmap1l6 40284. 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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ ✚ = (+_g‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_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 → (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 40280	Lemmma for hdmap1l6 40284. 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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ ✚ = (+_g‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_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 → (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 40281	Lemmma for hdmap1l6 40284. 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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ ✚ = (+_g‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_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 → (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 40282	Lemmma for hdmap1l6 40284. 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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ ✚ = (+_g‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_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 → (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 40283	Lemmma for hdmap1l6 40284. 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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ ✚ = (+_g‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_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 → (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 40284	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 20587. (Convert mapdh6N 40210 to use the function HDMap1.) (Contributed by NM, 17-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ + = (+_g‘𝑈) & ⊢ 0 = (0_g‘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 40285*	Lemma for hdmap1eu 40287. TODO: combine with hdmap1eu 40287 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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_g‘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((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)}))))) ⇒ ⊢ (𝜑 → ∃!𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩)))

Theorem	hdmap1eulemOLDN 40286*	Lemma for hdmap1euOLDN 40288. TODO: combine with hdmap1euOLDN 40288 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 = (0_g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-_g‘𝐶) & ⊢ 𝑄 = (0_g‘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((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)}))))) ⇒ ⊢ (𝜑 → ∃!𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩)))

Theorem	hdmap1eu 40287*	Convert mapdh9a 40252 to use the HDMap1 notation. (Contributed by NM, 15-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0_g‘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 40288*	Convert mapdh9aOLDN 40253 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 = (0_g‘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 40289*	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 40290*	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 40291*	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 39575). (𝐽‘𝐸) is a fixed reference functional determined by this vector (corresponding to u' on line 8; mapdhvmap 40232 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 40293. If a separate auxiliary vector is known, hdmapval2 40295 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 40292	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 40293	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 40294*	Lemma for hdmapval2 40295. (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 40295	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 40234 through hdmaplem4 40237, 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 40296	Value of map from vectors to functionals at zero. Note: we use dvh3dim 39909 for convenience, even though 3 dimensions aren't necessary at this point. TODO: I think either this or hdmapeq0 40307 could be derived from the other to shorten proof. (Contributed by NM, 17-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 0 = (0_g‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝑄 = (0_g‘𝐶) & ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → (𝑆‘ 0 ) = 𝑄)

Theorem	hdmapeveclem 40297	Lemma for hdmapevec 40298. TODO: combine with hdmapevec 40298 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 40298	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 40299	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 = (1_r‘𝑅) ⇒ ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐸) = 1 )

Theorem	hdmapval3lemN 40300	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 ∖ {(0_g‘𝑈)})) & ⊢ (𝜑 → 𝑥 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑥 ∈ (𝑁‘{𝐸, 𝑇})) ⇒ ⊢ (𝜑 → (𝑆‘𝑇) = (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑇⟩))

< Previous Next >

Page List Jump to page: Contents 1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47242

< Previous Next >