HomeHome Metamath Proof Explorer
Theorem List (p. 361 of 425)
< 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  Metamath Proof Explorer
(1-26947)
  Hilbert Space Explorer  Hilbert Space Explorer
(26948-28472)
  Users' Mathboxes  Users' Mathboxes
(28473-42426)
 

Theorem List for Metamath Proof Explorer - 36001-36100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhdmap1valc 36001* Connect the value of the preliminary map from vectors to functionals 𝐼 to the hypothesis 𝐿 used by earlier theorems. Note: the 𝑋 ∈ (𝑉 ∖ { 0 }) hypothesis could be the more general 𝑋𝑉 but the former will be easier to use. TODO: use the 𝐼 function directly in those theorems, so this theorem becomes unnecessary? TODO: The hdmap1cbv 36000 is probably unnecessary, but it would mean different $d's later on. (Contributed by NM, 15-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝑄 = (0g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐹𝐷)    &   (𝜑𝑌𝑉)    &   𝐿 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))       (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = (𝐿‘⟨𝑋, 𝐹, 𝑌⟩))
 
Theoremhdmap1cl 36002 Convert closure theorem mapdhcl 35924 to use HDMap1 function. (Contributed by NM, 15-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌𝑉)       (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷)
 
Theoremhdmap1eq2 36003 Convert mapdheq2 35926 to use HDMap1 function. (Contributed by NM, 16-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)       (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑋⟩) = 𝐹)
 
Theoremhdmap1eq4N 36004 Convert mapdheq4 35929 to use HDMap1 function. (Contributed by NM, 17-May-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)    &   (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐵)       (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑍⟩) = 𝐵)
 
Theoremhdmap1l6lem1 36005 Lemma for hdmap1l6 36019. Part (6) in [Baer] p. 47, lines 16-18. (Contributed by NM, 13-Apr-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &    = (+g𝐶)    &   𝑅 = (-g𝐶)    &   𝑄 = (0g𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))    &   (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)    &   (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)       (𝜑 → (𝑀‘(𝑁‘{(𝑋 (𝑌 + 𝑍))})) = (𝐿‘{(𝐹𝑅(𝐺 𝐸))}))
 
Theoremhdmap1l6lem2 36006 Lemma for hdmap1l6 36019. Part (6) in [Baer] p. 47, lines 20-22. (Contributed by NM, 13-Apr-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &    = (+g𝐶)    &   𝑅 = (-g𝐶)    &   𝑄 = (0g𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))    &   (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)    &   (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)       (𝜑 → (𝑀‘(𝑁‘{(𝑌 + 𝑍)})) = (𝐿‘{(𝐺 𝐸)}))
 
Theoremhdmap1l6a 36007 Lemma for hdmap1l6 36019. Part (6) in [Baer] p. 47, case 1. (Contributed by NM, 23-Apr-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &    = (+g𝐶)    &   𝑅 = (-g𝐶)    &   𝑄 = (0g𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))    &   (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)    &   (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)       (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
 
Theoremhdmap1l6b0N 36008 Lemmma for hdmap1l6 36019. (Contributed by NM, 23-Apr-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &    = (+g𝐶)    &   𝑅 = (-g𝐶)    &   𝑄 = (0g𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (𝜑𝑌𝑉)    &   (𝜑𝑍𝑉)    &   (𝜑 → ((𝑁‘{𝑋}) ∩ (𝑁‘{𝑌, 𝑍})) = { 0 })       (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
 
Theoremhdmap1l6b 36009 Lemmma for hdmap1l6 36019. (Contributed by NM, 24-Apr-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &    = (+g𝐶)    &   𝑅 = (-g𝐶)    &   𝑄 = (0g𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (𝜑𝑌 = 0 )    &   (𝜑𝑍𝑉)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))       (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
 
Theoremhdmap1l6c 36010 Lemmma for hdmap1l6 36019. (Contributed by NM, 24-Apr-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &    = (+g𝐶)    &   𝑅 = (-g𝐶)    &   𝑄 = (0g𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (𝜑𝑌𝑉)    &   (𝜑𝑍 = 0 )    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))       (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
 
Theoremhdmap1l6d 36011 Lemmma for hdmap1l6 36019. (Contributed by NM, 1-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &    = (+g𝐶)    &   𝑅 = (-g𝐶)    &   𝑄 = (0g𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑤 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))       (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑤 + (𝑌 + 𝑍))⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩)))
 
Theoremhdmap1l6e 36012 Lemmma for hdmap1l6 36019. Part (6) in [Baer] p. 47 line 38. (Contributed by NM, 1-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &    = (+g𝐶)    &   𝑅 = (-g𝐶)    &   𝑄 = (0g𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑤 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))       (𝜑 → (𝐼‘⟨𝑋, 𝐹, ((𝑤 + 𝑌) + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, (𝑤 + 𝑌)⟩) (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
 
Theoremhdmap1l6f 36013 Lemmma for hdmap1l6 36019. Part (6) in [Baer] p. 47 line 38. (Contributed by NM, 1-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &    = (+g𝐶)    &   𝑅 = (-g𝐶)    &   𝑄 = (0g𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑤 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))       (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑤 + 𝑌)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) (𝐼‘⟨𝑋, 𝐹, 𝑌⟩)))
 
Theoremhdmap1l6g 36014 Lemmma for hdmap1l6 36019. Part (6) of [Baer] p. 47 line 39. (Contributed by NM, 1-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &    = (+g𝐶)    &   𝑅 = (-g𝐶)    &   𝑄 = (0g𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑤 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))       (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩)) = (((𝐼‘⟨𝑋, 𝐹, 𝑤⟩) (𝐼‘⟨𝑋, 𝐹, 𝑌⟩)) (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
 
Theoremhdmap1l6h 36015 Lemmma for hdmap1l6 36019. Part (6) of [Baer] p. 48 line 2. (Contributed by NM, 1-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &    = (+g𝐶)    &   𝑅 = (-g𝐶)    &   𝑄 = (0g𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑤 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))       (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
 
Theoremhdmap1l6i 36016 Lemmma for hdmap1l6 36019. Eliminate auxiliary vector 𝑤. (Contributed by NM, 1-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &    = (+g𝐶)    &   𝑅 = (-g𝐶)    &   𝑄 = (0g𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍}))       (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
 
Theoremhdmap1l6j 36017 Lemmma for hdmap1l6 36019. Eliminate (𝑁 { Y } ) = ( N {𝑍}) hypothesis. (Contributed by NM, 1-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &    = (+g𝐶)    &   𝑅 = (-g𝐶)    &   𝑄 = (0g𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))       (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
 
Theoremhdmap1l6k 36018 Lemmma for hdmap1l6 36019. Eliminate nonzero vector requirement. (Contributed by NM, 1-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &    = (+g𝐶)    &   𝑅 = (-g𝐶)    &   𝑄 = (0g𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (𝜑𝑌𝑉)    &   (𝜑𝑍𝑉)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))       (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
 
Theoremhdmap1l6 36019 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 18851. (Contributed by NM, 17-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &    = (+g𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌𝑉)    &   (𝜑𝑍𝑉)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))       (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
 
Theoremhdmap1p6N 36020 (Convert mapdh6N 35944 to use HDMap1 function.) 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 18851. TODO: No longer used and should be deleted. Use hdmap1l6 36019 instead. Also delete unused mapdh6N 35944 etc. leading up to this. (Contributed by NM, 17-May-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &    = (+g𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌𝑉)    &   (𝜑𝑍𝑉)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))       (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))
 
Theoremhdmap1eulem 36021* Lemma for hdmap1eu 36023. TODO: combine with hdmap1eu 36023 or at least share some hypotheses. (Contributed by NM, 15-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝑄 = (0g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐹𝐷)    &   (𝜑𝑇𝑉)    &   𝐿 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))       (𝜑 → ∃!𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩)))
 
Theoremhdmap1eulemOLDN 36022* Lemma for hdmap1euOLDN 36024. TODO: combine with hdmap1euOLDN 36024 or at least share some hypotheses. (Contributed by NM, 15-May-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝑄 = (0g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐹𝐷)    &   (𝜑𝑇𝑉)    &   𝐿 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))       (𝜑 → ∃!𝑦𝐷𝑧𝑉𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩)))
 
Theoremhdmap1eu 36023* Convert mapdh9a 35987 to use the HDMap1 notation. (Contributed by NM, 15-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐹𝐷)    &   (𝜑𝑇𝑉)       (𝜑 → ∃!𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩)))
 
Theoremhdmap1euOLDN 36024* Convert mapdh9aOLDN 35988 to use the HDMap1 notation. (Contributed by NM, 15-May-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐹𝐷)    &   (𝜑𝑇𝑉)       (𝜑 → ∃!𝑦𝐷𝑧𝑉𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩)))
 
Theoremhdmap1neglem1N 36025 Lemma for hdmapneg 36046. TODO: Not used; delete. (Contributed by NM, 23-May-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑅 = (invg𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑆 = (invg𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)       (𝜑 → (𝐼‘⟨(𝑅𝑋), (𝑆𝐹), (𝑅𝑌)⟩) = (𝑆𝐺))
 
Theoremhdmapffval 36026* Map from vectors to functionals in the closed kernel dual space. (Contributed by NM, 15-May-2015.)
𝐻 = (LHyp‘𝐾)       (𝐾𝑋 → (HDMap‘𝐾) = (𝑤𝐻 ↦ {𝑎[⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑤))⟩ / 𝑒][((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘𝑢) / 𝑣][((HDMap1‘𝐾)‘𝑤) / 𝑖]𝑎 ∈ (𝑡𝑣 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑤))∀𝑧𝑣𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘⟨𝑧, (𝑖‘⟨𝑒, (((HVMap‘𝐾)‘𝑤)‘𝑒), 𝑧⟩), 𝑡⟩))))}))
 
Theoremhdmapfval 36027* Map from vectors to functionals in the closed kernel dual space. (Contributed by NM, 15-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝐽 = ((HVMap‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾𝐴𝑊𝐻))       (𝜑𝑆 = (𝑡𝑉 ↦ (𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩)))))
 
Theoremhdmapval 36028* 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 35309). (𝐽𝐸) is a fixed reference functional determined by this vector (corresponding to u' on line 8; mapdhvmap 35966 shows in Baer's notation (Fu)* = Gu'). Baer's independent vectors v and w on line 7 correspond to our 𝑧 that the 𝑧𝑉 ranges over. The middle term (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩) provides isolation to allow 𝐸 and 𝑇 to assume the same value without conflict. Closure is shown by hdmapcl 36030. If a separate auxiliary vector is known, hdmapval2 36032 provides a version without quantification. (Contributed by NM, 15-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝐽 = ((HVMap‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾𝐴𝑊𝐻))    &   (𝜑𝑇𝑉)       (𝜑 → (𝑆𝑇) = (𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑇⟩))))
 
TheoremhdmapfnN 36029 Functionality of map from vectors to functionals with closed kernels. (Contributed by NM, 30-May-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑆 Fn 𝑉)
 
Theoremhdmapcl 36030 Closure of map from vectors to functionals with closed kernels. (Contributed by NM, 15-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑇𝑉)       (𝜑 → (𝑆𝑇) ∈ 𝐷)
 
Theoremhdmapval2lem 36031* Lemma for hdmapval2 36032. (Contributed by NM, 15-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝐽 = ((HVMap‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑇𝑉)    &   (𝜑𝐹𝐷)       (𝜑 → ((𝑆𝑇) = 𝐹 ↔ ∀𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝐹 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑇⟩))))
 
Theoremhdmapval2 36032 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 35968 through hdmaplem4 35971, which would become obsolete. (Contributed by NM, 15-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝐽 = ((HVMap‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑇𝑉)    &   (𝜑𝑋𝑉)    &   (𝜑 → ¬ 𝑋 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})))       (𝜑 → (𝑆𝑇) = (𝐼‘⟨𝑋, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑋⟩), 𝑇⟩))
 
Theoremhdmapval0 36033 Value of map from vectors to functionals at zero. Note: we use dvh3dim 35643 for convenience, even though 3 dimensions aren't necessary at this point. TODO: I think either this or hdmapeq0 36044 could be derived from the other to shorten proof. (Contributed by NM, 17-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    0 = (0g𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑄 = (0g𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 → (𝑆0 ) = 𝑄)
 
Theoremhdmapeveclem 36034 Lemma for hdmapevec 36035. TODO: combine with hdmapevec 36035 if it shortens overall. (Contributed by NM, 16-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   𝐽 = ((HVMap‘𝐾)‘𝑊)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑𝑋𝑉)    &   (𝜑 → ¬ 𝑋 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝐸})))       (𝜑 → (𝑆𝐸) = (𝐽𝐸))
 
Theoremhdmapevec 36035 Value of map from vectors to functionals at the reference vector 𝐸. (Contributed by NM, 16-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   𝐽 = ((HVMap‘𝐾)‘𝑊)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 → (𝑆𝐸) = (𝐽𝐸))
 
Theoremhdmapevec2 36036 The inner product of the reference vector 𝐸 with itself is nonzero. This shows the inner product condition in the proof of Theorem 3.6 of [Holland95] p. 14 line 32, [ e , e ] ≠ 0 is satisfied. TODO: remove redundant hypothesis hdmapevec.j. (Contributed by NM, 1-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   𝐽 = ((HVMap‘𝐾)‘𝑊)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑅 = (Scalar‘𝑈)    &    1 = (1r𝑅)       (𝜑 → ((𝑆𝐸)‘𝐸) = 1 )
 
Theoremhdmapval3lemN 36037 Value of map from vectors to functionals at arguments not colinear with the reference vector 𝐸. (Contributed by NM, 17-May-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝐽 = ((HVMap‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑 → (𝑁‘{𝑇}) ≠ (𝑁‘{𝐸}))    &   (𝜑𝑇 ∈ (𝑉 ∖ {(0g𝑈)}))    &   (𝜑𝑥𝑉)    &   (𝜑 → ¬ 𝑥 ∈ (𝑁‘{𝐸, 𝑇}))       (𝜑 → (𝑆𝑇) = (𝐼‘⟨𝐸, (𝐽𝐸), 𝑇⟩))
 
Theoremhdmapval3N 36038 Value of map from vectors to functionals at arguments not colinear with the reference vector 𝐸. (Contributed by NM, 17-May-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝐽 = ((HVMap‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑 → (𝑁‘{𝑇}) ≠ (𝑁‘{𝐸}))    &   (𝜑𝑇𝑉)       (𝜑 → (𝑆𝑇) = (𝐼‘⟨𝐸, (𝐽𝐸), 𝑇⟩))
 
Theoremhdmap10lem 36039 Lemma for hdmap10 36040. (Contributed by NM, 17-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &    0 = (0g𝑈)    &   𝐷 = (Base‘𝐶)    &   𝐽 = ((HVMap‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑𝑇 ∈ (𝑉 ∖ { 0 }))       (𝜑 → (𝑀‘(𝑁‘{𝑇})) = (𝐿‘{(𝑆𝑇)}))
 
Theoremhdmap10 36040 Part 10 in [Baer] p. 48 line 33, (Ft)* = G(tS) in their notation (S = sigma). (Contributed by NM, 17-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑇𝑉)       (𝜑 → (𝑀‘(𝑁‘{𝑇})) = (𝐿‘{(𝑆𝑇)}))
 
Theoremhdmap11lem1 36041 Lemma for hdmapadd 36043. (Contributed by NM, 26-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = (+g𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐷 = (Base‘𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐽 = ((HVMap‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑𝑧𝑉)    &   (𝜑 → ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}))    &   (𝜑 → (𝑁‘{𝑧}) ≠ (𝑁‘{𝐸}))       (𝜑 → (𝑆‘(𝑋 + 𝑌)) = ((𝑆𝑋) (𝑆𝑌)))
 
Theoremhdmap11lem2 36042 Lemma for hdmapadd 36043. (Contributed by NM, 26-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = (+g𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐷 = (Base‘𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐽 = ((HVMap‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)       (𝜑 → (𝑆‘(𝑋 + 𝑌)) = ((𝑆𝑋) (𝑆𝑌)))
 
Theoremhdmapadd 36043 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)    &    + = (+g𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = (+g𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (𝑆‘(𝑋 + 𝑌)) = ((𝑆𝑋) (𝑆𝑌)))
 
Theoremhdmapeq0 36044 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 = (0g𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑄 = (0g𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑇𝑉)       (𝜑 → ((𝑆𝑇) = 𝑄𝑇 = 0 ))
 
Theoremhdmapnzcl 36045 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 = (0g𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑄 = (0g𝐶)    &   𝐷 = (Base‘𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑇 ∈ (𝑉 ∖ { 0 }))       (𝜑 → (𝑆𝑇) ∈ (𝐷 ∖ {𝑄}))
 
Theoremhdmapneg 36046 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 = (invg𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐼 = (invg𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑇𝑉)       (𝜑 → (𝑆‘(𝑀𝑇)) = (𝐼‘(𝑆𝑇)))
 
Theoremhdmapsub 36047 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)    &    = (-g𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑁 = (-g𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (𝑆‘(𝑋 𝑌)) = ((𝑆𝑋)𝑁(𝑆𝑌)))
 
Theoremhdmap11 36048 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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → ((𝑆𝑋) = (𝑆𝑌) ↔ 𝑋 = 𝑌))
 
Theoremhdmaprnlem1N 36049 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 ∧ 𝑊𝐻))    &   (𝜑𝑠 ∈ (𝐷 ∖ {𝑄}))    &   (𝜑𝑣𝑉)    &   (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))    &   (𝜑𝑢𝑉)    &   (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))       (𝜑 → (𝐿‘{(𝑆𝑢)}) ≠ (𝐿‘{𝑠}))
 
Theoremhdmaprnlem3N 36050 Part of proof of part 12 in [Baer] p. 49 line 15, T P. Our (𝑀‘(𝐿‘{((𝑆𝑢) 𝑠)})) is Baer's P, where P* = G(u'+s). (Contributed by NM, 27-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 ∧ 𝑊𝐻))    &   (𝜑𝑠 ∈ (𝐷 ∖ {𝑄}))    &   (𝜑𝑣𝑉)    &   (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))    &   (𝜑𝑢𝑉)    &   (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))    &   𝐷 = (Base‘𝐶)    &   𝑄 = (0g𝐶)    &    0 = (0g𝑈)    &    = (+g𝐶)       (𝜑 → (𝑁‘{𝑣}) ≠ (𝑀‘(𝐿‘{((𝑆𝑢) 𝑠)})))
 
Theoremhdmaprnlem3uN 36051 Part of proof of part 12 in [Baer] p. 49. (Contributed by NM, 29-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 ∧ 𝑊𝐻))    &   (𝜑𝑠 ∈ (𝐷 ∖ {𝑄}))    &   (𝜑𝑣𝑉)    &   (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))    &   (𝜑𝑢𝑉)    &   (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))    &   𝐷 = (Base‘𝐶)    &   𝑄 = (0g𝐶)    &    0 = (0g𝑈)    &    = (+g𝐶)       (𝜑 → (𝑁‘{𝑢}) ≠ (𝑀‘(𝐿‘{((𝑆𝑢) 𝑠)})))
 
Theoremhdmaprnlem4tN 36052 Lemma for hdmaprnN 36064. TODO: This lemma doesn't quite pay for itself even though used 4 times. Maybe prove this directly instead. (Contributed by NM, 27-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 ∧ 𝑊𝐻))    &   (𝜑𝑠 ∈ (𝐷 ∖ {𝑄}))    &   (𝜑𝑣𝑉)    &   (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))    &   (𝜑𝑢𝑉)    &   (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))    &   𝐷 = (Base‘𝐶)    &   𝑄 = (0g𝐶)    &    0 = (0g𝑈)    &    = (+g𝐶)    &   (𝜑𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }))       (𝜑𝑡𝑉)
 
Theoremhdmaprnlem4N 36053 Part of proof of part 12 in [Baer] p. 49 line 19. (T* =) (Ft)* = Gs. (Contributed by NM, 27-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 ∧ 𝑊𝐻))    &   (𝜑𝑠 ∈ (𝐷 ∖ {𝑄}))    &   (𝜑𝑣𝑉)    &   (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))    &   (𝜑𝑢𝑉)    &   (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))    &   𝐷 = (Base‘𝐶)    &   𝑄 = (0g𝐶)    &    0 = (0g𝑈)    &    = (+g𝐶)    &   (𝜑𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }))       (𝜑 → (𝑀‘(𝑁‘{𝑡})) = (𝐿‘{𝑠}))
 
Theoremhdmaprnlem6N 36054 Part of proof of part 12 in [Baer] p. 49 line 18, G(u'+s) = G(u'+t). (Contributed by NM, 27-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 ∧ 𝑊𝐻))    &   (𝜑𝑠 ∈ (𝐷 ∖ {𝑄}))    &   (𝜑𝑣𝑉)    &   (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))    &   (𝜑𝑢𝑉)    &   (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))    &   𝐷 = (Base‘𝐶)    &   𝑄 = (0g𝐶)    &    0 = (0g𝑈)    &    = (+g𝐶)    &   (𝜑𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }))    &    + = (+g𝑈)    &   (𝜑 → (𝐿‘{((𝑆𝑢) 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))       (𝜑 → (𝐿‘{((𝑆𝑢) 𝑠)}) = (𝐿‘{((𝑆𝑢) (𝑆𝑡))}))
 
Theoremhdmaprnlem7N 36055 Part of proof of part 12 in [Baer] p. 49 line 19, s-St G(u'+s) = P*. (Contributed by NM, 27-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 ∧ 𝑊𝐻))    &   (𝜑𝑠 ∈ (𝐷 ∖ {𝑄}))    &   (𝜑𝑣𝑉)    &   (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))    &   (𝜑𝑢𝑉)    &   (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))    &   𝐷 = (Base‘𝐶)    &   𝑄 = (0g𝐶)    &    0 = (0g𝑈)    &    = (+g𝐶)    &   (𝜑𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }))    &    + = (+g𝑈)    &   (𝜑 → (𝐿‘{((𝑆𝑢) 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))       (𝜑 → (𝑠(-g𝐶)(𝑆𝑡)) ∈ (𝐿‘{((𝑆𝑢) 𝑠)}))
 
Theoremhdmaprnlem8N 36056 Part of proof of part 12 in [Baer] p. 49 line 19, s-St (Ft)* = T*. (Contributed by NM, 27-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 ∧ 𝑊𝐻))    &   (𝜑𝑠 ∈ (𝐷 ∖ {𝑄}))    &   (𝜑𝑣𝑉)    &   (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))    &   (𝜑𝑢𝑉)    &   (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))    &   𝐷 = (Base‘𝐶)    &   𝑄 = (0g𝐶)    &    0 = (0g𝑈)    &    = (+g𝐶)    &   (𝜑𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }))    &    + = (+g𝑈)    &   (𝜑 → (𝐿‘{((𝑆𝑢) 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))       (𝜑 → (𝑠(-g𝐶)(𝑆𝑡)) ∈ (𝑀‘(𝑁‘{𝑡})))
 
Theoremhdmaprnlem9N 36057 Part of proof of part 12 in [Baer] p. 49 line 21, s=S(t). TODO: we seem to be going back and forth with mapd11 35836 and mapdcnv11N 35856. Use better hypotheses and/or theorems? (Contributed by NM, 27-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 ∧ 𝑊𝐻))    &   (𝜑𝑠 ∈ (𝐷 ∖ {𝑄}))    &   (𝜑𝑣𝑉)    &   (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))    &   (𝜑𝑢𝑉)    &   (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))    &   𝐷 = (Base‘𝐶)    &   𝑄 = (0g𝐶)    &    0 = (0g𝑈)    &    = (+g𝐶)    &   (𝜑𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }))    &    + = (+g𝑈)    &   (𝜑 → (𝐿‘{((𝑆𝑢) 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))       (𝜑𝑠 = (𝑆𝑡))
 
Theoremhdmaprnlem3eN 36058* Lemma for hdmaprnN 36064. (Contributed by NM, 29-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 ∧ 𝑊𝐻))    &   (𝜑𝑠 ∈ (𝐷 ∖ {𝑄}))    &   (𝜑𝑣𝑉)    &   (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))    &   (𝜑𝑢𝑉)    &   (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))    &   𝐷 = (Base‘𝐶)    &   𝑄 = (0g𝐶)    &    0 = (0g𝑈)    &    = (+g𝐶)    &    + = (+g𝑈)       (𝜑 → ∃𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })(𝐿‘{((𝑆𝑢) 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))
 
Theoremhdmaprnlem10N 36059* Lemma for hdmaprnN 36064. Show 𝑠 is in the range of 𝑆. (Contributed by NM, 29-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 ∧ 𝑊𝐻))    &   (𝜑𝑠 ∈ (𝐷 ∖ {𝑄}))    &   (𝜑𝑣𝑉)    &   (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))    &   (𝜑𝑢𝑉)    &   (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))    &   𝐷 = (Base‘𝐶)    &   𝑄 = (0g𝐶)    &    0 = (0g𝑈)    &    = (+g𝐶)    &    + = (+g𝑈)       (𝜑 → ∃𝑡𝑉 (𝑆𝑡) = 𝑠)
 
Theoremhdmaprnlem11N 36060* Lemma for hdmaprnN 36064. Show 𝑠 is in the range of 𝑆. (Contributed by NM, 29-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 ∧ 𝑊𝐻))    &   (𝜑𝑠 ∈ (𝐷 ∖ {𝑄}))    &   (𝜑𝑣𝑉)    &   (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))    &   (𝜑𝑢𝑉)    &   (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))    &   𝐷 = (Base‘𝐶)    &   𝑄 = (0g𝐶)    &    0 = (0g𝑈)    &    = (+g𝐶)    &    + = (+g𝑈)       (𝜑𝑠 ∈ ran 𝑆)
 
Theoremhdmaprnlem15N 36061* Lemma for hdmaprnN 36064. Eliminate 𝑢. (Contributed by NM, 30-May-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &    0 = (0g𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑠 ∈ (𝐷 ∖ { 0 }))    &   (𝜑𝑣𝑉)    &   (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))       (𝜑𝑠 ∈ ran 𝑆)
 
Theoremhdmaprnlem16N 36062 Lemma for hdmaprnN 36064. Eliminate 𝑣. (Contributed by NM, 30-May-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &    0 = (0g𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑠 ∈ (𝐷 ∖ { 0 }))       (𝜑𝑠 ∈ ran 𝑆)
 
Theoremhdmaprnlem17N 36063 Lemma for hdmaprnN 36064. Include zero. (Contributed by NM, 30-May-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &    0 = (0g𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑠𝐷)       (𝜑𝑠 ∈ ran 𝑆)
 
TheoremhdmaprnN 36064 Part of proof of part 12 in [Baer] p. 49 line 21, As=B. (Contributed by NM, 30-May-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 → ran 𝑆 = 𝐷)
 
Theoremhdmapf1oN 36065 Part 12 in [Baer] p. 49. The map from vectors to functionals with closed kernels maps one-to-one onto. Combined with hdmapadd 36043, this shows the map is an automorphism from the additive group of vectors to the additive group of functionals with closed kernels. (Contributed by NM, 30-May-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑆:𝑉1-1-onto𝐷)
 
Theoremhdmap14lem1a 36066 Prior to part 14 in [Baer] p. 49, line 25. (Contributed by NM, 31-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    · = ( ·𝑠𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = ( ·𝑠𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑃 = (Scalar‘𝐶)    &   𝐴 = (Base‘𝑃)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝐹𝐵)    &    0 = (0g𝑅)    &   (𝜑𝐹0 )       (𝜑 → (𝐿‘{(𝑆𝑋)}) = (𝐿‘{(𝑆‘(𝐹 · 𝑋))}))
 
Theoremhdmap14lem2a 36067* Prior to part 14 in [Baer] p. 49, line 25. TODO: fix to include 𝐹 = 0 so it can be used in hdmap14lem10 36077. (Contributed by NM, 31-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    · = ( ·𝑠𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = ( ·𝑠𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑃 = (Scalar‘𝐶)    &   𝐴 = (Base‘𝑃)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝐹𝐵)       (𝜑 → ∃𝑔𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 (𝑆𝑋)))
 
Theoremhdmap14lem1 36068 Prior to part 14 in [Baer] p. 49, line 25. (Contributed by NM, 31-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    · = ( ·𝑠𝑈)    &    0 = (0g𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝑍 = (0g𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = ( ·𝑠𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑃 = (Scalar‘𝐶)    &   𝐴 = (Base‘𝑃)    &   𝑄 = (0g𝑃)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐹 ∈ (𝐵 ∖ {𝑍}))       (𝜑 → (𝐿‘{(𝑆𝑋)}) = (𝐿‘{(𝑆‘(𝐹 · 𝑋))}))
 
Theoremhdmap14lem2N 36069* Prior to part 14 in [Baer] p. 49, line 25. TODO: fix to include 𝐹 = 𝑍 so it can be used in hdmap14lem10 36077. (Contributed by NM, 31-May-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    · = ( ·𝑠𝑈)    &    0 = (0g𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝑍 = (0g𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = ( ·𝑠𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑃 = (Scalar‘𝐶)    &   𝐴 = (Base‘𝑃)    &   𝑄 = (0g𝑃)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐹 ∈ (𝐵 ∖ {𝑍}))       (𝜑 → ∃𝑔 ∈ (𝐴 ∖ {𝑄})(𝑆‘(𝐹 · 𝑋)) = (𝑔 (𝑆𝑋)))
 
Theoremhdmap14lem3 36070* Prior to part 14 in [Baer] p. 49, line 26. (Contributed by NM, 31-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    · = ( ·𝑠𝑈)    &    0 = (0g𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝑍 = (0g𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = ( ·𝑠𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑃 = (Scalar‘𝐶)    &   𝐴 = (Base‘𝑃)    &   𝑄 = (0g𝑃)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐹 ∈ (𝐵 ∖ {𝑍}))       (𝜑 → ∃!𝑔 ∈ (𝐴 ∖ {𝑄})(𝑆‘(𝐹 · 𝑋)) = (𝑔 (𝑆𝑋)))
 
Theoremhdmap14lem4a 36071* Simplify (𝐴 ∖ {𝑄}) in hdmap14lem3 36070 to provide a slightly simpler definition later. (Contributed by NM, 31-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    · = ( ·𝑠𝑈)    &    0 = (0g𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝑍 = (0g𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = ( ·𝑠𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑃 = (Scalar‘𝐶)    &   𝐴 = (Base‘𝑃)    &   𝑄 = (0g𝑃)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐹 ∈ (𝐵 ∖ {𝑍}))       (𝜑 → (∃!𝑔 ∈ (𝐴 ∖ {𝑄})(𝑆‘(𝐹 · 𝑋)) = (𝑔 (𝑆𝑋)) ↔ ∃!𝑔𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 (𝑆𝑋))))
 
Theoremhdmap14lem4 36072* Simplify (𝐴 ∖ {𝑄}) in hdmap14lem3 36070 to provide a slightly simpler definition later. TODO: Use hdmap14lem4a 36071 if that one is also used directly elsewhere. Otherwise, merge hdmap14lem4a 36071 into this one. (Contributed by NM, 31-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    · = ( ·𝑠𝑈)    &    0 = (0g𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝑍 = (0g𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = ( ·𝑠𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑃 = (Scalar‘𝐶)    &   𝐴 = (Base‘𝑃)    &   𝑄 = (0g𝑃)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐹 ∈ (𝐵 ∖ {𝑍}))       (𝜑 → ∃!𝑔𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 (𝑆𝑋)))
 
Theoremhdmap14lem6 36073* Case where 𝐹 is zero. (Contributed by NM, 1-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    · = ( ·𝑠𝑈)    &    0 = (0g𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝑍 = (0g𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = ( ·𝑠𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑃 = (Scalar‘𝐶)    &   𝐴 = (Base‘𝑃)    &   𝑄 = (0g𝑃)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐹 = 𝑍)       (𝜑 → ∃!𝑔𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 (𝑆𝑋)))
 
Theoremhdmap14lem7 36074* Combine cases of 𝐹. TODO: Can this be done at once in hdmap14lem3 36070, in order to get rid of hdmap14lem6 36073? Perhaps modify lspsneu 18848 to become ∃!𝑘𝐾 instead of ∃!𝑘 ∈ (𝐾 ∖ { 0 })? (Contributed by NM, 1-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    · = ( ·𝑠𝑈)    &    0 = (0g𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = ( ·𝑠𝐶)    &   𝑃 = (Scalar‘𝐶)    &   𝐴 = (Base‘𝑃)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐹𝐵)       (𝜑 → ∃!𝑔𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 (𝑆𝑋)))
 
Theoremhdmap14lem8 36075 Part of proof of part 14 in [Baer] p. 49 lines 33-35. (Contributed by NM, 1-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    · = ( ·𝑠𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = (+g𝐶)    &    = ( ·𝑠𝐶)    &   𝑃 = (Scalar‘𝐶)    &   𝐴 = (Base‘𝑃)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐴)    &   (𝜑𝐼𝐴)    &   (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (𝐺 (𝑆𝑋)))    &   (𝜑 → (𝑆‘(𝐹 · 𝑌)) = (𝐼 (𝑆𝑌)))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑𝐽𝐴)    &   (𝜑 → (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝐽 (𝑆‘(𝑋 + 𝑌))))       (𝜑 → ((𝐽 (𝑆𝑋)) (𝐽 (𝑆𝑌))) = ((𝐺 (𝑆𝑋)) (𝐼 (𝑆𝑌))))
 
Theoremhdmap14lem9 36076 Part of proof of part 14 in [Baer] p. 49 line 38. (Contributed by NM, 1-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    · = ( ·𝑠𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = (+g𝐶)    &    = ( ·𝑠𝐶)    &   𝑃 = (Scalar‘𝐶)    &   𝐴 = (Base‘𝑃)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐴)    &   (𝜑𝐼𝐴)    &   (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (𝐺 (𝑆𝑋)))    &   (𝜑 → (𝑆‘(𝐹 · 𝑌)) = (𝐼 (𝑆𝑌)))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑𝐽𝐴)    &   (𝜑 → (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝐽 (𝑆‘(𝑋 + 𝑌))))       (𝜑𝐺 = 𝐼)
 
Theoremhdmap14lem10 36077 Part of proof of part 14 in [Baer] p. 49 line 38. (Contributed by NM, 3-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    · = ( ·𝑠𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = (+g𝐶)    &    = ( ·𝑠𝐶)    &   𝑃 = (Scalar‘𝐶)    &   𝐴 = (Base‘𝑃)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐴)    &   (𝜑𝐼𝐴)    &   (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (𝐺 (𝑆𝑋)))    &   (𝜑 → (𝑆‘(𝐹 · 𝑌)) = (𝐼 (𝑆𝑌)))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))       (𝜑𝐺 = 𝐼)
 
Theoremhdmap14lem11 36078 Part of proof of part 14 in [Baer] p. 50 line 3. (Contributed by NM, 3-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    · = ( ·𝑠𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = (+g𝐶)    &    = ( ·𝑠𝐶)    &   𝑃 = (Scalar‘𝐶)    &   𝐴 = (Base‘𝑃)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐴)    &   (𝜑𝐼𝐴)    &   (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (𝐺 (𝑆𝑋)))    &   (𝜑 → (𝑆‘(𝐹 · 𝑌)) = (𝐼 (𝑆𝑌)))       (𝜑𝐺 = 𝐼)
 
Theoremhdmap14lem12 36079* Lemma for proof of part 14 in [Baer] p. 50. (Contributed by NM, 6-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    · = ( ·𝑠𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = ( ·𝑠𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐵)    &   𝑃 = (Scalar‘𝐶)    &   𝐴 = (Base‘𝑃)    &    0 = (0g𝑈)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐺𝐴)       (𝜑 → ((𝑆‘(𝐹 · 𝑋)) = (𝐺 (𝑆𝑋)) ↔ ∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 (𝑆𝑦))))
 
Theoremhdmap14lem13 36080* Lemma for proof of part 14 in [Baer] p. 50. (Contributed by NM, 6-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    · = ( ·𝑠𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = ( ·𝑠𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐵)    &   𝑃 = (Scalar‘𝐶)    &   𝐴 = (Base‘𝑃)    &    0 = (0g𝑈)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐺𝐴)       (𝜑 → ((𝑆‘(𝐹 · 𝑋)) = (𝐺 (𝑆𝑋)) ↔ ∀𝑦𝑉 (𝑆‘(𝐹 · 𝑦)) = (𝐺 (𝑆𝑦))))
 
Theoremhdmap14lem14 36081* Part of proof of part 14 in [Baer] p. 50 line 3. (Contributed by NM, 6-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    · = ( ·𝑠𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = ( ·𝑠𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐵)    &   𝑃 = (Scalar‘𝐶)    &   𝐴 = (Base‘𝑃)       (𝜑 → ∃!𝑔𝐴𝑥𝑉 (𝑆‘(𝐹 · 𝑥)) = (𝑔 (𝑆𝑥)))
 
Theoremhdmap14lem15 36082* Part of proof of part 14 in [Baer] p. 50 line 3. Convert scalar base of dual to scalar base of vector space. (Contributed by NM, 6-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    · = ( ·𝑠𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = ( ·𝑠𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐵)       (𝜑 → ∃!𝑔𝐵𝑥𝑉 (𝑆‘(𝐹 · 𝑥)) = (𝑔 (𝑆𝑥)))
 
Syntaxchg 36083 Extend class notation with g-map.
class HGMap
 
Definitiondf-hgmap 36084* Define map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. (Contributed by NM, 25-Mar-2015.)
HGMap = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑎[((DVecH‘𝑘)‘𝑤) / 𝑢][(Base‘(Scalar‘𝑢)) / 𝑏][((HDMap‘𝑘)‘𝑤) / 𝑚]𝑎 ∈ (𝑥𝑏 ↦ (𝑦𝑏𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝑘)‘𝑤))(𝑚𝑣))))}))
 
Theoremhgmapffval 36085* Map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. (Contributed by NM, 25-Mar-2015.)
𝐻 = (LHyp‘𝐾)       (𝐾𝑋 → (HGMap‘𝐾) = (𝑤𝐻 ↦ {𝑎[((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘(Scalar‘𝑢)) / 𝑏][((HDMap‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥𝑏 ↦ (𝑦𝑏𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑤))(𝑚𝑣))))}))
 
Theoremhgmapfval 36086* Map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. (Contributed by NM, 25-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    · = ( ·𝑠𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = ( ·𝑠𝐶)    &   𝑀 = ((HDMap‘𝐾)‘𝑊)    &   𝐼 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾𝑌𝑊𝐻))       (𝜑𝐼 = (𝑥𝐵 ↦ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣)))))
 
Theoremhgmapval 36087* Value of map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. Function sigma of scalar f in part 14 of [Baer] p. 50 line 4. TODO: variable names are inherited from older version. Maybe make more consistent with hdmap14lem15 36082. (Contributed by NM, 25-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    · = ( ·𝑠𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = ( ·𝑠𝐶)    &   𝑀 = ((HDMap‘𝐾)‘𝑊)    &   𝐼 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾𝑌𝑊𝐻))    &   (𝜑𝑋𝐵)       (𝜑 → (𝐼𝑋) = (𝑦𝐵𝑣𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 (𝑀𝑣))))
 
TheoremhgmapfnN 36088 Functionality of scalar sigma map. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝐺 Fn 𝐵)
 
Theoremhgmapcl 36089 Closure of scalar sigma map i.e. the map from the vector space scalar base to the dual space scalar base. (Contributed by NM, 6-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐵)       (𝜑 → (𝐺𝐹) ∈ 𝐵)
 
Theoremhgmapdcl 36090 Closure of the vector space to dual space scalar map, in the scalar sigma map. (Contributed by NM, 6-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑄 = (Scalar‘𝐶)    &   𝐴 = (Base‘𝑄)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐵)       (𝜑 → (𝐺𝐹) ∈ 𝐴)
 
Theoremhgmapvs 36091 Part 15 of [Baer] p. 50 line 6. Also line 15 in [Holland95] p. 14. (Contributed by NM, 6-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    · = ( ·𝑠𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = ( ·𝑠𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝐹𝐵)       (𝜑 → (𝑆‘(𝐹 · 𝑋)) = ((𝐺𝐹) (𝑆𝑋)))
 
Theoremhgmapval0 36092 Value of the scalar sigma map at zero. (Contributed by NM, 12-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑅 = (Scalar‘𝑈)    &    0 = (0g𝑅)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 → (𝐺0 ) = 0 )
 
Theoremhgmapval1 36093 Value of the scalar sigma map at one. (Contributed by NM, 12-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑅 = (Scalar‘𝑈)    &    1 = (1r𝑅)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 → (𝐺1 ) = 1 )
 
Theoremhgmapadd 36094 Part 15 of [Baer] p. 50 line 13. (Contributed by NM, 6-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝐺‘(𝑋 + 𝑌)) = ((𝐺𝑋) + (𝐺𝑌)))
 
Theoremhgmapmul 36095 Part 15 of [Baer] p. 50 line 16. The multiplication is reversed after converting to the dual space scalar to the vector space scalar. (Contributed by NM, 7-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝐺‘(𝑋 · 𝑌)) = ((𝐺𝑌) · (𝐺𝑋)))
 
Theoremhgmaprnlem1N 36096 Lemma for hgmaprnN 36101. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &    · = ( ·𝑠𝑈)    &    0 = (0g𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑃 = (Scalar‘𝐶)    &   𝐴 = (Base‘𝑃)    &    = ( ·𝑠𝐶)    &   𝑄 = (0g𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑧𝐴)    &   (𝜑𝑡 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑠𝑉)    &   (𝜑 → (𝑆𝑠) = (𝑧 (𝑆𝑡)))    &   (𝜑𝑘𝐵)    &   (𝜑𝑠 = (𝑘 · 𝑡))       (𝜑𝑧 ∈ ran 𝐺)
 
Theoremhgmaprnlem2N 36097 Lemma for hgmaprnN 36101. Part 15 of [Baer] p. 50 line 20. We only require a subset relation, rather than equality, so that the case of zero 𝑧 is taken care of automatically. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &    · = ( ·𝑠𝑈)    &    0 = (0g𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑃 = (Scalar‘𝐶)    &   𝐴 = (Base‘𝑃)    &    = ( ·𝑠𝐶)    &   𝑄 = (0g𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑧𝐴)    &   (𝜑𝑡 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑠𝑉)    &   (𝜑 → (𝑆𝑠) = (𝑧 (𝑆𝑡)))    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑁 = (LSpan‘𝑈)    &   𝐿 = (LSpan‘𝐶)       (𝜑 → (𝑁‘{𝑠}) ⊆ (𝑁‘{𝑡}))
 
Theoremhgmaprnlem3N 36098* Lemma for hgmaprnN 36101. Eliminate 𝑘. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &    · = ( ·𝑠𝑈)    &    0 = (0g𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑃 = (Scalar‘𝐶)    &   𝐴 = (Base‘𝑃)    &    = ( ·𝑠𝐶)    &   𝑄 = (0g𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑧𝐴)    &   (𝜑𝑡 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑠𝑉)    &   (𝜑 → (𝑆𝑠) = (𝑧 (𝑆𝑡)))    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑁 = (LSpan‘𝑈)    &   𝐿 = (LSpan‘𝐶)       (𝜑𝑧 ∈ ran 𝐺)
 
Theoremhgmaprnlem4N 36099* Lemma for hgmaprnN 36101. Eliminate 𝑠. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &    · = ( ·𝑠𝑈)    &    0 = (0g𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑃 = (Scalar‘𝐶)    &   𝐴 = (Base‘𝑃)    &    = ( ·𝑠𝐶)    &   𝑄 = (0g𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑧𝐴)    &   (𝜑𝑡 ∈ (𝑉 ∖ { 0 }))       (𝜑𝑧 ∈ ran 𝐺)
 
Theoremhgmaprnlem5N 36100 Lemma for hgmaprnN 36101. Eliminate 𝑡. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝑅)    &    · = ( ·𝑠𝑈)    &    0 = (0g𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑃 = (Scalar‘𝐶)    &   𝐴 = (Base‘𝑃)    &    = ( ·𝑠𝐶)    &   𝑄 = (0g𝐶)    &   𝑆 = ((HDMap‘𝐾)‘𝑊)    &   𝐺 = ((HGMap‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑧𝐴)       (𝜑𝑧 ∈ ran 𝐺)
    < 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 42401-42426
  Copyright terms: Public domain < Previous  Next >