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Theorem List for Metamath Proof Explorer - 38801-38900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremhdmap1val 38801* Value of preliminary map from vectors to functionals in the closed kernel dual space. (Restatement of mapdhval 38727.) TODO: change 𝐼 = (𝑥 ∈ V ↦... to (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌 > ) =... in e.g. mapdh8 38791 to shorten proofs with no \$d on 𝑥. (Contributed by NM, 15-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝑄 = (0g𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾𝐴𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝐹𝐷)    &   (𝜑𝑌𝑉)       (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if(𝑌 = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐹𝑅)})))))

Theoremhdmap1val0 38802 Value of preliminary map from vectors to functionals at zero. (Restated mapdhval0 38728.) (Contributed by NM, 17-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑄 = (0g𝐶)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋𝑉)       (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = 𝑄)

Theoremhdmap1val2 38803* Value of preliminary map from vectors to functionals in the closed kernel dual space, for nonzero 𝑌. (Contributed by NM, 16-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝐹𝐷)    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))       (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐿‘{(𝐹𝑅)}))))

Theoremhdmap1eq 38804 The defining equation for h(x,x',y)=y' in part (2) in [Baer] p. 45 line 24. (Contributed by NM, 16-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝑅 = (-g𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐹𝐷)    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐺𝐷)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))       (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺 ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐿‘{(𝐹𝑅𝐺)}))))

Theoremhdmap1cbv 38805* Frequently used lemma to change bound variables in 𝐿 hypothesis. (Contributed by NM, 15-May-2015.)
𝐿 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))       𝐿 = (𝑦 ∈ V ↦ if((2nd𝑦) = 0 , 𝑄, (𝑖𝐷 ((𝑀‘(𝑁‘{(2nd𝑦)})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑦)) (2nd𝑦))})) = (𝐽‘{((2nd ‘(1st𝑦))𝑅𝑖)})))))

Theoremhdmap1valc 38806* 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 38805 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 38807 Convert closure theorem mapdhcl 38730 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 38808 Convert mapdheq2 38732 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 38809 Convert mapdheq4 38735 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 38810 Lemma for hdmap1l6 38824. 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 38811 Lemma for hdmap1l6 38824. 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 38812 Lemma for hdmap1l6 38824. 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 38813 Lemmma for hdmap1l6 38824. (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 38814 Lemmma for hdmap1l6 38824. (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 38815 Lemmma for hdmap1l6 38824. (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 38816 Lemmma for hdmap1l6 38824. (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 38817 Lemmma for hdmap1l6 38824. 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 38818 Lemmma for hdmap1l6 38824. 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 38819 Lemmma for hdmap1l6 38824. 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 38820 Lemmma for hdmap1l6 38824. 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 38821 Lemmma for hdmap1l6 38824. 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 38822 Lemmma for hdmap1l6 38824. 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 38823 Lemmma for hdmap1l6 38824. 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 38824 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 19820. (Convert mapdh6N 38750 to use the function HDMap1.) (Contributed by NM, 17-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &    = (+g𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌𝑉)    &   (𝜑𝑍𝑉)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))       (𝜑 → (𝐼‘⟨𝑋, 𝐹, (𝑌 + 𝑍)⟩) = ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) (𝐼‘⟨𝑋, 𝐹, 𝑍⟩)))

Theoremhdmap1eulem 38825* Lemma for hdmap1eu 38827. TODO: combine with hdmap1eu 38827 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 38826* Lemma for hdmap1euOLDN 38828. TODO: combine with hdmap1euOLDN 38828 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 38827* Convert mapdh9a 38792 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 38828* Convert mapdh9aOLDN 38793 to use the HDMap1 notation. (Contributed by NM, 15-May-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝐿 = (LSpan‘𝐶)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐼 = ((HDMap1‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹}))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐹𝐷)    &   (𝜑𝑇𝑉)       (𝜑 → ∃!𝑦𝐷𝑧𝑉𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩)))

Theoremhdmapffval 38829* 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 38830* 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 38831* 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 38115). (𝐽𝐸) is a fixed reference functional determined by this vector (corresponding to u' on line 8; mapdhvmap 38772 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 38833. If a separate auxiliary vector is known, hdmapval2 38835 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 38832 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 38833 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 38834* Lemma for hdmapval2 38835. (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 38835 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 38774 through hdmaplem4 38777, 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 38836 Value of map from vectors to functionals at zero. Note: we use dvh3dim 38449 for convenience, even though 3 dimensions aren't necessary at this point. TODO: I think either this or hdmapeq0 38847 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 38837 Lemma for hdmapevec 38838. TODO: combine with hdmapevec 38838 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 38838 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 38839 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 38840 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 38841 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 38842 Lemma for hdmap10 38843. (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 38843 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 38844 Lemma for hdmapadd 38846. (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 38845 Lemma for hdmapadd 38846. (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 38846 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 38847 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 38848 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 38849 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 38850 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 38851 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 38852 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 38853 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 38854 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 38855 Lemma for hdmaprnN 38867. TODO: This lemma doesn't quite pay for itself even though used six 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 38856 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 38857 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 38858 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 38859 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 38860 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 38642 and mapdcnv11N 38662. 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 38861* Lemma for hdmaprnN 38867. (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 38862* Lemma for hdmaprnN 38867. 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 38863* Lemma for hdmaprnN 38867. 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 38864* Lemma for hdmaprnN 38867. 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 38865 Lemma for hdmaprnN 38867. 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 38866 Lemma for hdmaprnN 38867. 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 38867 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 38868 Part 12 in [Baer] p. 49. The map from vectors to functionals with closed kernels maps one-to-one onto. Combined with hdmapadd 38846, 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 38869 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 38870* Prior to part 14 in [Baer] p. 49, line 25. TODO: fix to include 𝐹 = 0 so it can be used in hdmap14lem10 38880. (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 38871 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 38872* Prior to part 14 in [Baer] p. 49, line 25. TODO: fix to include 𝐹 = 𝑍 so it can be used in hdmap14lem10 38880. (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 38873* 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 38874* Simplify (𝐴 ∖ {𝑄}) in hdmap14lem3 38873 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 38875* Simplify (𝐴 ∖ {𝑄}) in hdmap14lem3 38873 to provide a slightly simpler definition later. TODO: Use hdmap14lem4a 38874 if that one is also used directly elsewhere. Otherwise, merge hdmap14lem4a 38874 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 38876* 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 38877* Combine cases of 𝐹. TODO: Can this be done at once in hdmap14lem3 38873, in order to get rid of hdmap14lem6 38876? Perhaps modify lspsneu 19817 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 38878 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 38879 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 38880 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 38881 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 38882* 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 38883* 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 38884* 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 38885* 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 38886 Extend class notation with g-map.
class HGMap

Definitiondf-hgmap 38887* 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 38888* 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 38889* 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 38890* 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 38885. (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 38891 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 38892 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 38893 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 38894 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 38895 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 38896 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 38897 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 38898 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 38899 Lemma for hgmaprnN 38904. (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 38900 Lemma for hgmaprnN 38904. 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}))

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