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Theorem List for Metamath Proof Explorer - 40001-40100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmapdcnvcl 40001 Closure of the converse of the map defined by df-mapd 39974. (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘ˆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ ran 𝑀)    β‡’   (πœ‘ β†’ (β—‘π‘€β€˜π‘‹) ∈ 𝑆)
 
Theoremmapdcl 40002 Closure the value of the map defined by df-mapd 39974. (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘ˆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑆)    β‡’   (πœ‘ β†’ (π‘€β€˜π‘‹) ∈ ran 𝑀)
 
Theoremmapdcnvid1N 40003 Converse of the value of the map defined by df-mapd 39974. (Contributed by NM, 13-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘ˆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑆)    β‡’   (πœ‘ β†’ (β—‘π‘€β€˜(π‘€β€˜π‘‹)) = 𝑋)
 
Theoremmapdsord 40004 Strong ordering property of themap defined by df-mapd 39974. (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘ˆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑆)    &   (πœ‘ β†’ π‘Œ ∈ 𝑆)    β‡’   (πœ‘ β†’ ((π‘€β€˜π‘‹) ⊊ (π‘€β€˜π‘Œ) ↔ 𝑋 ⊊ π‘Œ))
 
Theoremmapdcl2 40005 The mapping of a subspace of vector space H is a subspace in the dual space of functionals with closed kernels. (Contributed by NM, 31-Jan-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π‘‡ = (LSubSpβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑅 ∈ 𝑆)    β‡’   (πœ‘ β†’ (π‘€β€˜π‘…) ∈ 𝑇)
 
Theoremmapdcnvid2 40006 Value of the converse of the map defined by df-mapd 39974. (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ ran 𝑀)    β‡’   (πœ‘ β†’ (π‘€β€˜(β—‘π‘€β€˜π‘‹)) = 𝑋)
 
TheoremmapdcnvordN 40007 Ordering property of the converse of the map defined by df-mapd 39974. (Contributed by NM, 13-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ ran 𝑀)    &   (πœ‘ β†’ π‘Œ ∈ ran 𝑀)    β‡’   (πœ‘ β†’ ((β—‘π‘€β€˜π‘‹) βŠ† (β—‘π‘€β€˜π‘Œ) ↔ 𝑋 βŠ† π‘Œ))
 
Theoremmapdcnv11N 40008 The converse of the map defined by df-mapd 39974 is one-to-one. (Contributed by NM, 13-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ ran 𝑀)    &   (πœ‘ β†’ π‘Œ ∈ ran 𝑀)    β‡’   (πœ‘ β†’ ((β—‘π‘€β€˜π‘‹) = (β—‘π‘€β€˜π‘Œ) ↔ 𝑋 = π‘Œ))
 
Theoremmapdcv 40009 Covering property of the converse of the map defined by df-mapd 39974. (Contributed by NM, 14-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘ˆ)    &   πΆ = ( β‹–L β€˜π‘ˆ)    &   π· = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   πΈ = ( β‹–L β€˜π·)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑆)    &   (πœ‘ β†’ π‘Œ ∈ 𝑆)    β‡’   (πœ‘ β†’ (π‘‹πΆπ‘Œ ↔ (π‘€β€˜π‘‹)𝐸(π‘€β€˜π‘Œ)))
 
Theoremmapdincl 40010 Closure of dual subspace intersection for the map defined by df-mapd 39974. (Contributed by NM, 12-Apr-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ ran 𝑀)    &   (πœ‘ β†’ π‘Œ ∈ ran 𝑀)    β‡’   (πœ‘ β†’ (𝑋 ∩ π‘Œ) ∈ ran 𝑀)
 
Theoremmapdin 40011 Subspace intersection is preserved by the map defined by df-mapd 39974. Part of property (e) in [Baer] p. 40. (Contributed by NM, 12-Apr-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘ˆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑆)    &   (πœ‘ β†’ π‘Œ ∈ 𝑆)    β‡’   (πœ‘ β†’ (π‘€β€˜(𝑋 ∩ π‘Œ)) = ((π‘€β€˜π‘‹) ∩ (π‘€β€˜π‘Œ)))
 
Theoremmapdlsmcl 40012 Closure of dual subspace sum for the map defined by df-mapd 39974. (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &    βŠ• = (LSSumβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ ran 𝑀)    &   (πœ‘ β†’ π‘Œ ∈ ran 𝑀)    β‡’   (πœ‘ β†’ (𝑋 βŠ• π‘Œ) ∈ ran 𝑀)
 
Theoremmapdlsm 40013 Subspace sum is preserved by the map defined by df-mapd 39974. Part of property (e) in [Baer] p. 40. (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘ˆ)    &    βŠ• = (LSSumβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &    ✚ = (LSSumβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑆)    &   (πœ‘ β†’ π‘Œ ∈ 𝑆)    β‡’   (πœ‘ β†’ (π‘€β€˜(𝑋 βŠ• π‘Œ)) = ((π‘€β€˜π‘‹) ✚ (π‘€β€˜π‘Œ)))
 
Theoremmapd0 40014 Projectivity map of the zero subspace. Part of property (f) in [Baer] p. 40. TODO: does proof need to be this long for this simple fact? (Contributed by NM, 15-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‚ = (0gβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &    0 = (0gβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    β‡’   (πœ‘ β†’ (π‘€β€˜{𝑂}) = { 0 })
 
TheoremmapdcnvatN 40015 Atoms are preserved by the map defined by df-mapd 39974. (Contributed by NM, 29-May-2015.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π΄ = (LSAtomsβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π΅ = (LSAtomsβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑄 ∈ 𝐡)    β‡’   (πœ‘ β†’ (β—‘π‘€β€˜π‘„) ∈ 𝐴)
 
Theoremmapdat 40016 Atoms are preserved by the map defined by df-mapd 39974. Property (g) in [Baer] p. 41. (Contributed by NM, 14-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π΄ = (LSAtomsβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π΅ = (LSAtomsβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑄 ∈ 𝐴)    β‡’   (πœ‘ β†’ (π‘€β€˜π‘„) ∈ 𝐡)
 
Theoremmapdspex 40017* The map of a span equals the dual span of some vector (functional). (Contributed by NM, 15-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π΅ = (Baseβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    β‡’   (πœ‘ β†’ βˆƒπ‘” ∈ 𝐡 (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝑔}))
 
Theoremmapdn0 40018 Transfer nonzero property from domain to range of projectivity mapd. (Contributed by NM, 12-Apr-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π· = (Baseβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐹 ∈ 𝐷)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐹}))    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (0gβ€˜πΆ)    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    β‡’   (πœ‘ β†’ 𝐹 ∈ (𝐷 βˆ– {𝑍}))
 
Theoremmapdncol 40019 Transfer non-colinearity from domain to range of projectivity mapd. (Contributed by NM, 11-Apr-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π· = (Baseβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐹 ∈ 𝐷)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐹}))    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &   (πœ‘ β†’ 𝐺 ∈ 𝐷)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝐺}))    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    β‡’   (πœ‘ β†’ (π½β€˜{𝐹}) β‰  (π½β€˜{𝐺}))
 
Theoremmapdindp 40020 Transfer (part of) vector independence condition from domain to range of projectivity mapd. (Contributed by NM, 11-Apr-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π· = (Baseβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐹 ∈ 𝐷)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐹}))    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &   (πœ‘ β†’ 𝐺 ∈ 𝐷)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝐺}))    &   (πœ‘ β†’ 𝑍 ∈ 𝑉)    &   (πœ‘ β†’ 𝐸 ∈ 𝐷)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑍})) = (π½β€˜{𝐸}))    &   (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))    β‡’   (πœ‘ β†’ Β¬ 𝐹 ∈ (π½β€˜{𝐺, 𝐸}))
 
Theoremmapdpglem1 40021 Lemma for mapdpg 40055. Baer p. 44, last line: "(F(x-y))* <= (Fx)*+(Fy)*." (Contributed by NM, 15-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &    βŠ• = (LSSumβ€˜πΆ)    β‡’   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) βŠ† ((π‘€β€˜(π‘β€˜{𝑋})) βŠ• (π‘€β€˜(π‘β€˜{π‘Œ}))))
 
Theoremmapdpglem2 40022* Lemma for mapdpg 40055. Baer p. 45, lines 1 and 2: "we have (F(x-y))* = Gt where t necessarily belongs to (Fx)*+(Fy)*." (We scope $d π‘‘πœ‘ locally to avoid clashes with later substitutions into πœ‘.) (Contributed by NM, 15-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &    βŠ• = (LSSumβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    β‡’   (πœ‘ β†’ βˆƒπ‘‘ ∈ ((π‘€β€˜(π‘β€˜{𝑋})) βŠ• (π‘€β€˜(π‘β€˜{π‘Œ})))(π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{𝑑}))
 
Theoremmapdpglem2a 40023* Lemma for mapdpg 40055. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &    βŠ• = (LSSumβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   πΉ = (Baseβ€˜πΆ)    &   (πœ‘ β†’ 𝑑 ∈ ((π‘€β€˜(π‘β€˜{𝑋})) βŠ• (π‘€β€˜(π‘β€˜{π‘Œ}))))    β‡’   (πœ‘ β†’ 𝑑 ∈ 𝐹)
 
Theoremmapdpglem3 40024* Lemma for mapdpg 40055. Baer p. 45, line 3: "infer ... the existence of a number g in G and of an element z in (Fy)* such that t = gx'-z." (We scope $d π‘”π‘€π‘§πœ‘ locally to avoid clashes with later substitutions into πœ‘.) (Contributed by NM, 18-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &    βŠ• = (LSSumβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   πΉ = (Baseβ€˜πΆ)    &   (πœ‘ β†’ 𝑑 ∈ ((π‘€β€˜(π‘β€˜{𝑋})) βŠ• (π‘€β€˜(π‘β€˜{π‘Œ}))))    &   π΄ = (Scalarβ€˜π‘ˆ)    &   π΅ = (Baseβ€˜π΄)    &    Β· = ( ·𝑠 β€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))    β‡’   (πœ‘ β†’ βˆƒπ‘” ∈ 𝐡 βˆƒπ‘§ ∈ (π‘€β€˜(π‘β€˜{π‘Œ}))𝑑 = ((𝑔 Β· 𝐺)𝑅𝑧))
 
Theoremmapdpglem4N 40025* Lemma for mapdpg 40055. (Contributed by NM, 20-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &    βŠ• = (LSSumβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   πΉ = (Baseβ€˜πΆ)    &   (πœ‘ β†’ 𝑑 ∈ ((π‘€β€˜(π‘β€˜{𝑋})) βŠ• (π‘€β€˜(π‘β€˜{π‘Œ}))))    &   π΄ = (Scalarβ€˜π‘ˆ)    &   π΅ = (Baseβ€˜π΄)    &    Β· = ( ·𝑠 β€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))    &   π‘„ = (0gβ€˜π‘ˆ)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    β‡’   (πœ‘ β†’ (𝑋 βˆ’ π‘Œ) β‰  𝑄)
 
Theoremmapdpglem5N 40026* Lemma for mapdpg 40055. (Contributed by NM, 20-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &    βŠ• = (LSSumβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   πΉ = (Baseβ€˜πΆ)    &   (πœ‘ β†’ 𝑑 ∈ ((π‘€β€˜(π‘β€˜{𝑋})) βŠ• (π‘€β€˜(π‘β€˜{π‘Œ}))))    &   π΄ = (Scalarβ€˜π‘ˆ)    &   π΅ = (Baseβ€˜π΄)    &    Β· = ( ·𝑠 β€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))    &   π‘„ = (0gβ€˜π‘ˆ)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{𝑑}))    β‡’   (πœ‘ β†’ 𝑑 β‰  (0gβ€˜πΆ))
 
Theoremmapdpglem6 40027* Lemma for mapdpg 40055. Baer p. 45, line 4: "If g were 0, then t would be in (Fy)*..." (Contributed by NM, 18-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &    βŠ• = (LSSumβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   πΉ = (Baseβ€˜πΆ)    &   (πœ‘ β†’ 𝑑 ∈ ((π‘€β€˜(π‘β€˜{𝑋})) βŠ• (π‘€β€˜(π‘β€˜{π‘Œ}))))    &   π΄ = (Scalarβ€˜π‘ˆ)    &   π΅ = (Baseβ€˜π΄)    &    Β· = ( ·𝑠 β€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))    &   π‘„ = (0gβ€˜π‘ˆ)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{𝑑}))    &    0 = (0gβ€˜π΄)    &   (πœ‘ β†’ 𝑔 ∈ 𝐡)    &   (πœ‘ β†’ 𝑧 ∈ (π‘€β€˜(π‘β€˜{π‘Œ})))    &   (πœ‘ β†’ 𝑑 = ((𝑔 Β· 𝐺)𝑅𝑧))    &   (πœ‘ β†’ 𝑋 β‰  𝑄)    &   (πœ‘ β†’ 𝑔 = 0 )    β‡’   (πœ‘ β†’ 𝑑 ∈ (π‘€β€˜(π‘β€˜{π‘Œ})))
 
Theoremmapdpglem8 40028* Lemma for mapdpg 40055. Baer p. 45, line 4: "...so that (F(x-y))* <= (Fy)*. This would imply that F(x-y) <= F(y)..." (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &    βŠ• = (LSSumβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   πΉ = (Baseβ€˜πΆ)    &   (πœ‘ β†’ 𝑑 ∈ ((π‘€β€˜(π‘β€˜{𝑋})) βŠ• (π‘€β€˜(π‘β€˜{π‘Œ}))))    &   π΄ = (Scalarβ€˜π‘ˆ)    &   π΅ = (Baseβ€˜π΄)    &    Β· = ( ·𝑠 β€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))    &   π‘„ = (0gβ€˜π‘ˆ)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{𝑑}))    &    0 = (0gβ€˜π΄)    &   (πœ‘ β†’ 𝑔 ∈ 𝐡)    &   (πœ‘ β†’ 𝑧 ∈ (π‘€β€˜(π‘β€˜{π‘Œ})))    &   (πœ‘ β†’ 𝑑 = ((𝑔 Β· 𝐺)𝑅𝑧))    &   (πœ‘ β†’ 𝑋 β‰  𝑄)    &   (πœ‘ β†’ 𝑔 = 0 )    β‡’   (πœ‘ β†’ (π‘β€˜{(𝑋 βˆ’ π‘Œ)}) βŠ† (π‘β€˜{π‘Œ}))
 
Theoremmapdpglem9 40029* Lemma for mapdpg 40055. Baer p. 45, line 4: "...so that x would consequently belong to Fy." (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &    βŠ• = (LSSumβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   πΉ = (Baseβ€˜πΆ)    &   (πœ‘ β†’ 𝑑 ∈ ((π‘€β€˜(π‘β€˜{𝑋})) βŠ• (π‘€β€˜(π‘β€˜{π‘Œ}))))    &   π΄ = (Scalarβ€˜π‘ˆ)    &   π΅ = (Baseβ€˜π΄)    &    Β· = ( ·𝑠 β€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))    &   π‘„ = (0gβ€˜π‘ˆ)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{𝑑}))    &    0 = (0gβ€˜π΄)    &   (πœ‘ β†’ 𝑔 ∈ 𝐡)    &   (πœ‘ β†’ 𝑧 ∈ (π‘€β€˜(π‘β€˜{π‘Œ})))    &   (πœ‘ β†’ 𝑑 = ((𝑔 Β· 𝐺)𝑅𝑧))    &   (πœ‘ β†’ 𝑋 β‰  𝑄)    &   (πœ‘ β†’ 𝑔 = 0 )    β‡’   (πœ‘ β†’ 𝑋 ∈ (π‘β€˜{π‘Œ}))
 
Theoremmapdpglem10 40030* Lemma for mapdpg 40055. Baer p. 45, line 6: "Hence Fx=Fy, an impossibility." (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &    βŠ• = (LSSumβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   πΉ = (Baseβ€˜πΆ)    &   (πœ‘ β†’ 𝑑 ∈ ((π‘€β€˜(π‘β€˜{𝑋})) βŠ• (π‘€β€˜(π‘β€˜{π‘Œ}))))    &   π΄ = (Scalarβ€˜π‘ˆ)    &   π΅ = (Baseβ€˜π΄)    &    Β· = ( ·𝑠 β€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))    &   π‘„ = (0gβ€˜π‘ˆ)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{𝑑}))    &    0 = (0gβ€˜π΄)    &   (πœ‘ β†’ 𝑔 ∈ 𝐡)    &   (πœ‘ β†’ 𝑧 ∈ (π‘€β€˜(π‘β€˜{π‘Œ})))    &   (πœ‘ β†’ 𝑑 = ((𝑔 Β· 𝐺)𝑅𝑧))    &   (πœ‘ β†’ 𝑋 β‰  𝑄)    &   (πœ‘ β†’ 𝑔 = 0 )    β‡’   (πœ‘ β†’ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ}))
 
Theoremmapdpglem11 40031* Lemma for mapdpg 40055. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &    βŠ• = (LSSumβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   πΉ = (Baseβ€˜πΆ)    &   (πœ‘ β†’ 𝑑 ∈ ((π‘€β€˜(π‘β€˜{𝑋})) βŠ• (π‘€β€˜(π‘β€˜{π‘Œ}))))    &   π΄ = (Scalarβ€˜π‘ˆ)    &   π΅ = (Baseβ€˜π΄)    &    Β· = ( ·𝑠 β€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))    &   π‘„ = (0gβ€˜π‘ˆ)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{𝑑}))    &    0 = (0gβ€˜π΄)    &   (πœ‘ β†’ 𝑔 ∈ 𝐡)    &   (πœ‘ β†’ 𝑧 ∈ (π‘€β€˜(π‘β€˜{π‘Œ})))    &   (πœ‘ β†’ 𝑑 = ((𝑔 Β· 𝐺)𝑅𝑧))    &   (πœ‘ β†’ 𝑋 β‰  𝑄)    β‡’   (πœ‘ β†’ 𝑔 β‰  0 )
 
Theoremmapdpglem12 40032* Lemma for mapdpg 40055. TODO: Can some commonality with mapdpglem6 40027 through mapdpglem11 40031 be exploited? Also, some consolidation of small lemmas here could be done. (Contributed by NM, 18-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &    βŠ• = (LSSumβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   πΉ = (Baseβ€˜πΆ)    &   (πœ‘ β†’ 𝑑 ∈ ((π‘€β€˜(π‘β€˜{𝑋})) βŠ• (π‘€β€˜(π‘β€˜{π‘Œ}))))    &   π΄ = (Scalarβ€˜π‘ˆ)    &   π΅ = (Baseβ€˜π΄)    &    Β· = ( ·𝑠 β€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))    &   π‘„ = (0gβ€˜π‘ˆ)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{𝑑}))    &    0 = (0gβ€˜π΄)    &   (πœ‘ β†’ 𝑔 ∈ 𝐡)    &   (πœ‘ β†’ 𝑧 ∈ (π‘€β€˜(π‘β€˜{π‘Œ})))    &   (πœ‘ β†’ 𝑑 = ((𝑔 Β· 𝐺)𝑅𝑧))    &   (πœ‘ β†’ 𝑋 β‰  𝑄)    &   (πœ‘ β†’ π‘Œ β‰  𝑄)    &   (πœ‘ β†’ 𝑧 = (0gβ€˜πΆ))    β‡’   (πœ‘ β†’ 𝑑 ∈ (π‘€β€˜(π‘β€˜{𝑋})))
 
Theoremmapdpglem13 40033* Lemma for mapdpg 40055. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &    βŠ• = (LSSumβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   πΉ = (Baseβ€˜πΆ)    &   (πœ‘ β†’ 𝑑 ∈ ((π‘€β€˜(π‘β€˜{𝑋})) βŠ• (π‘€β€˜(π‘β€˜{π‘Œ}))))    &   π΄ = (Scalarβ€˜π‘ˆ)    &   π΅ = (Baseβ€˜π΄)    &    Β· = ( ·𝑠 β€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))    &   π‘„ = (0gβ€˜π‘ˆ)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{𝑑}))    &    0 = (0gβ€˜π΄)    &   (πœ‘ β†’ 𝑔 ∈ 𝐡)    &   (πœ‘ β†’ 𝑧 ∈ (π‘€β€˜(π‘β€˜{π‘Œ})))    &   (πœ‘ β†’ 𝑑 = ((𝑔 Β· 𝐺)𝑅𝑧))    &   (πœ‘ β†’ 𝑋 β‰  𝑄)    &   (πœ‘ β†’ π‘Œ β‰  𝑄)    &   (πœ‘ β†’ 𝑧 = (0gβ€˜πΆ))    β‡’   (πœ‘ β†’ (π‘β€˜{(𝑋 βˆ’ π‘Œ)}) βŠ† (π‘β€˜{𝑋}))
 
Theoremmapdpglem14 40034* Lemma for mapdpg 40055. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &    βŠ• = (LSSumβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   πΉ = (Baseβ€˜πΆ)    &   (πœ‘ β†’ 𝑑 ∈ ((π‘€β€˜(π‘β€˜{𝑋})) βŠ• (π‘€β€˜(π‘β€˜{π‘Œ}))))    &   π΄ = (Scalarβ€˜π‘ˆ)    &   π΅ = (Baseβ€˜π΄)    &    Β· = ( ·𝑠 β€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))    &   π‘„ = (0gβ€˜π‘ˆ)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{𝑑}))    &    0 = (0gβ€˜π΄)    &   (πœ‘ β†’ 𝑔 ∈ 𝐡)    &   (πœ‘ β†’ 𝑧 ∈ (π‘€β€˜(π‘β€˜{π‘Œ})))    &   (πœ‘ β†’ 𝑑 = ((𝑔 Β· 𝐺)𝑅𝑧))    &   (πœ‘ β†’ 𝑋 β‰  𝑄)    &   (πœ‘ β†’ π‘Œ β‰  𝑄)    &   (πœ‘ β†’ 𝑧 = (0gβ€˜πΆ))    β‡’   (πœ‘ β†’ π‘Œ ∈ (π‘β€˜{𝑋}))
 
Theoremmapdpglem15 40035* Lemma for mapdpg 40055. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &    βŠ• = (LSSumβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   πΉ = (Baseβ€˜πΆ)    &   (πœ‘ β†’ 𝑑 ∈ ((π‘€β€˜(π‘β€˜{𝑋})) βŠ• (π‘€β€˜(π‘β€˜{π‘Œ}))))    &   π΄ = (Scalarβ€˜π‘ˆ)    &   π΅ = (Baseβ€˜π΄)    &    Β· = ( ·𝑠 β€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))    &   π‘„ = (0gβ€˜π‘ˆ)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{𝑑}))    &    0 = (0gβ€˜π΄)    &   (πœ‘ β†’ 𝑔 ∈ 𝐡)    &   (πœ‘ β†’ 𝑧 ∈ (π‘€β€˜(π‘β€˜{π‘Œ})))    &   (πœ‘ β†’ 𝑑 = ((𝑔 Β· 𝐺)𝑅𝑧))    &   (πœ‘ β†’ 𝑋 β‰  𝑄)    &   (πœ‘ β†’ π‘Œ β‰  𝑄)    &   (πœ‘ β†’ 𝑧 = (0gβ€˜πΆ))    β‡’   (πœ‘ β†’ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ}))
 
Theoremmapdpglem16 40036* Lemma for mapdpg 40055. Baer p. 45, line 7: "Likewise we see that z =/= 0." (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &    βŠ• = (LSSumβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   πΉ = (Baseβ€˜πΆ)    &   (πœ‘ β†’ 𝑑 ∈ ((π‘€β€˜(π‘β€˜{𝑋})) βŠ• (π‘€β€˜(π‘β€˜{π‘Œ}))))    &   π΄ = (Scalarβ€˜π‘ˆ)    &   π΅ = (Baseβ€˜π΄)    &    Β· = ( ·𝑠 β€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))    &   π‘„ = (0gβ€˜π‘ˆ)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{𝑑}))    &    0 = (0gβ€˜π΄)    &   (πœ‘ β†’ 𝑔 ∈ 𝐡)    &   (πœ‘ β†’ 𝑧 ∈ (π‘€β€˜(π‘β€˜{π‘Œ})))    &   (πœ‘ β†’ 𝑑 = ((𝑔 Β· 𝐺)𝑅𝑧))    &   (πœ‘ β†’ 𝑋 β‰  𝑄)    &   (πœ‘ β†’ π‘Œ β‰  𝑄)    β‡’   (πœ‘ β†’ 𝑧 β‰  (0gβ€˜πΆ))
 
Theoremmapdpglem17N 40037* Lemma for mapdpg 40055. Baer p. 45, line 7: "Hence we may form y' = g^-1 z." (Contributed by NM, 20-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &    βŠ• = (LSSumβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   πΉ = (Baseβ€˜πΆ)    &   (πœ‘ β†’ 𝑑 ∈ ((π‘€β€˜(π‘β€˜{𝑋})) βŠ• (π‘€β€˜(π‘β€˜{π‘Œ}))))    &   π΄ = (Scalarβ€˜π‘ˆ)    &   π΅ = (Baseβ€˜π΄)    &    Β· = ( ·𝑠 β€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))    &   π‘„ = (0gβ€˜π‘ˆ)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{𝑑}))    &    0 = (0gβ€˜π΄)    &   (πœ‘ β†’ 𝑔 ∈ 𝐡)    &   (πœ‘ β†’ 𝑧 ∈ (π‘€β€˜(π‘β€˜{π‘Œ})))    &   (πœ‘ β†’ 𝑑 = ((𝑔 Β· 𝐺)𝑅𝑧))    &   (πœ‘ β†’ 𝑋 β‰  𝑄)    &   (πœ‘ β†’ π‘Œ β‰  𝑄)    &   πΈ = (((invrβ€˜π΄)β€˜π‘”) Β· 𝑧)    β‡’   (πœ‘ β†’ 𝐸 ∈ 𝐹)
 
Theoremmapdpglem18 40038* Lemma for mapdpg 40055. Baer p. 45, line 7: "Then y =/= 0..." (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &    βŠ• = (LSSumβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   πΉ = (Baseβ€˜πΆ)    &   (πœ‘ β†’ 𝑑 ∈ ((π‘€β€˜(π‘β€˜{𝑋})) βŠ• (π‘€β€˜(π‘β€˜{π‘Œ}))))    &   π΄ = (Scalarβ€˜π‘ˆ)    &   π΅ = (Baseβ€˜π΄)    &    Β· = ( ·𝑠 β€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))    &   π‘„ = (0gβ€˜π‘ˆ)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{𝑑}))    &    0 = (0gβ€˜π΄)    &   (πœ‘ β†’ 𝑔 ∈ 𝐡)    &   (πœ‘ β†’ 𝑧 ∈ (π‘€β€˜(π‘β€˜{π‘Œ})))    &   (πœ‘ β†’ 𝑑 = ((𝑔 Β· 𝐺)𝑅𝑧))    &   (πœ‘ β†’ 𝑋 β‰  𝑄)    &   (πœ‘ β†’ π‘Œ β‰  𝑄)    &   πΈ = (((invrβ€˜π΄)β€˜π‘”) Β· 𝑧)    β‡’   (πœ‘ β†’ 𝐸 β‰  (0gβ€˜πΆ))
 
Theoremmapdpglem19 40039* Lemma for mapdpg 40055. Baer p. 45, line 8: "...is in (Fy)*..." (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &    βŠ• = (LSSumβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   πΉ = (Baseβ€˜πΆ)    &   (πœ‘ β†’ 𝑑 ∈ ((π‘€β€˜(π‘β€˜{𝑋})) βŠ• (π‘€β€˜(π‘β€˜{π‘Œ}))))    &   π΄ = (Scalarβ€˜π‘ˆ)    &   π΅ = (Baseβ€˜π΄)    &    Β· = ( ·𝑠 β€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))    &   π‘„ = (0gβ€˜π‘ˆ)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{𝑑}))    &    0 = (0gβ€˜π΄)    &   (πœ‘ β†’ 𝑔 ∈ 𝐡)    &   (πœ‘ β†’ 𝑧 ∈ (π‘€β€˜(π‘β€˜{π‘Œ})))    &   (πœ‘ β†’ 𝑑 = ((𝑔 Β· 𝐺)𝑅𝑧))    &   (πœ‘ β†’ 𝑋 β‰  𝑄)    &   (πœ‘ β†’ π‘Œ β‰  𝑄)    &   πΈ = (((invrβ€˜π΄)β€˜π‘”) Β· 𝑧)    β‡’   (πœ‘ β†’ 𝐸 ∈ (π‘€β€˜(π‘β€˜{π‘Œ})))
 
Theoremmapdpglem20 40040* Lemma for mapdpg 40055. Baer p. 45, line 8: "...so that (Fy)*=Gy'." (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &    βŠ• = (LSSumβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   πΉ = (Baseβ€˜πΆ)    &   (πœ‘ β†’ 𝑑 ∈ ((π‘€β€˜(π‘β€˜{𝑋})) βŠ• (π‘€β€˜(π‘β€˜{π‘Œ}))))    &   π΄ = (Scalarβ€˜π‘ˆ)    &   π΅ = (Baseβ€˜π΄)    &    Β· = ( ·𝑠 β€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))    &   π‘„ = (0gβ€˜π‘ˆ)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{𝑑}))    &    0 = (0gβ€˜π΄)    &   (πœ‘ β†’ 𝑔 ∈ 𝐡)    &   (πœ‘ β†’ 𝑧 ∈ (π‘€β€˜(π‘β€˜{π‘Œ})))    &   (πœ‘ β†’ 𝑑 = ((𝑔 Β· 𝐺)𝑅𝑧))    &   (πœ‘ β†’ 𝑋 β‰  𝑄)    &   (πœ‘ β†’ π‘Œ β‰  𝑄)    &   πΈ = (((invrβ€˜π΄)β€˜π‘”) Β· 𝑧)    β‡’   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝐸}))
 
Theoremmapdpglem21 40041* Lemma for mapdpg 40055. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &    βŠ• = (LSSumβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   πΉ = (Baseβ€˜πΆ)    &   (πœ‘ β†’ 𝑑 ∈ ((π‘€β€˜(π‘β€˜{𝑋})) βŠ• (π‘€β€˜(π‘β€˜{π‘Œ}))))    &   π΄ = (Scalarβ€˜π‘ˆ)    &   π΅ = (Baseβ€˜π΄)    &    Β· = ( ·𝑠 β€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))    &   π‘„ = (0gβ€˜π‘ˆ)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{𝑑}))    &    0 = (0gβ€˜π΄)    &   (πœ‘ β†’ 𝑔 ∈ 𝐡)    &   (πœ‘ β†’ 𝑧 ∈ (π‘€β€˜(π‘β€˜{π‘Œ})))    &   (πœ‘ β†’ 𝑑 = ((𝑔 Β· 𝐺)𝑅𝑧))    &   (πœ‘ β†’ 𝑋 β‰  𝑄)    &   (πœ‘ β†’ π‘Œ β‰  𝑄)    &   πΈ = (((invrβ€˜π΄)β€˜π‘”) Β· 𝑧)    β‡’   (πœ‘ β†’ (((invrβ€˜π΄)β€˜π‘”) Β· 𝑑) = (𝐺𝑅𝐸))
 
Theoremmapdpglem22 40042* Lemma for mapdpg 40055. Baer p. 45, line 9: "(F(x-y))* = ... = G(x'-y')." (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &    βŠ• = (LSSumβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   πΉ = (Baseβ€˜πΆ)    &   (πœ‘ β†’ 𝑑 ∈ ((π‘€β€˜(π‘β€˜{𝑋})) βŠ• (π‘€β€˜(π‘β€˜{π‘Œ}))))    &   π΄ = (Scalarβ€˜π‘ˆ)    &   π΅ = (Baseβ€˜π΄)    &    Β· = ( ·𝑠 β€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))    &   π‘„ = (0gβ€˜π‘ˆ)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{𝑑}))    &    0 = (0gβ€˜π΄)    &   (πœ‘ β†’ 𝑔 ∈ 𝐡)    &   (πœ‘ β†’ 𝑧 ∈ (π‘€β€˜(π‘β€˜{π‘Œ})))    &   (πœ‘ β†’ 𝑑 = ((𝑔 Β· 𝐺)𝑅𝑧))    &   (πœ‘ β†’ 𝑋 β‰  𝑄)    &   (πœ‘ β†’ π‘Œ β‰  𝑄)    &   πΈ = (((invrβ€˜π΄)β€˜π‘”) Β· 𝑧)    β‡’   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝐸)}))
 
Theoremmapdpglem23 40043* Lemma for mapdpg 40055. Baer p. 45, line 10: "and so y' meets all our requirements." Our β„Ž is Baer's y'. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &    βŠ• = (LSSumβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   πΉ = (Baseβ€˜πΆ)    &   (πœ‘ β†’ 𝑑 ∈ ((π‘€β€˜(π‘β€˜{𝑋})) βŠ• (π‘€β€˜(π‘β€˜{π‘Œ}))))    &   π΄ = (Scalarβ€˜π‘ˆ)    &   π΅ = (Baseβ€˜π΄)    &    Β· = ( ·𝑠 β€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))    &   π‘„ = (0gβ€˜π‘ˆ)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{𝑑}))    &    0 = (0gβ€˜π΄)    &   (πœ‘ β†’ 𝑔 ∈ 𝐡)    &   (πœ‘ β†’ 𝑧 ∈ (π‘€β€˜(π‘β€˜{π‘Œ})))    &   (πœ‘ β†’ 𝑑 = ((𝑔 Β· 𝐺)𝑅𝑧))    &   (πœ‘ β†’ 𝑋 β‰  𝑄)    &   (πœ‘ β†’ π‘Œ β‰  𝑄)    &   πΈ = (((invrβ€˜π΄)β€˜π‘”) Β· 𝑧)    β‡’   (πœ‘ β†’ βˆƒβ„Ž ∈ 𝐹 ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})))
 
Theoremmapdpglem30a 40044 Lemma for mapdpg 40055. (Contributed by NM, 22-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   πΉ = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))    β‡’   (πœ‘ β†’ 𝐺 β‰  (0gβ€˜πΆ))
 
Theoremmapdpglem30b 40045 Lemma for mapdpg 40055. (Contributed by NM, 22-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   πΉ = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))    &   (πœ‘ β†’ (β„Ž ∈ 𝐹 ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)}))))    &   (πœ‘ β†’ (𝑖 ∈ 𝐹 ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)}))))    β‡’   (πœ‘ β†’ 𝑖 β‰  (0gβ€˜πΆ))
 
Theoremmapdpglem25 40046 Lemma for mapdpg 40055. Baer p. 45 line 12: "Then we have Gy' = Gy'' and G(x'-y') = G(x'-y'')." (Contributed by NM, 21-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   πΉ = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))    &   (πœ‘ β†’ (β„Ž ∈ 𝐹 ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)}))))    &   (πœ‘ β†’ (𝑖 ∈ 𝐹 ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)}))))    β‡’   (πœ‘ β†’ ((π½β€˜{β„Ž}) = (π½β€˜{𝑖}) ∧ (π½β€˜{(πΊπ‘…β„Ž)}) = (π½β€˜{(𝐺𝑅𝑖)})))
 
Theoremmapdpglem26 40047* Lemma for mapdpg 40055. Baer p. 45 line 14: "Consequently there exist numbers u,v in G neither of which is 0 such that y = uy'' and..." (We scope $d π‘’πœ‘ locally to avoid clashes with later substitutions into πœ‘.) (Contributed by NM, 22-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   πΉ = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))    &   (πœ‘ β†’ (β„Ž ∈ 𝐹 ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)}))))    &   (πœ‘ β†’ (𝑖 ∈ 𝐹 ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)}))))    &   π΄ = (Scalarβ€˜π‘ˆ)    &   π΅ = (Baseβ€˜π΄)    &    Β· = ( ·𝑠 β€˜πΆ)    &   π‘‚ = (0gβ€˜π΄)    β‡’   (πœ‘ β†’ βˆƒπ‘’ ∈ (𝐡 βˆ– {𝑂})β„Ž = (𝑒 Β· 𝑖))
 
Theoremmapdpglem27 40048* Lemma for mapdpg 40055. Baer p. 45 line 16: "v(x'-y'') = x'-y'" (with equality swapped). (Contributed by NM, 22-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   πΉ = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))    &   (πœ‘ β†’ (β„Ž ∈ 𝐹 ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)}))))    &   (πœ‘ β†’ (𝑖 ∈ 𝐹 ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)}))))    &   π΄ = (Scalarβ€˜π‘ˆ)    &   π΅ = (Baseβ€˜π΄)    &    Β· = ( ·𝑠 β€˜πΆ)    &   π‘‚ = (0gβ€˜π΄)    β‡’   (πœ‘ β†’ βˆƒπ‘£ ∈ (𝐡 βˆ– {𝑂})(πΊπ‘…β„Ž) = (𝑣 Β· (𝐺𝑅𝑖)))
 
Theoremmapdpglem29 40049* Lemma for mapdpg 40055. Baer p. 45 line 16: "But Gx' and Gy'' are distinct points and so x' and y'' are independent elements in B. (Contributed by NM, 22-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   πΉ = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))    &   (πœ‘ β†’ (β„Ž ∈ 𝐹 ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)}))))    &   (πœ‘ β†’ (𝑖 ∈ 𝐹 ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)}))))    &   π΄ = (Scalarβ€˜π‘ˆ)    &   π΅ = (Baseβ€˜π΄)    &    Β· = ( ·𝑠 β€˜πΆ)    &   π‘‚ = (0gβ€˜π΄)    &   (πœ‘ β†’ 𝑣 ∈ 𝐡)    &   (πœ‘ β†’ β„Ž = (𝑒 Β· 𝑖))    &   (πœ‘ β†’ (πΊπ‘…β„Ž) = (𝑣 Β· (𝐺𝑅𝑖)))    β‡’   (πœ‘ β†’ (π½β€˜{𝐺}) β‰  (π½β€˜{𝑖}))
 
Theoremmapdpglem28 40050* Lemma for mapdpg 40055. Baer p. 45 line 18: "vx'-vy'' = x'-uy''". (Contributed by NM, 22-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   πΉ = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))    &   (πœ‘ β†’ (β„Ž ∈ 𝐹 ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)}))))    &   (πœ‘ β†’ (𝑖 ∈ 𝐹 ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)}))))    &   π΄ = (Scalarβ€˜π‘ˆ)    &   π΅ = (Baseβ€˜π΄)    &    Β· = ( ·𝑠 β€˜πΆ)    &   π‘‚ = (0gβ€˜π΄)    &   (πœ‘ β†’ 𝑣 ∈ 𝐡)    &   (πœ‘ β†’ β„Ž = (𝑒 Β· 𝑖))    &   (πœ‘ β†’ (πΊπ‘…β„Ž) = (𝑣 Β· (𝐺𝑅𝑖)))    β‡’   (πœ‘ β†’ ((𝑣 Β· 𝐺)𝑅(𝑣 Β· 𝑖)) = (𝐺𝑅(𝑒 Β· 𝑖)))
 
Theoremmapdpglem30 40051* Lemma for mapdpg 40055. Baer p. 45 line 18: "Hence we deduce (from mapdpglem28 40050, using lvecindp2 20523) that v = 1 and v = u...". TODO: would it be shorter to have only the 𝑣 = (1rβ€˜π΄) part and use mapdpglem28.u2 in mapdpglem31 40052? (Contributed by NM, 22-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   πΉ = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))    &   (πœ‘ β†’ (β„Ž ∈ 𝐹 ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)}))))    &   (πœ‘ β†’ (𝑖 ∈ 𝐹 ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)}))))    &   π΄ = (Scalarβ€˜π‘ˆ)    &   π΅ = (Baseβ€˜π΄)    &    Β· = ( ·𝑠 β€˜πΆ)    &   π‘‚ = (0gβ€˜π΄)    &   (πœ‘ β†’ 𝑣 ∈ 𝐡)    &   (πœ‘ β†’ β„Ž = (𝑒 Β· 𝑖))    &   (πœ‘ β†’ (πΊπ‘…β„Ž) = (𝑣 Β· (𝐺𝑅𝑖)))    &   (πœ‘ β†’ 𝑒 ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝑣 = (1rβ€˜π΄) ∧ 𝑣 = 𝑒))
 
Theoremmapdpglem31 40052* Lemma for mapdpg 40055. Baer p. 45 line 19: "...and we have consequently that y' = y'', as we claimed." (Contributed by NM, 23-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   πΉ = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))    &   (πœ‘ β†’ (β„Ž ∈ 𝐹 ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)}))))    &   (πœ‘ β†’ (𝑖 ∈ 𝐹 ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)}))))    &   π΄ = (Scalarβ€˜π‘ˆ)    &   π΅ = (Baseβ€˜π΄)    &    Β· = ( ·𝑠 β€˜πΆ)    &   π‘‚ = (0gβ€˜π΄)    &   (πœ‘ β†’ 𝑣 ∈ 𝐡)    &   (πœ‘ β†’ β„Ž = (𝑒 Β· 𝑖))    &   (πœ‘ β†’ (πΊπ‘…β„Ž) = (𝑣 Β· (𝐺𝑅𝑖)))    &   (πœ‘ β†’ 𝑒 ∈ 𝐡)    β‡’   (πœ‘ β†’ β„Ž = 𝑖)
 
Theoremmapdpglem24 40053* Lemma for mapdpg 40055. Existence part - consolidate hypotheses in mapdpglem23 40043. (Contributed by NM, 21-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   πΉ = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))    β‡’   (πœ‘ β†’ βˆƒβ„Ž ∈ 𝐹 ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})))
 
Theoremmapdpglem32 40054* Lemma for mapdpg 40055. Uniqueness part - consolidate hypotheses in mapdpglem31 40052. (Contributed by NM, 23-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   πΉ = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))    β‡’   ((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) β†’ β„Ž = 𝑖)
 
Theoremmapdpg 40055* Part 1 of proof of the first fundamental theorem of projective geometry. Part (1) in [Baer] p. 44. Our notation corresponds to Baer's as follows: 𝑀 for *, π‘β€˜{} for F(), π½β€˜{} for G(), 𝑋 for x, 𝐺 for x', π‘Œ for y, β„Ž for y'. TODO: Rename variables per mapdhval 40073. (Contributed by NM, 22-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   πΉ = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))    β‡’   (πœ‘ β†’ βˆƒ!β„Ž ∈ 𝐹 ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})))
 
Theorembaerlem3lem1 40056 Lemma for baerlem3 40062. (Contributed by NM, 9-Apr-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    βˆ’ = (-gβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    βŠ• = (LSSumβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LVec)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))    &   (πœ‘ β†’ (π‘β€˜{π‘Œ}) β‰  (π‘β€˜{𝑍}))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))    &    + = (+gβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π‘… = (Scalarβ€˜π‘Š)    &   π΅ = (Baseβ€˜π‘…)    &    ⨣ = (+gβ€˜π‘…)    &   πΏ = (-gβ€˜π‘…)    &   π‘„ = (0gβ€˜π‘…)    &   πΌ = (invgβ€˜π‘…)    &   (πœ‘ β†’ π‘Ž ∈ 𝐡)    &   (πœ‘ β†’ 𝑏 ∈ 𝐡)    &   (πœ‘ β†’ 𝑑 ∈ 𝐡)    &   (πœ‘ β†’ 𝑒 ∈ 𝐡)    &   (πœ‘ β†’ 𝑗 = ((π‘Ž Β· π‘Œ) + (𝑏 Β· 𝑍)))    &   (πœ‘ β†’ 𝑗 = ((𝑑 Β· (𝑋 βˆ’ π‘Œ)) + (𝑒 Β· (𝑋 βˆ’ 𝑍))))    β‡’   (πœ‘ β†’ 𝑗 = (π‘Ž Β· (π‘Œ βˆ’ 𝑍)))
 
Theorembaerlem5alem1 40057 Lemma for baerlem5a 40063. (Contributed by NM, 13-Apr-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    βˆ’ = (-gβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    βŠ• = (LSSumβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LVec)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))    &   (πœ‘ β†’ (π‘β€˜{π‘Œ}) β‰  (π‘β€˜{𝑍}))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))    &    + = (+gβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π‘… = (Scalarβ€˜π‘Š)    &   π΅ = (Baseβ€˜π‘…)    &    ⨣ = (+gβ€˜π‘…)    &   πΏ = (-gβ€˜π‘…)    &   π‘„ = (0gβ€˜π‘…)    &   πΌ = (invgβ€˜π‘…)    &   (πœ‘ β†’ π‘Ž ∈ 𝐡)    &   (πœ‘ β†’ 𝑏 ∈ 𝐡)    &   (πœ‘ β†’ 𝑑 ∈ 𝐡)    &   (πœ‘ β†’ 𝑒 ∈ 𝐡)    &   (πœ‘ β†’ 𝑗 = ((π‘Ž Β· (𝑋 βˆ’ π‘Œ)) + (𝑏 Β· 𝑍)))    &   (πœ‘ β†’ 𝑗 = ((𝑑 Β· (𝑋 βˆ’ 𝑍)) + (𝑒 Β· π‘Œ)))    β‡’   (πœ‘ β†’ 𝑗 = (π‘Ž Β· (𝑋 βˆ’ (π‘Œ + 𝑍))))
 
Theorembaerlem5blem1 40058 Lemma for baerlem5b 40064. (Contributed by NM, 9-Apr-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    βˆ’ = (-gβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    βŠ• = (LSSumβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LVec)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))    &   (πœ‘ β†’ (π‘β€˜{π‘Œ}) β‰  (π‘β€˜{𝑍}))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))    &    + = (+gβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π‘… = (Scalarβ€˜π‘Š)    &   π΅ = (Baseβ€˜π‘…)    &    ⨣ = (+gβ€˜π‘…)    &   πΏ = (-gβ€˜π‘…)    &   π‘„ = (0gβ€˜π‘…)    &   πΌ = (invgβ€˜π‘…)    &   (πœ‘ β†’ π‘Ž ∈ 𝐡)    &   (πœ‘ β†’ 𝑏 ∈ 𝐡)    &   (πœ‘ β†’ 𝑑 ∈ 𝐡)    &   (πœ‘ β†’ 𝑒 ∈ 𝐡)    &   (πœ‘ β†’ 𝑗 = ((π‘Ž Β· π‘Œ) + (𝑏 Β· 𝑍)))    &   (πœ‘ β†’ 𝑗 = ((𝑑 Β· (𝑋 βˆ’ (π‘Œ + 𝑍))) + (𝑒 Β· 𝑋)))    β‡’   (πœ‘ β†’ 𝑗 = ((πΌβ€˜π‘‘) Β· (π‘Œ + 𝑍)))
 
Theorembaerlem3lem2 40059 Lemma for baerlem3 40062. (Contributed by NM, 9-Apr-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    βˆ’ = (-gβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    βŠ• = (LSSumβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LVec)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))    &   (πœ‘ β†’ (π‘β€˜{π‘Œ}) β‰  (π‘β€˜{𝑍}))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))    &    + = (+gβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π‘… = (Scalarβ€˜π‘Š)    &   π΅ = (Baseβ€˜π‘…)    &    ⨣ = (+gβ€˜π‘…)    &   πΏ = (-gβ€˜π‘…)    &   π‘„ = (0gβ€˜π‘…)    &   πΌ = (invgβ€˜π‘…)    β‡’   (πœ‘ β†’ (π‘β€˜{(π‘Œ βˆ’ 𝑍)}) = (((π‘β€˜{π‘Œ}) βŠ• (π‘β€˜{𝑍})) ∩ ((π‘β€˜{(𝑋 βˆ’ π‘Œ)}) βŠ• (π‘β€˜{(𝑋 βˆ’ 𝑍)}))))
 
Theorembaerlem5alem2 40060 Lemma for baerlem5a 40063. (Contributed by NM, 9-Apr-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    βˆ’ = (-gβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    βŠ• = (LSSumβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LVec)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))    &   (πœ‘ β†’ (π‘β€˜{π‘Œ}) β‰  (π‘β€˜{𝑍}))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))    &    + = (+gβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π‘… = (Scalarβ€˜π‘Š)    &   π΅ = (Baseβ€˜π‘…)    &    ⨣ = (+gβ€˜π‘…)    &   πΏ = (-gβ€˜π‘…)    &   π‘„ = (0gβ€˜π‘…)    &   πΌ = (invgβ€˜π‘…)    β‡’   (πœ‘ β†’ (π‘β€˜{(𝑋 βˆ’ (π‘Œ + 𝑍))}) = (((π‘β€˜{(𝑋 βˆ’ π‘Œ)}) βŠ• (π‘β€˜{𝑍})) ∩ ((π‘β€˜{(𝑋 βˆ’ 𝑍)}) βŠ• (π‘β€˜{π‘Œ}))))
 
Theorembaerlem5blem2 40061 Lemma for baerlem5b 40064. (Contributed by NM, 13-Apr-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    βˆ’ = (-gβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    βŠ• = (LSSumβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LVec)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))    &   (πœ‘ β†’ (π‘β€˜{π‘Œ}) β‰  (π‘β€˜{𝑍}))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))    &    + = (+gβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π‘… = (Scalarβ€˜π‘Š)    &   π΅ = (Baseβ€˜π‘…)    &    ⨣ = (+gβ€˜π‘…)    &   πΏ = (-gβ€˜π‘…)    &   π‘„ = (0gβ€˜π‘…)    &   πΌ = (invgβ€˜π‘…)    β‡’   (πœ‘ β†’ (π‘β€˜{(π‘Œ + 𝑍)}) = (((π‘β€˜{π‘Œ}) βŠ• (π‘β€˜{𝑍})) ∩ ((π‘β€˜{(𝑋 βˆ’ (π‘Œ + 𝑍))}) βŠ• (π‘β€˜{𝑋}))))
 
Theorembaerlem3 40062 An equality that holds when 𝑋, π‘Œ, 𝑍 are independent (non-colinear) vectors. Part (3) in [Baer] p. 45. TODO fix ref. (Contributed by NM, 9-Apr-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    βˆ’ = (-gβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    βŠ• = (LSSumβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LVec)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))    &   (πœ‘ β†’ (π‘β€˜{π‘Œ}) β‰  (π‘β€˜{𝑍}))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))    β‡’   (πœ‘ β†’ (π‘β€˜{(π‘Œ βˆ’ 𝑍)}) = (((π‘β€˜{π‘Œ}) βŠ• (π‘β€˜{𝑍})) ∩ ((π‘β€˜{(𝑋 βˆ’ π‘Œ)}) βŠ• (π‘β€˜{(𝑋 βˆ’ 𝑍)}))))
 
Theorembaerlem5a 40063 An equality that holds when 𝑋, π‘Œ, 𝑍 are independent (non-colinear) vectors. First equation of part (5) in [Baer] p. 46. (Contributed by NM, 10-Apr-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    βˆ’ = (-gβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    βŠ• = (LSSumβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LVec)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))    &   (πœ‘ β†’ (π‘β€˜{π‘Œ}) β‰  (π‘β€˜{𝑍}))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))    &    + = (+gβ€˜π‘Š)    β‡’   (πœ‘ β†’ (π‘β€˜{(𝑋 βˆ’ (π‘Œ + 𝑍))}) = (((π‘β€˜{(𝑋 βˆ’ π‘Œ)}) βŠ• (π‘β€˜{𝑍})) ∩ ((π‘β€˜{(𝑋 βˆ’ 𝑍)}) βŠ• (π‘β€˜{π‘Œ}))))
 
Theorembaerlem5b 40064 An equality that holds when 𝑋, π‘Œ, 𝑍 are independent (non-colinear) vectors. Second equation of part (5) in [Baer] p. 46. (Contributed by NM, 13-Apr-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    βˆ’ = (-gβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    βŠ• = (LSSumβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LVec)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))    &   (πœ‘ β†’ (π‘β€˜{π‘Œ}) β‰  (π‘β€˜{𝑍}))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))    &    + = (+gβ€˜π‘Š)    β‡’   (πœ‘ β†’ (π‘β€˜{(π‘Œ + 𝑍)}) = (((π‘β€˜{π‘Œ}) βŠ• (π‘β€˜{𝑍})) ∩ ((π‘β€˜{(𝑋 βˆ’ (π‘Œ + 𝑍))}) βŠ• (π‘β€˜{𝑋}))))
 
Theorembaerlem5amN 40065 An equality that holds when 𝑋, π‘Œ, 𝑍 are independent (non-colinear) vectors. Subtraction version of first equation of part (5) in [Baer] p. 46. TODO: This is the subtraction version, may not be needed. TODO: delete if baerlem5abmN 40067 is used. (Contributed by NM, 24-May-2015.) (New usage is discouraged.)
𝑉 = (Baseβ€˜π‘Š)    &    βˆ’ = (-gβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    βŠ• = (LSSumβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LVec)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))    &   (πœ‘ β†’ (π‘β€˜{π‘Œ}) β‰  (π‘β€˜{𝑍}))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))    &    + = (+gβ€˜π‘Š)    β‡’   (πœ‘ β†’ (π‘β€˜{(𝑋 βˆ’ (π‘Œ βˆ’ 𝑍))}) = (((π‘β€˜{(𝑋 βˆ’ π‘Œ)}) βŠ• (π‘β€˜{𝑍})) ∩ ((π‘β€˜{(𝑋 + 𝑍)}) βŠ• (π‘β€˜{π‘Œ}))))
 
Theorembaerlem5bmN 40066 An equality that holds when 𝑋, π‘Œ, 𝑍 are independent (non-colinear) vectors. Subtraction version of second equation of part (5) in [Baer] p. 46. TODO: This is the subtraction version, may not be needed. TODO: delete if baerlem5abmN 40067 is used. (Contributed by NM, 24-May-2015.) (New usage is discouraged.)
𝑉 = (Baseβ€˜π‘Š)    &    βˆ’ = (-gβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    βŠ• = (LSSumβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LVec)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))    &   (πœ‘ β†’ (π‘β€˜{π‘Œ}) β‰  (π‘β€˜{𝑍}))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))    &    + = (+gβ€˜π‘Š)    β‡’   (πœ‘ β†’ (π‘β€˜{(π‘Œ βˆ’ 𝑍)}) = (((π‘β€˜{π‘Œ}) βŠ• (π‘β€˜{𝑍})) ∩ ((π‘β€˜{(𝑋 βˆ’ (π‘Œ βˆ’ 𝑍))}) βŠ• (π‘β€˜{𝑋}))))
 
Theorembaerlem5abmN 40067 An equality that holds when 𝑋, π‘Œ, 𝑍 are independent (non-colinear) vectors. Subtraction versions of first and second equations of part (5) in [Baer] p. 46, conjoined to share commonality in their proofs. TODO: Delete if not needed. (Contributed by NM, 24-May-2015.) (New usage is discouraged.)
𝑉 = (Baseβ€˜π‘Š)    &    βˆ’ = (-gβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    βŠ• = (LSSumβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LVec)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))    &   (πœ‘ β†’ (π‘β€˜{π‘Œ}) β‰  (π‘β€˜{𝑍}))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))    &    + = (+gβ€˜π‘Š)    β‡’   (πœ‘ β†’ ((π‘β€˜{(𝑋 βˆ’ (π‘Œ βˆ’ 𝑍))}) = (((π‘β€˜{(𝑋 βˆ’ π‘Œ)}) βŠ• (π‘β€˜{𝑍})) ∩ ((π‘β€˜{(𝑋 + 𝑍)}) βŠ• (π‘β€˜{π‘Œ}))) ∧ (π‘β€˜{(π‘Œ βˆ’ 𝑍)}) = (((π‘β€˜{π‘Œ}) βŠ• (π‘β€˜{𝑍})) ∩ ((π‘β€˜{(𝑋 βˆ’ (π‘Œ βˆ’ 𝑍))}) βŠ• (π‘β€˜{𝑋})))))
 
Theoremmapdindp0 40068 Vector independence lemma. (Contributed by NM, 29-Apr-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LVec)    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑀 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ (π‘β€˜{π‘Œ}) = (π‘β€˜{𝑍}))    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ Β¬ 𝑀 ∈ (π‘β€˜{𝑋, π‘Œ}))    &   (πœ‘ β†’ (π‘Œ + 𝑍) β‰  0 )    β‡’   (πœ‘ β†’ (π‘β€˜{(π‘Œ + 𝑍)}) = (π‘β€˜{π‘Œ}))
 
Theoremmapdindp1 40069 Vector independence lemma. (Contributed by NM, 1-May-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LVec)    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑀 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ (π‘β€˜{π‘Œ}) = (π‘β€˜{𝑍}))    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ Β¬ 𝑀 ∈ (π‘β€˜{𝑋, π‘Œ}))    β‡’   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{(π‘Œ + 𝑍)}))
 
Theoremmapdindp2 40070 Vector independence lemma. (Contributed by NM, 1-May-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LVec)    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑀 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ (π‘β€˜{π‘Œ}) = (π‘β€˜{𝑍}))    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ Β¬ 𝑀 ∈ (π‘β€˜{𝑋, π‘Œ}))    β‡’   (πœ‘ β†’ Β¬ 𝑀 ∈ (π‘β€˜{𝑋, (π‘Œ + 𝑍)}))
 
Theoremmapdindp3 40071 Vector independence lemma. (Contributed by NM, 29-Apr-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LVec)    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑀 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ (π‘β€˜{π‘Œ}) = (π‘β€˜{𝑍}))    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ Β¬ 𝑀 ∈ (π‘β€˜{𝑋, π‘Œ}))    β‡’   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{(𝑀 + π‘Œ)}))
 
Theoremmapdindp4 40072 Vector independence lemma. (Contributed by NM, 29-Apr-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LVec)    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑀 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ (π‘β€˜{π‘Œ}) = (π‘β€˜{𝑍}))    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ Β¬ 𝑀 ∈ (π‘β€˜{𝑋, π‘Œ}))    β‡’   (πœ‘ β†’ Β¬ 𝑍 ∈ (π‘β€˜{𝑋, (𝑀 + π‘Œ)}))
 
Theoremmapdhval 40073* Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 6-May-2015.)
𝑄 = (0gβ€˜πΆ)    &   πΌ = (π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))    &   (πœ‘ β†’ 𝑋 ∈ 𝐴)    &   (πœ‘ β†’ 𝐹 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐸)    β‡’   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = if(π‘Œ = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΉπ‘…β„Ž)})))))
 
Theoremmapdhval0 40074* Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015.)
𝑄 = (0gβ€˜πΆ)    &   πΌ = (π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))    &    0 = (0gβ€˜π‘ˆ)    &   (πœ‘ β†’ 𝑋 ∈ 𝐴)    &   (πœ‘ β†’ 𝐹 ∈ 𝐡)    β‡’   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, 0 ⟩) = 𝑄)
 
Theoremmapdhval2 40075* Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015.)
𝑄 = (0gβ€˜πΆ)    &   πΌ = (π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))    &   (πœ‘ β†’ 𝑋 ∈ 𝐴)    &   (πœ‘ β†’ 𝐹 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    β‡’   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΉπ‘…β„Ž)}))))
 
Theoremmapdhcl 40076* Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015.)
𝑄 = (0gβ€˜πΆ)    &   πΌ = (π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))    &   π» = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π· = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐹 ∈ 𝐷)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐹}))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    β‡’   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) ∈ 𝐷)
 
Theoremmapdheq 40077* Lemmma for ~? mapdh . The defining equation for h(x,x',y)=y' in part (2) in [Baer] p. 45 line 24. (Contributed by NM, 4-Apr-2015.)
𝑄 = (0gβ€˜πΆ)    &   πΌ = (π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))    &   π» = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π· = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐹 ∈ 𝐷)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐹}))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝐺 ∈ 𝐷)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    β‡’   (πœ‘ β†’ ((πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = 𝐺 ↔ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝐺}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐹𝑅𝐺)}))))
 
Theoremmapdheq2 40078* Lemmma for ~? mapdh . One direction of part (2) in [Baer] p. 45. (Contributed by NM, 4-Apr-2015.)
𝑄 = (0gβ€˜πΆ)    &   πΌ = (π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))    &   π» = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π· = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐹 ∈ 𝐷)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐹}))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝐺 ∈ 𝐷)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    β‡’   (πœ‘ β†’ ((πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = 𝐺 β†’ (πΌβ€˜βŸ¨π‘Œ, 𝐺, π‘‹βŸ©) = 𝐹))
 
Theoremmapdheq2biN 40079* Lemmma for ~? mapdh . Part (2) in [Baer] p. 45. The bidirectional version of mapdheq2 40078 seems to require an additional hypothesis not mentioned in Baer. TODO fix ref. TODO: We probably don't need this; delete if never used. (Contributed by NM, 4-Apr-2015.) (New usage is discouraged.)
𝑄 = (0gβ€˜πΆ)    &   πΌ = (π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))    &   π» = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π· = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐹 ∈ 𝐷)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐹}))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝐺 ∈ 𝐷)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝐺}))    β‡’   (πœ‘ β†’ ((πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = 𝐺 ↔ (πΌβ€˜βŸ¨π‘Œ, 𝐺, π‘‹βŸ©) = 𝐹))
 
Theoremmapdheq4lem 40080* Lemma for mapdheq4 40081. Part (4) in [Baer] p. 46. (Contributed by NM, 12-Apr-2015.)
𝑄 = (0gβ€˜πΆ)    &   πΌ = (π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))    &   π» = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π· = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐹 ∈ 𝐷)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐹}))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))    &   (πœ‘ β†’ (π‘β€˜{π‘Œ}) β‰  (π‘β€˜{𝑍}))    &   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = 𝐺)    &   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©) = 𝐸)    β‡’   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(π‘Œ βˆ’ 𝑍)})) = (π½β€˜{(𝐺𝑅𝐸)}))
 
Theoremmapdheq4 40081* Lemma for ~? mapdh . Part (4) in [Baer] p. 46. (Contributed by NM, 12-Apr-2015.)
𝑄 = (0gβ€˜πΆ)    &   πΌ = (π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))    &   π» = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π· = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐹 ∈ 𝐷)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐹}))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))    &   (πœ‘ β†’ (π‘β€˜{π‘Œ}) β‰  (π‘β€˜{𝑍}))    &   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = 𝐺)    &   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©) = 𝐸)    β‡’   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘Œ, 𝐺, π‘βŸ©) = 𝐸)
 
Theoremmapdh6lem1N 40082* Lemma for mapdh6N 40096. Part (6) in [Baer] p. 47, lines 16-18. (Contributed by NM, 13-Apr-2015.) (New usage is discouraged.)
𝑄 = (0gβ€˜πΆ)    &   πΌ = (π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))    &   π» = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π· = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐹 ∈ 𝐷)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐹}))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &    + = (+gβ€˜π‘ˆ)    &    ✚ = (+gβ€˜πΆ)    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))    &   (πœ‘ β†’ (π‘β€˜{π‘Œ}) β‰  (π‘β€˜{𝑍}))    &   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = 𝐺)    &   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©) = 𝐸)    β‡’   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ (π‘Œ + 𝑍))})) = (π½β€˜{(𝐹𝑅(𝐺 ✚ 𝐸))}))
 
Theoremmapdh6lem2N 40083* Lemma for mapdh6N 40096. Part (6) in [Baer] p. 47, lines 20-22. (Contributed by NM, 13-Apr-2015.) (New usage is discouraged.)
𝑄 = (0gβ€˜πΆ)    &   πΌ = (π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))    &   π» = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π· = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐹 ∈ 𝐷)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐹}))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &    + = (+gβ€˜π‘ˆ)    &    ✚ = (+gβ€˜πΆ)    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))    &   (πœ‘ β†’ (π‘β€˜{π‘Œ}) β‰  (π‘β€˜{𝑍}))    &   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = 𝐺)    &   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©) = 𝐸)    β‡’   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(π‘Œ + 𝑍)})) = (π½β€˜{(𝐺 ✚ 𝐸)}))
 
Theoremmapdh6aN 40084* Lemma for mapdh6N 40096. Part (6) in [Baer] p. 47, case 1. (Contributed by NM, 23-Apr-2015.) (New usage is discouraged.)
𝑄 = (0gβ€˜πΆ)    &   πΌ = (π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))    &   π» = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π· = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐹 ∈ 𝐷)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐹}))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &    + = (+gβ€˜π‘ˆ)    &    ✚ = (+gβ€˜πΆ)    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))    &   (πœ‘ β†’ (π‘β€˜{π‘Œ}) β‰  (π‘β€˜{𝑍}))    &   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = 𝐺)    &   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©) = 𝐸)    β‡’   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, (π‘Œ + 𝑍)⟩) = ((πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) ✚ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©)))
 
Theoremmapdh6b0N 40085* Lemmma for mapdh6N 40096. (Contributed by NM, 23-Apr-2015.) (New usage is discouraged.)
𝑄 = (0gβ€˜πΆ)    &   πΌ = (π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))    &   π» = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π· = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐹 ∈ 𝐷)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐹}))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &    + = (+gβ€˜π‘ˆ)    &    ✚ = (+gβ€˜πΆ)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &   (πœ‘ β†’ 𝑍 ∈ 𝑉)    &   (πœ‘ β†’ ((π‘β€˜{𝑋}) ∩ (π‘β€˜{π‘Œ, 𝑍})) = { 0 })    β‡’   (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))
 
Theoremmapdh6bN 40086* Lemmma for mapdh6N 40096. (Contributed by NM, 24-Apr-2015.) (New usage is discouraged.)
𝑄 = (0gβ€˜πΆ)    &   πΌ = (π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))    &   π» = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π· = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐹 ∈ 𝐷)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐹}))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &    + = (+gβ€˜π‘ˆ)    &    ✚ = (+gβ€˜πΆ)    &   (πœ‘ β†’ π‘Œ = 0 )    &   (πœ‘ β†’ 𝑍 ∈ 𝑉)    &   (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))    β‡’   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, (π‘Œ + 𝑍)⟩) = ((πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) ✚ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©)))
 
Theoremmapdh6cN 40087* Lemmma for mapdh6N 40096. (Contributed by NM, 24-Apr-2015.) (New usage is discouraged.)
𝑄 = (0gβ€˜πΆ)    &   πΌ = (π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))    &   π» = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π· = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐹 ∈ 𝐷)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐹}))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &    + = (+gβ€˜π‘ˆ)    &    ✚ = (+gβ€˜πΆ)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &   (πœ‘ β†’ 𝑍 = 0 )    &   (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))    β‡’   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, (π‘Œ + 𝑍)⟩) = ((πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) ✚ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©)))
 
Theoremmapdh6dN 40088* Lemmma for mapdh6N 40096. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
𝑄 = (0gβ€˜πΆ)    &   πΌ = (π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))    &   π» = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π· = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐹 ∈ 𝐷)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐹}))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &    + = (+gβ€˜π‘ˆ)    &    ✚ = (+gβ€˜πΆ)    &   (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))    &   (πœ‘ β†’ (π‘β€˜{π‘Œ}) = (π‘β€˜{𝑍}))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑀 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ Β¬ 𝑀 ∈ (π‘β€˜{𝑋, π‘Œ}))    β‡’   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, (𝑀 + (π‘Œ + 𝑍))⟩) = ((πΌβ€˜βŸ¨π‘‹, 𝐹, π‘€βŸ©) ✚ (πΌβ€˜βŸ¨π‘‹, 𝐹, (π‘Œ + 𝑍)⟩)))
 
Theoremmapdh6eN 40089* Lemmma for mapdh6N 40096. Part (6) in [Baer] p. 47 line 38. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
𝑄 = (0gβ€˜πΆ)    &   πΌ = (π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))    &   π» = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π· = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐹 ∈ 𝐷)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐹}))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &    + = (+gβ€˜π‘ˆ)    &    ✚ = (+gβ€˜πΆ)    &   (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))    &   (πœ‘ β†’ (π‘β€˜{π‘Œ}) = (π‘β€˜{𝑍}))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑀 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ Β¬ 𝑀 ∈ (π‘β€˜{𝑋, π‘Œ}))    β‡’   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, ((𝑀 + π‘Œ) + 𝑍)⟩) = ((πΌβ€˜βŸ¨π‘‹, 𝐹, (𝑀 + π‘Œ)⟩) ✚ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©)))
 
Theoremmapdh6fN 40090* Lemmma for mapdh6N 40096. Part (6) in [Baer] p. 47 line 38. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
𝑄 = (0gβ€˜πΆ)    &   πΌ = (π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))    &   π» = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π· = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐹 ∈ 𝐷)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐹}))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &    + = (+gβ€˜π‘ˆ)    &    ✚ = (+gβ€˜πΆ)    &   (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))    &   (πœ‘ β†’ (π‘β€˜{π‘Œ}) = (π‘β€˜{𝑍}))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑀 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ Β¬ 𝑀 ∈ (π‘β€˜{𝑋, π‘Œ}))    β‡’   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, (𝑀 + π‘Œ)⟩) = ((πΌβ€˜βŸ¨π‘‹, 𝐹, π‘€βŸ©) ✚ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)))
 
Theoremmapdh6gN 40091* Lemmma for mapdh6N 40096. Part (6) of [Baer] p. 47 line 39. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
𝑄 = (0gβ€˜πΆ)    &   πΌ = (π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))    &   π» = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π· = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐹 ∈ 𝐷)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐹}))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &    + = (+gβ€˜π‘ˆ)    &    ✚ = (+gβ€˜πΆ)    &   (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))    &   (πœ‘ β†’ (π‘β€˜{π‘Œ}) = (π‘β€˜{𝑍}))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑀 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ Β¬ 𝑀 ∈ (π‘β€˜{𝑋, π‘Œ}))    β‡’   (πœ‘ β†’ ((πΌβ€˜βŸ¨π‘‹, 𝐹, π‘€βŸ©) ✚ (πΌβ€˜βŸ¨π‘‹, 𝐹, (π‘Œ + 𝑍)⟩)) = (((πΌβ€˜βŸ¨π‘‹, 𝐹, π‘€βŸ©) ✚ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)) ✚ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©)))
 
Theoremmapdh6hN 40092* Lemmma for mapdh6N 40096. Part (6) of [Baer] p. 48 line 2. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
𝑄 = (0gβ€˜πΆ)    &   πΌ = (π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))    &   π» = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π· = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐹 ∈ 𝐷)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐹}))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &    + = (+gβ€˜π‘ˆ)    &    ✚ = (+gβ€˜πΆ)    &   (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))    &   (πœ‘ β†’ (π‘β€˜{π‘Œ}) = (π‘β€˜{𝑍}))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑀 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ Β¬ 𝑀 ∈ (π‘β€˜{𝑋, π‘Œ}))    β‡’   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, (π‘Œ + 𝑍)⟩) = ((πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) ✚ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©)))
 
Theoremmapdh6iN 40093* Lemmma for mapdh6N 40096. Eliminate auxiliary vector 𝑀. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
𝑄 = (0gβ€˜πΆ)    &   πΌ = (π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))    &   π» = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π· = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐹 ∈ 𝐷)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐹}))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &    + = (+gβ€˜π‘ˆ)    &    ✚ = (+gβ€˜πΆ)    &   (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ (π‘β€˜{π‘Œ}) = (π‘β€˜{𝑍}))    β‡’   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, (π‘Œ + 𝑍)⟩) = ((πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) ✚ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©)))
 
Theoremmapdh6jN 40094* Lemmma for mapdh6N 40096. Eliminate (π‘β€˜{π‘Œ}) = (π‘β€˜{𝑍}) hypothesis. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
𝑄 = (0gβ€˜πΆ)    &   πΌ = (π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))    &   π» = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π· = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐹 ∈ 𝐷)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐹}))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &    + = (+gβ€˜π‘ˆ)    &    ✚ = (+gβ€˜πΆ)    &   (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))    &   (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))    β‡’   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, (π‘Œ + 𝑍)⟩) = ((πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) ✚ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©)))
 
Theoremmapdh6kN 40095* Lemmma for mapdh6N 40096. Eliminate nonzero vector requirement. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
𝑄 = (0gβ€˜πΆ)    &   πΌ = (π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))    &   π» = (LHypβ€˜πΎ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π· = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐹 ∈ 𝐷)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐹}))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &    + = (+gβ€˜π‘ˆ)    &    ✚ = (+gβ€˜πΆ)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &   (πœ‘ β†’ 𝑍 ∈ 𝑉)    &   (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))    β‡’   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, (π‘Œ + 𝑍)⟩) = ((πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) ✚ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©)))
 
Theoremmapdh6N 40096* Part (6) of [Baer] p. 47 line 6. Note that we use Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}) which is equivalent to Baer's "Fx ∩ (Fy + Fz)" by lspdisjb 20510. TODO: If disjoint variable conditions with 𝐼 and πœ‘ become a problem later, use cbv* theorems on 𝐼 variables here to get rid of them. Maybe reorder hypotheses in lemmas to the more consistent order of this theorem, so they can be shared with this theorem. TODO: may be deleted (with its lemmas), if not needed, in view of hdmap1l6 40170. (Contributed by NM, 1-May-2015.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    + = (+gβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π· = (Baseβ€˜πΆ)    &    ✚ = (+gβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π‘„ = (0gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   πΌ = (π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐹 ∈ 𝐷)    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &   (πœ‘ β†’ 𝑍 ∈ 𝑉)    &   (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐹}))    β‡’   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, (π‘Œ + 𝑍)⟩) = ((πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) ✚ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©)))
 
Theoremmapdh7eN 40097* Part (7) of [Baer] p. 48 line 10 (5 of 6 cases). (Note: 1 of 6 and 2 of 6 are hypotheses a and b.) (Contributed by NM, 2-May-2015.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π· = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π‘„ = (0gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   πΌ = (π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐹 ∈ 𝐷)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑒})) = (π½β€˜{𝐹}))    &   (πœ‘ β†’ 𝑒 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑣 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑀 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ (π‘β€˜{𝑒}) β‰  (π‘β€˜{𝑣}))    &   (πœ‘ β†’ Β¬ 𝑀 ∈ (π‘β€˜{𝑒, 𝑣}))    &   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘’, 𝐹, π‘€βŸ©) = 𝐸)    β‡’   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘€, 𝐸, π‘’βŸ©) = 𝐹)
 
Theoremmapdh7cN 40098* Part (7) of [Baer] p. 48 line 10 (3 of 6 cases). (Contributed by NM, 2-May-2015.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π· = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π‘„ = (0gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   πΌ = (π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐹 ∈ 𝐷)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑒})) = (π½β€˜{𝐹}))    &   (πœ‘ β†’ 𝑒 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑣 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑀 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ (π‘β€˜{𝑒}) β‰  (π‘β€˜{𝑣}))    &   (πœ‘ β†’ Β¬ 𝑀 ∈ (π‘β€˜{𝑒, 𝑣}))    &   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘’, 𝐹, π‘£βŸ©) = 𝐺)    β‡’   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘£, 𝐺, π‘’βŸ©) = 𝐹)
 
Theoremmapdh7dN 40099* Part (7) of [Baer] p. 48 line 10 (4 of 6 cases). (Contributed by NM, 2-May-2015.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π· = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π‘„ = (0gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   πΌ = (π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐹 ∈ 𝐷)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑒})) = (π½β€˜{𝐹}))    &   (πœ‘ β†’ 𝑒 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑣 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑀 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ (π‘β€˜{𝑒}) β‰  (π‘β€˜{𝑣}))    &   (πœ‘ β†’ Β¬ 𝑀 ∈ (π‘β€˜{𝑒, 𝑣}))    &   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘’, 𝐹, π‘£βŸ©) = 𝐺)    &   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘’, 𝐹, π‘€βŸ©) = 𝐸)    β‡’   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘£, 𝐺, π‘€βŸ©) = 𝐸)
 
Theoremmapdh7fN 40100* Part (7) of [Baer] p. 48 line 10 (6 of 6 cases). (Contributed by NM, 2-May-2015.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    βˆ’ = (-gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π· = (Baseβ€˜πΆ)    &   π‘… = (-gβ€˜πΆ)    &   π‘„ = (0gβ€˜πΆ)    &   π½ = (LSpanβ€˜πΆ)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   πΌ = (π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐹 ∈ 𝐷)    &   (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑒})) = (π½β€˜{𝐹}))    &   (πœ‘ β†’ 𝑒 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑣 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ 𝑀 ∈ (𝑉 βˆ– { 0 }))    &   (πœ‘ β†’ (π‘β€˜{𝑒}) β‰  (π‘β€˜{𝑣}))    &   (πœ‘ β†’ Β¬ 𝑀 ∈ (π‘β€˜{𝑒, 𝑣}))    &   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘’, 𝐹, π‘£βŸ©) = 𝐺)    &   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘’, 𝐹, π‘€βŸ©) = 𝐸)    β‡’   (πœ‘ β†’ (πΌβ€˜βŸ¨π‘€, 𝐸, π‘£βŸ©) = 𝐺)
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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 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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 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