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Theorem List for Metamath Proof Explorer - 38601-38700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlcfrlem41 38601* Lemma for lcfr 38603. Eliminate span condition. (Contributed by NM, 11-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    + = (+g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑄 = (LSubSp‘𝐷)    &   𝐶 = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   𝐸 = 𝑔𝐺 ( ‘(𝐿𝑔))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝑄)    &   (𝜑𝐺𝐶)    &   (𝜑𝑋𝐸)    &   (𝜑𝑌𝐸)    &    0 = (0g𝑈)    &   (𝜑𝑋0 )    &   (𝜑𝑌0 )       (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)
 
Theoremlcfrlem42 38602* Lemma for lcfr 38603. Eliminate nonzero condition. (Contributed by NM, 11-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    + = (+g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑄 = (LSubSp‘𝐷)    &   𝐶 = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   𝐸 = 𝑔𝐺 ( ‘(𝐿𝑔))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝑄)    &   (𝜑𝐺𝐶)    &   (𝜑𝑋𝐸)    &   (𝜑𝑌𝐸)       (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)
 
Theoremlcfr 38603* Reconstruction of a subspace from a dual subspace of functionals with closed kernels. Our proof was suggested by Mario Carneiro, 20-Feb-2015. (Contributed by NM, 5-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑇 = (LSubSp‘𝐷)    &   𝐶 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   𝑄 = 𝑔𝑅 ( ‘(𝐿𝑔))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑅𝑇)    &   (𝜑𝑅𝐶)       (𝜑𝑄𝑆)
 
Syntaxclcd 38604 Extend class notation with vector space of functionals with closed kernels.
class LCDual
 
Definitiondf-lcdual 38605* Dual vector space of functionals with closed kernels. Note: we could also define this directly without mapd by using mapdrn 38667. TODO: see if it makes sense to go back and replace some of the LDual stuff with this. TODO: We could simplify df-mapd 38643 using (Base‘((LCDual‘𝐾)‘𝑊)). (Contributed by NM, 13-Mar-2015.)
LCDual = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ ((LDual‘((DVecH‘𝑘)‘𝑤)) ↾s {𝑓 ∈ (LFnl‘((DVecH‘𝑘)‘𝑤)) ∣ (((ocH‘𝑘)‘𝑤)‘(((ocH‘𝑘)‘𝑤)‘((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓)})))
 
Theoremlcdfval 38606* Dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHyp‘𝐾)       (𝐾𝑋 → (LCDual‘𝐾) = (𝑤𝐻 ↦ ((LDual‘((DVecH‘𝐾)‘𝑤)) ↾s {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑤)) ∣ (((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)})))
 
Theoremlcdval 38607* Dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   (𝜑 → (𝐾𝑋𝑊𝐻))       (𝜑𝐶 = (𝐷s {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}))
 
Theoremlcdval2 38608* Dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   (𝜑 → (𝐾𝑋𝑊𝐻))    &   𝐵 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}       (𝜑𝐶 = (𝐷s 𝐵))
 
Theoremlcdlvec 38609 The dual vector space of functionals with closed kernels is a left vector space. (Contributed by NM, 14-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝐶 ∈ LVec)
 
Theoremlcdlmod 38610 The dual vector space of functionals with closed kernels is a left module. (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝐶 ∈ LMod)
 
Theoremlcdvbase 38611* Vector base set of a dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝐶)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐵 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑉 = 𝐵)
 
Theoremlcdvbasess 38612 The vector base set of the closed kernel dual space is a set of functionals. (Contributed by NM, 15-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝐶)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (LFnl‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑉𝐹)
 
Theoremlcdvbaselfl 38613 A vector in the base set of the closed kernel dual space is a functional. (Contributed by NM, 28-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝐶)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (LFnl‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)       (𝜑𝑋𝐹)
 
Theoremlcdvbasecl 38614 Closure of the value of a vector (functional) in the closed kernel dual space. (Contributed by NM, 28-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝑆)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐸 = (Base‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐸)    &   (𝜑𝑋𝑉)       (𝜑 → (𝐹𝑋) ∈ 𝑅)
 
Theoremlcdvadd 38615 Vector addition for the closed kernel vector space dual. (Contributed by NM, 10-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐷 = (LDual‘𝑈)    &    + = (+g𝐷)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = (+g𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 = + )
 
Theoremlcdvaddval 38616 The value of the value of vector addition in the closed kernel vector space dual. (Contributed by NM, 10-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑅 = (Scalar‘𝑈)    &    + = (+g𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &    = (+g𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝐺𝐷)    &   (𝜑𝑋𝑉)       (𝜑 → ((𝐹 𝐺)‘𝑋) = ((𝐹𝑋) + (𝐺𝑋)))
 
Theoremlcdsca 38617 The ring of scalars of the closed kernel dual space. (Contributed by NM, 16-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)    &   𝑂 = (oppr𝐹)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑅 = (Scalar‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑅 = 𝑂)
 
Theoremlcdsbase 38618 Base set of scalar ring for the closed kernel dual of a vector space. (Contributed by NM, 18-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)    &   𝐿 = (Base‘𝐹)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑆 = (Scalar‘𝐶)    &   𝑅 = (Base‘𝑆)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑅 = 𝐿)
 
Theoremlcdsadd 38619 Scalar addition for the closed kernel vector space dual. (Contributed by NM, 6-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)    &    + = (+g𝐹)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑆 = (Scalar‘𝐶)    &    = (+g𝑆)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 = + )
 
Theoremlcdsmul 38620 Scalar multiplication for the closed kernel vector space dual. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)    &   𝐿 = (Base‘𝐹)    &    · = (.r𝐹)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑆 = (Scalar‘𝐶)    &    = (.r𝑆)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝐿)    &   (𝜑𝑌𝐿)       (𝜑 → (𝑋 𝑌) = (𝑌 · 𝑋))
 
Theoremlcdvs 38621 Scalar product for the closed kernel vector space dual. (Contributed by NM, 28-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐷 = (LDual‘𝑈)    &    · = ( ·𝑠𝐷)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = ( ·𝑠𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 = · )
 
Theoremlcdvsval 38622 Value of scalar product operation value for the closed kernel vector space dual. (Contributed by NM, 28-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝑆)    &    · = (.r𝑆)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐹 = (Base‘𝐶)    &    = ( ·𝑠𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑅)    &   (𝜑𝐺𝐹)    &   (𝜑𝐴𝑉)       (𝜑 → ((𝑋 𝐺)‘𝐴) = ((𝐺𝐴) · 𝑋))
 
Theoremlcdvscl 38623 The scalar product operation value is a functional. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝑆)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝐶)    &    · = ( ·𝑠𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑅)    &   (𝜑𝐺𝑉)       (𝜑 → (𝑋 · 𝐺) ∈ 𝑉)
 
Theoremlcdlssvscl 38624 Closure of scalar product in a closed kernel dual vector space. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝐹)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝐶)    &    · = ( ·𝑠𝐶)    &   𝑆 = (LSubSp‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐿𝑆)    &   (𝜑𝑋𝑅)    &   (𝜑𝑌𝐿)       (𝜑 → (𝑋 · 𝑌) ∈ 𝐿)
 
Theoremlcdvsass 38625 Associative law for scalar product in a closed kernel dual vector space. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑅 = (Scalar‘𝑈)    &   𝐿 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐹 = (Base‘𝐶)    &    = ( ·𝑠𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝐿)    &   (𝜑𝑌𝐿)    &   (𝜑𝐺𝐹)       (𝜑 → ((𝑌 · 𝑋) 𝐺) = (𝑋 (𝑌 𝐺)))
 
Theoremlcd0 38626 The zero scalar of the closed kernel dual of a vector space. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)    &    0 = (0g𝐹)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑆 = (Scalar‘𝐶)    &   𝑂 = (0g𝑆)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑂 = 0 )
 
Theoremlcd1 38627 The unit scalar of the closed kernel dual of a vector space. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)    &    1 = (1r𝐹)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑆 = (Scalar‘𝐶)    &   𝐼 = (1r𝑆)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝐼 = 1 )
 
Theoremlcdneg 38628 The unit scalar of the closed kernel dual of a vector space. (Contributed by NM, 11-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑅 = (Scalar‘𝑈)    &   𝑀 = (invg𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑆 = (Scalar‘𝐶)    &   𝑁 = (invg𝑆)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑁 = 𝑀)
 
Theoremlcd0v 38629 The zero functional in the set of functionals with closed kernels. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑅 = (Scalar‘𝑈)    &    0 = (0g𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑂 = (0g𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑂 = (𝑉 × { 0 }))
 
Theoremlcd0v2 38630 The zero functional in the set of functionals with closed kernels. (Contributed by NM, 27-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐷 = (LDual‘𝑈)    &    0 = (0g𝐷)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑂 = (0g𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑂 = 0 )
 
Theoremlcd0vvalN 38631 Value of the zero functional at any vector. (Contributed by NM, 28-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑆 = (Scalar‘𝑈)    &    0 = (0g𝑆)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑂 = (0g𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)       (𝜑 → (𝑂𝑋) = 0 )
 
Theoremlcd0vcl 38632 Closure of the zero functional in the set of functionals with closed kernels. (Contributed by NM, 15-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝐶)    &   𝑂 = (0g𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑂𝑉)
 
Theoremlcd0vs 38633 A scalar zero times a functional is the zero functional. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑅 = (Scalar‘𝑈)    &    0 = (0g𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝐶)    &    · = ( ·𝑠𝐶)    &   𝑂 = (0g𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝑉)       (𝜑 → ( 0 · 𝐺) = 𝑂)
 
Theoremlcdvs0N 38634 A scalar times the zero functional is the zero functional. (Contributed by NM, 20-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝑆)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    · = ( ·𝑠𝐶)    &    0 = (0g𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑅)       (𝜑 → (𝑋 · 0 ) = 0 )
 
Theoremlcdvsub 38635 The value of vector subtraction in the closed kernel dual space. (Contributed by NM, 22-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (Scalar‘𝑈)    &   𝑁 = (invg𝑆)    &    1 = (1r𝑆)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝐶)    &    + = (+g𝐶)    &    · = ( ·𝑠𝐶)    &    = (-g𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝑉)    &   (𝜑𝐺𝑉)       (𝜑 → (𝐹 𝐺) = (𝐹 + ((𝑁1 ) · 𝐺)))
 
Theoremlcdvsubval 38636 The value of the value of vector addition in the closed kernel vector space dual. (Contributed by NM, 11-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝑆 = (-g𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &    = (-g𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝐺𝐷)    &   (𝜑𝑋𝑉)       (𝜑 → ((𝐹 𝐺)‘𝑋) = ((𝐹𝑋)𝑆(𝐺𝑋)))
 
Theoremlcdlss 38637* Subspaces of a dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝐶)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑇 = (LSubSp‘𝐷)    &   𝐵 = {𝑓𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑆 = (𝑇 ∩ 𝒫 𝐵))
 
Theoremlcdlss2N 38638 Subspaces of a dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝐶)    &   𝑉 = (Base‘𝐶)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐷 = (LDual‘𝑈)    &   𝑇 = (LSubSp‘𝐷)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑆 = (𝑇 ∩ 𝒫 𝑉))
 
Theoremlcdlsp 38639 Span in the set of functionals with closed kernels. (Contributed by NM, 28-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐷 = (LDual‘𝑈)    &   𝑀 = (LSpan‘𝐷)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐹 = (Base‘𝐶)    &   𝑁 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)       (𝜑 → (𝑁𝐺) = (𝑀𝐺))
 
TheoremlcdlkreqN 38640 Colinear functionals have equal kernels. (Contributed by NM, 28-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐿 = (LKer‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    0 = (0g𝐶)    &   𝑁 = (LSpan‘𝐶)    &   𝑉 = (Base‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐼𝑉)    &   (𝜑𝐺 ∈ (𝑁‘{𝐼}))    &   (𝜑𝐺0 )       (𝜑 → (𝐿𝐺) = (𝐿𝐼))
 
Theoremlcdlkreq2N 38641 Colinear functionals have equal kernels. (Contributed by NM, 28-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝑆)    &    0 = (0g𝑆)    &   𝐿 = (LKer‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝐶)    &    · = ( ·𝑠𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐴 ∈ (𝑅 ∖ { 0 }))    &   (𝜑𝐼𝑉)    &   (𝜑𝐺 = (𝐴 · 𝐼))       (𝜑 → (𝐿𝐺) = (𝐿𝐼))
 
Syntaxcmpd 38642 Extend class notation with projectivity from subspaces of vector space H to subspaces of functionals with closed kernels.
class mapd
 
Definitiondf-mapd 38643* Extend class notation with a one-to-one onto (mapd1o 38666), order-preserving (mapdord 38656) map, called a projectivity (definition of projectivity in [Baer] p. 40), from subspaces of vector space H to those subspaces of the dual space having functionals with closed kernels. (Contributed by NM, 25-Jan-2015.)
mapd = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑠 ∈ (LSubSp‘((DVecH‘𝑘)‘𝑤)) ↦ {𝑓 ∈ (LFnl‘((DVecH‘𝑘)‘𝑤)) ∣ ((((ocH‘𝑘)‘𝑤)‘(((ocH‘𝑘)‘𝑤)‘((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓) ∧ (((ocH‘𝑘)‘𝑤)‘((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓)) ⊆ 𝑠)})))
 
Theoremmapdffval 38644* Projectivity from vector space H to dual space. (Contributed by NM, 25-Jan-2015.)
𝐻 = (LHyp‘𝐾)       (𝐾𝑋 → (mapd‘𝐾) = (𝑤𝐻 ↦ (𝑠 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤)) ↦ {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑤)) ∣ ((((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓) ∧ (((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)) ⊆ 𝑠)})))
 
Theoremmapdfval 38645* Projectivity from vector space H to dual space. (Contributed by NM, 25-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)       ((𝐾𝑋𝑊𝐻) → 𝑀 = (𝑠𝑆 ↦ {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑠)}))
 
Theoremmapdval 38646* Value of projectivity from vector space H to dual space. (Contributed by NM, 27-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   (𝜑 → (𝐾𝑋𝑊𝐻))    &   (𝜑𝑇𝑆)       (𝜑 → (𝑀𝑇) = {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑇)})
 
Theoremmapdvalc 38647* Value of projectivity from vector space H to dual space. (Contributed by NM, 27-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   (𝜑 → (𝐾𝑋𝑊𝐻))    &   (𝜑𝑇𝑆)    &   𝐶 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) = (𝐿𝑔)}       (𝜑 → (𝑀𝑇) = {𝑓𝐶 ∣ (𝑂‘(𝐿𝑓)) ⊆ 𝑇})
 
Theoremmapdval2N 38648* Value of projectivity from vector space H to dual space. (Contributed by NM, 31-Jan-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑇𝑆)    &   𝐶 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) = (𝐿𝑔)}       (𝜑 → (𝑀𝑇) = {𝑓𝐶 ∣ ∃𝑣𝑇 (𝑂‘(𝐿𝑓)) = (𝑁‘{𝑣})})
 
Theoremmapdval3N 38649* Value of projectivity from vector space H to dual space. (Contributed by NM, 31-Jan-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑇𝑆)    &   𝐶 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) = (𝐿𝑔)}       (𝜑 → (𝑀𝑇) = 𝑣𝑇 {𝑓𝐶 ∣ (𝑂‘(𝐿𝑓)) = (𝑁‘{𝑣})})
 
Theoremmapdval4N 38650* Value of projectivity from vector space H to dual space. TODO: 1. This is shorter than others - make it the official def? (but is not as obvious that it is 𝐶) 2. The unneeded direction of lcfl8a 38521 has awkward - add another thm with only one direction of it? 3. Swap 𝑂‘{𝑣} and 𝐿𝑓? (Contributed by NM, 31-Jan-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑇𝑆)       (𝜑 → (𝑀𝑇) = {𝑓𝐹 ∣ ∃𝑣𝑇 (𝑂‘{𝑣}) = (𝐿𝑓)})
 
Theoremmapdval5N 38651* Value of projectivity from vector space H to dual space. (Contributed by NM, 31-Jan-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑇𝑆)       (𝜑 → (𝑀𝑇) = 𝑣𝑇 {𝑓𝐹 ∣ (𝑂‘{𝑣}) = (𝐿𝑓)})
 
Theoremmapdordlem1a 38652* Lemma for mapdord 38656. (Contributed by NM, 27-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑌 = (LSHyp‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝑇 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) ∈ 𝑌}    &   𝐶 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) = (𝐿𝑔)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 → (𝐽𝑇 ↔ (𝐽𝐶 ∧ (𝑂‘(𝑂‘(𝐿𝐽))) ∈ 𝑌)))
 
Theoremmapdordlem1bN 38653* Lemma for mapdord 38656. (Contributed by NM, 27-Jan-2015.) (New usage is discouraged.)
𝐶 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) = (𝐿𝑔)}       (𝐽𝐶 ↔ (𝐽𝐹 ∧ (𝑂‘(𝑂‘(𝐿𝐽))) = (𝐿𝐽)))
 
Theoremmapdordlem1 38654* Lemma for mapdord 38656. (Contributed by NM, 27-Jan-2015.)
𝑇 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) ∈ 𝑌}       (𝐽𝑇 ↔ (𝐽𝐹 ∧ (𝑂‘(𝑂‘(𝐿𝐽))) ∈ 𝑌))
 
Theoremmapdordlem2 38655* Lemma for mapdord 38656. Ordering property of projectivity 𝑀. TODO: This was proved using some hacked-up older proofs. Maybe simplify; get rid of the 𝑇 hypothesis. (Contributed by NM, 27-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑆)    &   (𝜑𝑌𝑆)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝐴 = (LSAtoms‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐽 = (LSHyp‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝑇 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) ∈ 𝐽}    &   𝐶 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) = (𝐿𝑔)}       (𝜑 → ((𝑀𝑋) ⊆ (𝑀𝑌) ↔ 𝑋𝑌))
 
Theoremmapdord 38656 Ordering property of the map defined by df-mapd 38643. Property (b) of [Baer] p. 40. (Contributed by NM, 27-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑆)    &   (𝜑𝑌𝑆)       (𝜑 → ((𝑀𝑋) ⊆ (𝑀𝑌) ↔ 𝑋𝑌))
 
Theoremmapd11 38657 The map defined by df-mapd 38643 is one-to-one. Property (c) of [Baer] p. 40. (Contributed by NM, 12-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑆)    &   (𝜑𝑌𝑆)       (𝜑 → ((𝑀𝑋) = (𝑀𝑌) ↔ 𝑋 = 𝑌))
 
TheoremmapddlssN 38658 The mapping of a subspace of vector space H to the dual space is a subspace of the dual space. TODO: Make this obsolete, use mapdcl2 38674 instead. (Contributed by NM, 31-Jan-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑇 = (LSubSp‘𝐷)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑅𝑆)       (𝜑 → (𝑀𝑅) ∈ 𝑇)
 
Theoremmapdsn 38659* Value of the map defined by df-mapd 38643 at the span of a singleton. (Contributed by NM, 16-Feb-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)       (𝜑 → (𝑀‘(𝑁‘{𝑋})) = {𝑓𝐹 ∣ (𝑂‘{𝑋}) ⊆ (𝐿𝑓)})
 
Theoremmapdsn2 38660* Value of the map defined by df-mapd 38643 at the span of a singleton. (Contributed by NM, 16-Feb-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑 → (𝐿𝐺) = (𝑂‘{𝑋}))       (𝜑 → (𝑀‘(𝑁‘{𝑋})) = {𝑓𝐹 ∣ (𝐿𝐺) ⊆ (𝐿𝑓)})
 
Theoremmapdsn3 38661 Value of the map defined by df-mapd 38643 at the span of a singleton. (Contributed by NM, 17-Feb-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑃 = (LSpan‘𝐷)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝐿𝐺) = (𝑂‘{𝑋}))       (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝑃‘{𝐺}))
 
Theoremmapd1dim2lem1N 38662* Value of the map defined by df-mapd 38643 at an atom. (Contributed by NM, 10-Feb-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐴 = (LSAtoms‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑄𝐴)       (𝜑 → (𝑀𝑄) = {𝑓𝐹 ∣ ∃𝑣𝑄 (𝑂‘{𝑣}) = (𝐿𝑓)})
 
Theoremmapdrvallem2 38663* Lemma for mapdrval 38665. TODO: very long antecedents are dragged through proof in some places - see if it shortens proof to remove unused conjuncts. (Contributed by NM, 2-Feb-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑇 = (LSubSp‘𝐷)    &   𝐶 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) = (𝐿𝑔)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑅𝑇)    &   (𝜑𝑅𝐶)    &   𝑄 = 𝑅 (𝑂‘(𝐿))    &   𝑉 = (Base‘𝑈)    &   𝐴 = (LSAtoms‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &    0 = (0g𝑈)    &   𝑌 = (0g𝐷)       (𝜑 → {𝑓𝐶 ∣ (𝑂‘(𝐿𝑓)) ⊆ 𝑄} ⊆ 𝑅)
 
Theoremmapdrvallem3 38664* Lemma for mapdrval 38665. (Contributed by NM, 2-Feb-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑇 = (LSubSp‘𝐷)    &   𝐶 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) = (𝐿𝑔)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑅𝑇)    &   (𝜑𝑅𝐶)    &   𝑄 = 𝑅 (𝑂‘(𝐿))    &   𝑉 = (Base‘𝑈)    &   𝐴 = (LSAtoms‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &    0 = (0g𝑈)    &   𝑌 = (0g𝐷)       (𝜑 → {𝑓𝐶 ∣ (𝑂‘(𝐿𝑓)) ⊆ 𝑄} = 𝑅)
 
Theoremmapdrval 38665* Given a dual subspace 𝑅 (of functionals with closed kernels), reconstruct the subspace 𝑄 that maps to it. (Contributed by NM, 12-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑇 = (LSubSp‘𝐷)    &   𝐶 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) = (𝐿𝑔)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑅𝑇)    &   (𝜑𝑅𝐶)    &   𝑄 = 𝑅 (𝑂‘(𝐿))       (𝜑 → (𝑀𝑄) = 𝑅)
 
Theoremmapd1o 38666* The map defined by df-mapd 38643 is one-to-one and onto the set of dual subspaces of functionals with closed kernels. This shows 𝑀 satisfies part of the definition of projectivity of [Baer] p. 40. TODO: change theorems leading to this (lcfr 38603, mapdrval 38665, lclkrs 38557, lclkr 38551,...) to use 𝑇 ∩ 𝒫 𝐶? TODO: maybe get rid of $d's for 𝑔 versus 𝐾𝑈𝑊; propagate to mapdrn 38667 and any others. (Contributed by NM, 12-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑇 = (LSubSp‘𝐷)    &   𝐶 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) = (𝐿𝑔)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑀:𝑆1-1-onto→(𝑇 ∩ 𝒫 𝐶))
 
Theoremmapdrn 38667* Range of the map defined by df-mapd 38643. (Contributed by NM, 12-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑇 = (LSubSp‘𝐷)    &   𝐶 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) = (𝐿𝑔)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 → ran 𝑀 = (𝑇 ∩ 𝒫 𝐶))
 
TheoremmapdunirnN 38668* Union of the range of the map defined by df-mapd 38643. (Contributed by NM, 13-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐶 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) = (𝐿𝑔)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 ran 𝑀 = 𝐶)
 
Theoremmapdrn2 38669 Range of the map defined by df-mapd 38643. TODO: this seems to be needed a lot in hdmaprnlem3eN 38876 etc. Would it be better to change df-mapd 38643 theorems to use LSubSp‘𝐶 instead of ran 𝑀? (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑇 = (LSubSp‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 → ran 𝑀 = 𝑇)
 
Theoremmapdcnvcl 38670 Closure of the converse of the map defined by df-mapd 38643. (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝑀)       (𝜑 → (𝑀𝑋) ∈ 𝑆)
 
Theoremmapdcl 38671 Closure the value of the map defined by df-mapd 38643. (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑆)       (𝜑 → (𝑀𝑋) ∈ ran 𝑀)
 
Theoremmapdcnvid1N 38672 Converse of the value of the map defined by df-mapd 38643. (Contributed by NM, 13-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑆)       (𝜑 → (𝑀‘(𝑀𝑋)) = 𝑋)
 
Theoremmapdsord 38673 Strong ordering property of themap defined by df-mapd 38643. (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑆)    &   (𝜑𝑌𝑆)       (𝜑 → ((𝑀𝑋) ⊊ (𝑀𝑌) ↔ 𝑋𝑌))
 
Theoremmapdcl2 38674 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 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑇 = (LSubSp‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑅𝑆)       (𝜑 → (𝑀𝑅) ∈ 𝑇)
 
Theoremmapdcnvid2 38675 Value of the converse of the map defined by df-mapd 38643. (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝑀)       (𝜑 → (𝑀‘(𝑀𝑋)) = 𝑋)
 
TheoremmapdcnvordN 38676 Ordering property of the converse of the map defined by df-mapd 38643. (Contributed by NM, 13-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝑀)    &   (𝜑𝑌 ∈ ran 𝑀)       (𝜑 → ((𝑀𝑋) ⊆ (𝑀𝑌) ↔ 𝑋𝑌))
 
Theoremmapdcnv11N 38677 The converse of the map defined by df-mapd 38643 is one-to-one. (Contributed by NM, 13-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝑀)    &   (𝜑𝑌 ∈ ran 𝑀)       (𝜑 → ((𝑀𝑋) = (𝑀𝑌) ↔ 𝑋 = 𝑌))
 
Theoremmapdcv 38678 Covering property of the converse of the map defined by df-mapd 38643. (Contributed by NM, 14-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝐶 = ( ⋖L𝑈)    &   𝐷 = ((LCDual‘𝐾)‘𝑊)    &   𝐸 = ( ⋖L𝐷)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑆)    &   (𝜑𝑌𝑆)       (𝜑 → (𝑋𝐶𝑌 ↔ (𝑀𝑋)𝐸(𝑀𝑌)))
 
Theoremmapdincl 38679 Closure of dual subspace intersection for the map defined by df-mapd 38643. (Contributed by NM, 12-Apr-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝑀)    &   (𝜑𝑌 ∈ ran 𝑀)       (𝜑 → (𝑋𝑌) ∈ ran 𝑀)
 
Theoremmapdin 38680 Subspace intersection is preserved by the map defined by df-mapd 38643. Part of property (e) in [Baer] p. 40. (Contributed by NM, 12-Apr-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑆)    &   (𝜑𝑌𝑆)       (𝜑 → (𝑀‘(𝑋𝑌)) = ((𝑀𝑋) ∩ (𝑀𝑌)))
 
Theoremmapdlsmcl 38681 Closure of dual subspace sum for the map defined by df-mapd 38643. (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = (LSSum‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝑀)    &   (𝜑𝑌 ∈ ran 𝑀)       (𝜑 → (𝑋 𝑌) ∈ ran 𝑀)
 
Theoremmapdlsm 38682 Subspace sum is preserved by the map defined by df-mapd 38643. Part of property (e) in [Baer] p. 40. (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &    = (LSSum‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = (LSSum‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑆)    &   (𝜑𝑌𝑆)       (𝜑 → (𝑀‘(𝑋 𝑌)) = ((𝑀𝑋) (𝑀𝑌)))
 
Theoremmapd0 38683 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 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑂 = (0g𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    0 = (0g𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 → (𝑀‘{𝑂}) = { 0 })
 
TheoremmapdcnvatN 38684 Atoms are preserved by the map defined by df-mapd 38643. (Contributed by NM, 29-May-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐴 = (LSAtoms‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐵 = (LSAtoms‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑄𝐵)       (𝜑 → (𝑀𝑄) ∈ 𝐴)
 
Theoremmapdat 38685 Atoms are preserved by the map defined by df-mapd 38643. Property (g) in [Baer] p. 41. (Contributed by NM, 14-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐴 = (LSAtoms‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐵 = (LSAtoms‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑄𝐴)       (𝜑 → (𝑀𝑄) ∈ 𝐵)
 
Theoremmapdspex 38686* The map of a span equals the dual span of some vector (functional). (Contributed by NM, 15-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐵 = (Base‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)       (𝜑 → ∃𝑔𝐵 (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝑔}))
 
Theoremmapdn0 38687 Transfer nonzero property from domain to range of projectivity mapd. (Contributed by NM, 12-Apr-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &    0 = (0g𝑈)    &   𝑍 = (0g𝐶)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))       (𝜑𝐹 ∈ (𝐷 ∖ {𝑍}))
 
Theoremmapdncol 38688 Transfer non-colinearity from domain to range of projectivity mapd. (Contributed by NM, 11-Apr-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝐺𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))       (𝜑 → (𝐽‘{𝐹}) ≠ (𝐽‘{𝐺}))
 
Theoremmapdindp 38689 Transfer (part of) vector independence condition from domain to range of projectivity mapd. (Contributed by NM, 11-Apr-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝐺𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}))    &   (𝜑𝑍𝑉)    &   (𝜑𝐸𝐷)    &   (𝜑 → (𝑀‘(𝑁‘{𝑍})) = (𝐽‘{𝐸}))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))       (𝜑 → ¬ 𝐹 ∈ (𝐽‘{𝐺, 𝐸}))
 
Theoremmapdpglem1 38690 Lemma for mapdpg 38724. Baer p. 44, last line: "(F(x-y))* <= (Fx)*+(Fy)*." (Contributed by NM, 15-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)       (𝜑 → (𝑀‘(𝑁‘{(𝑋 𝑌)})) ⊆ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌}))))
 
Theoremmapdpglem2 38691* Lemma for mapdpg 38724. Baer p. 45, lines 1 and 2: "we have (F(x-y))* = Gt where t necessarily belongs to (Fx)*+(Fy)*." (We scope $d 𝑡𝜑 locally to avoid clashes with later substitutions into 𝜑.) (Contributed by NM, 15-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)    &   𝐽 = (LSpan‘𝐶)       (𝜑 → ∃𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌})))(𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{𝑡}))
 
Theoremmapdpglem2a 38692* Lemma for mapdpg 38724. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝐹 = (Base‘𝐶)    &   (𝜑𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌}))))       (𝜑𝑡𝐹)
 
Theoremmapdpglem3 38693* Lemma for mapdpg 38724. 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 𝑔𝑤𝑧𝜑 locally to avoid clashes with later substitutions into 𝜑.) (Contributed by NM, 18-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝐹 = (Base‘𝐶)    &   (𝜑𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑅 = (-g𝐶)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))       (𝜑 → ∃𝑔𝐵𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = ((𝑔 · 𝐺)𝑅𝑧))
 
Theoremmapdpglem4N 38694* Lemma for mapdpg 38724. (Contributed by NM, 20-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝐹 = (Base‘𝐶)    &   (𝜑𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑅 = (-g𝐶)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   𝑄 = (0g𝑈)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))       (𝜑 → (𝑋 𝑌) ≠ 𝑄)
 
Theoremmapdpglem5N 38695* Lemma for mapdpg 38724. (Contributed by NM, 20-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝐹 = (Base‘𝐶)    &   (𝜑𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑅 = (-g𝐶)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   𝑄 = (0g𝑈)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{𝑡}))       (𝜑𝑡 ≠ (0g𝐶))
 
Theoremmapdpglem6 38696* Lemma for mapdpg 38724. Baer p. 45, line 4: "If g were 0, then t would be in (Fy)*..." (Contributed by NM, 18-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝐹 = (Base‘𝐶)    &   (𝜑𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑅 = (-g𝐶)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   𝑄 = (0g𝑈)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{𝑡}))    &    0 = (0g𝐴)    &   (𝜑𝑔𝐵)    &   (𝜑𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))    &   (𝜑𝑡 = ((𝑔 · 𝐺)𝑅𝑧))    &   (𝜑𝑋𝑄)    &   (𝜑𝑔 = 0 )       (𝜑𝑡 ∈ (𝑀‘(𝑁‘{𝑌})))
 
Theoremmapdpglem8 38697* Lemma for mapdpg 38724. 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 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝐹 = (Base‘𝐶)    &   (𝜑𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑅 = (-g𝐶)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   𝑄 = (0g𝑈)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{𝑡}))    &    0 = (0g𝐴)    &   (𝜑𝑔𝐵)    &   (𝜑𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))    &   (𝜑𝑡 = ((𝑔 · 𝐺)𝑅𝑧))    &   (𝜑𝑋𝑄)    &   (𝜑𝑔 = 0 )       (𝜑 → (𝑁‘{(𝑋 𝑌)}) ⊆ (𝑁‘{𝑌}))
 
Theoremmapdpglem9 38698* Lemma for mapdpg 38724. Baer p. 45, line 4: "...so that x would consequently belong to Fy." (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝐹 = (Base‘𝐶)    &   (𝜑𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑅 = (-g𝐶)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   𝑄 = (0g𝑈)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{𝑡}))    &    0 = (0g𝐴)    &   (𝜑𝑔𝐵)    &   (𝜑𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))    &   (𝜑𝑡 = ((𝑔 · 𝐺)𝑅𝑧))    &   (𝜑𝑋𝑄)    &   (𝜑𝑔 = 0 )       (𝜑𝑋 ∈ (𝑁‘{𝑌}))
 
Theoremmapdpglem10 38699* Lemma for mapdpg 38724. Baer p. 45, line 6: "Hence Fx=Fy, an impossibility." (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝐹 = (Base‘𝐶)    &   (𝜑𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑅 = (-g𝐶)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   𝑄 = (0g𝑈)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{𝑡}))    &    0 = (0g𝐴)    &   (𝜑𝑔𝐵)    &   (𝜑𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))    &   (𝜑𝑡 = ((𝑔 · 𝐺)𝑅𝑧))    &   (𝜑𝑋𝑄)    &   (𝜑𝑔 = 0 )       (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))
 
Theoremmapdpglem11 38700* Lemma for mapdpg 38724. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑀 = ((mapd‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = (-g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (LSSum‘𝐶)    &   𝐽 = (LSpan‘𝐶)    &   𝐹 = (Base‘𝐶)    &   (𝜑𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) (𝑀‘(𝑁‘{𝑌}))))    &   𝐴 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐶)    &   𝑅 = (-g𝐶)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))    &   𝑄 = (0g𝑈)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{𝑡}))    &    0 = (0g𝐴)    &   (𝜑𝑔𝐵)    &   (𝜑𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))    &   (𝜑𝑡 = ((𝑔 · 𝐺)𝑅𝑧))    &   (𝜑𝑋𝑄)       (𝜑𝑔0 )
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