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Theorem List for Metamath Proof Explorer - 40401-40500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlclkrlem2x 40401 Lemma for lclkr 40404. Eliminate by cases the hypotheses of lclkrlem2u 40398, lclkrlem2u 40398 and lclkrlem2w 40400. (Contributed by NM, 18-Jan-2015.)
𝐿 = (LKerβ€˜π‘ˆ)    &   π» = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &    + = (+gβ€˜π·)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &   (πœ‘ β†’ 𝐸 ∈ 𝐹)    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ (πΏβ€˜πΈ) = ( βŠ₯ β€˜{𝑋}))    &   (πœ‘ β†’ (πΏβ€˜πΊ) = ( βŠ₯ β€˜{π‘Œ}))    β‡’   (πœ‘ β†’ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜(𝐸 + 𝐺)))) = (πΏβ€˜(𝐸 + 𝐺)))
 
Theoremlclkrlem2y 40402 Lemma for lclkr 40404. Restate the hypotheses for 𝐸 and 𝐺 to say their kernels are closed, in order to eliminate the generating vectors 𝑋 and π‘Œ. (Contributed by NM, 18-Jan-2015.)
𝐿 = (LKerβ€˜π‘ˆ)    &   π» = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &    + = (+gβ€˜π·)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐸 ∈ 𝐹)    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜πΈ))) = (πΏβ€˜πΈ))    &   (πœ‘ β†’ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜πΊ))) = (πΏβ€˜πΊ))    β‡’   (πœ‘ β†’ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜(𝐸 + 𝐺)))) = (πΏβ€˜(𝐸 + 𝐺)))
 
Theoremlclkrlem2 40403* The set of functionals having closed kernels is closed under vector (functional) addition. Lemmas lclkrlem2a 40378 through lclkrlem2y 40402 are used for the proof. Here we express lclkrlem2y 40402 in terms of membership in the set 𝐢 of functionals with closed kernels. (Contributed by NM, 18-Jan-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &    + = (+gβ€˜π·)    &   πΆ = {𝑓 ∈ 𝐹 ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)}    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐸 ∈ 𝐢)    &   (πœ‘ β†’ 𝐺 ∈ 𝐢)    β‡’   (πœ‘ β†’ (𝐸 + 𝐺) ∈ 𝐢)
 
Theoremlclkr 40404* The set of functionals with closed kernels is a subspace. Part of proof of Theorem 3.6 of [Holland95] p. 218, line 20, stating "The fM that arise this way generate a subspace F of E'". Our proof was suggested by Mario Carneiro, 5-Jan-2015. (Contributed by NM, 18-Jan-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   π‘† = (LSubSpβ€˜π·)    &   πΆ = {𝑓 ∈ 𝐹 ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)}    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    β‡’   (πœ‘ β†’ 𝐢 ∈ 𝑆)
 
Theoremlcfls1lem 40405* Property of a functional with a closed kernel. (Contributed by NM, 27-Jan-2015.)
𝐢 = {𝑓 ∈ 𝐹 ∣ (( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“) ∧ ( βŠ₯ β€˜(πΏβ€˜π‘“)) βŠ† 𝑄)}    β‡’   (𝐺 ∈ 𝐢 ↔ (𝐺 ∈ 𝐹 ∧ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜πΊ))) = (πΏβ€˜πΊ) ∧ ( βŠ₯ β€˜(πΏβ€˜πΊ)) βŠ† 𝑄))
 
Theoremlcfls1N 40406* Property of a functional with a closed kernel. (Contributed by NM, 27-Jan-2015.) (New usage is discouraged.)
𝐢 = {𝑓 ∈ 𝐹 ∣ (( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“) ∧ ( βŠ₯ β€˜(πΏβ€˜π‘“)) βŠ† 𝑄)}    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    β‡’   (πœ‘ β†’ (𝐺 ∈ 𝐢 ↔ (( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜πΊ))) = (πΏβ€˜πΊ) ∧ ( βŠ₯ β€˜(πΏβ€˜πΊ)) βŠ† 𝑄)))
 
Theoremlcfls1c 40407* Property of a functional with a closed kernel. (Contributed by NM, 28-Jan-2015.)
𝐢 = {𝑓 ∈ 𝐹 ∣ (( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“) ∧ ( βŠ₯ β€˜(πΏβ€˜π‘“)) βŠ† 𝑄)}    &   π· = {𝑓 ∈ 𝐹 ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)}    β‡’   (𝐺 ∈ 𝐢 ↔ (𝐺 ∈ 𝐷 ∧ ( βŠ₯ β€˜(πΏβ€˜πΊ)) βŠ† 𝑄))
 
Theoremlclkrslem1 40408* The set of functionals having closed kernels and majorizing the orthocomplement of a given subspace 𝑄 is closed under scalar product. (Contributed by NM, 27-Jan-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘ˆ)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   π‘… = (Scalarβ€˜π‘ˆ)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = ( ·𝑠 β€˜π·)    &   πΆ = {𝑓 ∈ 𝐹 ∣ (( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“) ∧ ( βŠ₯ β€˜(πΏβ€˜π‘“)) βŠ† 𝑄)}    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑄 ∈ 𝑆)    &   (πœ‘ β†’ 𝐺 ∈ 𝐢)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝑋 Β· 𝐺) ∈ 𝐢)
 
Theoremlclkrslem2 40409* The set of functionals having closed kernels and majorizing the orthocomplement of a given subspace 𝑄 is closed under scalar product. (Contributed by NM, 28-Jan-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘ˆ)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   π‘… = (Scalarβ€˜π‘ˆ)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = ( ·𝑠 β€˜π·)    &   πΆ = {𝑓 ∈ 𝐹 ∣ (( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“) ∧ ( βŠ₯ β€˜(πΏβ€˜π‘“)) βŠ† 𝑄)}    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑄 ∈ 𝑆)    &   (πœ‘ β†’ 𝐺 ∈ 𝐢)    &    + = (+gβ€˜π·)    &   (πœ‘ β†’ 𝐸 ∈ 𝐢)    β‡’   (πœ‘ β†’ (𝐸 + 𝐺) ∈ 𝐢)
 
Theoremlclkrs 40410* The set of functionals having closed kernels and majorizing the orthocomplement of a given subspace 𝑅 is a subspace of the dual space. TODO: This proof repeats large parts of the lclkr 40404 proof. Do we achieve overall shortening by breaking them out as subtheorems? Or make lclkr 40404 a special case of this? (Contributed by NM, 29-Jan-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘ˆ)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   π‘‡ = (LSubSpβ€˜π·)    &   πΆ = {𝑓 ∈ 𝐹 ∣ (( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“) ∧ ( βŠ₯ β€˜(πΏβ€˜π‘“)) βŠ† 𝑅)}    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑅 ∈ 𝑆)    β‡’   (πœ‘ β†’ 𝐢 ∈ 𝑇)
 
Theoremlclkrs2 40411* The set of functionals with closed kernels and majorizing the orthocomplement of a given subspace 𝑄 is a subspace of the dual space containing functionals with closed kernels. Note that 𝑅 is the value given by mapdval 40499. (Contributed by NM, 12-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘ˆ)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   π‘‡ = (LSubSpβ€˜π·)    &   πΆ = {𝑓 ∈ 𝐹 ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)}    &   π‘… = {𝑔 ∈ 𝐹 ∣ (( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘”))) = (πΏβ€˜π‘”) ∧ ( βŠ₯ β€˜(πΏβ€˜π‘”)) βŠ† 𝑄)}    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑄 ∈ 𝑆)    β‡’   (πœ‘ β†’ (𝑅 ∈ 𝑇 ∧ 𝑅 βŠ† 𝐢))
 
TheoremlcfrvalsnN 40412* Reconstruction from the dual space span of a singleton. (Contributed by NM, 19-Feb-2015.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π·)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   π‘„ = βˆͺ 𝑓 ∈ 𝑅 ( βŠ₯ β€˜(πΏβ€˜π‘“))    &   π‘… = (π‘β€˜{𝐺})    β‡’   (πœ‘ β†’ 𝑄 = ( βŠ₯ β€˜(πΏβ€˜πΊ)))
 
Theoremlcfrlem1 40413 Lemma for lcfr 40456. Note that 𝑋 is z in Mario's notes. (Contributed by NM, 27-Feb-2015.)
𝑉 = (Baseβ€˜π‘ˆ)    &   π‘† = (Scalarβ€˜π‘ˆ)    &    Γ— = (.rβ€˜π‘†)    &    0 = (0gβ€˜π‘†)    &   πΌ = (invrβ€˜π‘†)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &    Β· = ( ·𝑠 β€˜π·)    &    βˆ’ = (-gβ€˜π·)    &   (πœ‘ β†’ π‘ˆ ∈ LVec)    &   (πœ‘ β†’ 𝐸 ∈ 𝐹)    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ (πΊβ€˜π‘‹) β‰  0 )    &   π» = (𝐸 βˆ’ (((πΌβ€˜(πΊβ€˜π‘‹)) Γ— (πΈβ€˜π‘‹)) Β· 𝐺))    β‡’   (πœ‘ β†’ (π»β€˜π‘‹) = 0 )
 
Theoremlcfrlem2 40414 Lemma for lcfr 40456. (Contributed by NM, 27-Feb-2015.)
𝑉 = (Baseβ€˜π‘ˆ)    &   π‘† = (Scalarβ€˜π‘ˆ)    &    Γ— = (.rβ€˜π‘†)    &    0 = (0gβ€˜π‘†)    &   πΌ = (invrβ€˜π‘†)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &    Β· = ( ·𝑠 β€˜π·)    &    βˆ’ = (-gβ€˜π·)    &   (πœ‘ β†’ π‘ˆ ∈ LVec)    &   (πœ‘ β†’ 𝐸 ∈ 𝐹)    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ (πΊβ€˜π‘‹) β‰  0 )    &   π» = (𝐸 βˆ’ (((πΌβ€˜(πΊβ€˜π‘‹)) Γ— (πΈβ€˜π‘‹)) Β· 𝐺))    &   πΏ = (LKerβ€˜π‘ˆ)    β‡’   (πœ‘ β†’ ((πΏβ€˜πΈ) ∩ (πΏβ€˜πΊ)) βŠ† (πΏβ€˜π»))
 
Theoremlcfrlem3 40415 Lemma for lcfr 40456. (Contributed by NM, 27-Feb-2015.)
𝑉 = (Baseβ€˜π‘ˆ)    &   π‘† = (Scalarβ€˜π‘ˆ)    &    Γ— = (.rβ€˜π‘†)    &    0 = (0gβ€˜π‘†)    &   πΌ = (invrβ€˜π‘†)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &    Β· = ( ·𝑠 β€˜π·)    &    βˆ’ = (-gβ€˜π·)    &   (πœ‘ β†’ π‘ˆ ∈ LVec)    &   (πœ‘ β†’ 𝐸 ∈ 𝐹)    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ (πΊβ€˜π‘‹) β‰  0 )    &   π» = (𝐸 βˆ’ (((πΌβ€˜(πΊβ€˜π‘‹)) Γ— (πΈβ€˜π‘‹)) Β· 𝐺))    &   πΏ = (LKerβ€˜π‘ˆ)    β‡’   (πœ‘ β†’ 𝑋 ∈ (πΏβ€˜π»))
 
Theoremlcfrlem4 40416* Lemma for lcfr 40456. (Contributed by NM, 10-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   π‘„ = (LSubSpβ€˜π·)    &   πΈ = βˆͺ 𝑔 ∈ 𝐺 ( βŠ₯ β€˜(πΏβ€˜π‘”))    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐺 ∈ 𝑄)    &   (πœ‘ β†’ 𝑋 ∈ 𝐸)    β‡’   (πœ‘ β†’ 𝑋 ∈ 𝑉)
 
Theoremlcfrlem5 40417* Lemma for lcfr 40456. The set of functionals having closed kernels and majorizing the orthocomplement of a given subspace 𝑄 is closed under scalar product. TODO: share hypotheses with others. Use more consistent variable names here or elsewhere when possible. (Contributed by NM, 5-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   π‘† = (LSubSpβ€˜π·)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑅 ∈ 𝑆)    &   π‘„ = βˆͺ 𝑓 ∈ 𝑅 ( βŠ₯ β€˜(πΏβ€˜π‘“))    &   (πœ‘ β†’ 𝑋 ∈ 𝑄)    &   πΆ = (Scalarβ€˜π‘ˆ)    &   π΅ = (Baseβ€˜πΆ)    &    Β· = ( ·𝑠 β€˜π‘ˆ)    &   (πœ‘ β†’ 𝐴 ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝐴 Β· 𝑋) ∈ 𝑄)
 
Theoremlcfrlem6 40418* Lemma for lcfr 40456. Closure of vector sum with colinear vectors. TODO: Move down 𝑁 definition so top hypotheses can be shared. (Contributed by NM, 10-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &    + = (+gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   π‘„ = (LSubSpβ€˜π·)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐺 ∈ 𝑄)    &   πΈ = βˆͺ 𝑔 ∈ 𝐺 ( βŠ₯ β€˜(πΏβ€˜π‘”))    &   (πœ‘ β†’ 𝑋 ∈ 𝐸)    &   (πœ‘ β†’ π‘Œ ∈ 𝐸)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ}))    β‡’   (πœ‘ β†’ (𝑋 + π‘Œ) ∈ 𝐸)
 
Theoremlcfrlem7 40419* Lemma for lcfr 40456. Closure of vector sum when one vector is zero. TODO: share hypotheses with others. (Contributed by NM, 11-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &    + = (+gβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   π‘„ = (LSubSpβ€˜π·)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐺 ∈ 𝑄)    &   πΈ = βˆͺ 𝑔 ∈ 𝐺 ( βŠ₯ β€˜(πΏβ€˜π‘”))    &   (πœ‘ β†’ 𝑋 ∈ 𝐸)    &    0 = (0gβ€˜π‘ˆ)    &   (πœ‘ β†’ π‘Œ = 0 )    β‡’   (πœ‘ β†’ (𝑋 + π‘Œ) ∈ 𝐸)
 
Theoremlcfrlem8 40420* Lemma for lcf1o 40422 and lcfr 40456. (Contributed by NM, 21-Feb-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    + = (+gβ€˜π‘ˆ)    &    Β· = ( ·𝑠 β€˜π‘ˆ)    &   π‘† = (Scalarβ€˜π‘ˆ)    &   π‘… = (Baseβ€˜π‘†)    &    0 = (0gβ€˜π‘ˆ)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   π‘„ = (0gβ€˜π·)    &   πΆ = {𝑓 ∈ 𝐹 ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)}    &   π½ = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{π‘₯})𝑣 = (𝑀 + (π‘˜ Β· π‘₯)))))    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    β‡’   (πœ‘ β†’ (π½β€˜π‘‹) = (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{𝑋})𝑣 = (𝑀 + (π‘˜ Β· 𝑋)))))
 
Theoremlcfrlem9 40421* Lemma for lcf1o 40422. (This part has undesirable $d's on 𝐽 and πœ‘ that we remove in lcf1o 40422.) TODO: ugly proof; maybe have better subtheorems or abbreviate some β„©π‘˜ expansions with π½β€˜π‘§? TODO: Some redundant $d's? (Contributed by NM, 22-Feb-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    + = (+gβ€˜π‘ˆ)    &    Β· = ( ·𝑠 β€˜π‘ˆ)    &   π‘† = (Scalarβ€˜π‘ˆ)    &   π‘… = (Baseβ€˜π‘†)    &    0 = (0gβ€˜π‘ˆ)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   π‘„ = (0gβ€˜π·)    &   πΆ = {𝑓 ∈ 𝐹 ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)}    &   π½ = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{π‘₯})𝑣 = (𝑀 + (π‘˜ Β· π‘₯)))))    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    β‡’   (πœ‘ β†’ 𝐽:(𝑉 βˆ– { 0 })–1-1-ontoβ†’(𝐢 βˆ– {𝑄}))
 
Theoremlcf1o 40422* Define a function 𝐽 that provides a bijection from nonzero vectors 𝑉 to nonzero functionals with closed kernels 𝐢. (Contributed by NM, 22-Feb-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    + = (+gβ€˜π‘ˆ)    &    Β· = ( ·𝑠 β€˜π‘ˆ)    &   π‘† = (Scalarβ€˜π‘ˆ)    &   π‘… = (Baseβ€˜π‘†)    &    0 = (0gβ€˜π‘ˆ)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   π‘„ = (0gβ€˜π·)    &   πΆ = {𝑓 ∈ 𝐹 ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)}    &   π½ = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{π‘₯})𝑣 = (𝑀 + (π‘˜ Β· π‘₯)))))    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    β‡’   (πœ‘ β†’ 𝐽:(𝑉 βˆ– { 0 })–1-1-ontoβ†’(𝐢 βˆ– {𝑄}))
 
Theoremlcfrlem10 40423* Lemma for lcfr 40456. (Contributed by NM, 23-Feb-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    + = (+gβ€˜π‘ˆ)    &    Β· = ( ·𝑠 β€˜π‘ˆ)    &   π‘† = (Scalarβ€˜π‘ˆ)    &   π‘… = (Baseβ€˜π‘†)    &    0 = (0gβ€˜π‘ˆ)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   π‘„ = (0gβ€˜π·)    &   πΆ = {𝑓 ∈ 𝐹 ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)}    &   π½ = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{π‘₯})𝑣 = (𝑀 + (π‘˜ Β· π‘₯)))))    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    β‡’   (πœ‘ β†’ (π½β€˜π‘‹) ∈ 𝐹)
 
Theoremlcfrlem11 40424* Lemma for lcfr 40456. (Contributed by NM, 23-Feb-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    + = (+gβ€˜π‘ˆ)    &    Β· = ( ·𝑠 β€˜π‘ˆ)    &   π‘† = (Scalarβ€˜π‘ˆ)    &   π‘… = (Baseβ€˜π‘†)    &    0 = (0gβ€˜π‘ˆ)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   π‘„ = (0gβ€˜π·)    &   πΆ = {𝑓 ∈ 𝐹 ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)}    &   π½ = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{π‘₯})𝑣 = (𝑀 + (π‘˜ Β· π‘₯)))))    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    β‡’   (πœ‘ β†’ (πΏβ€˜(π½β€˜π‘‹)) = ( βŠ₯ β€˜{𝑋}))
 
Theoremlcfrlem12N 40425* Lemma for lcfr 40456. (Contributed by NM, 23-Feb-2015.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    + = (+gβ€˜π‘ˆ)    &    Β· = ( ·𝑠 β€˜π‘ˆ)    &   π‘† = (Scalarβ€˜π‘ˆ)    &   π‘… = (Baseβ€˜π‘†)    &    0 = (0gβ€˜π‘ˆ)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   π‘„ = (0gβ€˜π·)    &   πΆ = {𝑓 ∈ 𝐹 ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)}    &   π½ = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{π‘₯})𝑣 = (𝑀 + (π‘˜ Β· π‘₯)))))    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &   π΅ = (0gβ€˜π‘†)    &   (πœ‘ β†’ π‘Œ ∈ ( βŠ₯ β€˜{𝑋}))    β‡’   (πœ‘ β†’ ((π½β€˜π‘‹)β€˜π‘Œ) = 𝐡)
 
Theoremlcfrlem13 40426* Lemma for lcfr 40456. (Contributed by NM, 8-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    + = (+gβ€˜π‘ˆ)    &    Β· = ( ·𝑠 β€˜π‘ˆ)    &   π‘† = (Scalarβ€˜π‘ˆ)    &   π‘… = (Baseβ€˜π‘†)    &    0 = (0gβ€˜π‘ˆ)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   π‘„ = (0gβ€˜π·)    &   πΆ = {𝑓 ∈ 𝐹 ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)}    &   π½ = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{π‘₯})𝑣 = (𝑀 + (π‘˜ Β· π‘₯)))))    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    β‡’   (πœ‘ β†’ (π½β€˜π‘‹) ∈ (𝐢 βˆ– {𝑄}))
 
Theoremlcfrlem14 40427* Lemma for lcfr 40456. (Contributed by NM, 10-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    + = (+gβ€˜π‘ˆ)    &    Β· = ( ·𝑠 β€˜π‘ˆ)    &   π‘† = (Scalarβ€˜π‘ˆ)    &   π‘… = (Baseβ€˜π‘†)    &    0 = (0gβ€˜π‘ˆ)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   π‘„ = (0gβ€˜π·)    &   πΆ = {𝑓 ∈ 𝐹 ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)}    &   π½ = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{π‘₯})𝑣 = (𝑀 + (π‘˜ Β· π‘₯)))))    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    &   π‘ = (LSpanβ€˜π‘ˆ)    β‡’   (πœ‘ β†’ ( βŠ₯ β€˜(πΏβ€˜(π½β€˜π‘‹))) = (π‘β€˜{𝑋}))
 
Theoremlcfrlem15 40428* Lemma for lcfr 40456. (Contributed by NM, 9-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    + = (+gβ€˜π‘ˆ)    &    Β· = ( ·𝑠 β€˜π‘ˆ)    &   π‘† = (Scalarβ€˜π‘ˆ)    &   π‘… = (Baseβ€˜π‘†)    &    0 = (0gβ€˜π‘ˆ)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   π‘„ = (0gβ€˜π·)    &   πΆ = {𝑓 ∈ 𝐹 ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)}    &   π½ = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{π‘₯})𝑣 = (𝑀 + (π‘˜ Β· π‘₯)))))    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))    β‡’   (πœ‘ β†’ 𝑋 ∈ ( βŠ₯ β€˜(πΏβ€˜(π½β€˜π‘‹))))
 
Theoremlcfrlem16 40429* Lemma for lcfr 40456. (Contributed by NM, 8-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    + = (+gβ€˜π‘ˆ)    &    Β· = ( ·𝑠 β€˜π‘ˆ)    &   π‘† = (Scalarβ€˜π‘ˆ)    &   π‘… = (Baseβ€˜π‘†)    &    0 = (0gβ€˜π‘ˆ)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   π‘„ = (0gβ€˜π·)    &   πΆ = {𝑓 ∈ 𝐹 ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)}    &   π½ = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{π‘₯})𝑣 = (𝑀 + (π‘˜ Β· π‘₯)))))    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   π‘ƒ = (LSubSpβ€˜π·)    &   (πœ‘ β†’ 𝐺 ∈ 𝑃)    &   (πœ‘ β†’ 𝐺 βŠ† 𝐢)    &   πΈ = βˆͺ 𝑔 ∈ 𝐺 ( βŠ₯ β€˜(πΏβ€˜π‘”))    &   (πœ‘ β†’ 𝑋 ∈ (𝐸 βˆ– { 0 }))    β‡’   (πœ‘ β†’ (π½β€˜π‘‹) ∈ 𝐺)
 
Theoremlcfrlem17 40430 Lemma for lcfr 40456. Condition needed more than once. (Contributed by NM, 11-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    + = (+gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   π΄ = (LSAtomsβ€˜π‘ˆ)    &   (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†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    β‡’   (πœ‘ β†’ (𝑋 + π‘Œ) ∈ (𝑉 βˆ– { 0 }))
 
Theoremlcfrlem18 40431 Lemma for lcfr 40456. (Contributed by NM, 24-Feb-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    + = (+gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   π΄ = (LSAtomsβ€˜π‘ˆ)    &   (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†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    β‡’   (πœ‘ β†’ ( βŠ₯ β€˜{𝑋, π‘Œ}) = (( βŠ₯ β€˜{𝑋}) ∩ ( βŠ₯ β€˜{π‘Œ})))
 
Theoremlcfrlem19 40432 Lemma for lcfr 40456. (Contributed by NM, 11-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    + = (+gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   π΄ = (LSAtomsβ€˜π‘ˆ)    &   (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†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    β‡’   (πœ‘ β†’ (Β¬ 𝑋 ∈ ( βŠ₯ β€˜{(𝑋 + π‘Œ)}) ∨ Β¬ π‘Œ ∈ ( βŠ₯ β€˜{(𝑋 + π‘Œ)})))
 
Theoremlcfrlem20 40433 Lemma for lcfr 40456. (Contributed by NM, 11-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    + = (+gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   π΄ = (LSAtomsβ€˜π‘ˆ)    &   (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†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   (πœ‘ β†’ Β¬ 𝑋 ∈ ( βŠ₯ β€˜{(𝑋 + π‘Œ)}))    β‡’   (πœ‘ β†’ ((π‘β€˜{𝑋, π‘Œ}) ∩ ( βŠ₯ β€˜{(𝑋 + π‘Œ)})) ∈ 𝐴)
 
Theoremlcfrlem21 40434 Lemma for lcfr 40456. (Contributed by NM, 11-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    + = (+gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   π΄ = (LSAtomsβ€˜π‘ˆ)    &   (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†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    β‡’   (πœ‘ β†’ ((π‘β€˜{𝑋, π‘Œ}) ∩ ( βŠ₯ β€˜{(𝑋 + π‘Œ)})) ∈ 𝐴)
 
Theoremlcfrlem22 40435 Lemma for lcfr 40456. (Contributed by NM, 24-Feb-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    + = (+gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   π΄ = (LSAtomsβ€˜π‘ˆ)    &   (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†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   π΅ = ((π‘β€˜{𝑋, π‘Œ}) ∩ ( βŠ₯ β€˜{(𝑋 + π‘Œ)}))    β‡’   (πœ‘ β†’ 𝐡 ∈ 𝐴)
 
Theoremlcfrlem23 40436 Lemma for lcfr 40456. TODO: this proof was built from other proof pieces that may change π‘β€˜{𝑋, π‘Œ} into subspace sum and back unnecessarily, or similar things. (Contributed by NM, 1-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    + = (+gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   π΄ = (LSAtomsβ€˜π‘ˆ)    &   (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†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   π΅ = ((π‘β€˜{𝑋, π‘Œ}) ∩ ( βŠ₯ β€˜{(𝑋 + π‘Œ)}))    &    βŠ• = (LSSumβ€˜π‘ˆ)    β‡’   (πœ‘ β†’ (( βŠ₯ β€˜{𝑋, π‘Œ}) βŠ• 𝐡) = ( βŠ₯ β€˜{(𝑋 + π‘Œ)}))
 
Theoremlcfrlem24 40437* Lemma for lcfr 40456. (Contributed by NM, 24-Feb-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    + = (+gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   π΄ = (LSAtomsβ€˜π‘ˆ)    &   (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†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   π΅ = ((π‘β€˜{𝑋, π‘Œ}) ∩ ( βŠ₯ β€˜{(𝑋 + π‘Œ)}))    &    Β· = ( ·𝑠 β€˜π‘ˆ)    &   π‘† = (Scalarβ€˜π‘ˆ)    &   π‘„ = (0gβ€˜π‘†)    &   π‘… = (Baseβ€˜π‘†)    &   π½ = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{π‘₯})𝑣 = (𝑀 + (π‘˜ Β· π‘₯)))))    &   (πœ‘ β†’ 𝐼 ∈ 𝐡)    &   πΏ = (LKerβ€˜π‘ˆ)    β‡’   (πœ‘ β†’ ( βŠ₯ β€˜{𝑋, π‘Œ}) = ((πΏβ€˜(π½β€˜π‘‹)) ∩ (πΏβ€˜(π½β€˜π‘Œ))))
 
Theoremlcfrlem25 40438* Lemma for lcfr 40456. Special case of lcfrlem35 40448 when ((π½β€˜π‘Œ)β€˜πΌ) is zero. (Contributed by NM, 11-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    + = (+gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   π΄ = (LSAtomsβ€˜π‘ˆ)    &   (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†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   π΅ = ((π‘β€˜{𝑋, π‘Œ}) ∩ ( βŠ₯ β€˜{(𝑋 + π‘Œ)}))    &    Β· = ( ·𝑠 β€˜π‘ˆ)    &   π‘† = (Scalarβ€˜π‘ˆ)    &   π‘„ = (0gβ€˜π‘†)    &   π‘… = (Baseβ€˜π‘†)    &   π½ = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{π‘₯})𝑣 = (𝑀 + (π‘˜ Β· π‘₯)))))    &   (πœ‘ β†’ 𝐼 ∈ 𝐡)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   (πœ‘ β†’ ((π½β€˜π‘Œ)β€˜πΌ) = 𝑄)    &   (πœ‘ β†’ 𝐼 β‰  0 )    β‡’   (πœ‘ β†’ ( βŠ₯ β€˜{(𝑋 + π‘Œ)}) = (πΏβ€˜(π½β€˜π‘Œ)))
 
Theoremlcfrlem26 40439* Lemma for lcfr 40456. Special case of lcfrlem36 40449 when ((π½β€˜π‘Œ)β€˜πΌ) is zero. (Contributed by NM, 11-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    + = (+gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   π΄ = (LSAtomsβ€˜π‘ˆ)    &   (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†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   π΅ = ((π‘β€˜{𝑋, π‘Œ}) ∩ ( βŠ₯ β€˜{(𝑋 + π‘Œ)}))    &    Β· = ( ·𝑠 β€˜π‘ˆ)    &   π‘† = (Scalarβ€˜π‘ˆ)    &   π‘„ = (0gβ€˜π‘†)    &   π‘… = (Baseβ€˜π‘†)    &   π½ = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{π‘₯})𝑣 = (𝑀 + (π‘˜ Β· π‘₯)))))    &   (πœ‘ β†’ 𝐼 ∈ 𝐡)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   (πœ‘ β†’ ((π½β€˜π‘Œ)β€˜πΌ) = 𝑄)    &   (πœ‘ β†’ 𝐼 β‰  0 )    β‡’   (πœ‘ β†’ (𝑋 + π‘Œ) ∈ ( βŠ₯ β€˜(πΏβ€˜(π½β€˜π‘Œ))))
 
Theoremlcfrlem27 40440* Lemma for lcfr 40456. Special case of lcfrlem37 40450 when ((π½β€˜π‘Œ)β€˜πΌ) is zero. (Contributed by NM, 11-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    + = (+gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   π΄ = (LSAtomsβ€˜π‘ˆ)    &   (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†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   π΅ = ((π‘β€˜{𝑋, π‘Œ}) ∩ ( βŠ₯ β€˜{(𝑋 + π‘Œ)}))    &    Β· = ( ·𝑠 β€˜π‘ˆ)    &   π‘† = (Scalarβ€˜π‘ˆ)    &   π‘„ = (0gβ€˜π‘†)    &   π‘… = (Baseβ€˜π‘†)    &   π½ = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{π‘₯})𝑣 = (𝑀 + (π‘˜ Β· π‘₯)))))    &   (πœ‘ β†’ 𝐼 ∈ 𝐡)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   (πœ‘ β†’ ((π½β€˜π‘Œ)β€˜πΌ) = 𝑄)    &   (πœ‘ β†’ 𝐼 β‰  0 )    &   (πœ‘ β†’ 𝐺 ∈ (LSubSpβ€˜π·))    &   (πœ‘ β†’ 𝐺 βŠ† {𝑓 ∈ (LFnlβ€˜π‘ˆ) ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)})    &   πΈ = βˆͺ 𝑔 ∈ 𝐺 ( βŠ₯ β€˜(πΏβ€˜π‘”))    &   (πœ‘ β†’ 𝑋 ∈ 𝐸)    &   (πœ‘ β†’ π‘Œ ∈ 𝐸)    β‡’   (πœ‘ β†’ (𝑋 + π‘Œ) ∈ 𝐸)
 
Theoremlcfrlem28 40441* Lemma for lcfr 40456. TODO: This can be a hypothesis since the zero version of (π½β€˜π‘Œ)β€˜πΌ needs it. (Contributed by NM, 9-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    + = (+gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   π΄ = (LSAtomsβ€˜π‘ˆ)    &   (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†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   π΅ = ((π‘β€˜{𝑋, π‘Œ}) ∩ ( βŠ₯ β€˜{(𝑋 + π‘Œ)}))    &    Β· = ( ·𝑠 β€˜π‘ˆ)    &   π‘† = (Scalarβ€˜π‘ˆ)    &   π‘„ = (0gβ€˜π‘†)    &   π‘… = (Baseβ€˜π‘†)    &   π½ = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{π‘₯})𝑣 = (𝑀 + (π‘˜ Β· π‘₯)))))    &   (πœ‘ β†’ 𝐼 ∈ 𝐡)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   (πœ‘ β†’ ((π½β€˜π‘Œ)β€˜πΌ) β‰  𝑄)    β‡’   (πœ‘ β†’ 𝐼 β‰  0 )
 
Theoremlcfrlem29 40442* Lemma for lcfr 40456. (Contributed by NM, 9-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    + = (+gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   π΄ = (LSAtomsβ€˜π‘ˆ)    &   (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†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   π΅ = ((π‘β€˜{𝑋, π‘Œ}) ∩ ( βŠ₯ β€˜{(𝑋 + π‘Œ)}))    &    Β· = ( ·𝑠 β€˜π‘ˆ)    &   π‘† = (Scalarβ€˜π‘ˆ)    &   π‘„ = (0gβ€˜π‘†)    &   π‘… = (Baseβ€˜π‘†)    &   π½ = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{π‘₯})𝑣 = (𝑀 + (π‘˜ Β· π‘₯)))))    &   (πœ‘ β†’ 𝐼 ∈ 𝐡)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   (πœ‘ β†’ ((π½β€˜π‘Œ)β€˜πΌ) β‰  𝑄)    &   πΉ = (invrβ€˜π‘†)    β‡’   (πœ‘ β†’ ((πΉβ€˜((π½β€˜π‘Œ)β€˜πΌ))(.rβ€˜π‘†)((π½β€˜π‘‹)β€˜πΌ)) ∈ 𝑅)
 
Theoremlcfrlem30 40443* Lemma for lcfr 40456. (Contributed by NM, 6-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    + = (+gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   π΄ = (LSAtomsβ€˜π‘ˆ)    &   (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†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   π΅ = ((π‘β€˜{𝑋, π‘Œ}) ∩ ( βŠ₯ β€˜{(𝑋 + π‘Œ)}))    &    Β· = ( ·𝑠 β€˜π‘ˆ)    &   π‘† = (Scalarβ€˜π‘ˆ)    &   π‘„ = (0gβ€˜π‘†)    &   π‘… = (Baseβ€˜π‘†)    &   π½ = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{π‘₯})𝑣 = (𝑀 + (π‘˜ Β· π‘₯)))))    &   (πœ‘ β†’ 𝐼 ∈ 𝐡)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   (πœ‘ β†’ ((π½β€˜π‘Œ)β€˜πΌ) β‰  𝑄)    &   πΉ = (invrβ€˜π‘†)    &    βˆ’ = (-gβ€˜π·)    &   πΆ = ((π½β€˜π‘‹) βˆ’ (((πΉβ€˜((π½β€˜π‘Œ)β€˜πΌ))(.rβ€˜π‘†)((π½β€˜π‘‹)β€˜πΌ))( ·𝑠 β€˜π·)(π½β€˜π‘Œ)))    β‡’   (πœ‘ β†’ 𝐢 ∈ (LFnlβ€˜π‘ˆ))
 
Theoremlcfrlem31 40444* Lemma for lcfr 40456. (Contributed by NM, 10-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    + = (+gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   π΄ = (LSAtomsβ€˜π‘ˆ)    &   (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†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   π΅ = ((π‘β€˜{𝑋, π‘Œ}) ∩ ( βŠ₯ β€˜{(𝑋 + π‘Œ)}))    &    Β· = ( ·𝑠 β€˜π‘ˆ)    &   π‘† = (Scalarβ€˜π‘ˆ)    &   π‘„ = (0gβ€˜π‘†)    &   π‘… = (Baseβ€˜π‘†)    &   π½ = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{π‘₯})𝑣 = (𝑀 + (π‘˜ Β· π‘₯)))))    &   (πœ‘ β†’ 𝐼 ∈ 𝐡)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   (πœ‘ β†’ ((π½β€˜π‘Œ)β€˜πΌ) β‰  𝑄)    &   πΉ = (invrβ€˜π‘†)    &    βˆ’ = (-gβ€˜π·)    &   πΆ = ((π½β€˜π‘‹) βˆ’ (((πΉβ€˜((π½β€˜π‘Œ)β€˜πΌ))(.rβ€˜π‘†)((π½β€˜π‘‹)β€˜πΌ))( ·𝑠 β€˜π·)(π½β€˜π‘Œ)))    &   (πœ‘ β†’ ((π½β€˜π‘‹)β€˜πΌ) β‰  𝑄)    &   (πœ‘ β†’ 𝐢 = (0gβ€˜π·))    β‡’   (πœ‘ β†’ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ}))
 
Theoremlcfrlem32 40445* Lemma for lcfr 40456. (Contributed by NM, 10-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    + = (+gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   π΄ = (LSAtomsβ€˜π‘ˆ)    &   (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†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   π΅ = ((π‘β€˜{𝑋, π‘Œ}) ∩ ( βŠ₯ β€˜{(𝑋 + π‘Œ)}))    &    Β· = ( ·𝑠 β€˜π‘ˆ)    &   π‘† = (Scalarβ€˜π‘ˆ)    &   π‘„ = (0gβ€˜π‘†)    &   π‘… = (Baseβ€˜π‘†)    &   π½ = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{π‘₯})𝑣 = (𝑀 + (π‘˜ Β· π‘₯)))))    &   (πœ‘ β†’ 𝐼 ∈ 𝐡)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   (πœ‘ β†’ ((π½β€˜π‘Œ)β€˜πΌ) β‰  𝑄)    &   πΉ = (invrβ€˜π‘†)    &    βˆ’ = (-gβ€˜π·)    &   πΆ = ((π½β€˜π‘‹) βˆ’ (((πΉβ€˜((π½β€˜π‘Œ)β€˜πΌ))(.rβ€˜π‘†)((π½β€˜π‘‹)β€˜πΌ))( ·𝑠 β€˜π·)(π½β€˜π‘Œ)))    &   (πœ‘ β†’ ((π½β€˜π‘‹)β€˜πΌ) β‰  𝑄)    β‡’   (πœ‘ β†’ 𝐢 β‰  (0gβ€˜π·))
 
Theoremlcfrlem33 40446* Lemma for lcfr 40456. (Contributed by NM, 10-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    + = (+gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   π΄ = (LSAtomsβ€˜π‘ˆ)    &   (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†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   π΅ = ((π‘β€˜{𝑋, π‘Œ}) ∩ ( βŠ₯ β€˜{(𝑋 + π‘Œ)}))    &    Β· = ( ·𝑠 β€˜π‘ˆ)    &   π‘† = (Scalarβ€˜π‘ˆ)    &   π‘„ = (0gβ€˜π‘†)    &   π‘… = (Baseβ€˜π‘†)    &   π½ = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{π‘₯})𝑣 = (𝑀 + (π‘˜ Β· π‘₯)))))    &   (πœ‘ β†’ 𝐼 ∈ 𝐡)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   (πœ‘ β†’ ((π½β€˜π‘Œ)β€˜πΌ) β‰  𝑄)    &   πΉ = (invrβ€˜π‘†)    &    βˆ’ = (-gβ€˜π·)    &   πΆ = ((π½β€˜π‘‹) βˆ’ (((πΉβ€˜((π½β€˜π‘Œ)β€˜πΌ))(.rβ€˜π‘†)((π½β€˜π‘‹)β€˜πΌ))( ·𝑠 β€˜π·)(π½β€˜π‘Œ)))    &   (πœ‘ β†’ ((π½β€˜π‘‹)β€˜πΌ) = 𝑄)    β‡’   (πœ‘ β†’ 𝐢 β‰  (0gβ€˜π·))
 
Theoremlcfrlem34 40447* Lemma for lcfr 40456. (Contributed by NM, 10-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    + = (+gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   π΄ = (LSAtomsβ€˜π‘ˆ)    &   (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†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   π΅ = ((π‘β€˜{𝑋, π‘Œ}) ∩ ( βŠ₯ β€˜{(𝑋 + π‘Œ)}))    &    Β· = ( ·𝑠 β€˜π‘ˆ)    &   π‘† = (Scalarβ€˜π‘ˆ)    &   π‘„ = (0gβ€˜π‘†)    &   π‘… = (Baseβ€˜π‘†)    &   π½ = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{π‘₯})𝑣 = (𝑀 + (π‘˜ Β· π‘₯)))))    &   (πœ‘ β†’ 𝐼 ∈ 𝐡)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   (πœ‘ β†’ ((π½β€˜π‘Œ)β€˜πΌ) β‰  𝑄)    &   πΉ = (invrβ€˜π‘†)    &    βˆ’ = (-gβ€˜π·)    &   πΆ = ((π½β€˜π‘‹) βˆ’ (((πΉβ€˜((π½β€˜π‘Œ)β€˜πΌ))(.rβ€˜π‘†)((π½β€˜π‘‹)β€˜πΌ))( ·𝑠 β€˜π·)(π½β€˜π‘Œ)))    β‡’   (πœ‘ β†’ 𝐢 β‰  (0gβ€˜π·))
 
Theoremlcfrlem35 40448* Lemma for lcfr 40456. (Contributed by NM, 2-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    + = (+gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   π΄ = (LSAtomsβ€˜π‘ˆ)    &   (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†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   π΅ = ((π‘β€˜{𝑋, π‘Œ}) ∩ ( βŠ₯ β€˜{(𝑋 + π‘Œ)}))    &    Β· = ( ·𝑠 β€˜π‘ˆ)    &   π‘† = (Scalarβ€˜π‘ˆ)    &   π‘„ = (0gβ€˜π‘†)    &   π‘… = (Baseβ€˜π‘†)    &   π½ = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{π‘₯})𝑣 = (𝑀 + (π‘˜ Β· π‘₯)))))    &   (πœ‘ β†’ 𝐼 ∈ 𝐡)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   (πœ‘ β†’ ((π½β€˜π‘Œ)β€˜πΌ) β‰  𝑄)    &   πΉ = (invrβ€˜π‘†)    &    βˆ’ = (-gβ€˜π·)    &   πΆ = ((π½β€˜π‘‹) βˆ’ (((πΉβ€˜((π½β€˜π‘Œ)β€˜πΌ))(.rβ€˜π‘†)((π½β€˜π‘‹)β€˜πΌ))( ·𝑠 β€˜π·)(π½β€˜π‘Œ)))    β‡’   (πœ‘ β†’ ( βŠ₯ β€˜{(𝑋 + π‘Œ)}) = (πΏβ€˜πΆ))
 
Theoremlcfrlem36 40449* Lemma for lcfr 40456. (Contributed by NM, 6-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    + = (+gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   π΄ = (LSAtomsβ€˜π‘ˆ)    &   (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†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   π΅ = ((π‘β€˜{𝑋, π‘Œ}) ∩ ( βŠ₯ β€˜{(𝑋 + π‘Œ)}))    &    Β· = ( ·𝑠 β€˜π‘ˆ)    &   π‘† = (Scalarβ€˜π‘ˆ)    &   π‘„ = (0gβ€˜π‘†)    &   π‘… = (Baseβ€˜π‘†)    &   π½ = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{π‘₯})𝑣 = (𝑀 + (π‘˜ Β· π‘₯)))))    &   (πœ‘ β†’ 𝐼 ∈ 𝐡)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   (πœ‘ β†’ ((π½β€˜π‘Œ)β€˜πΌ) β‰  𝑄)    &   πΉ = (invrβ€˜π‘†)    &    βˆ’ = (-gβ€˜π·)    &   πΆ = ((π½β€˜π‘‹) βˆ’ (((πΉβ€˜((π½β€˜π‘Œ)β€˜πΌ))(.rβ€˜π‘†)((π½β€˜π‘‹)β€˜πΌ))( ·𝑠 β€˜π·)(π½β€˜π‘Œ)))    β‡’   (πœ‘ β†’ (𝑋 + π‘Œ) ∈ ( βŠ₯ β€˜(πΏβ€˜πΆ)))
 
Theoremlcfrlem37 40450* Lemma for lcfr 40456. (Contributed by NM, 8-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    + = (+gβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘ˆ)    &   π‘ = (LSpanβ€˜π‘ˆ)    &   π΄ = (LSAtomsβ€˜π‘ˆ)    &   (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†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   π΅ = ((π‘β€˜{𝑋, π‘Œ}) ∩ ( βŠ₯ β€˜{(𝑋 + π‘Œ)}))    &    Β· = ( ·𝑠 β€˜π‘ˆ)    &   π‘† = (Scalarβ€˜π‘ˆ)    &   π‘„ = (0gβ€˜π‘†)    &   π‘… = (Baseβ€˜π‘†)    &   π½ = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{π‘₯})𝑣 = (𝑀 + (π‘˜ Β· π‘₯)))))    &   (πœ‘ β†’ 𝐼 ∈ 𝐡)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   (πœ‘ β†’ ((π½β€˜π‘Œ)β€˜πΌ) β‰  𝑄)    &   πΉ = (invrβ€˜π‘†)    &    βˆ’ = (-gβ€˜π·)    &   πΆ = ((π½β€˜π‘‹) βˆ’ (((πΉβ€˜((π½β€˜π‘Œ)β€˜πΌ))(.rβ€˜π‘†)((π½β€˜π‘‹)β€˜πΌ))( ·𝑠 β€˜π·)(π½β€˜π‘Œ)))    &   (πœ‘ β†’ 𝐺 ∈ (LSubSpβ€˜π·))    &   (πœ‘ β†’ 𝐺 βŠ† {𝑓 ∈ (LFnlβ€˜π‘ˆ) ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)})    &   πΈ = βˆͺ 𝑔 ∈ 𝐺 ( βŠ₯ β€˜(πΏβ€˜π‘”))    &   (πœ‘ β†’ 𝑋 ∈ 𝐸)    &   (πœ‘ β†’ π‘Œ ∈ 𝐸)    β‡’   (πœ‘ β†’ (𝑋 + π‘Œ) ∈ 𝐸)
 
Theoremlcfrlem38 40451* Lemma for lcfr 40456. Combine lcfrlem27 40440 and lcfrlem37 40450. (Contributed by NM, 11-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &    + = (+gβ€˜π‘ˆ)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   π‘„ = (LSubSpβ€˜π·)    &   πΆ = {𝑓 ∈ (LFnlβ€˜π‘ˆ) ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)}    &   πΈ = βˆͺ 𝑔 ∈ 𝐺 ( βŠ₯ β€˜(πΏβ€˜π‘”))    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐺 ∈ 𝑄)    &   (πœ‘ β†’ 𝐺 βŠ† 𝐢)    &   (πœ‘ β†’ 𝑋 ∈ 𝐸)    &   (πœ‘ β†’ π‘Œ ∈ 𝐸)    &    0 = (0gβ€˜π‘ˆ)    &   (πœ‘ β†’ 𝑋 β‰  0 )    &   (πœ‘ β†’ π‘Œ β‰  0 )    &   π‘ = (LSpanβ€˜π‘ˆ)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   π΅ = ((π‘β€˜{𝑋, π‘Œ}) ∩ ( βŠ₯ β€˜{(𝑋 + π‘Œ)}))    &   (πœ‘ β†’ 𝐼 ∈ 𝐡)    &   (πœ‘ β†’ 𝐼 β‰  0 )    &   π‘‰ = (Baseβ€˜π‘ˆ)    &    Β· = ( ·𝑠 β€˜π‘ˆ)    &   π‘† = (Scalarβ€˜π‘ˆ)    &   π‘… = (Baseβ€˜π‘†)    &   π½ = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (β„©π‘˜ ∈ 𝑅 βˆƒπ‘€ ∈ ( βŠ₯ β€˜{π‘₯})𝑣 = (𝑀 + (π‘˜ Β· π‘₯)))))    β‡’   (πœ‘ β†’ (𝑋 + π‘Œ) ∈ 𝐸)
 
Theoremlcfrlem39 40452* Lemma for lcfr 40456. Eliminate 𝐽. (Contributed by NM, 11-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &    + = (+gβ€˜π‘ˆ)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   π‘„ = (LSubSpβ€˜π·)    &   πΆ = {𝑓 ∈ (LFnlβ€˜π‘ˆ) ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)}    &   πΈ = βˆͺ 𝑔 ∈ 𝐺 ( βŠ₯ β€˜(πΏβ€˜π‘”))    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐺 ∈ 𝑄)    &   (πœ‘ β†’ 𝐺 βŠ† 𝐢)    &   (πœ‘ β†’ 𝑋 ∈ 𝐸)    &   (πœ‘ β†’ π‘Œ ∈ 𝐸)    &    0 = (0gβ€˜π‘ˆ)    &   (πœ‘ β†’ 𝑋 β‰  0 )    &   (πœ‘ β†’ π‘Œ β‰  0 )    &   π‘ = (LSpanβ€˜π‘ˆ)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    &   π΅ = ((π‘β€˜{𝑋, π‘Œ}) ∩ ( βŠ₯ β€˜{(𝑋 + π‘Œ)}))    &   (πœ‘ β†’ 𝐼 ∈ 𝐡)    &   (πœ‘ β†’ 𝐼 β‰  0 )    β‡’   (πœ‘ β†’ (𝑋 + π‘Œ) ∈ 𝐸)
 
Theoremlcfrlem40 40453* Lemma for lcfr 40456. Eliminate 𝐡 and 𝐼. (Contributed by NM, 11-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &    + = (+gβ€˜π‘ˆ)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   π‘„ = (LSubSpβ€˜π·)    &   πΆ = {𝑓 ∈ (LFnlβ€˜π‘ˆ) ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)}    &   πΈ = βˆͺ 𝑔 ∈ 𝐺 ( βŠ₯ β€˜(πΏβ€˜π‘”))    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐺 ∈ 𝑄)    &   (πœ‘ β†’ 𝐺 βŠ† 𝐢)    &   (πœ‘ β†’ 𝑋 ∈ 𝐸)    &   (πœ‘ β†’ π‘Œ ∈ 𝐸)    &    0 = (0gβ€˜π‘ˆ)    &   (πœ‘ β†’ 𝑋 β‰  0 )    &   (πœ‘ β†’ π‘Œ β‰  0 )    &   π‘ = (LSpanβ€˜π‘ˆ)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    β‡’   (πœ‘ β†’ (𝑋 + π‘Œ) ∈ 𝐸)
 
Theoremlcfrlem41 40454* Lemma for lcfr 40456. Eliminate span condition. (Contributed by NM, 11-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &    + = (+gβ€˜π‘ˆ)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   π‘„ = (LSubSpβ€˜π·)    &   πΆ = {𝑓 ∈ (LFnlβ€˜π‘ˆ) ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)}    &   πΈ = βˆͺ 𝑔 ∈ 𝐺 ( βŠ₯ β€˜(πΏβ€˜π‘”))    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐺 ∈ 𝑄)    &   (πœ‘ β†’ 𝐺 βŠ† 𝐢)    &   (πœ‘ β†’ 𝑋 ∈ 𝐸)    &   (πœ‘ β†’ π‘Œ ∈ 𝐸)    &    0 = (0gβ€˜π‘ˆ)    &   (πœ‘ β†’ 𝑋 β‰  0 )    &   (πœ‘ β†’ π‘Œ β‰  0 )    β‡’   (πœ‘ β†’ (𝑋 + π‘Œ) ∈ 𝐸)
 
Theoremlcfrlem42 40455* Lemma for lcfr 40456. Eliminate nonzero condition. (Contributed by NM, 11-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &    + = (+gβ€˜π‘ˆ)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   π‘„ = (LSubSpβ€˜π·)    &   πΆ = {𝑓 ∈ (LFnlβ€˜π‘ˆ) ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)}    &   πΈ = βˆͺ 𝑔 ∈ 𝐺 ( βŠ₯ β€˜(πΏβ€˜π‘”))    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐺 ∈ 𝑄)    &   (πœ‘ β†’ 𝐺 βŠ† 𝐢)    &   (πœ‘ β†’ 𝑋 ∈ 𝐸)    &   (πœ‘ β†’ π‘Œ ∈ 𝐸)    β‡’   (πœ‘ β†’ (𝑋 + π‘Œ) ∈ 𝐸)
 
Theoremlcfr 40456* 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 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘ˆ)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   π‘‡ = (LSubSpβ€˜π·)    &   πΆ = {𝑓 ∈ 𝐹 ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)}    &   π‘„ = βˆͺ 𝑔 ∈ 𝑅 ( βŠ₯ β€˜(πΏβ€˜π‘”))    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑅 ∈ 𝑇)    &   (πœ‘ β†’ 𝑅 βŠ† 𝐢)    β‡’   (πœ‘ β†’ 𝑄 ∈ 𝑆)
 
Syntaxclcd 40457 Extend class notation with vector space of functionals with closed kernels.
class LCDual
 
Definitiondf-lcdual 40458* Dual vector space of functionals with closed kernels. Note: we could also define this directly without mapd by using mapdrn 40520. TODO: see if it makes sense to go back and replace some of the LDual stuff with this. TODO: We could simplify df-mapd 40496 using (Baseβ€˜((LCDualβ€˜πΎ)β€˜π‘Š)). (Contributed by NM, 13-Mar-2015.)
LCDual = (π‘˜ ∈ V ↦ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ ((LDualβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)) β†Ύs {𝑓 ∈ (LFnlβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)) ∣ (((ocHβ€˜π‘˜)β€˜π‘€)β€˜(((ocHβ€˜π‘˜)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))β€˜π‘“))) = ((LKerβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))β€˜π‘“)})))
 
Theoremlcdfval 40459* Dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    β‡’   (𝐾 ∈ 𝑋 β†’ (LCDualβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ ((LDualβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) β†Ύs {𝑓 ∈ (LFnlβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ∣ (((ocHβ€˜πΎ)β€˜π‘€)β€˜(((ocHβ€˜πΎ)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“))) = ((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“)})))
 
Theoremlcdval 40460* Dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   (πœ‘ β†’ (𝐾 ∈ 𝑋 ∧ π‘Š ∈ 𝐻))    β‡’   (πœ‘ β†’ 𝐢 = (𝐷 β†Ύs {𝑓 ∈ 𝐹 ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)}))
 
Theoremlcdval2 40461* Dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   (πœ‘ β†’ (𝐾 ∈ 𝑋 ∧ π‘Š ∈ 𝐻))    &   π΅ = {𝑓 ∈ 𝐹 ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)}    β‡’   (πœ‘ β†’ 𝐢 = (𝐷 β†Ύs 𝐡))
 
Theoremlcdlvec 40462 The dual vector space of functionals with closed kernels is a left vector space. (Contributed by NM, 14-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    β‡’   (πœ‘ β†’ 𝐢 ∈ LVec)
 
Theoremlcdlmod 40463 The dual vector space of functionals with closed kernels is a left module. (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    β‡’   (πœ‘ β†’ 𝐢 ∈ LMod)
 
Theoremlcdvbase 40464* Vector base set of a dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &    βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜πΆ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π΅ = {𝑓 ∈ 𝐹 ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)}    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    β‡’   (πœ‘ β†’ 𝑉 = 𝐡)
 
Theoremlcdvbasess 40465 The vector base set of the closed kernel dual space is a set of functionals. (Contributed by NM, 15-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜πΆ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    β‡’   (πœ‘ β†’ 𝑉 βŠ† 𝐹)
 
Theoremlcdvbaselfl 40466 A vector in the base set of the closed kernel dual space is a functional. (Contributed by NM, 28-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜πΆ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    β‡’   (πœ‘ β†’ 𝑋 ∈ 𝐹)
 
Theoremlcdvbasecl 40467 Closure of the value of a vector (functional) in the closed kernel dual space. (Contributed by NM, 28-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &   π‘† = (Scalarβ€˜π‘ˆ)    &   π‘… = (Baseβ€˜π‘†)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   πΈ = (Baseβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐹 ∈ 𝐸)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    β‡’   (πœ‘ β†’ (πΉβ€˜π‘‹) ∈ 𝑅)
 
Theoremlcdvadd 40468 Vector addition for the closed kernel vector space dual. (Contributed by NM, 10-Jun-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π· = (LDualβ€˜π‘ˆ)    &    + = (+gβ€˜π·)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &    ✚ = (+gβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    β‡’   (πœ‘ β†’ ✚ = + )
 
Theoremlcdvaddval 40469 The value of the value of vector addition in the closed kernel vector space dual. (Contributed by NM, 10-Jun-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &   π‘… = (Scalarβ€˜π‘ˆ)    &    + = (+gβ€˜π‘…)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π· = (Baseβ€˜πΆ)    &    ✚ = (+gβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐹 ∈ 𝐷)    &   (πœ‘ β†’ 𝐺 ∈ 𝐷)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    β‡’   (πœ‘ β†’ ((𝐹 ✚ 𝐺)β€˜π‘‹) = ((πΉβ€˜π‘‹) + (πΊβ€˜π‘‹)))
 
Theoremlcdsca 40470 The ring of scalars of the closed kernel dual space. (Contributed by NM, 16-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘ˆ)    &   π‘‚ = (opprβ€˜πΉ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π‘… = (Scalarβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    β‡’   (πœ‘ β†’ 𝑅 = 𝑂)
 
Theoremlcdsbase 40471 Base set of scalar ring for the closed kernel dual of a vector space. (Contributed by NM, 18-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘ˆ)    &   πΏ = (Baseβ€˜πΉ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π‘† = (Scalarβ€˜πΆ)    &   π‘… = (Baseβ€˜π‘†)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    β‡’   (πœ‘ β†’ 𝑅 = 𝐿)
 
Theoremlcdsadd 40472 Scalar addition for the closed kernel vector space dual. (Contributed by NM, 6-Jun-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘ˆ)    &    + = (+gβ€˜πΉ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π‘† = (Scalarβ€˜πΆ)    &    ✚ = (+gβ€˜π‘†)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    β‡’   (πœ‘ β†’ ✚ = + )
 
Theoremlcdsmul 40473 Scalar multiplication for the closed kernel vector space dual. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘ˆ)    &   πΏ = (Baseβ€˜πΉ)    &    Β· = (.rβ€˜πΉ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π‘† = (Scalarβ€˜πΆ)    &    βˆ™ = (.rβ€˜π‘†)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝐿)    &   (πœ‘ β†’ π‘Œ ∈ 𝐿)    β‡’   (πœ‘ β†’ (𝑋 βˆ™ π‘Œ) = (π‘Œ Β· 𝑋))
 
Theoremlcdvs 40474 Scalar product for the closed kernel vector space dual. (Contributed by NM, 28-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π· = (LDualβ€˜π‘ˆ)    &    Β· = ( ·𝑠 β€˜π·)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &    βˆ™ = ( ·𝑠 β€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    β‡’   (πœ‘ β†’ βˆ™ = Β· )
 
Theoremlcdvsval 40475 Value of scalar product operation value for the closed kernel vector space dual. (Contributed by NM, 28-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &   π‘† = (Scalarβ€˜π‘ˆ)    &   π‘… = (Baseβ€˜π‘†)    &    Β· = (.rβ€˜π‘†)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   πΉ = (Baseβ€˜πΆ)    &    βˆ™ = ( ·𝑠 β€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑅)    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    β‡’   (πœ‘ β†’ ((𝑋 βˆ™ 𝐺)β€˜π΄) = ((πΊβ€˜π΄) Β· 𝑋))
 
Theoremlcdvscl 40476 The scalar product operation value is a functional. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘† = (Scalarβ€˜π‘ˆ)    &   π‘… = (Baseβ€˜π‘†)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜πΆ)    &    Β· = ( ·𝑠 β€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑅)    &   (πœ‘ β†’ 𝐺 ∈ 𝑉)    β‡’   (πœ‘ β†’ (𝑋 Β· 𝐺) ∈ 𝑉)
 
Theoremlcdlssvscl 40477 Closure of scalar product in a closed kernel dual vector space. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘ˆ)    &   π‘… = (Baseβ€˜πΉ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜πΆ)    &    Β· = ( ·𝑠 β€˜πΆ)    &   π‘† = (LSubSpβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐿 ∈ 𝑆)    &   (πœ‘ β†’ 𝑋 ∈ 𝑅)    &   (πœ‘ β†’ π‘Œ ∈ 𝐿)    β‡’   (πœ‘ β†’ (𝑋 Β· π‘Œ) ∈ 𝐿)
 
Theoremlcdvsass 40478 Associative law for scalar product in a closed kernel dual vector space. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘… = (Scalarβ€˜π‘ˆ)    &   πΏ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   πΉ = (Baseβ€˜πΆ)    &    βˆ™ = ( ·𝑠 β€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝐿)    &   (πœ‘ β†’ π‘Œ ∈ 𝐿)    &   (πœ‘ β†’ 𝐺 ∈ 𝐹)    β‡’   (πœ‘ β†’ ((π‘Œ Β· 𝑋) βˆ™ 𝐺) = (𝑋 βˆ™ (π‘Œ βˆ™ 𝐺)))
 
Theoremlcd0 40479 The zero scalar of the closed kernel dual of a vector space. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘ˆ)    &    0 = (0gβ€˜πΉ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π‘† = (Scalarβ€˜πΆ)    &   π‘‚ = (0gβ€˜π‘†)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    β‡’   (πœ‘ β†’ 𝑂 = 0 )
 
Theoremlcd1 40480 The unit scalar of the closed kernel dual of a vector space. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘ˆ)    &    1 = (1rβ€˜πΉ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π‘† = (Scalarβ€˜πΆ)    &   πΌ = (1rβ€˜π‘†)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    β‡’   (πœ‘ β†’ 𝐼 = 1 )
 
Theoremlcdneg 40481 The unit scalar of the closed kernel dual of a vector space. (Contributed by NM, 11-Jun-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘… = (Scalarβ€˜π‘ˆ)    &   π‘€ = (invgβ€˜π‘…)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π‘† = (Scalarβ€˜πΆ)    &   π‘ = (invgβ€˜π‘†)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    β‡’   (πœ‘ β†’ 𝑁 = 𝑀)
 
Theoremlcd0v 40482 The zero functional in the set of functionals with closed kernels. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &   π‘… = (Scalarβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘…)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π‘‚ = (0gβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    β‡’   (πœ‘ β†’ 𝑂 = (𝑉 Γ— { 0 }))
 
Theoremlcd0v2 40483 The zero functional in the set of functionals with closed kernels. (Contributed by NM, 27-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π· = (LDualβ€˜π‘ˆ)    &    0 = (0gβ€˜π·)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π‘‚ = (0gβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    β‡’   (πœ‘ β†’ 𝑂 = 0 )
 
Theoremlcd0vvalN 40484 Value of the zero functional at any vector. (Contributed by NM, 28-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &   π‘† = (Scalarβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘†)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π‘‚ = (0gβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    β‡’   (πœ‘ β†’ (π‘‚β€˜π‘‹) = 0 )
 
Theoremlcd0vcl 40485 Closure of the zero functional in the set of functionals with closed kernels. (Contributed by NM, 15-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜πΆ)    &   π‘‚ = (0gβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    β‡’   (πœ‘ β†’ 𝑂 ∈ 𝑉)
 
Theoremlcd0vs 40486 A scalar zero times a functional is the zero functional. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘… = (Scalarβ€˜π‘ˆ)    &    0 = (0gβ€˜π‘…)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜πΆ)    &    Β· = ( ·𝑠 β€˜πΆ)    &   π‘‚ = (0gβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐺 ∈ 𝑉)    β‡’   (πœ‘ β†’ ( 0 Β· 𝐺) = 𝑂)
 
Theoremlcdvs0N 40487 A scalar times the zero functional is the zero functional. (Contributed by NM, 20-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘† = (Scalarβ€˜π‘ˆ)    &   π‘… = (Baseβ€˜π‘†)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &    Β· = ( ·𝑠 β€˜πΆ)    &    0 = (0gβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑋 ∈ 𝑅)    β‡’   (πœ‘ β†’ (𝑋 Β· 0 ) = 0 )
 
Theoremlcdvsub 40488 The value of vector subtraction in the closed kernel dual space. (Contributed by NM, 22-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘† = (Scalarβ€˜π‘ˆ)    &   π‘ = (invgβ€˜π‘†)    &    1 = (1rβ€˜π‘†)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜πΆ)    &    + = (+gβ€˜πΆ)    &    Β· = ( ·𝑠 β€˜πΆ)    &    βˆ’ = (-gβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐹 ∈ 𝑉)    &   (πœ‘ β†’ 𝐺 ∈ 𝑉)    β‡’   (πœ‘ β†’ (𝐹 βˆ’ 𝐺) = (𝐹 + ((π‘β€˜ 1 ) Β· 𝐺)))
 
Theoremlcdvsubval 40489 The value of the value of vector addition in the closed kernel vector space dual. (Contributed by NM, 11-Jun-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘ˆ)    &   π‘… = (Scalarβ€˜π‘ˆ)    &   π‘† = (-gβ€˜π‘…)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π· = (Baseβ€˜πΆ)    &    βˆ’ = (-gβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐹 ∈ 𝐷)    &   (πœ‘ β†’ 𝐺 ∈ 𝐷)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    β‡’   (πœ‘ β†’ ((𝐹 βˆ’ 𝐺)β€˜π‘‹) = ((πΉβ€˜π‘‹)𝑆(πΊβ€˜π‘‹)))
 
Theoremlcdlss 40490* Subspaces of a dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‚ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π‘† = (LSubSpβ€˜πΆ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π· = (LDualβ€˜π‘ˆ)    &   π‘‡ = (LSubSpβ€˜π·)    &   π΅ = {𝑓 ∈ 𝐹 ∣ (π‘‚β€˜(π‘‚β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)}    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    β‡’   (πœ‘ β†’ 𝑆 = (𝑇 ∩ 𝒫 𝐡))
 
Theoremlcdlss2N 40491 Subspaces of a dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π‘† = (LSubSpβ€˜πΆ)    &   π‘‰ = (Baseβ€˜πΆ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π· = (LDualβ€˜π‘ˆ)    &   π‘‡ = (LSubSpβ€˜π·)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    β‡’   (πœ‘ β†’ 𝑆 = (𝑇 ∩ 𝒫 𝑉))
 
Theoremlcdlsp 40492 Span in the set of functionals with closed kernels. (Contributed by NM, 28-Mar-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π· = (LDualβ€˜π‘ˆ)    &   π‘€ = (LSpanβ€˜π·)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   πΉ = (Baseβ€˜πΆ)    &   π‘ = (LSpanβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐺 βŠ† 𝐹)    β‡’   (πœ‘ β†’ (π‘β€˜πΊ) = (π‘€β€˜πΊ))
 
TheoremlcdlkreqN 40493 Colinear functionals have equal kernels. (Contributed by NM, 28-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   πΏ = (LKerβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &    0 = (0gβ€˜πΆ)    &   π‘ = (LSpanβ€˜πΆ)    &   π‘‰ = (Baseβ€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝐺 ∈ (π‘β€˜{𝐼}))    &   (πœ‘ β†’ 𝐺 β‰  0 )    β‡’   (πœ‘ β†’ (πΏβ€˜πΊ) = (πΏβ€˜πΌ))
 
Theoremlcdlkreq2N 40494 Colinear functionals have equal kernels. (Contributed by NM, 28-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘† = (Scalarβ€˜π‘ˆ)    &   π‘… = (Baseβ€˜π‘†)    &    0 = (0gβ€˜π‘†)    &   πΏ = (LKerβ€˜π‘ˆ)    &   πΆ = ((LCDualβ€˜πΎ)β€˜π‘Š)    &   π‘‰ = (Baseβ€˜πΆ)    &    Β· = ( ·𝑠 β€˜πΆ)    &   (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝐴 ∈ (𝑅 βˆ– { 0 }))    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝐺 = (𝐴 Β· 𝐼))    β‡’   (πœ‘ β†’ (πΏβ€˜πΊ) = (πΏβ€˜πΌ))
 
Syntaxcmpd 40495 Extend class notation with projectivity from subspaces of vector space H to subspaces of functionals with closed kernels.
class mapd
 
Definitiondf-mapd 40496* Extend class notation with a one-to-one onto (mapd1o 40519), order-preserving (mapdord 40509) 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 = (π‘˜ ∈ V ↦ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ (𝑠 ∈ (LSubSpβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)) ↦ {𝑓 ∈ (LFnlβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)) ∣ ((((ocHβ€˜π‘˜)β€˜π‘€)β€˜(((ocHβ€˜π‘˜)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))β€˜π‘“))) = ((LKerβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))β€˜π‘“) ∧ (((ocHβ€˜π‘˜)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))β€˜π‘“)) βŠ† 𝑠)})))
 
Theoremmapdffval 40497* Projectivity from vector space H to dual space. (Contributed by NM, 25-Jan-2015.)
𝐻 = (LHypβ€˜πΎ)    β‡’   (𝐾 ∈ 𝑋 β†’ (mapdβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ (𝑠 ∈ (LSubSpβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ↦ {𝑓 ∈ (LFnlβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ∣ ((((ocHβ€˜πΎ)β€˜π‘€)β€˜(((ocHβ€˜πΎ)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“))) = ((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“) ∧ (((ocHβ€˜πΎ)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“)) βŠ† 𝑠)})))
 
Theoremmapdfval 40498* Projectivity from vector space H to dual space. (Contributed by NM, 25-Jan-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘ˆ)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π‘‚ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    β‡’   ((𝐾 ∈ 𝑋 ∧ π‘Š ∈ 𝐻) β†’ 𝑀 = (𝑠 ∈ 𝑆 ↦ {𝑓 ∈ 𝐹 ∣ ((π‘‚β€˜(π‘‚β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“) ∧ (π‘‚β€˜(πΏβ€˜π‘“)) βŠ† 𝑠)}))
 
Theoremmapdval 40499* Value of projectivity from vector space H to dual space. (Contributed by NM, 27-Jan-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘ˆ)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π‘‚ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   (πœ‘ β†’ (𝐾 ∈ 𝑋 ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑇 ∈ 𝑆)    β‡’   (πœ‘ β†’ (π‘€β€˜π‘‡) = {𝑓 ∈ 𝐹 ∣ ((π‘‚β€˜(π‘‚β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“) ∧ (π‘‚β€˜(πΏβ€˜π‘“)) βŠ† 𝑇)})
 
Theoremmapdvalc 40500* Value of projectivity from vector space H to dual space. (Contributed by NM, 27-Jan-2015.)
𝐻 = (LHypβ€˜πΎ)    &   π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘ˆ)    &   πΉ = (LFnlβ€˜π‘ˆ)    &   πΏ = (LKerβ€˜π‘ˆ)    &   π‘‚ = ((ocHβ€˜πΎ)β€˜π‘Š)    &   π‘€ = ((mapdβ€˜πΎ)β€˜π‘Š)    &   (πœ‘ β†’ (𝐾 ∈ 𝑋 ∧ π‘Š ∈ 𝐻))    &   (πœ‘ β†’ 𝑇 ∈ 𝑆)    &   πΆ = {𝑔 ∈ 𝐹 ∣ (π‘‚β€˜(π‘‚β€˜(πΏβ€˜π‘”))) = (πΏβ€˜π‘”)}    β‡’   (πœ‘ β†’ (π‘€β€˜π‘‡) = {𝑓 ∈ 𝐢 ∣ (π‘‚β€˜(πΏβ€˜π‘“)) βŠ† 𝑇})
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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47852
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