P. 401 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 40001-40100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	lclkrslem2 40001*	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 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝐷)    &   ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ (( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ ( ⊥ ‘(𝐿‘𝑓)) ⊆ 𝑄)}    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑄 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐶)    &   ⊢ + = (+g‘𝐷)    &   ⊢ (𝜑 → 𝐸 ∈ 𝐶)    ⇒   ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐶)
Theorem	lclkrs 40002*	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 39996 proof. Do we achieve overall shortening by breaking them out as subtheorems? Or make lclkr 39996 a special case of this? (Contributed by NM, 29-Jan-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 = {𝑓 ∈ 𝐹 ∣ (( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ ( ⊥ ‘(𝐿‘𝑓)) ⊆ 𝑅)}    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑅 ∈ 𝑆)    ⇒   ⊢ (𝜑 → 𝐶 ∈ 𝑇)
Theorem	lclkrs2 40003*	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 40091. (Contributed by NM, 12-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 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ 𝑅 = {𝑔 ∈ 𝐹 ∣ (( ⊥ ‘( ⊥ ‘(𝐿‘𝑔))) = (𝐿‘𝑔) ∧ ( ⊥ ‘(𝐿‘𝑔)) ⊆ 𝑄)}    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑄 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝑅 ∈ 𝑇 ∧ 𝑅 ⊆ 𝐶))
Theorem	lcfrvalsnN 40004*	Reconstruction from the dual space span of a singleton. (Contributed by NM, 19-Feb-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝐷)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ 𝑄 = ∪ 𝑓 ∈ 𝑅 ( ⊥ ‘(𝐿‘𝑓))    &   ⊢ 𝑅 = (𝑁‘{𝐺})    ⇒   ⊢ (𝜑 → 𝑄 = ( ⊥ ‘(𝐿‘𝐺)))
Theorem	lcfrlem1 40005	Lemma for lcfr 40048. Note that 𝑋 is z in Mario's notes. (Contributed by NM, 27-Feb-2015.)
⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ × = (.r‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 𝐼 = (invr‘𝑆)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝐷)    &   ⊢ − = (-g‘𝐷)    &   ⊢ (𝜑 → 𝑈 ∈ LVec)    &   ⊢ (𝜑 → 𝐸 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → (𝐺‘𝑋) ≠ 0 )    &   ⊢ 𝐻 = (𝐸 − (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺))    ⇒   ⊢ (𝜑 → (𝐻‘𝑋) = 0 )
Theorem	lcfrlem2 40006	Lemma for lcfr 40048. (Contributed by NM, 27-Feb-2015.)
⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ × = (.r‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 𝐼 = (invr‘𝑆)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝐷)    &   ⊢ − = (-g‘𝐷)    &   ⊢ (𝜑 → 𝑈 ∈ LVec)    &   ⊢ (𝜑 → 𝐸 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → (𝐺‘𝑋) ≠ 0 )    &   ⊢ 𝐻 = (𝐸 − (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺))    &   ⊢ 𝐿 = (LKer‘𝑈)    ⇒   ⊢ (𝜑 → ((𝐿‘𝐸) ∩ (𝐿‘𝐺)) ⊆ (𝐿‘𝐻))
Theorem	lcfrlem3 40007	Lemma for lcfr 40048. (Contributed by NM, 27-Feb-2015.)
⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ × = (.r‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 𝐼 = (invr‘𝑆)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝐷)    &   ⊢ − = (-g‘𝐷)    &   ⊢ (𝜑 → 𝑈 ∈ LVec)    &   ⊢ (𝜑 → 𝐸 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → (𝐺‘𝑋) ≠ 0 )    &   ⊢ 𝐻 = (𝐸 − (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺))    &   ⊢ 𝐿 = (LKer‘𝑈)    ⇒   ⊢ (𝜑 → 𝑋 ∈ (𝐿‘𝐻))
Theorem	lcfrlem4 40008*	Lemma for lcfr 40048. (Contributed by NM, 10-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ 𝑄 = (LSubSp‘𝐷)    &   ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝑄)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐸)    ⇒   ⊢ (𝜑 → 𝑋 ∈ 𝑉)
Theorem	lcfrlem5 40009*	Lemma for lcfr 40048. 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 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝐷)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑅 ∈ 𝑆)    &   ⊢ 𝑄 = ∪ 𝑓 ∈ 𝑅 ( ⊥ ‘(𝐿‘𝑓))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑄)    &   ⊢ 𝐶 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 · 𝑋) ∈ 𝑄)
Theorem	lcfrlem6 40010*	Lemma for lcfr 40048. 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 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ 𝑄 = (LSubSp‘𝐷)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝑄)    &   ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐸)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐸)    &   ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)
Theorem	lcfrlem7 40011*	Lemma for lcfr 40048. Closure of vector sum when one vector is zero. TODO: share hypotheses with others. (Contributed by NM, 11-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ 𝑄 = (LSubSp‘𝐷)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝑄)    &   ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐸)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ (𝜑 → 𝑌 = 0 )    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)
Theorem	lcfrlem8 40012*	Lemma for lcf1o 40014 and lcfr 40048. (Contributed by NM, 21-Feb-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ 𝑄 = (0g‘𝐷)    &   ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝐽‘𝑋) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))))
Theorem	lcfrlem9 40013*	Lemma for lcf1o 40014. (This part has undesirable $d's on 𝐽 and 𝜑 that we remove in lcf1o 40014.) TODO: ugly proof; maybe have better subtheorems or abbreviate some ℩𝑘 expansions with 𝐽‘𝑧? TODO: Some redundant $d's? (Contributed by NM, 22-Feb-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ 𝑄 = (0g‘𝐷)    &   ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝐽:(𝑉 ∖ { 0 })–1-1-onto→(𝐶 ∖ {𝑄}))
Theorem	lcf1o 40014*	Define a function 𝐽 that provides a bijection from nonzero vectors 𝑉 to nonzero functionals with closed kernels 𝐶. (Contributed by NM, 22-Feb-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ 𝑄 = (0g‘𝐷)    &   ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝐽:(𝑉 ∖ { 0 })–1-1-onto→(𝐶 ∖ {𝑄}))
Theorem	lcfrlem10 40015*	Lemma for lcfr 40048. (Contributed by NM, 23-Feb-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ 𝑄 = (0g‘𝐷)    &   ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝐽‘𝑋) ∈ 𝐹)
Theorem	lcfrlem11 40016*	Lemma for lcfr 40048. (Contributed by NM, 23-Feb-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ 𝑄 = (0g‘𝐷)    &   ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝐿‘(𝐽‘𝑋)) = ( ⊥ ‘{𝑋}))
Theorem	lcfrlem12N 40017*	Lemma for lcfr 40048. (Contributed by NM, 23-Feb-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ 𝑄 = (0g‘𝐷)    &   ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ 𝐵 = (0g‘𝑆)    &   ⊢ (𝜑 → 𝑌 ∈ ( ⊥ ‘{𝑋}))    ⇒   ⊢ (𝜑 → ((𝐽‘𝑋)‘𝑌) = 𝐵)
Theorem	lcfrlem13 40018*	Lemma for lcfr 40048. (Contributed by NM, 8-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ 𝑄 = (0g‘𝐷)    &   ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝐽‘𝑋) ∈ (𝐶 ∖ {𝑄}))
Theorem	lcfrlem14 40019*	Lemma for lcfr 40048. (Contributed by NM, 10-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ 𝑄 = (0g‘𝐷)    &   ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ 𝑁 = (LSpan‘𝑈)    ⇒   ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐽‘𝑋))) = (𝑁‘{𝑋}))
Theorem	lcfrlem15 40020*	Lemma for lcfr 40048. (Contributed by NM, 9-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ 𝑄 = (0g‘𝐷)    &   ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → 𝑋 ∈ ( ⊥ ‘(𝐿‘(𝐽‘𝑋))))
Theorem	lcfrlem16 40021*	Lemma for lcfr 40048. (Contributed by NM, 8-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ 𝑄 = (0g‘𝐷)    &   ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ 𝑃 = (LSubSp‘𝐷)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐺 ⊆ 𝐶)    &   ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐸 ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝐽‘𝑋) ∈ 𝐺)
Theorem	lcfrlem17 40022	Lemma for lcfr 40048. Condition needed more than once. (Contributed by NM, 11-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝑉 ∖ { 0 }))
Theorem	lcfrlem18 40023	Lemma for lcfr 40048. (Contributed by NM, 24-Feb-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    ⇒   ⊢ (𝜑 → ( ⊥ ‘{𝑋, 𝑌}) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})))
Theorem	lcfrlem19 40024	Lemma for lcfr 40048. (Contributed by NM, 11-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    ⇒   ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{(𝑋 + 𝑌)})))
Theorem	lcfrlem20 40025	Lemma for lcfr 40048. (Contributed by NM, 11-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   ⊢ (𝜑 → ¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}))    ⇒   ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ 𝐴)
Theorem	lcfrlem21 40026	Lemma for lcfr 40048. (Contributed by NM, 11-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    ⇒   ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ 𝐴)
Theorem	lcfrlem22 40027	Lemma for lcfr 40048. (Contributed by NM, 24-Feb-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))    ⇒   ⊢ (𝜑 → 𝐵 ∈ 𝐴)
Theorem	lcfrlem23 40028	Lemma for lcfr 40048. TODO: this proof was built from other proof pieces that may change 𝑁‘{𝑋, 𝑌} into subspace sum and back unnecessarily, or similar things. (Contributed by NM, 1-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))    &   ⊢ ⊕ = (LSSum‘𝑈)    ⇒   ⊢ (𝜑 → (( ⊥ ‘{𝑋, 𝑌}) ⊕ 𝐵) = ( ⊥ ‘{(𝑋 + 𝑌)}))
Theorem	lcfrlem24 40029*	Lemma for lcfr 40048. (Contributed by NM, 24-Feb-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑄 = (0g‘𝑆)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    &   ⊢ 𝐿 = (LKer‘𝑈)    ⇒   ⊢ (𝜑 → ( ⊥ ‘{𝑋, 𝑌}) = ((𝐿‘(𝐽‘𝑋)) ∩ (𝐿‘(𝐽‘𝑌))))
Theorem	lcfrlem25 40030*	Lemma for lcfr 40048. Special case of lcfrlem35 40040 when ((𝐽‘𝑌)‘𝐼) is zero. (Contributed by NM, 11-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑄 = (0g‘𝑆)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) = 𝑄)    &   ⊢ (𝜑 → 𝐼 ≠ 0 )    ⇒   ⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) = (𝐿‘(𝐽‘𝑌)))
Theorem	lcfrlem26 40031*	Lemma for lcfr 40048. Special case of lcfrlem36 40041 when ((𝐽‘𝑌)‘𝐼) is zero. (Contributed by NM, 11-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑄 = (0g‘𝑆)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) = 𝑄)    &   ⊢ (𝜑 → 𝐼 ≠ 0 )    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘(𝐽‘𝑌))))
Theorem	lcfrlem27 40032*	Lemma for lcfr 40048. Special case of lcfrlem37 40042 when ((𝐽‘𝑌)‘𝐼) is zero. (Contributed by NM, 11-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑄 = (0g‘𝑆)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) = 𝑄)    &   ⊢ (𝜑 → 𝐼 ≠ 0 )    &   ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝐷))    &   ⊢ (𝜑 → 𝐺 ⊆ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})    &   ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐸)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐸)    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)
Theorem	lcfrlem28 40033*	Lemma for lcfr 40048. TODO: This can be a hypothesis since the zero version of (𝐽‘𝑌)‘𝐼 needs it. (Contributed by NM, 9-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑄 = (0g‘𝑆)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ≠ 𝑄)    ⇒   ⊢ (𝜑 → 𝐼 ≠ 0 )
Theorem	lcfrlem29 40034*	Lemma for lcfr 40048. (Contributed by NM, 9-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑄 = (0g‘𝑆)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ≠ 𝑄)    &   ⊢ 𝐹 = (invr‘𝑆)    ⇒   ⊢ (𝜑 → ((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼)) ∈ 𝑅)
Theorem	lcfrlem30 40035*	Lemma for lcfr 40048. (Contributed by NM, 6-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑄 = (0g‘𝑆)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ≠ 𝑄)    &   ⊢ 𝐹 = (invr‘𝑆)    &   ⊢ − = (-g‘𝐷)    &   ⊢ 𝐶 = ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌)))    ⇒   ⊢ (𝜑 → 𝐶 ∈ (LFnl‘𝑈))
Theorem	lcfrlem31 40036*	Lemma for lcfr 40048. (Contributed by NM, 10-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑄 = (0g‘𝑆)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ≠ 𝑄)    &   ⊢ 𝐹 = (invr‘𝑆)    &   ⊢ − = (-g‘𝐷)    &   ⊢ 𝐶 = ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌)))    &   ⊢ (𝜑 → ((𝐽‘𝑋)‘𝐼) ≠ 𝑄)    &   ⊢ (𝜑 → 𝐶 = (0g‘𝐷))    ⇒   ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))
Theorem	lcfrlem32 40037*	Lemma for lcfr 40048. (Contributed by NM, 10-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑄 = (0g‘𝑆)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ≠ 𝑄)    &   ⊢ 𝐹 = (invr‘𝑆)    &   ⊢ − = (-g‘𝐷)    &   ⊢ 𝐶 = ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌)))    &   ⊢ (𝜑 → ((𝐽‘𝑋)‘𝐼) ≠ 𝑄)    ⇒   ⊢ (𝜑 → 𝐶 ≠ (0g‘𝐷))
Theorem	lcfrlem33 40038*	Lemma for lcfr 40048. (Contributed by NM, 10-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑄 = (0g‘𝑆)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ≠ 𝑄)    &   ⊢ 𝐹 = (invr‘𝑆)    &   ⊢ − = (-g‘𝐷)    &   ⊢ 𝐶 = ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌)))    &   ⊢ (𝜑 → ((𝐽‘𝑋)‘𝐼) = 𝑄)    ⇒   ⊢ (𝜑 → 𝐶 ≠ (0g‘𝐷))
Theorem	lcfrlem34 40039*	Lemma for lcfr 40048. (Contributed by NM, 10-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑄 = (0g‘𝑆)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ≠ 𝑄)    &   ⊢ 𝐹 = (invr‘𝑆)    &   ⊢ − = (-g‘𝐷)    &   ⊢ 𝐶 = ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌)))    ⇒   ⊢ (𝜑 → 𝐶 ≠ (0g‘𝐷))
Theorem	lcfrlem35 40040*	Lemma for lcfr 40048. (Contributed by NM, 2-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑄 = (0g‘𝑆)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ≠ 𝑄)    &   ⊢ 𝐹 = (invr‘𝑆)    &   ⊢ − = (-g‘𝐷)    &   ⊢ 𝐶 = ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌)))    ⇒   ⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) = (𝐿‘𝐶))
Theorem	lcfrlem36 40041*	Lemma for lcfr 40048. (Contributed by NM, 6-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑄 = (0g‘𝑆)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ≠ 𝑄)    &   ⊢ 𝐹 = (invr‘𝑆)    &   ⊢ − = (-g‘𝐷)    &   ⊢ 𝐶 = ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌)))    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝐶)))
Theorem	lcfrlem37 40042*	Lemma for lcfr 40048. (Contributed by NM, 8-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑄 = (0g‘𝑆)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ≠ 𝑄)    &   ⊢ 𝐹 = (invr‘𝑆)    &   ⊢ − = (-g‘𝐷)    &   ⊢ 𝐶 = ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(.r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·𝑠 ‘𝐷)(𝐽‘𝑌)))    &   ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝐷))    &   ⊢ (𝜑 → 𝐺 ⊆ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})    &   ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐸)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐸)    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)
Theorem	lcfrlem38 40043*	Lemma for lcfr 40048. Combine lcfrlem27 40032 and lcfrlem37 40042. (Contributed by NM, 11-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ 𝑄 = (LSubSp‘𝐷)    &   ⊢ 𝐶 = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝑄)    &   ⊢ (𝜑 → 𝐺 ⊆ 𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐸)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐸)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ (𝜑 → 𝑋 ≠ 0 )    &   ⊢ (𝜑 → 𝑌 ≠ 0 )    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐼 ≠ 0 )    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)
Theorem	lcfrlem39 40044*	Lemma for lcfr 40048. Eliminate 𝐽. (Contributed by NM, 11-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ 𝑄 = (LSubSp‘𝐷)    &   ⊢ 𝐶 = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝑄)    &   ⊢ (𝜑 → 𝐺 ⊆ 𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐸)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐸)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ (𝜑 → 𝑋 ≠ 0 )    &   ⊢ (𝜑 → 𝑌 ≠ 0 )    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐼 ≠ 0 )    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)
Theorem	lcfrlem40 40045*	Lemma for lcfr 40048. Eliminate 𝐵 and 𝐼. (Contributed by NM, 11-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ 𝑄 = (LSubSp‘𝐷)    &   ⊢ 𝐶 = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝑄)    &   ⊢ (𝜑 → 𝐺 ⊆ 𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐸)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐸)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ (𝜑 → 𝑋 ≠ 0 )    &   ⊢ (𝜑 → 𝑌 ≠ 0 )    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)
Theorem	lcfrlem41 40046*	Lemma for lcfr 40048. Eliminate span condition. (Contributed by NM, 11-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ 𝑄 = (LSubSp‘𝐷)    &   ⊢ 𝐶 = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝑄)    &   ⊢ (𝜑 → 𝐺 ⊆ 𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐸)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐸)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ (𝜑 → 𝑋 ≠ 0 )    &   ⊢ (𝜑 → 𝑌 ≠ 0 )    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)
Theorem	lcfrlem42 40047*	Lemma for lcfr 40048. Eliminate nonzero condition. (Contributed by NM, 11-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ 𝑄 = (LSubSp‘𝐷)    &   ⊢ 𝐶 = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝑄)    &   ⊢ (𝜑 → 𝐺 ⊆ 𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐸)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐸)    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)
Theorem	lcfr 40048*	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 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ 𝑄 = ∪ 𝑔 ∈ 𝑅 ( ⊥ ‘(𝐿‘𝑔))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑅 ∈ 𝑇)    &   ⊢ (𝜑 → 𝑅 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝑄 ∈ 𝑆)
Syntax	clcd 40049	Extend class notation with vector space of functionals with closed kernels.
class LCDual
Definition	df-lcdual 40050*	Dual vector space of functionals with closed kernels. Note: we could also define this directly without mapd by using mapdrn 40112. TODO: see if it makes sense to go back and replace some of the LDual stuff with this. TODO: We could simplify df-mapd 40088 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)})))
Theorem	lcdfval 40051*	Dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝑋 → (LCDual‘𝐾) = (𝑤 ∈ 𝐻 ↦ ((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)})))
Theorem	lcdval 40052*	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 → (𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝐶 = (𝐷 ↾s {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}))
Theorem	lcdval2 40053*	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 → (𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻))    &   ⊢ 𝐵 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    ⇒   ⊢ (𝜑 → 𝐶 = (𝐷 ↾s 𝐵))
Theorem	lcdlvec 40054	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 ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝐶 ∈ LVec)
Theorem	lcdlmod 40055	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 ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝐶 ∈ LMod)
Theorem	lcdvbase 40056*	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 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝑉 = 𝐵)
Theorem	lcdvbasess 40057	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 ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝑉 ⊆ 𝐹)
Theorem	lcdvbaselfl 40058	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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝑋 ∈ 𝐹)
Theorem	lcdvbasecl 40059	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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐸)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝑅)
Theorem	lcdvadd 40060	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)    &   ⊢ + = (+g‘𝐷)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → ✚ = + )
Theorem	lcdvaddval 40061	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)    &   ⊢ + = (+g‘𝑅)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝐹 ✚ 𝐺)‘𝑋) = ((𝐹‘𝑋) + (𝐺‘𝑋)))
Theorem	lcdsca 40062	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 = (oppr‘𝐹)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝐶)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝑅 = 𝑂)
Theorem	lcdsbase 40063	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 ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝑅 = 𝐿)
Theorem	lcdsadd 40064	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)    &   ⊢ + = (+g‘𝐹)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝐶)    &   ⊢ ✚ = (+g‘𝑆)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → ✚ = + )
Theorem	lcdsmul 40065	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)    &   ⊢ · = (.r‘𝐹)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝐶)    &   ⊢ ∙ = (.r‘𝑆)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐿)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐿)    ⇒   ⊢ (𝜑 → (𝑋 ∙ 𝑌) = (𝑌 · 𝑋))
Theorem	lcdvs 40066	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)    &   ⊢ · = ( ·𝑠 ‘𝐷)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ ∙ = ( ·𝑠 ‘𝐶)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → ∙ = · )
Theorem	lcdvsval 40067	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)    &   ⊢ · = (.r‘𝑆)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐹 = (Base‘𝐶)    &   ⊢ ∙ = ( ·𝑠 ‘𝐶)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑅)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝑋 ∙ 𝐺)‘𝐴) = ((𝐺‘𝐴) · 𝑋))
Theorem	lcdvscl 40068	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)    &   ⊢ · = ( ·𝑠 ‘𝐶)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑅)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑋 · 𝐺) ∈ 𝑉)
Theorem	lcdlssvscl 40069	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)    &   ⊢ · = ( ·𝑠 ‘𝐶)    &   ⊢ 𝑆 = (LSubSp‘𝐶)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐿 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑅)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐿)    ⇒   ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐿)
Theorem	lcdvsass 40070	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)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐹 = (Base‘𝐶)    &   ⊢ ∙ = ( ·𝑠 ‘𝐶)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐿)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐿)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → ((𝑌 · 𝑋) ∙ 𝐺) = (𝑋 ∙ (𝑌 ∙ 𝐺)))
Theorem	lcd0 40071	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 = (0g‘𝐹)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝐶)    &   ⊢ 𝑂 = (0g‘𝑆)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝑂 = 0 )
Theorem	lcd1 40072	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 = (1r‘𝐹)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝐶)    &   ⊢ 𝐼 = (1r‘𝑆)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝐼 = 1 )
Theorem	lcdneg 40073	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 = (invg‘𝑅)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝐶)    &   ⊢ 𝑁 = (invg‘𝑆)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝑁 = 𝑀)
Theorem	lcd0v 40074	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 = (0g‘𝑅)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝑂 = (0g‘𝐶)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝑂 = (𝑉 × { 0 }))
Theorem	lcd0v2 40075	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 = (0g‘𝐷)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝑂 = (0g‘𝐶)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝑂 = 0 )
Theorem	lcd0vvalN 40076	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 = (0g‘𝑆)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝑂 = (0g‘𝐶)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑂‘𝑋) = 0 )
Theorem	lcd0vcl 40077	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 = (0g‘𝐶)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝑂 ∈ 𝑉)
Theorem	lcd0vs 40078	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 = (0g‘𝑅)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝐶)    &   ⊢ · = ( ·𝑠 ‘𝐶)    &   ⊢ 𝑂 = (0g‘𝐶)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ( 0 · 𝐺) = 𝑂)
Theorem	lcdvs0N 40079	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)    &   ⊢ · = ( ·𝑠 ‘𝐶)    &   ⊢ 0 = (0g‘𝐶)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑅)    ⇒   ⊢ (𝜑 → (𝑋 · 0 ) = 0 )
Theorem	lcdvsub 40080	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 = (invg‘𝑆)    &   ⊢ 1 = (1r‘𝑆)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝐶)    &   ⊢ + = (+g‘𝐶)    &   ⊢ · = ( ·𝑠 ‘𝐶)    &   ⊢ − = (-g‘𝐶)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐹 − 𝐺) = (𝐹 + ((𝑁‘ 1 ) · 𝐺)))
Theorem	lcdvsubval 40081	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 = (-g‘𝑅)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ − = (-g‘𝐶)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝐹 − 𝐺)‘𝑋) = ((𝐹‘𝑋)𝑆(𝐺‘𝑋)))
Theorem	lcdlss 40082*	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 = {𝑓 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝑆 = (𝑇 ∩ 𝒫 𝐵))
Theorem	lcdlss2N 40083	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 ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝑆 = (𝑇 ∩ 𝒫 𝑉))
Theorem	lcdlsp 40084	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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ⊆ 𝐹)    ⇒   ⊢ (𝜑 → (𝑁‘𝐺) = (𝑀‘𝐺))
Theorem	lcdlkreqN 40085	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 = (0g‘𝐶)    &   ⊢ 𝑁 = (LSpan‘𝐶)    &   ⊢ 𝑉 = (Base‘𝐶)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐺 ∈ (𝑁‘{𝐼}))    &   ⊢ (𝜑 → 𝐺 ≠ 0 )    ⇒   ⊢ (𝜑 → (𝐿‘𝐺) = (𝐿‘𝐼))
Theorem	lcdlkreq2N 40086	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 = (0g‘𝑆)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝐶)    &   ⊢ · = ( ·𝑠 ‘𝐶)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐴 ∈ (𝑅 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐺 = (𝐴 · 𝐼))    ⇒   ⊢ (𝜑 → (𝐿‘𝐺) = (𝐿‘𝐼))
Syntax	cmpd 40087	Extend class notation with projectivity from subspaces of vector space H to subspaces of functionals with closed kernels.
class mapd
Definition	df-mapd 40088*	Extend class notation with a one-to-one onto (mapd1o 40111), order-preserving (mapdord 40101) 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)})))
Theorem	mapdffval 40089*	Projectivity from vector space H to dual space. (Contributed by NM, 25-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝑋 → (mapd‘𝐾) = (𝑤 ∈ 𝐻 ↦ (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)})))
Theorem	mapdfval 40090*	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)    ⇒   ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → 𝑀 = (𝑠 ∈ 𝑆 ↦ {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑠)}))
Theorem	mapdval 40091*	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 → (𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝑀‘𝑇) = {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)})
Theorem	mapdvalc 40092*	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 → (𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ 𝐶 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝐿‘𝑔)}    ⇒   ⊢ (𝜑 → (𝑀‘𝑇) = {𝑓 ∈ 𝐶 ∣ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇})
Theorem	mapdval2N 40093*	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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ 𝐶 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝐿‘𝑔)}    ⇒   ⊢ (𝜑 → (𝑀‘𝑇) = {𝑓 ∈ 𝐶 ∣ ∃𝑣 ∈ 𝑇 (𝑂‘(𝐿‘𝑓)) = (𝑁‘{𝑣})})
Theorem	mapdval3N 40094*	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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ 𝐶 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝐿‘𝑔)}    ⇒   ⊢ (𝜑 → (𝑀‘𝑇) = ∪ 𝑣 ∈ 𝑇 {𝑓 ∈ 𝐶 ∣ (𝑂‘(𝐿‘𝑓)) = (𝑁‘{𝑣})})
Theorem	mapdval4N 40095*	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 39966 has awkward ∃- add another thm with only one direction of it? 3. Swap 𝑂‘{𝑣} and 𝐿‘𝑓? (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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝑀‘𝑇) = {𝑓 ∈ 𝐹 ∣ ∃𝑣 ∈ 𝑇 (𝑂‘{𝑣}) = (𝐿‘𝑓)})
Theorem	mapdval5N 40096*	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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝑀‘𝑇) = ∪ 𝑣 ∈ 𝑇 {𝑓 ∈ 𝐹 ∣ (𝑂‘{𝑣}) = (𝐿‘𝑓)})
Theorem	mapdordlem1a 40097*	Lemma for mapdord 40101. (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 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) ∈ 𝑌}    &   ⊢ 𝐶 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝐿‘𝑔)}    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → (𝐽 ∈ 𝑇 ↔ (𝐽 ∈ 𝐶 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌)))
Theorem	mapdordlem1bN 40098*	Lemma for mapdord 40101. (Contributed by NM, 27-Jan-2015.) (New usage is discouraged.)
⊢ 𝐶 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝐿‘𝑔)}    ⇒   ⊢ (𝐽 ∈ 𝐶 ↔ (𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) = (𝐿‘𝐽)))
Theorem	mapdordlem1 40099*	Lemma for mapdord 40101. (Contributed by NM, 27-Jan-2015.)
⊢ 𝑇 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) ∈ 𝑌}    ⇒   ⊢ (𝐽 ∈ 𝑇 ↔ (𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌))
Theorem	mapdordlem2 40100*	Lemma for mapdord 40101. 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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑆)    &   ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐽 = (LSHyp‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝑇 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) ∈ 𝐽}    &   ⊢ 𝐶 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝐿‘𝑔)}    ⇒   ⊢ (𝜑 → ((𝑀‘𝑋) ⊆ (𝑀‘𝑌) ↔ 𝑋 ⊆ 𝑌))