P. 372 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37101-37200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	lcfl4N 37101*	Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑌 = (LSHyp‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 ∨ (𝐿‘𝐺) = 𝑉)))
Theorem	lcfl5 37102*	Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ (𝐿‘𝐺) ∈ ran 𝐼))
Theorem	lcfl5a 37103	Property of a functional with a closed kernel. TODO: Make lcfl5 37102 etc. obsolete and rewrite w/out 𝐶 hypothesis? (Contributed by NM, 29-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ↔ (𝐿‘𝐺) ∈ ran 𝐼))
Theorem	lcfl6lem 37104*	Lemma for lcfl6 37106. A functional 𝐺 (whose kernel is closed by dochsnkr 37078) is comletely determined by a vector 𝑋 in the orthocomplement in its kernel at which the functional value is 1. Note that the ∖ { 0 } in the 𝑋 hypothesis is redundant by the last hypothesis but allows easier use of other theorems. (Contributed by NM, 3-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 1 = (1r‘𝑆)    &   ⊢ 𝑅 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 }))    &   ⊢ (𝜑 → (𝐺‘𝑋) = 1 )    ⇒   ⊢ (𝜑 → 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))))
Theorem	lcfl7lem 37105*	Lemma for lcfl7N 37107. If two functionals 𝐺 and 𝐽 are equal, they are determined by the same vector. (Contributed by NM, 4-Jan-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)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))    &   ⊢ 𝐽 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑌})𝑣 = (𝑤 + (𝑘 · 𝑌))))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐺 = 𝐽)    ⇒   ⊢ (𝜑 → 𝑋 = 𝑌)
Theorem	lcfl6 37106*	Property of a functional with a closed kernel. Note that (𝐿‘𝐺) = 𝑉 means the functional is zero by lkr0f 34699. (Contributed by NM, 3-Jan-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)    &   ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ ((𝐿‘𝐺) = 𝑉 ∨ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))))
Theorem	lcfl7N 37107*	Property of a functional with a closed kernel. Every nonzero functional is determined by a unique nonzero vector. Note that (𝐿‘𝐺) = 𝑉 means the functional is zero by lkr0f 34699. (Contributed by NM, 4-Jan-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)    &   ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ ((𝐿‘𝐺) = 𝑉 ∨ ∃!𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))))
Theorem	lcfl8 37108*	Property of a functional with a closed kernel. (Contributed by NM, 17-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝑉 (𝐿‘𝐺) = ( ⊥ ‘{𝑥})))
Theorem	lcfl8a 37109*	Property of a functional with a closed kernel. (Contributed by NM, 17-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ↔ ∃𝑥 ∈ 𝑉 (𝐿‘𝐺) = ( ⊥ ‘{𝑥})))
Theorem	lcfl8b 37110*	Property of a nonzero functional with a closed kernel. (Contributed by NM, 4-Feb-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ 𝑌 = (0g‘𝐷)    &   ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐶 ∖ {𝑌}))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ (𝑉 ∖ { 0 })( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥}))
Theorem	lcfl9a 37111	Property implying that a functional has a closed kernel. (Contributed by NM, 16-Feb-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ⊆ (𝐿‘𝐺))    ⇒   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))
Theorem	lclkrlem1 37112*	The set of functionals having closed kernels is closed under scalar product. (Contributed by NM, 28-Dec-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝐷)    &   ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐶)    ⇒   ⊢ (𝜑 → (𝑋 · 𝐺) ∈ 𝐶)
Theorem	lclkrlem2a 37113	Lemma for lclkr 37139. Use lshpat 34661 to show that the intersection of a hyperplane with a noncomparable sum of atoms is an atom. (Contributed by NM, 16-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ≠ ( ⊥ ‘{𝑌}))    &   ⊢ (𝜑 → ¬ 𝑋 ∈ ( ⊥ ‘{𝐵}))    ⇒   ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵})) ∈ 𝐴)
Theorem	lclkrlem2b 37114	Lemma for lclkr 37139. (Contributed by NM, 17-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ≠ ( ⊥ ‘{𝑌}))    &   ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))    ⇒   ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘{𝐵})) ∈ 𝐴)
Theorem	lclkrlem2c 37115	Lemma for lclkr 37139. (Contributed by NM, 16-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ≠ ( ⊥ ‘{𝑌}))    &   ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))    &   ⊢ 𝐽 = (LSHyp‘𝑈)    ⇒   ⊢ (𝜑 → ((( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})) ⊕ (𝑁‘{𝐵})) ∈ 𝐽)
Theorem	lclkrlem2d 37116	Lemma for lclkr 37139. (Contributed by NM, 16-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ≠ ( ⊥ ‘{𝑌}))    &   ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ (𝜑 → ((( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})) ⊕ (𝑁‘{𝐵})) ∈ ran 𝐼)
Theorem	lclkrlem2e 37117	Lemma for lclkr 37139. The kernel of the sum is closed when the kernels of the summands are equal and closed. (Contributed by NM, 17-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ + = (+g‘𝐷)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐸 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))    &   ⊢ (𝜑 → (𝐿‘𝐸) = (𝐿‘𝐺))    ⇒   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
Theorem	lclkrlem2f 37118	Lemma for lclkr 37139. Construct a closed hyperplane under the kernel of the sum. (Contributed by NM, 16-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑄 = (0g‘𝑆)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐽 = (LSHyp‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ + = (+g‘𝐷)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐸 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))    &   ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))    &   ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄)    &   ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝐿‘𝐸) ≠ (𝐿‘𝐺))    &   ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽)    ⇒   ⊢ (𝜑 → (((𝐿‘𝐸) ∩ (𝐿‘𝐺)) ⊕ (𝑁‘{𝐵})) ⊆ (𝐿‘(𝐸 + 𝐺)))
Theorem	lclkrlem2g 37119	Lemma for lclkr 37139. Comparable hyperplanes are equal, so the kernel of the sum is closed. (Contributed by NM, 16-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑄 = (0g‘𝑆)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐽 = (LSHyp‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ + = (+g‘𝐷)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐸 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))    &   ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))    &   ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄)    &   ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝐿‘𝐸) ≠ (𝐿‘𝐺))    &   ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽)    ⇒   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
Theorem	lclkrlem2h 37120	Lemma for lclkr 37139. Eliminate the (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽 hypothesis. (Contributed by NM, 16-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑄 = (0g‘𝑆)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐽 = (LSHyp‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ + = (+g‘𝐷)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐸 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))    &   ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))    &   ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄)    &   ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → (𝐿‘𝐸) ≠ (𝐿‘𝐺))    ⇒   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
Theorem	lclkrlem2i 37121	Lemma for lclkr 37139. Eliminate the (𝐿‘𝐸) ≠ (𝐿‘𝐺) hypothesis. (Contributed by NM, 17-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑄 = (0g‘𝑆)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐽 = (LSHyp‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ + = (+g‘𝐷)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐸 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))    &   ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))    &   ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄)    &   ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
Theorem	lclkrlem2j 37122	Lemma for lclkr 37139. Kernel closure when 𝑌 is zero. (Contributed by NM, 18-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑄 = (0g‘𝑆)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐽 = (LSHyp‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ + = (+g‘𝐷)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐸 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))    &   ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))    &   ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄)    &   ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 = 0 )    ⇒   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
Theorem	lclkrlem2k 37123	Lemma for lclkr 37139. Kernel closure when 𝑋 is zero. (Contributed by NM, 18-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑄 = (0g‘𝑆)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐽 = (LSHyp‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ + = (+g‘𝐷)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐸 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))    &   ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))    &   ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄)    &   ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))    &   ⊢ (𝜑 → 𝑋 = 0 )    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
Theorem	lclkrlem2l 37124	Lemma for lclkr 37139. Eliminate the 𝑋 ≠ 0, 𝑌 ≠ 0 hypotheses. (Contributed by NM, 18-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ 𝑄 = (0g‘𝑆)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐽 = (LSHyp‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ + = (+g‘𝐷)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐸 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))    &   ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))    &   ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄)    &   ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
Theorem	lclkrlem2m 37125	Lemma for lclkr 37139. Construct a vector 𝐵 that makes the sum of functionals zero. Combine with 𝐵 ∈ 𝑉 to shorten overall proof. (Contributed by NM, 17-Jan-2015.)
⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ × = (.r‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 𝐼 = (invr‘𝑆)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ + = (+g‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → 𝑈 ∈ LVec)    &   ⊢ 𝐵 = (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌))    &   ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑌) ≠ 0 )    ⇒   ⊢ (𝜑 → (𝐵 ∈ 𝑉 ∧ ((𝐸 + 𝐺)‘𝐵) = 0 ))
Theorem	lclkrlem2n 37126	Lemma for lclkr 37139. (Contributed by NM, 12-Jan-2015.)
⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ × = (.r‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 𝐼 = (invr‘𝑆)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ + = (+g‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ LVec)    &   ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑋) = 0 )    &   ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑌) = 0 )    ⇒   ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ (𝐿‘(𝐸 + 𝐺)))
Theorem	lclkrlem2o 37127	Lemma for lclkr 37139. When 𝐵 is nonzero, the vectors 𝑋 and 𝑌 can't both belong to the hyperplane generated by 𝐵. (Contributed by NM, 17-Jan-2015.)
⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ × = (.r‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 𝐼 = (invr‘𝑆)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ + = (+g‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ 𝐵 = (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌))    &   ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑌) ≠ 0 )    &   ⊢ (𝜑 → 𝐵 ≠ (0g‘𝑈))    ⇒   ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))
Theorem	lclkrlem2p 37128	Lemma for lclkr 37139. When 𝐵 is zero, 𝑋 and 𝑌 must colinear, so their orthocomplements must be comparable. (Contributed by NM, 17-Jan-2015.)
⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ × = (.r‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 𝐼 = (invr‘𝑆)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ + = (+g‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ 𝐵 = (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌))    &   ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑌) ≠ 0 )    &   ⊢ (𝜑 → 𝐵 = (0g‘𝑈))    ⇒   ⊢ (𝜑 → ( ⊥ ‘{𝑌}) ⊆ ( ⊥ ‘{𝑋}))
Theorem	lclkrlem2q 37129	Lemma for lclkr 37139. The sum has a closed kernel when 𝐵 is nonzero. (Contributed by NM, 18-Jan-2015.)
⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ × = (.r‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 𝐼 = (invr‘𝑆)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ + = (+g‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))    &   ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))    &   ⊢ 𝐵 = (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌))    &   ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑌) ≠ 0 )    &   ⊢ (𝜑 → 𝐵 ≠ (0g‘𝑈))    ⇒   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
Theorem	lclkrlem2r 37130	Lemma for lclkr 37139. When 𝐵 is zero, i.e. when 𝑋 and 𝑌 are colinear, the intersection of the kernels of 𝐸 and 𝐺 equal the kernel of 𝐺, so the kernels of 𝐺 and the sum are comparable. (Contributed by NM, 18-Jan-2015.)
⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ × = (.r‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 𝐼 = (invr‘𝑆)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ + = (+g‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))    &   ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))    &   ⊢ 𝐵 = (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌))    &   ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑌) ≠ 0 )    &   ⊢ (𝜑 → 𝐵 = (0g‘𝑈))    ⇒   ⊢ (𝜑 → (𝐿‘𝐺) ⊆ (𝐿‘(𝐸 + 𝐺)))
Theorem	lclkrlem2s 37131	Lemma for lclkr 37139. Thus, the sum has a closed kernel when 𝐵 is zero. (Contributed by NM, 18-Jan-2015.)
⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ × = (.r‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 𝐼 = (invr‘𝑆)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ + = (+g‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))    &   ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))    &   ⊢ 𝐵 = (𝑋 − ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌))    &   ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑌) ≠ 0 )    &   ⊢ (𝜑 → 𝐵 = (0g‘𝑈))    ⇒   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
Theorem	lclkrlem2t 37132	Lemma for lclkr 37139. We eliminate all hypotheses with 𝐵 here. (Contributed by NM, 18-Jan-2015.)
⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ × = (.r‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 𝐼 = (invr‘𝑆)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ + = (+g‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))    &   ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))    &   ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑌) ≠ 0 )    ⇒   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
Theorem	lclkrlem2u 37133	Lemma for lclkr 37139. lclkrlem2t 37132 with 𝑋 and 𝑌 swapped. (Contributed by NM, 18-Jan-2015.)
⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ × = (.r‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 𝐼 = (invr‘𝑆)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ + = (+g‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))    &   ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))    &   ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑋) ≠ 0 )    ⇒   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
Theorem	lclkrlem2v 37134	Lemma for lclkr 37139. When the hypotheses of lclkrlem2u 37133 and lclkrlem2u 37133 are negated, the functional sum must be zero, so the kernel is the vector space. We make use of the law of excluded middle, dochexmid 37074, which requires the orthomodular law dihoml4 36983 (Lemma 3.3 of [Holland95] p. 214). (Contributed by NM, 16-Jan-2015.)
⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ × = (.r‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 𝐼 = (invr‘𝑆)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ + = (+g‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))    &   ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))    &   ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑋) = 0 )    &   ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑌) = 0 )    ⇒   ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) = 𝑉)
Theorem	lclkrlem2w 37135	Lemma for lclkr 37139. This is the same as lclkrlem2u 37133 and lclkrlem2u 37133 with the inequality hypotheses negated. When the sum of two functionals is zero at each generating vector, the kernel is the vector space and therefore closed. (Contributed by NM, 16-Jan-2015.)
⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = (Scalar‘𝑈)    &   ⊢ × = (.r‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 𝐼 = (invr‘𝑆)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ + = (+g‘𝐷)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))    &   ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))    &   ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑋) = 0 )    &   ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝑌) = 0 )    ⇒   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
Theorem	lclkrlem2x 37136	Lemma for lclkr 37139. Eliminate by cases the hypotheses of lclkrlem2u 37133, lclkrlem2u 37133 and lclkrlem2w 37135. (Contributed by NM, 18-Jan-2015.)
⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ + = (+g‘𝐷)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))    &   ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))    ⇒   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
Theorem	lclkrlem2y 37137	Lemma for lclkr 37139. Restate the hypotheses for 𝐸 and 𝐺 to say their kernels are closed, in order to eliminate the generating vectors 𝑋 and 𝑌. (Contributed by NM, 18-Jan-2015.)
⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ + = (+g‘𝐷)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐸 ∈ 𝐹)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    &   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐸))) = (𝐿‘𝐸))    &   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))    ⇒   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))
Theorem	lclkrlem2 37138*	The set of functionals having closed kernels is closed under vector (functional) addition. Lemmas lclkrlem2a 37113 through lclkrlem2y 37137 are used for the proof. Here we express lclkrlem2y 37137 in terms of membership in the set 𝐶 of functionals with closed kernels. (Contributed by NM, 18-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ + = (+g‘𝐷)    &   ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐸 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐶)    ⇒   ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐶)
Theorem	lclkr 37139*	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 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝐿 = (LKer‘𝑈)    &   ⊢ 𝐷 = (LDual‘𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝐷)    &   ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝐶 ∈ 𝑆)
Theorem	lcfls1lem 37140*	Property of a functional with a closed kernel. (Contributed by NM, 27-Jan-2015.)
⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ (( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ ( ⊥ ‘(𝐿‘𝑓)) ⊆ 𝑄)}    ⇒   ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄))
Theorem	lcfls1N 37141*	Property of a functional with a closed kernel. (Contributed by NM, 27-Jan-2015.) (New usage is discouraged.)
⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ (( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ ( ⊥ ‘(𝐿‘𝑓)) ⊆ 𝑄)}    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄)))
Theorem	lcfls1c 37142*	Property of a functional with a closed kernel. (Contributed by NM, 28-Jan-2015.)
⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ (( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ ( ⊥ ‘(𝐿‘𝑓)) ⊆ 𝑄)}    &   ⊢ 𝐷 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}    ⇒   ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐷 ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄))
Theorem	lclkrslem1 37143*	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 = (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)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 · 𝐺) ∈ 𝐶)
Theorem	lclkrslem2 37144*	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 37145*	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 37139 proof. Do we achieve overall shortening by breaking them out as subtheorems? Or make lclkr 37139 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 37146*	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 37234. (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 37147*	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 37148	Lemma for lcfr 37191. 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 37149	Lemma for lcfr 37191. (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 37150	Lemma for lcfr 37191. (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 37151*	Lemma for lcfr 37191. (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 37152*	Lemma for lcfr 37191. 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 37153*	Lemma for lcfr 37191. 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 37154*	Lemma for lcfr 37191. 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 37155*	Lemma for lcf1o 37157 and lcfr 37191. (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 37156*	Lemma for lcf1o 37157. (This part has undesirable $d's on 𝐽 and 𝜑 that we remove in lcf1o 37157.) 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 37157*	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 37158*	Lemma for lcfr 37191. (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 37159*	Lemma for lcfr 37191. (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 37160*	Lemma for lcfr 37191. (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 37161*	Lemma for lcfr 37191. (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 37162*	Lemma for lcfr 37191. (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 37163*	Lemma for lcfr 37191. (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 37164*	Lemma for lcfr 37191. (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 37165	Lemma for lcfr 37191. 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 37166	Lemma for lcfr 37191. (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 37167	Lemma for lcfr 37191. (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 37168	Lemma for lcfr 37191. (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 37169	Lemma for lcfr 37191. (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 37170	Lemma for lcfr 37191. (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 37171	Lemma for lcfr 37191. 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 37172*	Lemma for lcfr 37191. (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 37173*	Lemma for lcfr 37191. Special case of lcfrlem35 37183 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 37174*	Lemma for lcfr 37191. Special case of lcfrlem36 37184 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 37175*	Lemma for lcfr 37191. Special case of lcfrlem37 37185 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 37176*	Lemma for lcfr 37191. 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 37177*	Lemma for lcfr 37191. (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 37178*	Lemma for lcfr 37191. (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 37179*	Lemma for lcfr 37191. (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 37180*	Lemma for lcfr 37191. (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 37181*	Lemma for lcfr 37191. (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 37182*	Lemma for lcfr 37191. (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 37183*	Lemma for lcfr 37191. (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 37184*	Lemma for lcfr 37191. (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 37185*	Lemma for lcfr 37191. (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 37186*	Lemma for lcfr 37191. Combine lcfrlem27 37175 and lcfrlem37 37185. (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 37187*	Lemma for lcfr 37191. 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 37188*	Lemma for lcfr 37191. 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 37189*	Lemma for lcfr 37191. 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 37190*	Lemma for lcfr 37191. 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 37191*	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 37192	Extend class notation with vector space of functionals with closed kernels.
class LCDual
Definition	df-lcdual 37193*	Dual vector space of functionals with closed kernels. Note: we could also define this directly without mapd by using mapdrn 37255. TODO: see if it makes sense to go back and replace some of the LDual stuff with this. TODO: We could simplify df-mapd 37231 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 37194*	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 37195*	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 37196*	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 37197	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 37198	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 37199*	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 37200	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)