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

Theoremlcfl4N 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 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)       (𝜑 → (𝐺𝐶 ↔ (( ‘( ‘(𝐿𝐺))) ∈ 𝑌 ∨ (𝐿𝐺) = 𝑉)))

Theoremlcfl5 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 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)       (𝜑 → (𝐺𝐶 ↔ (𝐿𝐺) ∈ ran 𝐼))

Theoremlcfl5a 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 ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)       (𝜑 → (( ‘( ‘(𝐿𝐺))) = (𝐿𝐺) ↔ (𝐿𝐺) ∈ ran 𝐼))

Theoremlcfl6lem 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)    &    + = (+g𝑈)    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &    1 = (1r𝑆)    &   𝑅 = (Base‘𝑆)    &    0 = (0g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)    &   (𝜑𝑋 ∈ (( ‘(𝐿𝐺)) ∖ { 0 }))    &   (𝜑 → (𝐺𝑋) = 1 )       (𝜑𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))))

Theoremlcfl7lem 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)    &    + = (+g𝑈)    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝑆)    &    0 = (0g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))    &   𝐽 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑌})𝑣 = (𝑤 + (𝑘 · 𝑌))))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐺 = 𝐽)       (𝜑𝑋 = 𝑌)

Theoremlcfl6 37106* Property of a functional with a closed kernel. Note that (𝐿𝐺) = 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)    &    + = (+g𝑈)    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝑆)    &    0 = (0g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐶 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)       (𝜑 → (𝐺𝐶 ↔ ((𝐿𝐺) = 𝑉 ∨ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))))

Theoremlcfl7N 37107* Property of a functional with a closed kernel. Every nonzero functional is determined by a unique nonzero vector. Note that (𝐿𝐺) = 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)    &    + = (+g𝑈)    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝑆)    &    0 = (0g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐶 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)       (𝜑 → (𝐺𝐶 ↔ ((𝐿𝐺) = 𝑉 ∨ ∃!𝑥 ∈ (𝑉 ∖ { 0 })𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))))

Theoremlcfl8 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 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)       (𝜑 → (𝐺𝐶 ↔ ∃𝑥𝑉 (𝐿𝐺) = ( ‘{𝑥})))

Theoremlcfl8a 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 ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)       (𝜑 → (( ‘( ‘(𝐿𝐺))) = (𝐿𝐺) ↔ ∃𝑥𝑉 (𝐿𝐺) = ( ‘{𝑥})))

Theoremlcfl8b 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 = (0g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑌 = (0g𝐷)    &   𝐶 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺 ∈ (𝐶 ∖ {𝑌}))       (𝜑 → ∃𝑥 ∈ (𝑉 ∖ { 0 })( ‘(𝐿𝐺)) = (𝑁‘{𝑥}))

Theoremlcfl9a 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 ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)    &   (𝜑𝑋𝑉)    &   (𝜑 → ( ‘{𝑋}) ⊆ (𝐿𝐺))       (𝜑 → ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺))

Theoremlclkrlem1 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)    &    · = ( ·𝑠𝐷)    &   𝐶 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝐵)    &   (𝜑𝐺𝐶)       (𝜑 → (𝑋 · 𝐺) ∈ 𝐶)

Theoremlclkrlem2a 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 = (0g𝑈)    &    = (LSSum‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐴 = (LSAtoms‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐵 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → ( ‘{𝑋}) ≠ ( ‘{𝑌}))    &   (𝜑 → ¬ 𝑋 ∈ ( ‘{𝐵}))       (𝜑 → (((𝑁‘{𝑋}) (𝑁‘{𝑌})) ∩ ( ‘{𝐵})) ∈ 𝐴)

Theoremlclkrlem2b 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 = (0g𝑈)    &    = (LSSum‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐴 = (LSAtoms‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐵 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → ( ‘{𝑋}) ≠ ( ‘{𝑌}))    &   (𝜑 → (¬ 𝑋 ∈ ( ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ‘{𝐵})))       (𝜑 → (((𝑁‘{𝑋}) (𝑁‘{𝑌})) ∩ ( ‘{𝐵})) ∈ 𝐴)

Theoremlclkrlem2c 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 = (0g𝑈)    &    = (LSSum‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐴 = (LSAtoms‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐵 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → ( ‘{𝑋}) ≠ ( ‘{𝑌}))    &   (𝜑 → (¬ 𝑋 ∈ ( ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ‘{𝐵})))    &   𝐽 = (LSHyp‘𝑈)       (𝜑 → ((( ‘{𝑋}) ∩ ( ‘{𝑌})) (𝑁‘{𝐵})) ∈ 𝐽)

Theoremlclkrlem2d 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 = (0g𝑈)    &    = (LSSum‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐴 = (LSAtoms‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐵 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → ( ‘{𝑋}) ≠ ( ‘{𝑌}))    &   (𝜑 → (¬ 𝑋 ∈ ( ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ‘{𝐵})))    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       (𝜑 → ((( ‘{𝑋}) ∩ ( ‘{𝑌})) (𝑁‘{𝐵})) ∈ ran 𝐼)

Theoremlclkrlem2e 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 = (0g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &    + = (+g𝐷)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐸𝐹)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝐿𝐸) = ( ‘{𝑋}))    &   (𝜑 → (𝐿𝐸) = (𝐿𝐺))       (𝜑 → ( ‘( ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

Theoremlclkrlem2f 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 = (0g𝑆)    &    0 = (0g𝑈)    &    = (LSSum‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐽 = (LSHyp‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &    + = (+g𝐷)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐵 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐸𝐹)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝐿𝐸) = ( ‘{𝑋}))    &   (𝜑 → (𝐿𝐺) = ( ‘{𝑌}))    &   (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄)    &   (𝜑 → (¬ 𝑋 ∈ ( ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ‘{𝐵})))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝐿𝐸) ≠ (𝐿𝐺))    &   (𝜑 → (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽)       (𝜑 → (((𝐿𝐸) ∩ (𝐿𝐺)) (𝑁‘{𝐵})) ⊆ (𝐿‘(𝐸 + 𝐺)))

Theoremlclkrlem2g 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 = (0g𝑆)    &    0 = (0g𝑈)    &    = (LSSum‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐽 = (LSHyp‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &    + = (+g𝐷)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐵 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐸𝐹)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝐿𝐸) = ( ‘{𝑋}))    &   (𝜑 → (𝐿𝐺) = ( ‘{𝑌}))    &   (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄)    &   (𝜑 → (¬ 𝑋 ∈ ( ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ‘{𝐵})))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝐿𝐸) ≠ (𝐿𝐺))    &   (𝜑 → (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽)       (𝜑 → ( ‘( ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

Theoremlclkrlem2h 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 = (0g𝑆)    &    0 = (0g𝑈)    &    = (LSSum‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐽 = (LSHyp‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &    + = (+g𝐷)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐵 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐸𝐹)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝐿𝐸) = ( ‘{𝑋}))    &   (𝜑 → (𝐿𝐺) = ( ‘{𝑌}))    &   (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄)    &   (𝜑 → (¬ 𝑋 ∈ ( ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ‘{𝐵})))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝐿𝐸) ≠ (𝐿𝐺))       (𝜑 → ( ‘( ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

Theoremlclkrlem2i 37121 Lemma for lclkr 37139. Eliminate the (𝐿𝐸) ≠ (𝐿𝐺) hypothesis. (Contributed by NM, 17-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑄 = (0g𝑆)    &    0 = (0g𝑈)    &    = (LSSum‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐽 = (LSHyp‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &    + = (+g𝐷)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐵 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐸𝐹)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝐿𝐸) = ( ‘{𝑋}))    &   (𝜑 → (𝐿𝐺) = ( ‘{𝑌}))    &   (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄)    &   (𝜑 → (¬ 𝑋 ∈ ( ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ‘{𝐵})))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))       (𝜑 → ( ‘( ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

Theoremlclkrlem2j 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 = (0g𝑆)    &    0 = (0g𝑈)    &    = (LSSum‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐽 = (LSHyp‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &    + = (+g𝐷)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐵 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐸𝐹)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝐿𝐸) = ( ‘{𝑋}))    &   (𝜑 → (𝐿𝐺) = ( ‘{𝑌}))    &   (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄)    &   (𝜑 → (¬ 𝑋 ∈ ( ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ‘{𝐵})))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌 = 0 )       (𝜑 → ( ‘( ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

Theoremlclkrlem2k 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 = (0g𝑆)    &    0 = (0g𝑈)    &    = (LSSum‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐽 = (LSHyp‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &    + = (+g𝐷)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐵 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐸𝐹)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝐿𝐸) = ( ‘{𝑋}))    &   (𝜑 → (𝐿𝐺) = ( ‘{𝑌}))    &   (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄)    &   (𝜑 → (¬ 𝑋 ∈ ( ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ‘{𝐵})))    &   (𝜑𝑋 = 0 )    &   (𝜑𝑌𝑉)       (𝜑 → ( ‘( ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

Theoremlclkrlem2l 37124 Lemma for lclkr 37139. Eliminate the 𝑋0, 𝑌0 hypotheses. (Contributed by NM, 18-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑄 = (0g𝑆)    &    0 = (0g𝑈)    &    = (LSSum‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐽 = (LSHyp‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &    + = (+g𝐷)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐵 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐸𝐹)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝐿𝐸) = ( ‘{𝑋}))    &   (𝜑 → (𝐿𝐺) = ( ‘{𝑌}))    &   (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄)    &   (𝜑 → (¬ 𝑋 ∈ ( ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ‘{𝐵})))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → ( ‘( ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

Theoremlclkrlem2m 37125 Lemma for lclkr 37139. Construct a vector 𝐵 that makes the sum of functionals zero. Combine with 𝐵𝑉 to shorten overall proof. (Contributed by NM, 17-Jan-2015.)
𝑉 = (Base‘𝑈)    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &    × = (.r𝑆)    &    0 = (0g𝑆)    &   𝐼 = (invr𝑆)    &    = (-g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐷 = (LDual‘𝑈)    &    + = (+g𝐷)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝐸𝐹)    &   (𝜑𝐺𝐹)    &   (𝜑𝑈 ∈ LVec)    &   𝐵 = (𝑋 ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌))    &   (𝜑 → ((𝐸 + 𝐺)‘𝑌) ≠ 0 )       (𝜑 → (𝐵𝑉 ∧ ((𝐸 + 𝐺)‘𝐵) = 0 ))

Theoremlclkrlem2n 37126 Lemma for lclkr 37139. (Contributed by NM, 12-Jan-2015.)
𝑉 = (Base‘𝑈)    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &    × = (.r𝑆)    &    0 = (0g𝑆)    &   𝐼 = (invr𝑆)    &    = (-g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐷 = (LDual‘𝑈)    &    + = (+g𝐷)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝐸𝐹)    &   (𝜑𝐺𝐹)    &   𝑁 = (LSpan‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   (𝜑𝑈 ∈ LVec)    &   (𝜑 → ((𝐸 + 𝐺)‘𝑋) = 0 )    &   (𝜑 → ((𝐸 + 𝐺)‘𝑌) = 0 )       (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ (𝐿‘(𝐸 + 𝐺)))

Theoremlclkrlem2o 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)    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &    × = (.r𝑆)    &    0 = (0g𝑆)    &   𝐼 = (invr𝑆)    &    = (-g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐷 = (LDual‘𝑈)    &    + = (+g𝐷)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝐸𝐹)    &   (𝜑𝐺𝐹)    &   𝑁 = (LSpan‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   𝐵 = (𝑋 ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌))    &   (𝜑 → ((𝐸 + 𝐺)‘𝑌) ≠ 0 )    &   (𝜑𝐵 ≠ (0g𝑈))       (𝜑 → (¬ 𝑋 ∈ ( ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ‘{𝐵})))

Theoremlclkrlem2p 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)    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &    × = (.r𝑆)    &    0 = (0g𝑆)    &   𝐼 = (invr𝑆)    &    = (-g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐷 = (LDual‘𝑈)    &    + = (+g𝐷)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝐸𝐹)    &   (𝜑𝐺𝐹)    &   𝑁 = (LSpan‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   𝐵 = (𝑋 ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌))    &   (𝜑 → ((𝐸 + 𝐺)‘𝑌) ≠ 0 )    &   (𝜑𝐵 = (0g𝑈))       (𝜑 → ( ‘{𝑌}) ⊆ ( ‘{𝑋}))

Theoremlclkrlem2q 37129 Lemma for lclkr 37139. The sum has a closed kernel when 𝐵 is nonzero. (Contributed by NM, 18-Jan-2015.)
𝑉 = (Base‘𝑈)    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &    × = (.r𝑆)    &    0 = (0g𝑆)    &   𝐼 = (invr𝑆)    &    = (-g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐷 = (LDual‘𝑈)    &    + = (+g𝐷)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝐸𝐹)    &   (𝜑𝐺𝐹)    &   𝑁 = (LSpan‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑 → (𝐿𝐸) = ( ‘{𝑋}))    &   (𝜑 → (𝐿𝐺) = ( ‘{𝑌}))    &   𝐵 = (𝑋 ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌))    &   (𝜑 → ((𝐸 + 𝐺)‘𝑌) ≠ 0 )    &   (𝜑𝐵 ≠ (0g𝑈))       (𝜑 → ( ‘( ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

Theoremlclkrlem2r 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)    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &    × = (.r𝑆)    &    0 = (0g𝑆)    &   𝐼 = (invr𝑆)    &    = (-g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐷 = (LDual‘𝑈)    &    + = (+g𝐷)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝐸𝐹)    &   (𝜑𝐺𝐹)    &   𝑁 = (LSpan‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑 → (𝐿𝐸) = ( ‘{𝑋}))    &   (𝜑 → (𝐿𝐺) = ( ‘{𝑌}))    &   𝐵 = (𝑋 ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌))    &   (𝜑 → ((𝐸 + 𝐺)‘𝑌) ≠ 0 )    &   (𝜑𝐵 = (0g𝑈))       (𝜑 → (𝐿𝐺) ⊆ (𝐿‘(𝐸 + 𝐺)))

Theoremlclkrlem2s 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)    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &    × = (.r𝑆)    &    0 = (0g𝑆)    &   𝐼 = (invr𝑆)    &    = (-g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐷 = (LDual‘𝑈)    &    + = (+g𝐷)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝐸𝐹)    &   (𝜑𝐺𝐹)    &   𝑁 = (LSpan‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑 → (𝐿𝐸) = ( ‘{𝑋}))    &   (𝜑 → (𝐿𝐺) = ( ‘{𝑌}))    &   𝐵 = (𝑋 ((((𝐸 + 𝐺)‘𝑋) × (𝐼‘((𝐸 + 𝐺)‘𝑌))) · 𝑌))    &   (𝜑 → ((𝐸 + 𝐺)‘𝑌) ≠ 0 )    &   (𝜑𝐵 = (0g𝑈))       (𝜑 → ( ‘( ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

Theoremlclkrlem2t 37132 Lemma for lclkr 37139. We eliminate all hypotheses with 𝐵 here. (Contributed by NM, 18-Jan-2015.)
𝑉 = (Base‘𝑈)    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &    × = (.r𝑆)    &    0 = (0g𝑆)    &   𝐼 = (invr𝑆)    &    = (-g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐷 = (LDual‘𝑈)    &    + = (+g𝐷)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝐸𝐹)    &   (𝜑𝐺𝐹)    &   𝑁 = (LSpan‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑 → (𝐿𝐸) = ( ‘{𝑋}))    &   (𝜑 → (𝐿𝐺) = ( ‘{𝑌}))    &   (𝜑 → ((𝐸 + 𝐺)‘𝑌) ≠ 0 )       (𝜑 → ( ‘( ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

Theoremlclkrlem2u 37133 Lemma for lclkr 37139. lclkrlem2t 37132 with 𝑋 and 𝑌 swapped. (Contributed by NM, 18-Jan-2015.)
𝑉 = (Base‘𝑈)    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &    × = (.r𝑆)    &    0 = (0g𝑆)    &   𝐼 = (invr𝑆)    &    = (-g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐷 = (LDual‘𝑈)    &    + = (+g𝐷)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝐸𝐹)    &   (𝜑𝐺𝐹)    &   𝑁 = (LSpan‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑 → (𝐿𝐸) = ( ‘{𝑋}))    &   (𝜑 → (𝐿𝐺) = ( ‘{𝑌}))    &   (𝜑 → ((𝐸 + 𝐺)‘𝑋) ≠ 0 )       (𝜑 → ( ‘( ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

Theoremlclkrlem2v 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)    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &    × = (.r𝑆)    &    0 = (0g𝑆)    &   𝐼 = (invr𝑆)    &    = (-g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐷 = (LDual‘𝑈)    &    + = (+g𝐷)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝐸𝐹)    &   (𝜑𝐺𝐹)    &   𝑁 = (LSpan‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑 → (𝐿𝐸) = ( ‘{𝑋}))    &   (𝜑 → (𝐿𝐺) = ( ‘{𝑌}))    &   (𝜑 → ((𝐸 + 𝐺)‘𝑋) = 0 )    &   (𝜑 → ((𝐸 + 𝐺)‘𝑌) = 0 )       (𝜑 → (𝐿‘(𝐸 + 𝐺)) = 𝑉)

Theoremlclkrlem2w 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)    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &    × = (.r𝑆)    &    0 = (0g𝑆)    &   𝐼 = (invr𝑆)    &    = (-g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐷 = (LDual‘𝑈)    &    + = (+g𝐷)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝐸𝐹)    &   (𝜑𝐺𝐹)    &   𝑁 = (LSpan‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑 → (𝐿𝐸) = ( ‘{𝑋}))    &   (𝜑 → (𝐿𝐺) = ( ‘{𝑌}))    &   (𝜑 → ((𝐸 + 𝐺)‘𝑋) = 0 )    &   (𝜑 → ((𝐸 + 𝐺)‘𝑌) = 0 )       (𝜑 → ( ‘( ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

Theoremlclkrlem2x 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)    &    + = (+g𝐷)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝐸𝐹)    &   (𝜑𝐺𝐹)    &   (𝜑 → (𝐿𝐸) = ( ‘{𝑋}))    &   (𝜑 → (𝐿𝐺) = ( ‘{𝑌}))       (𝜑 → ( ‘( ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

Theoremlclkrlem2y 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)    &    + = (+g𝐷)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐸𝐹)    &   (𝜑𝐺𝐹)    &   (𝜑 → ( ‘( ‘(𝐿𝐸))) = (𝐿𝐸))    &   (𝜑 → ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺))       (𝜑 → ( ‘( ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

Theoremlclkrlem2 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)    &    + = (+g𝐷)    &   𝐶 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐸𝐶)    &   (𝜑𝐺𝐶)       (𝜑 → (𝐸 + 𝐺) ∈ 𝐶)

Theoremlclkr 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 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝐶𝑆)

Theoremlcfls1lem 37140* Property of a functional with a closed kernel. (Contributed by NM, 27-Jan-2015.)
𝐶 = {𝑓𝐹 ∣ (( ‘( ‘(𝐿𝑓))) = (𝐿𝑓) ∧ ( ‘(𝐿𝑓)) ⊆ 𝑄)}       (𝐺𝐶 ↔ (𝐺𝐹 ∧ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺) ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄))

Theoremlcfls1N 37141* Property of a functional with a closed kernel. (Contributed by NM, 27-Jan-2015.) (New usage is discouraged.)
𝐶 = {𝑓𝐹 ∣ (( ‘( ‘(𝐿𝑓))) = (𝐿𝑓) ∧ ( ‘(𝐿𝑓)) ⊆ 𝑄)}    &   (𝜑𝐺𝐹)       (𝜑 → (𝐺𝐶 ↔ (( ‘( ‘(𝐿𝐺))) = (𝐿𝐺) ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄)))

Theoremlcfls1c 37142* Property of a functional with a closed kernel. (Contributed by NM, 28-Jan-2015.)
𝐶 = {𝑓𝐹 ∣ (( ‘( ‘(𝐿𝑓))) = (𝐿𝑓) ∧ ( ‘(𝐿𝑓)) ⊆ 𝑄)}    &   𝐷 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}       (𝐺𝐶 ↔ (𝐺𝐷 ∧ ( ‘(𝐿𝐺)) ⊆ 𝑄))

Theoremlclkrslem1 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)    &    · = ( ·𝑠𝐷)    &   𝐶 = {𝑓𝐹 ∣ (( ‘( ‘(𝐿𝑓))) = (𝐿𝑓) ∧ ( ‘(𝐿𝑓)) ⊆ 𝑄)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑄𝑆)    &   (𝜑𝐺𝐶)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑋 · 𝐺) ∈ 𝐶)

Theoremlclkrslem2 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)    &    · = ( ·𝑠𝐷)    &   𝐶 = {𝑓𝐹 ∣ (( ‘( ‘(𝐿𝑓))) = (𝐿𝑓) ∧ ( ‘(𝐿𝑓)) ⊆ 𝑄)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑄𝑆)    &   (𝜑𝐺𝐶)    &    + = (+g𝐷)    &   (𝜑𝐸𝐶)       (𝜑 → (𝐸 + 𝐺) ∈ 𝐶)

Theoremlclkrs 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 = {𝑓𝐹 ∣ (( ‘( ‘(𝐿𝑓))) = (𝐿𝑓) ∧ ( ‘(𝐿𝑓)) ⊆ 𝑅)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑅𝑆)       (𝜑𝐶𝑇)

Theoremlclkrs2 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 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   𝑅 = {𝑔𝐹 ∣ (( ‘( ‘(𝐿𝑔))) = (𝐿𝑔) ∧ ( ‘(𝐿𝑔)) ⊆ 𝑄)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑄𝑆)       (𝜑 → (𝑅𝑇𝑅𝐶))

TheoremlcfrvalsnN 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 ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)    &   𝑄 = 𝑓𝑅 ( ‘(𝐿𝑓))    &   𝑅 = (𝑁‘{𝐺})       (𝜑𝑄 = ( ‘(𝐿𝐺)))

Theoremlcfrlem1 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)    &    × = (.r𝑆)    &    0 = (0g𝑆)    &   𝐼 = (invr𝑆)    &   𝐹 = (LFnl‘𝑈)    &   𝐷 = (LDual‘𝑈)    &    · = ( ·𝑠𝐷)    &    = (-g𝐷)    &   (𝜑𝑈 ∈ LVec)    &   (𝜑𝐸𝐹)    &   (𝜑𝐺𝐹)    &   (𝜑𝑋𝑉)    &   (𝜑 → (𝐺𝑋) ≠ 0 )    &   𝐻 = (𝐸 (((𝐼‘(𝐺𝑋)) × (𝐸𝑋)) · 𝐺))       (𝜑 → (𝐻𝑋) = 0 )

Theoremlcfrlem2 37149 Lemma for lcfr 37191. (Contributed by NM, 27-Feb-2015.)
𝑉 = (Base‘𝑈)    &   𝑆 = (Scalar‘𝑈)    &    × = (.r𝑆)    &    0 = (0g𝑆)    &   𝐼 = (invr𝑆)    &   𝐹 = (LFnl‘𝑈)    &   𝐷 = (LDual‘𝑈)    &    · = ( ·𝑠𝐷)    &    = (-g𝐷)    &   (𝜑𝑈 ∈ LVec)    &   (𝜑𝐸𝐹)    &   (𝜑𝐺𝐹)    &   (𝜑𝑋𝑉)    &   (𝜑 → (𝐺𝑋) ≠ 0 )    &   𝐻 = (𝐸 (((𝐼‘(𝐺𝑋)) × (𝐸𝑋)) · 𝐺))    &   𝐿 = (LKer‘𝑈)       (𝜑 → ((𝐿𝐸) ∩ (𝐿𝐺)) ⊆ (𝐿𝐻))

Theoremlcfrlem3 37150 Lemma for lcfr 37191. (Contributed by NM, 27-Feb-2015.)
𝑉 = (Base‘𝑈)    &   𝑆 = (Scalar‘𝑈)    &    × = (.r𝑆)    &    0 = (0g𝑆)    &   𝐼 = (invr𝑆)    &   𝐹 = (LFnl‘𝑈)    &   𝐷 = (LDual‘𝑈)    &    · = ( ·𝑠𝐷)    &    = (-g𝐷)    &   (𝜑𝑈 ∈ LVec)    &   (𝜑𝐸𝐹)    &   (𝜑𝐺𝐹)    &   (𝜑𝑋𝑉)    &   (𝜑 → (𝐺𝑋) ≠ 0 )    &   𝐻 = (𝐸 (((𝐼‘(𝐺𝑋)) × (𝐸𝑋)) · 𝐺))    &   𝐿 = (LKer‘𝑈)       (𝜑𝑋 ∈ (𝐿𝐻))

Theoremlcfrlem4 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 = 𝑔𝐺 ( ‘(𝐿𝑔))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝑄)    &   (𝜑𝑋𝐸)       (𝜑𝑋𝑉)

Theoremlcfrlem5 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 ∧ 𝑊𝐻))    &   (𝜑𝑅𝑆)    &   𝑄 = 𝑓𝑅 ( ‘(𝐿𝑓))    &   (𝜑𝑋𝑄)    &   𝐶 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐶)    &    · = ( ·𝑠𝑈)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐴 · 𝑋) ∈ 𝑄)

Theoremlcfrlem6 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)    &    + = (+g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑄 = (LSubSp‘𝐷)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝑄)    &   𝐸 = 𝑔𝐺 ( ‘(𝐿𝑔))    &   (𝜑𝑋𝐸)    &   (𝜑𝑌𝐸)    &   (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))       (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Theoremlcfrlem7 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)    &    + = (+g𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑄 = (LSubSp‘𝐷)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝑄)    &   𝐸 = 𝑔𝐺 ( ‘(𝐿𝑔))    &   (𝜑𝑋𝐸)    &    0 = (0g𝑈)    &   (𝜑𝑌 = 0 )       (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Theoremlcfrlem8 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)    &    + = (+g𝑈)    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝑆)    &    0 = (0g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑄 = (0g𝐷)    &   𝐶 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))       (𝜑 → (𝐽𝑋) = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))))

Theoremlcfrlem9 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 𝐽𝑧? TODO: Some redundant \$d's? (Contributed by NM, 22-Feb-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝑆)    &    0 = (0g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑄 = (0g𝐷)    &   𝐶 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝐽:(𝑉 ∖ { 0 })–1-1-onto→(𝐶 ∖ {𝑄}))

Theoremlcf1o 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)    &    + = (+g𝑈)    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝑆)    &    0 = (0g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑄 = (0g𝐷)    &   𝐶 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝐽:(𝑉 ∖ { 0 })–1-1-onto→(𝐶 ∖ {𝑄}))

Theoremlcfrlem10 37158* Lemma for lcfr 37191. (Contributed by NM, 23-Feb-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝑆)    &    0 = (0g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑄 = (0g𝐷)    &   𝐶 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))       (𝜑 → (𝐽𝑋) ∈ 𝐹)

Theoremlcfrlem11 37159* Lemma for lcfr 37191. (Contributed by NM, 23-Feb-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝑆)    &    0 = (0g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑄 = (0g𝐷)    &   𝐶 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))       (𝜑 → (𝐿‘(𝐽𝑋)) = ( ‘{𝑋}))

Theoremlcfrlem12N 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)    &    + = (+g𝑈)    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝑆)    &    0 = (0g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑄 = (0g𝐷)    &   𝐶 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   𝐵 = (0g𝑆)    &   (𝜑𝑌 ∈ ( ‘{𝑋}))       (𝜑 → ((𝐽𝑋)‘𝑌) = 𝐵)

Theoremlcfrlem13 37161* Lemma for lcfr 37191. (Contributed by NM, 8-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝑆)    &    0 = (0g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑄 = (0g𝐷)    &   𝐶 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))       (𝜑 → (𝐽𝑋) ∈ (𝐶 ∖ {𝑄}))

Theoremlcfrlem14 37162* Lemma for lcfr 37191. (Contributed by NM, 10-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝑆)    &    0 = (0g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑄 = (0g𝐷)    &   𝐶 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   𝑁 = (LSpan‘𝑈)       (𝜑 → ( ‘(𝐿‘(𝐽𝑋))) = (𝑁‘{𝑋}))

Theoremlcfrlem15 37163* Lemma for lcfr 37191. (Contributed by NM, 9-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝑆)    &    0 = (0g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑄 = (0g𝐷)    &   𝐶 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))       (𝜑𝑋 ∈ ( ‘(𝐿‘(𝐽𝑋))))

Theoremlcfrlem16 37164* Lemma for lcfr 37191. (Contributed by NM, 8-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝑆)    &    0 = (0g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑄 = (0g𝐷)    &   𝐶 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   𝑃 = (LSubSp‘𝐷)    &   (𝜑𝐺𝑃)    &   (𝜑𝐺𝐶)    &   𝐸 = 𝑔𝐺 ( ‘(𝐿𝑔))    &   (𝜑𝑋 ∈ (𝐸 ∖ { 0 }))       (𝜑 → (𝐽𝑋) ∈ 𝐺)

Theoremlcfrlem17 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)    &    + = (+g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐴 = (LSAtoms‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))       (𝜑 → (𝑋 + 𝑌) ∈ (𝑉 ∖ { 0 }))

Theoremlcfrlem18 37166 Lemma for lcfr 37191. (Contributed by NM, 24-Feb-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐴 = (LSAtoms‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))       (𝜑 → ( ‘{𝑋, 𝑌}) = (( ‘{𝑋}) ∩ ( ‘{𝑌})))

Theoremlcfrlem19 37167 Lemma for lcfr 37191. (Contributed by NM, 11-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐴 = (LSAtoms‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))       (𝜑 → (¬ 𝑋 ∈ ( ‘{(𝑋 + 𝑌)}) ∨ ¬ 𝑌 ∈ ( ‘{(𝑋 + 𝑌)})))

Theoremlcfrlem20 37168 Lemma for lcfr 37191. (Contributed by NM, 11-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐴 = (LSAtoms‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → ¬ 𝑋 ∈ ( ‘{(𝑋 + 𝑌)}))       (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ‘{(𝑋 + 𝑌)})) ∈ 𝐴)

Theoremlcfrlem21 37169 Lemma for lcfr 37191. (Contributed by NM, 11-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐴 = (LSAtoms‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))       (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ‘{(𝑋 + 𝑌)})) ∈ 𝐴)

Theoremlcfrlem22 37170 Lemma for lcfr 37191. (Contributed by NM, 24-Feb-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐴 = (LSAtoms‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ‘{(𝑋 + 𝑌)}))       (𝜑𝐵𝐴)

Theoremlcfrlem23 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)    &    + = (+g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐴 = (LSAtoms‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ‘{(𝑋 + 𝑌)}))    &    = (LSSum‘𝑈)       (𝜑 → (( ‘{𝑋, 𝑌}) 𝐵) = ( ‘{(𝑋 + 𝑌)}))

Theoremlcfrlem24 37172* Lemma for lcfr 37191. (Contributed by NM, 24-Feb-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐴 = (LSAtoms‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ‘{(𝑋 + 𝑌)}))    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑄 = (0g𝑆)    &   𝑅 = (Base‘𝑆)    &   𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   (𝜑𝐼𝐵)    &   𝐿 = (LKer‘𝑈)       (𝜑 → ( ‘{𝑋, 𝑌}) = ((𝐿‘(𝐽𝑋)) ∩ (𝐿‘(𝐽𝑌))))

Theoremlcfrlem25 37173* Lemma for lcfr 37191. Special case of lcfrlem35 37183 when ((𝐽𝑌)‘𝐼) is zero. (Contributed by NM, 11-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐴 = (LSAtoms‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ‘{(𝑋 + 𝑌)}))    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑄 = (0g𝑆)    &   𝑅 = (Base‘𝑆)    &   𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   (𝜑𝐼𝐵)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   (𝜑 → ((𝐽𝑌)‘𝐼) = 𝑄)    &   (𝜑𝐼0 )       (𝜑 → ( ‘{(𝑋 + 𝑌)}) = (𝐿‘(𝐽𝑌)))

Theoremlcfrlem26 37174* Lemma for lcfr 37191. Special case of lcfrlem36 37184 when ((𝐽𝑌)‘𝐼) is zero. (Contributed by NM, 11-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐴 = (LSAtoms‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ‘{(𝑋 + 𝑌)}))    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑄 = (0g𝑆)    &   𝑅 = (Base‘𝑆)    &   𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   (𝜑𝐼𝐵)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   (𝜑 → ((𝐽𝑌)‘𝐼) = 𝑄)    &   (𝜑𝐼0 )       (𝜑 → (𝑋 + 𝑌) ∈ ( ‘(𝐿‘(𝐽𝑌))))

Theoremlcfrlem27 37175* Lemma for lcfr 37191. Special case of lcfrlem37 37185 when ((𝐽𝑌)‘𝐼) is zero. (Contributed by NM, 11-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐴 = (LSAtoms‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ‘{(𝑋 + 𝑌)}))    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑄 = (0g𝑆)    &   𝑅 = (Base‘𝑆)    &   𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   (𝜑𝐼𝐵)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   (𝜑 → ((𝐽𝑌)‘𝐼) = 𝑄)    &   (𝜑𝐼0 )    &   (𝜑𝐺 ∈ (LSubSp‘𝐷))    &   (𝜑𝐺 ⊆ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)})    &   𝐸 = 𝑔𝐺 ( ‘(𝐿𝑔))    &   (𝜑𝑋𝐸)    &   (𝜑𝑌𝐸)       (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Theoremlcfrlem28 37176* Lemma for lcfr 37191. TODO: This can be a hypothesis since the zero version of (𝐽𝑌)‘𝐼 needs it. (Contributed by NM, 9-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐴 = (LSAtoms‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ‘{(𝑋 + 𝑌)}))    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑄 = (0g𝑆)    &   𝑅 = (Base‘𝑆)    &   𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   (𝜑𝐼𝐵)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   (𝜑 → ((𝐽𝑌)‘𝐼) ≠ 𝑄)       (𝜑𝐼0 )

Theoremlcfrlem29 37177* Lemma for lcfr 37191. (Contributed by NM, 9-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐴 = (LSAtoms‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ‘{(𝑋 + 𝑌)}))    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑄 = (0g𝑆)    &   𝑅 = (Base‘𝑆)    &   𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   (𝜑𝐼𝐵)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   (𝜑 → ((𝐽𝑌)‘𝐼) ≠ 𝑄)    &   𝐹 = (invr𝑆)       (𝜑 → ((𝐹‘((𝐽𝑌)‘𝐼))(.r𝑆)((𝐽𝑋)‘𝐼)) ∈ 𝑅)

Theoremlcfrlem30 37178* Lemma for lcfr 37191. (Contributed by NM, 6-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐴 = (LSAtoms‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ‘{(𝑋 + 𝑌)}))    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑄 = (0g𝑆)    &   𝑅 = (Base‘𝑆)    &   𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   (𝜑𝐼𝐵)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   (𝜑 → ((𝐽𝑌)‘𝐼) ≠ 𝑄)    &   𝐹 = (invr𝑆)    &    = (-g𝐷)    &   𝐶 = ((𝐽𝑋) (((𝐹‘((𝐽𝑌)‘𝐼))(.r𝑆)((𝐽𝑋)‘𝐼))( ·𝑠𝐷)(𝐽𝑌)))       (𝜑𝐶 ∈ (LFnl‘𝑈))

Theoremlcfrlem31 37179* Lemma for lcfr 37191. (Contributed by NM, 10-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐴 = (LSAtoms‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ‘{(𝑋 + 𝑌)}))    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑄 = (0g𝑆)    &   𝑅 = (Base‘𝑆)    &   𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   (𝜑𝐼𝐵)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   (𝜑 → ((𝐽𝑌)‘𝐼) ≠ 𝑄)    &   𝐹 = (invr𝑆)    &    = (-g𝐷)    &   𝐶 = ((𝐽𝑋) (((𝐹‘((𝐽𝑌)‘𝐼))(.r𝑆)((𝐽𝑋)‘𝐼))( ·𝑠𝐷)(𝐽𝑌)))    &   (𝜑 → ((𝐽𝑋)‘𝐼) ≠ 𝑄)    &   (𝜑𝐶 = (0g𝐷))       (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))

Theoremlcfrlem32 37180* Lemma for lcfr 37191. (Contributed by NM, 10-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐴 = (LSAtoms‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ‘{(𝑋 + 𝑌)}))    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑄 = (0g𝑆)    &   𝑅 = (Base‘𝑆)    &   𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   (𝜑𝐼𝐵)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   (𝜑 → ((𝐽𝑌)‘𝐼) ≠ 𝑄)    &   𝐹 = (invr𝑆)    &    = (-g𝐷)    &   𝐶 = ((𝐽𝑋) (((𝐹‘((𝐽𝑌)‘𝐼))(.r𝑆)((𝐽𝑋)‘𝐼))( ·𝑠𝐷)(𝐽𝑌)))    &   (𝜑 → ((𝐽𝑋)‘𝐼) ≠ 𝑄)       (𝜑𝐶 ≠ (0g𝐷))

Theoremlcfrlem33 37181* Lemma for lcfr 37191. (Contributed by NM, 10-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐴 = (LSAtoms‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ‘{(𝑋 + 𝑌)}))    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑄 = (0g𝑆)    &   𝑅 = (Base‘𝑆)    &   𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   (𝜑𝐼𝐵)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   (𝜑 → ((𝐽𝑌)‘𝐼) ≠ 𝑄)    &   𝐹 = (invr𝑆)    &    = (-g𝐷)    &   𝐶 = ((𝐽𝑋) (((𝐹‘((𝐽𝑌)‘𝐼))(.r𝑆)((𝐽𝑋)‘𝐼))( ·𝑠𝐷)(𝐽𝑌)))    &   (𝜑 → ((𝐽𝑋)‘𝐼) = 𝑄)       (𝜑𝐶 ≠ (0g𝐷))

Theoremlcfrlem34 37182* Lemma for lcfr 37191. (Contributed by NM, 10-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐴 = (LSAtoms‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ‘{(𝑋 + 𝑌)}))    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑄 = (0g𝑆)    &   𝑅 = (Base‘𝑆)    &   𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   (𝜑𝐼𝐵)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   (𝜑 → ((𝐽𝑌)‘𝐼) ≠ 𝑄)    &   𝐹 = (invr𝑆)    &    = (-g𝐷)    &   𝐶 = ((𝐽𝑋) (((𝐹‘((𝐽𝑌)‘𝐼))(.r𝑆)((𝐽𝑋)‘𝐼))( ·𝑠𝐷)(𝐽𝑌)))       (𝜑𝐶 ≠ (0g𝐷))

Theoremlcfrlem35 37183* Lemma for lcfr 37191. (Contributed by NM, 2-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐴 = (LSAtoms‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ‘{(𝑋 + 𝑌)}))    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑄 = (0g𝑆)    &   𝑅 = (Base‘𝑆)    &   𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   (𝜑𝐼𝐵)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   (𝜑 → ((𝐽𝑌)‘𝐼) ≠ 𝑄)    &   𝐹 = (invr𝑆)    &    = (-g𝐷)    &   𝐶 = ((𝐽𝑋) (((𝐹‘((𝐽𝑌)‘𝐼))(.r𝑆)((𝐽𝑋)‘𝐼))( ·𝑠𝐷)(𝐽𝑌)))       (𝜑 → ( ‘{(𝑋 + 𝑌)}) = (𝐿𝐶))

Theoremlcfrlem36 37184* Lemma for lcfr 37191. (Contributed by NM, 6-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐴 = (LSAtoms‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ‘{(𝑋 + 𝑌)}))    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑄 = (0g𝑆)    &   𝑅 = (Base‘𝑆)    &   𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   (𝜑𝐼𝐵)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   (𝜑 → ((𝐽𝑌)‘𝐼) ≠ 𝑄)    &   𝐹 = (invr𝑆)    &    = (-g𝐷)    &   𝐶 = ((𝐽𝑋) (((𝐹‘((𝐽𝑌)‘𝐼))(.r𝑆)((𝐽𝑋)‘𝐼))( ·𝑠𝐷)(𝐽𝑌)))       (𝜑 → (𝑋 + 𝑌) ∈ ( ‘(𝐿𝐶)))

Theoremlcfrlem37 37185* Lemma for lcfr 37191. (Contributed by NM, 8-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    + = (+g𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐴 = (LSAtoms‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ‘{(𝑋 + 𝑌)}))    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑄 = (0g𝑆)    &   𝑅 = (Base‘𝑆)    &   𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))    &   (𝜑𝐼𝐵)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   (𝜑 → ((𝐽𝑌)‘𝐼) ≠ 𝑄)    &   𝐹 = (invr𝑆)    &    = (-g𝐷)    &   𝐶 = ((𝐽𝑋) (((𝐹‘((𝐽𝑌)‘𝐼))(.r𝑆)((𝐽𝑋)‘𝐼))( ·𝑠𝐷)(𝐽𝑌)))    &   (𝜑𝐺 ∈ (LSubSp‘𝐷))    &   (𝜑𝐺 ⊆ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)})    &   𝐸 = 𝑔𝐺 ( ‘(𝐿𝑔))    &   (𝜑𝑋𝐸)    &   (𝜑𝑌𝐸)       (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Theoremlcfrlem38 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)    &    + = (+g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑄 = (LSubSp‘𝐷)    &   𝐶 = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   𝐸 = 𝑔𝐺 ( ‘(𝐿𝑔))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝑄)    &   (𝜑𝐺𝐶)    &   (𝜑𝑋𝐸)    &   (𝜑𝑌𝐸)    &    0 = (0g𝑈)    &   (𝜑𝑋0 )    &   (𝜑𝑌0 )    &   𝑁 = (LSpan‘𝑈)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ‘{(𝑋 + 𝑌)}))    &   (𝜑𝐼𝐵)    &   (𝜑𝐼0 )    &   𝑉 = (Base‘𝑈)    &    · = ( ·𝑠𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝑆)    &   𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))       (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Theoremlcfrlem39 37187* Lemma for lcfr 37191. Eliminate 𝐽. (Contributed by NM, 11-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    + = (+g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑄 = (LSubSp‘𝐷)    &   𝐶 = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   𝐸 = 𝑔𝐺 ( ‘(𝐿𝑔))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝑄)    &   (𝜑𝐺𝐶)    &   (𝜑𝑋𝐸)    &   (𝜑𝑌𝐸)    &    0 = (0g𝑈)    &   (𝜑𝑋0 )    &   (𝜑𝑌0 )    &   𝑁 = (LSpan‘𝑈)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ‘{(𝑋 + 𝑌)}))    &   (𝜑𝐼𝐵)    &   (𝜑𝐼0 )       (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Theoremlcfrlem40 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)    &    + = (+g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑄 = (LSubSp‘𝐷)    &   𝐶 = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   𝐸 = 𝑔𝐺 ( ‘(𝐿𝑔))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝑄)    &   (𝜑𝐺𝐶)    &   (𝜑𝑋𝐸)    &   (𝜑𝑌𝐸)    &    0 = (0g𝑈)    &   (𝜑𝑋0 )    &   (𝜑𝑌0 )    &   𝑁 = (LSpan‘𝑈)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))       (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Theoremlcfrlem41 37189* Lemma for lcfr 37191. Eliminate span condition. (Contributed by NM, 11-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    + = (+g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑄 = (LSubSp‘𝐷)    &   𝐶 = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   𝐸 = 𝑔𝐺 ( ‘(𝐿𝑔))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝑄)    &   (𝜑𝐺𝐶)    &   (𝜑𝑋𝐸)    &   (𝜑𝑌𝐸)    &    0 = (0g𝑈)    &   (𝜑𝑋0 )    &   (𝜑𝑌0 )       (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Theoremlcfrlem42 37190* Lemma for lcfr 37191. Eliminate nonzero condition. (Contributed by NM, 11-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    + = (+g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑄 = (LSubSp‘𝐷)    &   𝐶 = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   𝐸 = 𝑔𝐺 ( ‘(𝐿𝑔))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝑄)    &   (𝜑𝐺𝐶)    &   (𝜑𝑋𝐸)    &   (𝜑𝑌𝐸)       (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Theoremlcfr 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 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   𝑄 = 𝑔𝑅 ( ‘(𝐿𝑔))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑅𝑇)    &   (𝜑𝑅𝐶)       (𝜑𝑄𝑆)

Syntaxclcd 37192 Extend class notation with vector space of functionals with closed kernels.
class LCDual

Definitiondf-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)})))

Theoremlcdfval 37194* Dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHyp‘𝐾)       (𝐾𝑋 → (LCDual‘𝐾) = (𝑤𝐻 ↦ ((LDual‘((DVecH‘𝐾)‘𝑤)) ↾s {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑤)) ∣ (((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)})))

Theoremlcdval 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 → (𝐾𝑋𝑊𝐻))       (𝜑𝐶 = (𝐷s {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}))

Theoremlcdval2 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 → (𝐾𝑋𝑊𝐻))    &   𝐵 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}       (𝜑𝐶 = (𝐷s 𝐵))

Theoremlcdlvec 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 ∧ 𝑊𝐻))       (𝜑𝐶 ∈ LVec)

Theoremlcdlmod 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 ∧ 𝑊𝐻))       (𝜑𝐶 ∈ LMod)

Theoremlcdvbase 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 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑉 = 𝐵)

Theoremlcdvbasess 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 ∧ 𝑊𝐻))       (𝜑𝑉𝐹)

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