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Theorem List for Metamath Proof Explorer - 35801-35900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlcfl3 35801* Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝐴 = (LSAtoms‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐶 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)       (𝜑 → (𝐺𝐶 ↔ (( ‘(𝐿𝐺)) ∈ 𝐴 ∨ (𝐿𝐺) = 𝑉)))

Theoremlcfl4N 35802* 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 35803* 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 35804 Property of a functional with a closed kernel. TODO: Make lcfl5 35803 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 35805* Lemma for lcfl6 35807. A functional 𝐺 (whose kernel is closed by dochsnkr 35779) 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 35806* Lemma for lcfl7N 35808. 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 35807* Property of a functional with a closed kernel. Note that (𝐿𝐺) = 𝑉 means the functional is zero by lkr0f 33399. (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 35808* 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 33399. (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 35809* 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 35810* 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 35811* 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 35812 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 35813* 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 35814 Lemma for lclkr 35840. Use lshpat 33361 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 35815 Lemma for lclkr 35840. (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 35816 Lemma for lclkr 35840. (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 35817 Lemma for lclkr 35840. (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 35818 Lemma for lclkr 35840. 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 35819 Lemma for lclkr 35840. 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 35820 Lemma for lclkr 35840. 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 35821 Lemma for lclkr 35840. 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 35822 Lemma for lclkr 35840. 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 35823 Lemma for lclkr 35840. 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 35824 Lemma for lclkr 35840. 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 35825 Lemma for lclkr 35840. 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 35826 Lemma for lclkr 35840. 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 35827 Lemma for lclkr 35840. (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 35828 Lemma for lclkr 35840. 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 35829 Lemma for lclkr 35840. 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 35830 Lemma for lclkr 35840. 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 35831 Lemma for lclkr 35840. 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 35832 Lemma for lclkr 35840. 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 35833 Lemma for lclkr 35840. 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 35834 Lemma for lclkr 35840. lclkrlem2t 35833 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 35835 Lemma for lclkr 35840. When the hypotheses of lclkrlem2u 35834 and lclkrlem2u 35834 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 35775, which requires the orthomodular law dihoml4 35684 (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 35836 Lemma for lclkr 35840. This is the same as lclkrlem2u 35834 and lclkrlem2u 35834 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 35837 Lemma for lclkr 35840. Eliminate by cases the hypotheses of lclkrlem2u 35834, lclkrlem2u 35834 and lclkrlem2w 35836. (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 35838 Lemma for lclkr 35840. 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 35839* The set of functionals having closed kernels is closed under vector (functional) addition. Lemmas lclkrlem2a 35814 through lclkrlem2y 35838 are used for the proof. Here we express lclkrlem2y 35838 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 35840* 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 35841* 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 35842* 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 35843* 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 35844* 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 35845* 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 35846* 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 35840 proof. Do we achieve overall shortening by breaking them out as subtheorems? Or make lclkr 35840 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 35847* 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 35935. (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 35848* 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 35849 Lemma for lcfr 35892. 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 35850 Lemma for lcfr 35892. (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 35851 Lemma for lcfr 35892. (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 35852* Lemma for lcfr 35892. (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 35853* Lemma for lcfr 35892. 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 35854* Lemma for lcfr 35892. 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 35855* Lemma for lcfr 35892. 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 35856* Lemma for lcf1o 35858 and lcfr 35892. (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 35857* Lemma for lcf1o 35858. (This part has undesirable \$d's on 𝐽 and 𝜑 that we remove in lcf1o 35858.) 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 35858* 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 35859* Lemma for lcfr 35892. (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 35860* Lemma for lcfr 35892. (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 35861* Lemma for lcfr 35892. (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 35862* Lemma for lcfr 35892. (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 35863* Lemma for lcfr 35892. (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 35864* Lemma for lcfr 35892. (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 35865* Lemma for lcfr 35892. (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 35866 Lemma for lcfr 35892. 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 35867 Lemma for lcfr 35892. (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 35868 Lemma for lcfr 35892. (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 35869 Lemma for lcfr 35892. (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 35870 Lemma for lcfr 35892. (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 35871 Lemma for lcfr 35892. (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 35872 Lemma for lcfr 35892. 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 35873* Lemma for lcfr 35892. (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 35874* Lemma for lcfr 35892. Special case of lcfrlem35 35884 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 35875* Lemma for lcfr 35892. Special case of lcfrlem36 35885 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 35876* Lemma for lcfr 35892. Special case of lcfrlem37 35886 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 35877* Lemma for lcfr 35892. 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 35878* Lemma for lcfr 35892. (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 35879* Lemma for lcfr 35892. (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 35880* Lemma for lcfr 35892. (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 35881* Lemma for lcfr 35892. (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 35882* Lemma for lcfr 35892. (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 35883* Lemma for lcfr 35892. (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 35884* Lemma for lcfr 35892. (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 35885* Lemma for lcfr 35892. (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 35886* Lemma for lcfr 35892. (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 35887* Lemma for lcfr 35892. Combine lcfrlem27 35876 and lcfrlem37 35886. (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 35888* Lemma for lcfr 35892. 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 35889* Lemma for lcfr 35892. 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 35890* Lemma for lcfr 35892. 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 35891* Lemma for lcfr 35892. 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 35892* 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 35893 Extend class notation with vector space of functionals with closed kernels.
class LCDual

Definitiondf-lcdual 35894* Dual vector space of functionals with closed kernels. Note: we could also define this directly without mapd by using mapdrn 35956. TODO: see if it makes sense to go back and replace some of the LDual stuff with this. TODO: We could simplify df-mapd 35932 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 35895* 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 35896* 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 35897* 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 35898 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 35899 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 35900* 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)

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