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Theorem List for Metamath Proof Explorer - 35701-35800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlcfl8b 35701* 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 35702 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 35703* 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 35704 Lemma for lclkr 35730. Use lshpat 33251 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 35705 Lemma for lclkr 35730. (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 35706 Lemma for lclkr 35730. (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 35707 Lemma for lclkr 35730. (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 35708 Lemma for lclkr 35730. 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 35709 Lemma for lclkr 35730. 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 35710 Lemma for lclkr 35730. 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 35711 Lemma for lclkr 35730. 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 35712 Lemma for lclkr 35730. 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 35713 Lemma for lclkr 35730. 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 35714 Lemma for lclkr 35730. 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 35715 Lemma for lclkr 35730. 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 35716 Lemma for lclkr 35730. 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 35717 Lemma for lclkr 35730. (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 35718 Lemma for lclkr 35730. 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 35719 Lemma for lclkr 35730. 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 35720 Lemma for lclkr 35730. 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 35721 Lemma for lclkr 35730. 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 35722 Lemma for lclkr 35730. 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 35723 Lemma for lclkr 35730. 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 35724 Lemma for lclkr 35730. lclkrlem2t 35723 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 35725 Lemma for lclkr 35730. When the hypotheses of lclkrlem2u 35724 and lclkrlem2u 35724 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 35665, which requires the orthomodular law dihoml4 35574 (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 35726 Lemma for lclkr 35730. This is the same as lclkrlem2u 35724 and lclkrlem2u 35724 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 35727 Lemma for lclkr 35730. Eliminate by cases the hypotheses of lclkrlem2u 35724, lclkrlem2u 35724 and lclkrlem2w 35726. (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 35728 Lemma for lclkr 35730. 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 35729* The set of functionals having closed kernels is closed under vector (functional) addition. Lemmas lclkrlem2a 35704 through lclkrlem2y 35728 are used for the proof. Here we express lclkrlem2y 35728 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 35730* 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 35731* 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 35732* 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 35733* 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 35734* 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 35735* 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 35736* 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 35730 proof. Do we achieve overall shortening by breaking them out as subtheorems? Or make lclkr 35730 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 35737* 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 35825. (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 35738* 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 35739 Lemma for lcfr 35782. 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 35740 Lemma for lcfr 35782. (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 35741 Lemma for lcfr 35782. (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 35742* Lemma for lcfr 35782. (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 35743* Lemma for lcfr 35782. 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 35744* Lemma for lcfr 35782. 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 35745* Lemma for lcfr 35782. 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 35746* Lemma for lcf1o 35748 and lcfr 35782. (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 35747* Lemma for lcf1o 35748. (This part has undesirable \$d's on 𝐽 and 𝜑 that we remove in lcf1o 35748.) 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 35748* 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 35749* Lemma for lcfr 35782. (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 35750* Lemma for lcfr 35782. (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 35751* Lemma for lcfr 35782. (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 35752* Lemma for lcfr 35782. (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 35753* Lemma for lcfr 35782. (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 35754* Lemma for lcfr 35782. (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 35755* Lemma for lcfr 35782. (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 35756 Lemma for lcfr 35782. 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 35757 Lemma for lcfr 35782. (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 35758 Lemma for lcfr 35782. (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 35759 Lemma for lcfr 35782. (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 35760 Lemma for lcfr 35782. (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 35761 Lemma for lcfr 35782. (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 35762 Lemma for lcfr 35782. 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 35763* Lemma for lcfr 35782. (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 35764* Lemma for lcfr 35782. Special case of lcfrlem35 35774 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 35765* Lemma for lcfr 35782. Special case of lcfrlem36 35775 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 35766* Lemma for lcfr 35782. Special case of lcfrlem37 35776 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 35767* Lemma for lcfr 35782. 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 35768* Lemma for lcfr 35782. (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 35769* Lemma for lcfr 35782. (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 35770* Lemma for lcfr 35782. (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 35771* Lemma for lcfr 35782. (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 35772* Lemma for lcfr 35782. (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 35773* Lemma for lcfr 35782. (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 35774* Lemma for lcfr 35782. (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 35775* Lemma for lcfr 35782. (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 35776* Lemma for lcfr 35782. (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 35777* Lemma for lcfr 35782. Combine lcfrlem27 35766 and lcfrlem37 35776. (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 35778* Lemma for lcfr 35782. 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 35779* Lemma for lcfr 35782. 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 35780* Lemma for lcfr 35782. 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 35781* Lemma for lcfr 35782. 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 35782* 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 35783 Extend class notation with vector space of functionals with closed kernels.
class LCDual

Definitiondf-lcdual 35784* Dual vector space of functionals with closed kernels. Note: we could also define this directly without mapd by using mapdrn 35846. TODO: see if it makes sense to go back and replace some of the LDual stuff with this. TODO: We could simplify df-mapd 35822 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 35785* 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 35786* 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 35787* 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 35788 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 35789 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 35790* 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 35791 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)

Theoremlcdvbaselfl 35792 A vector in the base set of the closed kernel dual space is a functional. (Contributed by NM, 28-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝐶)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (LFnl‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)       (𝜑𝑋𝐹)

Theoremlcdvbasecl 35793 Closure of the value of a vector (functional) in the closed kernel dual space. (Contributed by NM, 28-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝑆)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐸 = (Base‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐸)    &   (𝜑𝑋𝑉)       (𝜑 → (𝐹𝑋) ∈ 𝑅)

Theoremlcdvadd 35794 Vector addition for the closed kernel vector space dual. (Contributed by NM, 10-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐷 = (LDual‘𝑈)    &    + = (+g𝐷)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = (+g𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 = + )

Theoremlcdvaddval 35795 The value of the value of vector addition in the closed kernel vector space dual. (Contributed by NM, 10-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑅 = (Scalar‘𝑈)    &    + = (+g𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &    = (+g𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝐺𝐷)    &   (𝜑𝑋𝑉)       (𝜑 → ((𝐹 𝐺)‘𝑋) = ((𝐹𝑋) + (𝐺𝑋)))

Theoremlcdsca 35796 The ring of scalars of the closed kernel dual space. (Contributed by NM, 16-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)    &   𝑂 = (oppr𝐹)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑅 = (Scalar‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑅 = 𝑂)

Theoremlcdsbase 35797 Base set of scalar ring for the closed kernel dual of a vector space. (Contributed by NM, 18-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)    &   𝐿 = (Base‘𝐹)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑆 = (Scalar‘𝐶)    &   𝑅 = (Base‘𝑆)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑅 = 𝐿)

Theoremlcdsadd 35798 Scalar addition for the closed kernel vector space dual. (Contributed by NM, 6-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)    &    + = (+g𝐹)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑆 = (Scalar‘𝐶)    &    = (+g𝑆)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 = + )

Theoremlcdsmul 35799 Scalar multiplication for the closed kernel vector space dual. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)    &   𝐿 = (Base‘𝐹)    &    · = (.r𝐹)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑆 = (Scalar‘𝐶)    &    = (.r𝑆)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝐿)    &   (𝜑𝑌𝐿)       (𝜑 → (𝑋 𝑌) = (𝑌 · 𝑋))

Theoremlcdvs 35800 Scalar product for the closed kernel vector space dual. (Contributed by NM, 28-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐷 = (LDual‘𝑈)    &    · = ( ·𝑠𝐷)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    = ( ·𝑠𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 = · )

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