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Theorem List for Metamath Proof Explorer - 37601-37700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlclkrlem2s 37601 Lemma for lclkr 37609. 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 37602 Lemma for lclkr 37609. 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 37603 Lemma for lclkr 37609. lclkrlem2t 37602 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 37604 Lemma for lclkr 37609. When the hypotheses of lclkrlem2u 37603 and lclkrlem2u 37603 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 37544, which requires the orthomodular law dihoml4 37453 (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 37605 Lemma for lclkr 37609. This is the same as lclkrlem2u 37603 and lclkrlem2u 37603 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 37606 Lemma for lclkr 37609. Eliminate by cases the hypotheses of lclkrlem2u 37603, lclkrlem2u 37603 and lclkrlem2w 37605. (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 37607 Lemma for lclkr 37609. 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 37608* The set of functionals having closed kernels is closed under vector (functional) addition. Lemmas lclkrlem2a 37583 through lclkrlem2y 37607 are used for the proof. Here we express lclkrlem2y 37607 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 37609* 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 37610* 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 37611* 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 37612* 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 37613* 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 37614* 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 37615* 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 37609 proof. Do we achieve overall shortening by breaking them out as subtheorems? Or make lclkr 37609 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 37616* 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 37704. (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 37617* 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 37618 Lemma for lcfr 37661. 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 37619 Lemma for lcfr 37661. (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 37620 Lemma for lcfr 37661. (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 37621* Lemma for lcfr 37661. (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 37622* Lemma for lcfr 37661. 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 37623* Lemma for lcfr 37661. 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 37624* Lemma for lcfr 37661. 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 37625* Lemma for lcf1o 37627 and lcfr 37661. (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 37626* Lemma for lcf1o 37627. (This part has undesirable \$d's on 𝐽 and 𝜑 that we remove in lcf1o 37627.) 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 37627* 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 37628* Lemma for lcfr 37661. (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 37629* Lemma for lcfr 37661. (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 37630* Lemma for lcfr 37661. (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 37631* Lemma for lcfr 37661. (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 37632* Lemma for lcfr 37661. (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 37633* Lemma for lcfr 37661. (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 37634* Lemma for lcfr 37661. (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 37635 Lemma for lcfr 37661. 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 37636 Lemma for lcfr 37661. (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 37637 Lemma for lcfr 37661. (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 37638 Lemma for lcfr 37661. (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 37639 Lemma for lcfr 37661. (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 37640 Lemma for lcfr 37661. (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 37641 Lemma for lcfr 37661. 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 37642* Lemma for lcfr 37661. (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 37643* Lemma for lcfr 37661. Special case of lcfrlem35 37653 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 37644* Lemma for lcfr 37661. Special case of lcfrlem36 37654 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 37645* Lemma for lcfr 37661. Special case of lcfrlem37 37655 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 37646* Lemma for lcfr 37661. 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 37647* Lemma for lcfr 37661. (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 37648* Lemma for lcfr 37661. (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 37649* Lemma for lcfr 37661. (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 37650* Lemma for lcfr 37661. (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 37651* Lemma for lcfr 37661. (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 37652* Lemma for lcfr 37661. (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 37653* Lemma for lcfr 37661. (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 37654* Lemma for lcfr 37661. (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 37655* Lemma for lcfr 37661. (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 37656* Lemma for lcfr 37661. Combine lcfrlem27 37645 and lcfrlem37 37655. (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 37657* Lemma for lcfr 37661. 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 37658* Lemma for lcfr 37661. 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 37659* Lemma for lcfr 37661. 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 37660* Lemma for lcfr 37661. 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 37661* 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 37662 Extend class notation with vector space of functionals with closed kernels.
class LCDual

Definitiondf-lcdual 37663* Dual vector space of functionals with closed kernels. Note: we could also define this directly without mapd by using mapdrn 37725. TODO: see if it makes sense to go back and replace some of the LDual stuff with this. TODO: We could simplify df-mapd 37701 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 37664* 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 37665* 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 37666* 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 37667 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 37668 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 37669* 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 37670 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 37671 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 37672 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 37673 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 37674 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 37675 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 37676 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 37677 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 37678 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 37679 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 = · )

Theoremlcdvsval 37680 Value of scalar product operation value for the closed kernel vector space dual. (Contributed by NM, 28-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑆 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝑆)    &    · = (.r𝑆)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐹 = (Base‘𝐶)    &    = ( ·𝑠𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑅)    &   (𝜑𝐺𝐹)    &   (𝜑𝐴𝑉)       (𝜑 → ((𝑋 𝐺)‘𝐴) = ((𝐺𝐴) · 𝑋))

Theoremlcdvscl 37681 The scalar product operation value is a functional. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝑆)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝐶)    &    · = ( ·𝑠𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑅)    &   (𝜑𝐺𝑉)       (𝜑 → (𝑋 · 𝐺) ∈ 𝑉)

Theoremlcdlssvscl 37682 Closure of scalar product in a closed kernel dual vector space. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝐹)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝐶)    &    · = ( ·𝑠𝐶)    &   𝑆 = (LSubSp‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐿𝑆)    &   (𝜑𝑋𝑅)    &   (𝜑𝑌𝐿)       (𝜑 → (𝑋 · 𝑌) ∈ 𝐿)

Theoremlcdvsass 37683 Associative law for scalar product in a closed kernel dual vector space. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑅 = (Scalar‘𝑈)    &   𝐿 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐹 = (Base‘𝐶)    &    = ( ·𝑠𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝐿)    &   (𝜑𝑌𝐿)    &   (𝜑𝐺𝐹)       (𝜑 → ((𝑌 · 𝑋) 𝐺) = (𝑋 (𝑌 𝐺)))

Theoremlcd0 37684 The zero scalar of the closed kernel dual of a vector space. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)    &    0 = (0g𝐹)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑆 = (Scalar‘𝐶)    &   𝑂 = (0g𝑆)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑂 = 0 )

Theoremlcd1 37685 The unit scalar of the closed kernel dual of a vector space. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)    &    1 = (1r𝐹)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑆 = (Scalar‘𝐶)    &   𝐼 = (1r𝑆)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝐼 = 1 )

Theoremlcdneg 37686 The unit scalar of the closed kernel dual of a vector space. (Contributed by NM, 11-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑅 = (Scalar‘𝑈)    &   𝑀 = (invg𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑆 = (Scalar‘𝐶)    &   𝑁 = (invg𝑆)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑁 = 𝑀)

Theoremlcd0v 37687 The zero functional in the set of functionals with closed kernels. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑅 = (Scalar‘𝑈)    &    0 = (0g𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑂 = (0g𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑂 = (𝑉 × { 0 }))

Theoremlcd0v2 37688 The zero functional in the set of functionals with closed kernels. (Contributed by NM, 27-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐷 = (LDual‘𝑈)    &    0 = (0g𝐷)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑂 = (0g𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑂 = 0 )

Theoremlcd0vvalN 37689 Value of the zero functional at any vector. (Contributed by NM, 28-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑆 = (Scalar‘𝑈)    &    0 = (0g𝑆)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑂 = (0g𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)       (𝜑 → (𝑂𝑋) = 0 )

Theoremlcd0vcl 37690 Closure of the zero functional in the set of functionals with closed kernels. (Contributed by NM, 15-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝐶)    &   𝑂 = (0g𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑂𝑉)

Theoremlcd0vs 37691 A scalar zero times a functional is the zero functional. (Contributed by NM, 20-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑅 = (Scalar‘𝑈)    &    0 = (0g𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝐶)    &    · = ( ·𝑠𝐶)    &   𝑂 = (0g𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝑉)       (𝜑 → ( 0 · 𝐺) = 𝑂)

Theoremlcdvs0N 37692 A scalar times the zero functional is the zero functional. (Contributed by NM, 20-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝑆)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    · = ( ·𝑠𝐶)    &    0 = (0g𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑅)       (𝜑 → (𝑋 · 0 ) = 0 )

Theoremlcdvsub 37693 The value of vector subtraction in the closed kernel dual space. (Contributed by NM, 22-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (Scalar‘𝑈)    &   𝑁 = (invg𝑆)    &    1 = (1r𝑆)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝐶)    &    + = (+g𝐶)    &    · = ( ·𝑠𝐶)    &    = (-g𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝑉)    &   (𝜑𝐺𝑉)       (𝜑 → (𝐹 𝐺) = (𝐹 + ((𝑁1 ) · 𝐺)))

Theoremlcdvsubval 37694 The value of the value of vector addition in the closed kernel vector space dual. (Contributed by NM, 11-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑅 = (Scalar‘𝑈)    &   𝑆 = (-g𝑅)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐷 = (Base‘𝐶)    &    = (-g𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐹𝐷)    &   (𝜑𝐺𝐷)    &   (𝜑𝑋𝑉)       (𝜑 → ((𝐹 𝐺)‘𝑋) = ((𝐹𝑋)𝑆(𝐺𝑋)))

Theoremlcdlss 37695* Subspaces of a dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑂 = ((ocH‘𝐾)‘𝑊)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝐶)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (LDual‘𝑈)    &   𝑇 = (LSubSp‘𝐷)    &   𝐵 = {𝑓𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑆 = (𝑇 ∩ 𝒫 𝐵))

Theoremlcdlss2N 37696 Subspaces of a dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝐶)    &   𝑉 = (Base‘𝐶)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐷 = (LDual‘𝑈)    &   𝑇 = (LSubSp‘𝐷)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝑆 = (𝑇 ∩ 𝒫 𝑉))

Theoremlcdlsp 37697 Span in the set of functionals with closed kernels. (Contributed by NM, 28-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐷 = (LDual‘𝑈)    &   𝑀 = (LSpan‘𝐷)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝐹 = (Base‘𝐶)    &   𝑁 = (LSpan‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)       (𝜑 → (𝑁𝐺) = (𝑀𝐺))

TheoremlcdlkreqN 37698 Colinear functionals have equal kernels. (Contributed by NM, 28-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐿 = (LKer‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &    0 = (0g𝐶)    &   𝑁 = (LSpan‘𝐶)    &   𝑉 = (Base‘𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐼𝑉)    &   (𝜑𝐺 ∈ (𝑁‘{𝐼}))    &   (𝜑𝐺0 )       (𝜑 → (𝐿𝐺) = (𝐿𝐼))

Theoremlcdlkreq2N 37699 Colinear functionals have equal kernels. (Contributed by NM, 28-Mar-2015.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝑆)    &    0 = (0g𝑆)    &   𝐿 = (LKer‘𝑈)    &   𝐶 = ((LCDual‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝐶)    &    · = ( ·𝑠𝐶)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐴 ∈ (𝑅 ∖ { 0 }))    &   (𝜑𝐼𝑉)    &   (𝜑𝐺 = (𝐴 · 𝐼))       (𝜑 → (𝐿𝐺) = (𝐿𝐼))

Syntaxcmpd 37700 Extend class notation with projectivity from subspaces of vector space H to subspaces of functionals with closed kernels.
class mapd

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