P. 369 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 36801-36900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	lclkrlem2f 36801	Lemma for lclkr 36822. Construct a closed hyperplane under the kernel of the sum. (Contributed by NM, 16-Jan-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- S = (Scalar ` U )   &    |- Q = ( 0g ` S )   &    |- .0. = ( 0g ` U )   &    |- .(+) = ( LSSum ` U )   &    |- N = ( LSpan ` U )   &    |- F = (LFnl ` U )   &    |- J = (LSHyp ` U )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- .+ = ( +g ` D )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> B e. ( V \ { .0. } ) )   &    |- ( ph -> E e. F )   &    |- ( ph -> G e. F )   &    |- ( ph -> ( L ` E ) = ( ._|_ ` { X } ) )   &    |- ( ph -> ( L ` G ) = ( ._|_ ` { Y } ) )   &    |- ( ph -> ( ( E .+ G ) ` B ) = Q )   &    |- ( ph -> ( -. X e. ( ._|_ ` { B } ) \/ -. Y e. ( ._|_ ` { B } ) ) )   &    |- ( ph -> X e. ( V \ { .0. } ) )   &    |- ( ph -> Y e. ( V \ { .0. } ) )   &    |- ( ph -> ( L ` E ) =/= ( L ` G ) )   &    |- ( ph -> ( L ` ( E .+ G ) ) e. J )   =>    |- ( ph -> ( ( ( L ` E ) i^i ( L ` G ) ) .(+) ( N ` { B } ) ) C_ ( L ` ( E .+ G ) ) )
Theorem	lclkrlem2g 36802	Lemma for lclkr 36822. Comparable hyperplanes are equal, so the kernel of the sum is closed. (Contributed by NM, 16-Jan-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- S = (Scalar ` U )   &    |- Q = ( 0g ` S )   &    |- .0. = ( 0g ` U )   &    |- .(+) = ( LSSum ` U )   &    |- N = ( LSpan ` U )   &    |- F = (LFnl ` U )   &    |- J = (LSHyp ` U )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- .+ = ( +g ` D )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> B e. ( V \ { .0. } ) )   &    |- ( ph -> E e. F )   &    |- ( ph -> G e. F )   &    |- ( ph -> ( L ` E ) = ( ._|_ ` { X } ) )   &    |- ( ph -> ( L ` G ) = ( ._|_ ` { Y } ) )   &    |- ( ph -> ( ( E .+ G ) ` B ) = Q )   &    |- ( ph -> ( -. X e. ( ._|_ ` { B } ) \/ -. Y e. ( ._|_ ` { B } ) ) )   &    |- ( ph -> X e. ( V \ { .0. } ) )   &    |- ( ph -> Y e. ( V \ { .0. } ) )   &    |- ( ph -> ( L ` E ) =/= ( L ` G ) )   &    |- ( ph -> ( L ` ( E .+ G ) ) e. J )   =>    |- ( ph -> ( ._|_ ` ( ._|_ ` ( L ` ( E .+ G ) ) ) ) = ( L ` ( E .+ G ) ) )
Theorem	lclkrlem2h 36803	Lemma for lclkr 36822. Eliminate the ( L ` ( E .+ G ) ) e. J hypothesis. (Contributed by NM, 16-Jan-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- S = (Scalar ` U )   &    |- Q = ( 0g ` S )   &    |- .0. = ( 0g ` U )   &    |- .(+) = ( LSSum ` U )   &    |- N = ( LSpan ` U )   &    |- F = (LFnl ` U )   &    |- J = (LSHyp ` U )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- .+ = ( +g ` D )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> B e. ( V \ { .0. } ) )   &    |- ( ph -> E e. F )   &    |- ( ph -> G e. F )   &    |- ( ph -> ( L ` E ) = ( ._|_ ` { X } ) )   &    |- ( ph -> ( L ` G ) = ( ._|_ ` { Y } ) )   &    |- ( ph -> ( ( E .+ G ) ` B ) = Q )   &    |- ( ph -> ( -. X e. ( ._|_ ` { B } ) \/ -. Y e. ( ._|_ ` { B } ) ) )   &    |- ( ph -> X e. ( V \ { .0. } ) )   &    |- ( ph -> Y e. ( V \ { .0. } ) )   &    |- ( ph -> ( L ` E ) =/= ( L ` G ) )   =>    |- ( ph -> ( ._|_ ` ( ._|_ ` ( L ` ( E .+ G ) ) ) ) = ( L ` ( E .+ G ) ) )
Theorem	lclkrlem2i 36804	Lemma for lclkr 36822. Eliminate the ( L ` E ) =/= ( L ` G ) hypothesis. (Contributed by NM, 17-Jan-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- S = (Scalar ` U )   &    |- Q = ( 0g ` S )   &    |- .0. = ( 0g ` U )   &    |- .(+) = ( LSSum ` U )   &    |- N = ( LSpan ` U )   &    |- F = (LFnl ` U )   &    |- J = (LSHyp ` U )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- .+ = ( +g ` D )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> B e. ( V \ { .0. } ) )   &    |- ( ph -> E e. F )   &    |- ( ph -> G e. F )   &    |- ( ph -> ( L ` E ) = ( ._|_ ` { X } ) )   &    |- ( ph -> ( L ` G ) = ( ._|_ ` { Y } ) )   &    |- ( ph -> ( ( E .+ G ) ` B ) = Q )   &    |- ( ph -> ( -. X e. ( ._|_ ` { B } ) \/ -. Y e. ( ._|_ ` { B } ) ) )   &    |- ( ph -> X e. ( V \ { .0. } ) )   &    |- ( ph -> Y e. ( V \ { .0. } ) )   =>    |- ( ph -> ( ._|_ ` ( ._|_ ` ( L ` ( E .+ G ) ) ) ) = ( L ` ( E .+ G ) ) )
Theorem	lclkrlem2j 36805	Lemma for lclkr 36822. Kernel closure when Y is zero. (Contributed by NM, 18-Jan-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- S = (Scalar ` U )   &    |- Q = ( 0g ` S )   &    |- .0. = ( 0g ` U )   &    |- .(+) = ( LSSum ` U )   &    |- N = ( LSpan ` U )   &    |- F = (LFnl ` U )   &    |- J = (LSHyp ` U )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- .+ = ( +g ` D )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> B e. ( V \ { .0. } ) )   &    |- ( ph -> E e. F )   &    |- ( ph -> G e. F )   &    |- ( ph -> ( L ` E ) = ( ._|_ ` { X } ) )   &    |- ( ph -> ( L ` G ) = ( ._|_ ` { Y } ) )   &    |- ( ph -> ( ( E .+ G ) ` B ) = Q )   &    |- ( ph -> ( -. X e. ( ._|_ ` { B } ) \/ -. Y e. ( ._|_ ` { B } ) ) )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y = .0. )   =>    |- ( ph -> ( ._|_ ` ( ._|_ ` ( L ` ( E .+ G ) ) ) ) = ( L ` ( E .+ G ) ) )
Theorem	lclkrlem2k 36806	Lemma for lclkr 36822. Kernel closure when X is zero. (Contributed by NM, 18-Jan-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- S = (Scalar ` U )   &    |- Q = ( 0g ` S )   &    |- .0. = ( 0g ` U )   &    |- .(+) = ( LSSum ` U )   &    |- N = ( LSpan ` U )   &    |- F = (LFnl ` U )   &    |- J = (LSHyp ` U )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- .+ = ( +g ` D )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> B e. ( V \ { .0. } ) )   &    |- ( ph -> E e. F )   &    |- ( ph -> G e. F )   &    |- ( ph -> ( L ` E ) = ( ._|_ ` { X } ) )   &    |- ( ph -> ( L ` G ) = ( ._|_ ` { Y } ) )   &    |- ( ph -> ( ( E .+ G ) ` B ) = Q )   &    |- ( ph -> ( -. X e. ( ._|_ ` { B } ) \/ -. Y e. ( ._|_ ` { B } ) ) )   &    |- ( ph -> X = .0. )   &    |- ( ph -> Y e. V )   =>    |- ( ph -> ( ._|_ ` ( ._|_ ` ( L ` ( E .+ G ) ) ) ) = ( L ` ( E .+ G ) ) )
Theorem	lclkrlem2l 36807	Lemma for lclkr 36822. Eliminate the X =/= .0., Y =/= .0. hypotheses. (Contributed by NM, 18-Jan-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- S = (Scalar ` U )   &    |- Q = ( 0g ` S )   &    |- .0. = ( 0g ` U )   &    |- .(+) = ( LSSum ` U )   &    |- N = ( LSpan ` U )   &    |- F = (LFnl ` U )   &    |- J = (LSHyp ` U )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- .+ = ( +g ` D )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> B e. ( V \ { .0. } ) )   &    |- ( ph -> E e. F )   &    |- ( ph -> G e. F )   &    |- ( ph -> ( L ` E ) = ( ._|_ ` { X } ) )   &    |- ( ph -> ( L ` G ) = ( ._|_ ` { Y } ) )   &    |- ( ph -> ( ( E .+ G ) ` B ) = Q )   &    |- ( ph -> ( -. X e. ( ._|_ ` { B } ) \/ -. Y e. ( ._|_ ` { B } ) ) )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   =>    |- ( ph -> ( ._|_ ` ( ._|_ ` ( L ` ( E .+ G ) ) ) ) = ( L ` ( E .+ G ) ) )
Theorem	lclkrlem2m 36808	Lemma for lclkr 36822. Construct a vector B that makes the sum of functionals zero. Combine with B e. V to shorten overall proof. (Contributed by NM, 17-Jan-2015.)
|- V = ( Base ` U )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- .X. = ( .r ` S )   &    |- .0. = ( 0g ` S )   &    |- I = ( invr ` S )   &    |- .- = ( -g ` U )   &    |- F = (LFnl ` U )   &    |- D = (LDual ` U )   &    |- .+ = ( +g ` D )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ph -> E e. F )   &    |- ( ph -> G e. F )   &    |- ( ph -> U e. LVec )   &    |- B = ( X .- ( ( ( ( E .+ G ) ` X ) .X. ( I ` ( ( E .+ G ) ` Y ) ) ) .x. Y ) )   &    |- ( ph -> ( ( E .+ G ) ` Y ) =/= .0. )   =>    |- ( ph -> ( B e. V /\ ( ( E .+ G ) ` B ) = .0. ) )
Theorem	lclkrlem2n 36809	Lemma for lclkr 36822. (Contributed by NM, 12-Jan-2015.)
|- V = ( Base ` U )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- .X. = ( .r ` S )   &    |- .0. = ( 0g ` S )   &    |- I = ( invr ` S )   &    |- .- = ( -g ` U )   &    |- F = (LFnl ` U )   &    |- D = (LDual ` U )   &    |- .+ = ( +g ` D )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ph -> E e. F )   &    |- ( ph -> G e. F )   &    |- N = ( LSpan ` U )   &    |- L = (LKer ` U )   &    |- ( ph -> U e. LVec )   &    |- ( ph -> ( ( E .+ G ) ` X ) = .0. )   &    |- ( ph -> ( ( E .+ G ) ` Y ) = .0. )   =>    |- ( ph -> ( N ` { X , Y } ) C_ ( L ` ( E .+ G ) ) )
Theorem	lclkrlem2o 36810	Lemma for lclkr 36822. 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 )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- .X. = ( .r ` S )   &    |- .0. = ( 0g ` S )   &    |- I = ( invr ` S )   &    |- .- = ( -g ` U )   &    |- F = (LFnl ` U )   &    |- D = (LDual ` U )   &    |- .+ = ( +g ` D )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ph -> E e. F )   &    |- ( ph -> G e. F )   &    |- N = ( LSpan ` U )   &    |- L = (LKer ` U )   &    |- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- .(+) = ( LSSum ` U )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- B = ( X .- ( ( ( ( E .+ G ) ` X ) .X. ( I ` ( ( E .+ G ) ` Y ) ) ) .x. Y ) )   &    |- ( ph -> ( ( E .+ G ) ` Y ) =/= .0. )   &    |- ( ph -> B =/= ( 0g ` U ) )   =>    |- ( ph -> ( -. X e. ( ._|_ ` { B } ) \/ -. Y e. ( ._|_ ` { B } ) ) )
Theorem	lclkrlem2p 36811	Lemma for lclkr 36822. When B is zero, X and Y must colinear, so their orthocomplements must be comparable. (Contributed by NM, 17-Jan-2015.)
|- V = ( Base ` U )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- .X. = ( .r ` S )   &    |- .0. = ( 0g ` S )   &    |- I = ( invr ` S )   &    |- .- = ( -g ` U )   &    |- F = (LFnl ` U )   &    |- D = (LDual ` U )   &    |- .+ = ( +g ` D )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ph -> E e. F )   &    |- ( ph -> G e. F )   &    |- N = ( LSpan ` U )   &    |- L = (LKer ` U )   &    |- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- .(+) = ( LSSum ` U )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- B = ( X .- ( ( ( ( E .+ G ) ` X ) .X. ( I ` ( ( E .+ G ) ` Y ) ) ) .x. Y ) )   &    |- ( ph -> ( ( E .+ G ) ` Y ) =/= .0. )   &    |- ( ph -> B = ( 0g ` U ) )   =>    |- ( ph -> ( ._|_ ` { Y } ) C_ ( ._|_ ` { X } ) )
Theorem	lclkrlem2q 36812	Lemma for lclkr 36822. The sum has a closed kernel when B is nonzero. (Contributed by NM, 18-Jan-2015.)
|- V = ( Base ` U )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- .X. = ( .r ` S )   &    |- .0. = ( 0g ` S )   &    |- I = ( invr ` S )   &    |- .- = ( -g ` U )   &    |- F = (LFnl ` U )   &    |- D = (LDual ` U )   &    |- .+ = ( +g ` D )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ph -> E e. F )   &    |- ( ph -> G e. F )   &    |- N = ( LSpan ` U )   &    |- L = (LKer ` U )   &    |- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- .(+) = ( LSSum ` U )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> ( L ` E ) = ( ._|_ ` { X } ) )   &    |- ( ph -> ( L ` G ) = ( ._|_ ` { Y } ) )   &    |- B = ( X .- ( ( ( ( E .+ G ) ` X ) .X. ( I ` ( ( E .+ G ) ` Y ) ) ) .x. Y ) )   &    |- ( ph -> ( ( E .+ G ) ` Y ) =/= .0. )   &    |- ( ph -> B =/= ( 0g ` U ) )   =>    |- ( ph -> ( ._|_ ` ( ._|_ ` ( L ` ( E .+ G ) ) ) ) = ( L ` ( E .+ G ) ) )
Theorem	lclkrlem2r 36813	Lemma for lclkr 36822. 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 )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- .X. = ( .r ` S )   &    |- .0. = ( 0g ` S )   &    |- I = ( invr ` S )   &    |- .- = ( -g ` U )   &    |- F = (LFnl ` U )   &    |- D = (LDual ` U )   &    |- .+ = ( +g ` D )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ph -> E e. F )   &    |- ( ph -> G e. F )   &    |- N = ( LSpan ` U )   &    |- L = (LKer ` U )   &    |- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- .(+) = ( LSSum ` U )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> ( L ` E ) = ( ._|_ ` { X } ) )   &    |- ( ph -> ( L ` G ) = ( ._|_ ` { Y } ) )   &    |- B = ( X .- ( ( ( ( E .+ G ) ` X ) .X. ( I ` ( ( E .+ G ) ` Y ) ) ) .x. Y ) )   &    |- ( ph -> ( ( E .+ G ) ` Y ) =/= .0. )   &    |- ( ph -> B = ( 0g ` U ) )   =>    |- ( ph -> ( L ` G ) C_ ( L ` ( E .+ G ) ) )
Theorem	lclkrlem2s 36814	Lemma for lclkr 36822. Thus, the sum has a closed kernel when B is zero. (Contributed by NM, 18-Jan-2015.)
|- V = ( Base ` U )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- .X. = ( .r ` S )   &    |- .0. = ( 0g ` S )   &    |- I = ( invr ` S )   &    |- .- = ( -g ` U )   &    |- F = (LFnl ` U )   &    |- D = (LDual ` U )   &    |- .+ = ( +g ` D )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ph -> E e. F )   &    |- ( ph -> G e. F )   &    |- N = ( LSpan ` U )   &    |- L = (LKer ` U )   &    |- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- .(+) = ( LSSum ` U )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> ( L ` E ) = ( ._|_ ` { X } ) )   &    |- ( ph -> ( L ` G ) = ( ._|_ ` { Y } ) )   &    |- B = ( X .- ( ( ( ( E .+ G ) ` X ) .X. ( I ` ( ( E .+ G ) ` Y ) ) ) .x. Y ) )   &    |- ( ph -> ( ( E .+ G ) ` Y ) =/= .0. )   &    |- ( ph -> B = ( 0g ` U ) )   =>    |- ( ph -> ( ._|_ ` ( ._|_ ` ( L ` ( E .+ G ) ) ) ) = ( L ` ( E .+ G ) ) )
Theorem	lclkrlem2t 36815	Lemma for lclkr 36822. We eliminate all hypotheses with B here. (Contributed by NM, 18-Jan-2015.)
|- V = ( Base ` U )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- .X. = ( .r ` S )   &    |- .0. = ( 0g ` S )   &    |- I = ( invr ` S )   &    |- .- = ( -g ` U )   &    |- F = (LFnl ` U )   &    |- D = (LDual ` U )   &    |- .+ = ( +g ` D )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ph -> E e. F )   &    |- ( ph -> G e. F )   &    |- N = ( LSpan ` U )   &    |- L = (LKer ` U )   &    |- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- .(+) = ( LSSum ` U )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> ( L ` E ) = ( ._|_ ` { X } ) )   &    |- ( ph -> ( L ` G ) = ( ._|_ ` { Y } ) )   &    |- ( ph -> ( ( E .+ G ) ` Y ) =/= .0. )   =>    |- ( ph -> ( ._|_ ` ( ._|_ ` ( L ` ( E .+ G ) ) ) ) = ( L ` ( E .+ G ) ) )
Theorem	lclkrlem2u 36816	Lemma for lclkr 36822. lclkrlem2t 36815 with X and Y swapped. (Contributed by NM, 18-Jan-2015.)
|- V = ( Base ` U )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- .X. = ( .r ` S )   &    |- .0. = ( 0g ` S )   &    |- I = ( invr ` S )   &    |- .- = ( -g ` U )   &    |- F = (LFnl ` U )   &    |- D = (LDual ` U )   &    |- .+ = ( +g ` D )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ph -> E e. F )   &    |- ( ph -> G e. F )   &    |- N = ( LSpan ` U )   &    |- L = (LKer ` U )   &    |- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- .(+) = ( LSSum ` U )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> ( L ` E ) = ( ._|_ ` { X } ) )   &    |- ( ph -> ( L ` G ) = ( ._|_ ` { Y } ) )   &    |- ( ph -> ( ( E .+ G ) ` X ) =/= .0. )   =>    |- ( ph -> ( ._|_ ` ( ._|_ ` ( L ` ( E .+ G ) ) ) ) = ( L ` ( E .+ G ) ) )
Theorem	lclkrlem2v 36817	Lemma for lclkr 36822. When the hypotheses of lclkrlem2u 36816 and lclkrlem2u 36816 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 36757, which requires the orthomodular law dihoml4 36666 (Lemma 3.3 of [Holland95] p. 214). (Contributed by NM, 16-Jan-2015.)
|- V = ( Base ` U )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- .X. = ( .r ` S )   &    |- .0. = ( 0g ` S )   &    |- I = ( invr ` S )   &    |- .- = ( -g ` U )   &    |- F = (LFnl ` U )   &    |- D = (LDual ` U )   &    |- .+ = ( +g ` D )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ph -> E e. F )   &    |- ( ph -> G e. F )   &    |- N = ( LSpan ` U )   &    |- L = (LKer ` U )   &    |- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- .(+) = ( LSSum ` U )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> ( L ` E ) = ( ._|_ ` { X } ) )   &    |- ( ph -> ( L ` G ) = ( ._|_ ` { Y } ) )   &    |- ( ph -> ( ( E .+ G ) ` X ) = .0. )   &    |- ( ph -> ( ( E .+ G ) ` Y ) = .0. )   =>    |- ( ph -> ( L ` ( E .+ G ) ) = V )
Theorem	lclkrlem2w 36818	Lemma for lclkr 36822. This is the same as lclkrlem2u 36816 and lclkrlem2u 36816 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 )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- .X. = ( .r ` S )   &    |- .0. = ( 0g ` S )   &    |- I = ( invr ` S )   &    |- .- = ( -g ` U )   &    |- F = (LFnl ` U )   &    |- D = (LDual ` U )   &    |- .+ = ( +g ` D )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ph -> E e. F )   &    |- ( ph -> G e. F )   &    |- N = ( LSpan ` U )   &    |- L = (LKer ` U )   &    |- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- .(+) = ( LSSum ` U )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> ( L ` E ) = ( ._|_ ` { X } ) )   &    |- ( ph -> ( L ` G ) = ( ._|_ ` { Y } ) )   &    |- ( ph -> ( ( E .+ G ) ` X ) = .0. )   &    |- ( ph -> ( ( E .+ G ) ` Y ) = .0. )   =>    |- ( ph -> ( ._|_ ` ( ._|_ ` ( L ` ( E .+ G ) ) ) ) = ( L ` ( E .+ G ) ) )
Theorem	lclkrlem2x 36819	Lemma for lclkr 36822. Eliminate by cases the hypotheses of lclkrlem2u 36816, lclkrlem2u 36816 and lclkrlem2w 36818. (Contributed by NM, 18-Jan-2015.)
|- L = (LKer ` U )   &    |- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- F = (LFnl ` U )   &    |- D = (LDual ` U )   &    |- .+ = ( +g ` D )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. V )   &    |- ( ph -> E e. F )   &    |- ( ph -> G e. F )   &    |- ( ph -> ( L ` E ) = ( ._|_ ` { X } ) )   &    |- ( ph -> ( L ` G ) = ( ._|_ ` { Y } ) )   =>    |- ( ph -> ( ._|_ ` ( ._|_ ` ( L ` ( E .+ G ) ) ) ) = ( L ` ( E .+ G ) ) )
Theorem	lclkrlem2y 36820	Lemma for lclkr 36822. Restate the hypotheses for E and G to say their kernels are closed, in order to eliminate the generating vectors X and Y. (Contributed by NM, 18-Jan-2015.)
|- L = (LKer ` U )   &    |- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- F = (LFnl ` U )   &    |- D = (LDual ` U )   &    |- .+ = ( +g ` D )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> E e. F )   &    |- ( ph -> G e. F )   &    |- ( ph -> ( ._|_ ` ( ._|_ ` ( L ` E ) ) ) = ( L ` E ) )   &    |- ( ph -> ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) )   =>    |- ( ph -> ( ._|_ ` ( ._|_ ` ( L ` ( E .+ G ) ) ) ) = ( L ` ( E .+ G ) ) )
Theorem	lclkrlem2 36821*	The set of functionals having closed kernels is closed under vector (functional) addition. Lemmas lclkrlem2a 36796 through lclkrlem2y 36820 are used for the proof. Here we express lclkrlem2y 36820 in terms of membership in the set C of functionals with closed kernels. (Contributed by NM, 18-Jan-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- F = (LFnl ` U )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- .+ = ( +g ` D )   &    |- C = { f e. F | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> E e. C )   &    |- ( ph -> G e. C )   =>    |- ( ph -> ( E .+ G ) e. C )
Theorem	lclkr 36822*	The set of functionals with closed kernels is a subspace. Part of proof of Theorem 3.6 of [Holland95] p. 218, line 20, stating "The fM that arise this way generate a subspace F of E'". Our proof was suggested by Mario Carneiro, 5-Jan-2015. (Contributed by NM, 18-Jan-2015.)
|- H = ( LHyp ` K )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- F = (LFnl ` U )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- S = ( LSubSp ` D )   &    |- C = { f e. F | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }   &    |- ( ph -> ( K e. HL /\ W e. H ) )   =>    |- ( ph -> C e. S )
Theorem	lcfls1lem 36823*	Property of a functional with a closed kernel. (Contributed by NM, 27-Jan-2015.)
|- C = { f e. F | ( ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) /\ ( ._|_ ` ( L ` f ) ) C_ Q ) }   =>    |- ( G e. C <-> ( G e. F /\ ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) /\ ( ._|_ ` ( L ` G ) ) C_ Q ) )
Theorem	lcfls1N 36824*	Property of a functional with a closed kernel. (Contributed by NM, 27-Jan-2015.) (New usage is discouraged.)
|- C = { f e. F | ( ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) /\ ( ._|_ ` ( L ` f ) ) C_ Q ) }   &    |- ( ph -> G e. F )   =>    |- ( ph -> ( G e. C <-> ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) /\ ( ._|_ ` ( L ` G ) ) C_ Q ) ) )
Theorem	lcfls1c 36825*	Property of a functional with a closed kernel. (Contributed by NM, 28-Jan-2015.)
|- C = { f e. F | ( ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) /\ ( ._|_ ` ( L ` f ) ) C_ Q ) }   &    |- D = { f e. F | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }   =>    |- ( G e. C <-> ( G e. D /\ ( ._|_ ` ( L ` G ) ) C_ Q ) )
Theorem	lclkrslem1 36826*	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 )   &    |- .x. = ( .s ` D )   &    |- C = { f e. F | ( ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) /\ ( ._|_ ` ( L ` f ) ) C_ Q ) }   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> Q e. S )   &    |- ( ph -> G e. C )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( X .x. G ) e. C )
Theorem	lclkrslem2 36827*	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 )   &    |- .x. = ( .s ` D )   &    |- C = { f e. F | ( ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) /\ ( ._|_ ` ( L ` f ) ) C_ Q ) }   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> Q e. S )   &    |- ( ph -> G e. C )   &    |- .+ = ( +g ` D )   &    |- ( ph -> E e. C )   =>    |- ( ph -> ( E .+ G ) e. C )
Theorem	lclkrs 36828*	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 36822 proof. Do we achieve overall shortening by breaking them out as subtheorems? Or make lclkr 36822 a special case of this? (Contributed by NM, 29-Jan-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- S = ( LSubSp ` U )   &    |- F = (LFnl ` U )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- T = ( LSubSp ` D )   &    |- C = { f e. F | ( ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) /\ ( ._|_ ` ( L ` f ) ) C_ R ) }   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> R e. S )   =>    |- ( ph -> C e. T )
Theorem	lclkrs2 36829*	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 36917. (Contributed by NM, 12-Mar-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- S = ( LSubSp ` U )   &    |- F = (LFnl ` U )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- T = ( LSubSp ` D )   &    |- C = { f e. F | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }   &    |- R = { g e. F | ( ( ._|_ ` ( ._|_ ` ( L ` g ) ) ) = ( L ` g ) /\ ( ._|_ ` ( L ` g ) ) C_ Q ) }   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> Q e. S )   =>    |- ( ph -> ( R e. T /\ R C_ C ) )
Theorem	lcfrvalsnN 36830*	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 )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> G e. F )   &    |- Q = U_ f e. R ( ._|_ ` ( L ` f ) )   &    |- R = ( N ` { G } )   =>    |- ( ph -> Q = ( ._|_ ` ( L ` G ) ) )
Theorem	lcfrlem1 36831	Lemma for lcfr 36874. Note that X is z in Mario's notes. (Contributed by NM, 27-Feb-2015.)
|- V = ( Base ` U )   &    |- S = (Scalar ` U )   &    |- .X. = ( .r ` S )   &    |- .0. = ( 0g ` S )   &    |- I = ( invr ` S )   &    |- F = (LFnl ` U )   &    |- D = (LDual ` U )   &    |- .x. = ( .s ` D )   &    |- .- = ( -g ` D )   &    |- ( ph -> U e. LVec )   &    |- ( ph -> E e. F )   &    |- ( ph -> G e. F )   &    |- ( ph -> X e. V )   &    |- ( ph -> ( G ` X ) =/= .0. )   &    |- H = ( E .- ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) )   =>    |- ( ph -> ( H ` X ) = .0. )
Theorem	lcfrlem2 36832	Lemma for lcfr 36874. (Contributed by NM, 27-Feb-2015.)
|- V = ( Base ` U )   &    |- S = (Scalar ` U )   &    |- .X. = ( .r ` S )   &    |- .0. = ( 0g ` S )   &    |- I = ( invr ` S )   &    |- F = (LFnl ` U )   &    |- D = (LDual ` U )   &    |- .x. = ( .s ` D )   &    |- .- = ( -g ` D )   &    |- ( ph -> U e. LVec )   &    |- ( ph -> E e. F )   &    |- ( ph -> G e. F )   &    |- ( ph -> X e. V )   &    |- ( ph -> ( G ` X ) =/= .0. )   &    |- H = ( E .- ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) )   &    |- L = (LKer ` U )   =>    |- ( ph -> ( ( L ` E ) i^i ( L ` G ) ) C_ ( L ` H ) )
Theorem	lcfrlem3 36833	Lemma for lcfr 36874. (Contributed by NM, 27-Feb-2015.)
|- V = ( Base ` U )   &    |- S = (Scalar ` U )   &    |- .X. = ( .r ` S )   &    |- .0. = ( 0g ` S )   &    |- I = ( invr ` S )   &    |- F = (LFnl ` U )   &    |- D = (LDual ` U )   &    |- .x. = ( .s ` D )   &    |- .- = ( -g ` D )   &    |- ( ph -> U e. LVec )   &    |- ( ph -> E e. F )   &    |- ( ph -> G e. F )   &    |- ( ph -> X e. V )   &    |- ( ph -> ( G ` X ) =/= .0. )   &    |- H = ( E .- ( ( ( I ` ( G ` X ) ) .X. ( E ` X ) ) .x. G ) )   &    |- L = (LKer ` U )   =>    |- ( ph -> X e. ( L ` H ) )
Theorem	lcfrlem4 36834*	Lemma for lcfr 36874. (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 = U_ g e. G ( ._|_ ` ( L ` g ) )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> G e. Q )   &    |- ( ph -> X e. E )   =>    |- ( ph -> X e. V )
Theorem	lcfrlem5 36835*	Lemma for lcfr 36874. 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 )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> R e. S )   &    |- Q = U_ f e. R ( ._|_ ` ( L ` f ) )   &    |- ( ph -> X e. Q )   &    |- C = (Scalar ` U )   &    |- B = ( Base ` C )   &    |- .x. = ( .s ` U )   &    |- ( ph -> A e. B )   =>    |- ( ph -> ( A .x. X ) e. Q )
Theorem	lcfrlem6 36836*	Lemma for lcfr 36874. Closure of vector sum with colinear vectors. TODO: Move down N definition so top hypotheses can be shared. (Contributed by NM, 10-Mar-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- .+ = ( +g ` U )   &    |- N = ( LSpan ` U )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- Q = ( LSubSp ` D )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> G e. Q )   &    |- E = U_ g e. G ( ._|_ ` ( L ` g ) )   &    |- ( ph -> X e. E )   &    |- ( ph -> Y e. E )   &    |- ( ph -> ( N ` { X } ) = ( N ` { Y } ) )   =>    |- ( ph -> ( X .+ Y ) e. E )
Theorem	lcfrlem7 36837*	Lemma for lcfr 36874. Closure of vector sum when one vector is zero. TODO: share hypotheses with others. (Contributed by NM, 11-Mar-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- .+ = ( +g ` U )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- Q = ( LSubSp ` D )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> G e. Q )   &    |- E = U_ g e. G ( ._|_ ` ( L ` g ) )   &    |- ( ph -> X e. E )   &    |- .0. = ( 0g ` U )   &    |- ( ph -> Y = .0. )   =>    |- ( ph -> ( X .+ Y ) e. E )
Theorem	lcfrlem8 36838*	Lemma for lcf1o 36840 and lcfr 36874. (Contributed by NM, 21-Feb-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- .+ = ( +g ` U )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- R = ( Base ` S )   &    |- .0. = ( 0g ` U )   &    |- F = (LFnl ` U )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- Q = ( 0g ` D )   &    |- C = { f e. F | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }   &    |- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. ( V \ { .0. } ) )   =>    |- ( ph -> ( J ` X ) = ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { X } ) v = ( w .+ ( k .x. X ) ) ) ) )
Theorem	lcfrlem9 36839*	Lemma for lcf1o 36840. (This part has undesirable $d's on J and ph that we remove in lcf1o 36840.) TODO: ugly proof; maybe have better subtheorems or abbreviate some iota_ k expansions with J ` z? TODO: Some redundant $d's? (Contributed by NM, 22-Feb-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- .+ = ( +g ` U )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- R = ( Base ` S )   &    |- .0. = ( 0g ` U )   &    |- F = (LFnl ` U )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- Q = ( 0g ` D )   &    |- C = { f e. F | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }   &    |- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   =>    |- ( ph -> J : ( V \ { .0. } ) -1-1-onto-> ( C \ { Q } ) )
Theorem	lcf1o 36840*	Define a function J that provides a bijection from nonzero vectors V to nonzero functionals with closed kernels C. (Contributed by NM, 22-Feb-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- .+ = ( +g ` U )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- R = ( Base ` S )   &    |- .0. = ( 0g ` U )   &    |- F = (LFnl ` U )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- Q = ( 0g ` D )   &    |- C = { f e. F | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }   &    |- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   =>    |- ( ph -> J : ( V \ { .0. } ) -1-1-onto-> ( C \ { Q } ) )
Theorem	lcfrlem10 36841*	Lemma for lcfr 36874. (Contributed by NM, 23-Feb-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- .+ = ( +g ` U )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- R = ( Base ` S )   &    |- .0. = ( 0g ` U )   &    |- F = (LFnl ` U )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- Q = ( 0g ` D )   &    |- C = { f e. F | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }   &    |- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. ( V \ { .0. } ) )   =>    |- ( ph -> ( J ` X ) e. F )
Theorem	lcfrlem11 36842*	Lemma for lcfr 36874. (Contributed by NM, 23-Feb-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- .+ = ( +g ` U )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- R = ( Base ` S )   &    |- .0. = ( 0g ` U )   &    |- F = (LFnl ` U )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- Q = ( 0g ` D )   &    |- C = { f e. F | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }   &    |- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. ( V \ { .0. } ) )   =>    |- ( ph -> ( L ` ( J ` X ) ) = ( ._|_ ` { X } ) )
Theorem	lcfrlem12N 36843*	Lemma for lcfr 36874. (Contributed by NM, 23-Feb-2015.) (New usage is discouraged.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- .+ = ( +g ` U )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- R = ( Base ` S )   &    |- .0. = ( 0g ` U )   &    |- F = (LFnl ` U )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- Q = ( 0g ` D )   &    |- C = { f e. F | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }   &    |- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. ( V \ { .0. } ) )   &    |- B = ( 0g ` S )   &    |- ( ph -> Y e. ( ._|_ ` { X } ) )   =>    |- ( ph -> ( ( J ` X ) ` Y ) = B )
Theorem	lcfrlem13 36844*	Lemma for lcfr 36874. (Contributed by NM, 8-Mar-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- .+ = ( +g ` U )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- R = ( Base ` S )   &    |- .0. = ( 0g ` U )   &    |- F = (LFnl ` U )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- Q = ( 0g ` D )   &    |- C = { f e. F | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }   &    |- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. ( V \ { .0. } ) )   =>    |- ( ph -> ( J ` X ) e. ( C \ { Q } ) )
Theorem	lcfrlem14 36845*	Lemma for lcfr 36874. (Contributed by NM, 10-Mar-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- .+ = ( +g ` U )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- R = ( Base ` S )   &    |- .0. = ( 0g ` U )   &    |- F = (LFnl ` U )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- Q = ( 0g ` D )   &    |- C = { f e. F | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }   &    |- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. ( V \ { .0. } ) )   &    |- N = ( LSpan ` U )   =>    |- ( ph -> ( ._|_ ` ( L ` ( J ` X ) ) ) = ( N ` { X } ) )
Theorem	lcfrlem15 36846*	Lemma for lcfr 36874. (Contributed by NM, 9-Mar-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- .+ = ( +g ` U )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- R = ( Base ` S )   &    |- .0. = ( 0g ` U )   &    |- F = (LFnl ` U )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- Q = ( 0g ` D )   &    |- C = { f e. F | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }   &    |- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. ( V \ { .0. } ) )   =>    |- ( ph -> X e. ( ._|_ ` ( L ` ( J ` X ) ) ) )
Theorem	lcfrlem16 36847*	Lemma for lcfr 36874. (Contributed by NM, 8-Mar-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- .+ = ( +g ` U )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- R = ( Base ` S )   &    |- .0. = ( 0g ` U )   &    |- F = (LFnl ` U )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- Q = ( 0g ` D )   &    |- C = { f e. F | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }   &    |- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- P = ( LSubSp ` D )   &    |- ( ph -> G e. P )   &    |- ( ph -> G C_ C )   &    |- E = U_ g e. G ( ._|_ ` ( L ` g ) )   &    |- ( ph -> X e. ( E \ { .0. } ) )   =>    |- ( ph -> ( J ` X ) e. G )
Theorem	lcfrlem17 36848	Lemma for lcfr 36874. Condition needed more than once. (Contributed by NM, 11-Mar-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- .+ = ( +g ` U )   &    |- .0. = ( 0g ` U )   &    |- N = ( LSpan ` U )   &    |- A = (LSAtoms ` U )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. ( V \ { .0. } ) )   &    |- ( ph -> Y e. ( V \ { .0. } ) )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )   =>    |- ( ph -> ( X .+ Y ) e. ( V \ { .0. } ) )
Theorem	lcfrlem18 36849	Lemma for lcfr 36874. (Contributed by NM, 24-Feb-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- .+ = ( +g ` U )   &    |- .0. = ( 0g ` U )   &    |- N = ( LSpan ` U )   &    |- A = (LSAtoms ` U )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. ( V \ { .0. } ) )   &    |- ( ph -> Y e. ( V \ { .0. } ) )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )   =>    |- ( ph -> ( ._|_ ` { X , Y } ) = ( ( ._|_ ` { X } ) i^i ( ._|_ ` { Y } ) ) )
Theorem	lcfrlem19 36850	Lemma for lcfr 36874. (Contributed by NM, 11-Mar-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- .+ = ( +g ` U )   &    |- .0. = ( 0g ` U )   &    |- N = ( LSpan ` U )   &    |- A = (LSAtoms ` U )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. ( V \ { .0. } ) )   &    |- ( ph -> Y e. ( V \ { .0. } ) )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )   =>    |- ( ph -> ( -. X e. ( ._|_ ` { ( X .+ Y ) } ) \/ -. Y e. ( ._|_ ` { ( X .+ Y ) } ) ) )
Theorem	lcfrlem20 36851	Lemma for lcfr 36874. (Contributed by NM, 11-Mar-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- .+ = ( +g ` U )   &    |- .0. = ( 0g ` U )   &    |- N = ( LSpan ` U )   &    |- A = (LSAtoms ` U )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. ( V \ { .0. } ) )   &    |- ( ph -> Y e. ( V \ { .0. } ) )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )   &    |- ( ph -> -. X e. ( ._|_ ` { ( X .+ Y ) } ) )   =>    |- ( ph -> ( ( N ` { X , Y } ) i^i ( ._|_ ` { ( X .+ Y ) } ) ) e. A )
Theorem	lcfrlem21 36852	Lemma for lcfr 36874. (Contributed by NM, 11-Mar-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- .+ = ( +g ` U )   &    |- .0. = ( 0g ` U )   &    |- N = ( LSpan ` U )   &    |- A = (LSAtoms ` U )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. ( V \ { .0. } ) )   &    |- ( ph -> Y e. ( V \ { .0. } ) )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )   =>    |- ( ph -> ( ( N ` { X , Y } ) i^i ( ._|_ ` { ( X .+ Y ) } ) ) e. A )
Theorem	lcfrlem22 36853	Lemma for lcfr 36874. (Contributed by NM, 24-Feb-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- .+ = ( +g ` U )   &    |- .0. = ( 0g ` U )   &    |- N = ( LSpan ` U )   &    |- A = (LSAtoms ` U )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. ( V \ { .0. } ) )   &    |- ( ph -> Y e. ( V \ { .0. } ) )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )   &    |- B = ( ( N ` { X , Y } ) i^i ( ._|_ ` { ( X .+ Y ) } ) )   =>    |- ( ph -> B e. A )
Theorem	lcfrlem23 36854	Lemma for lcfr 36874. TODO: this proof was built from other proof pieces that may change N ` { X , Y } into subspace sum and back unnecessarily, or similar things. (Contributed by NM, 1-Mar-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- .+ = ( +g ` U )   &    |- .0. = ( 0g ` U )   &    |- N = ( LSpan ` U )   &    |- A = (LSAtoms ` U )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. ( V \ { .0. } ) )   &    |- ( ph -> Y e. ( V \ { .0. } ) )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )   &    |- B = ( ( N ` { X , Y } ) i^i ( ._|_ ` { ( X .+ Y ) } ) )   &    |- .(+) = ( LSSum ` U )   =>    |- ( ph -> ( ( ._|_ ` { X , Y } ) .(+) B ) = ( ._|_ ` { ( X .+ Y ) } ) )
Theorem	lcfrlem24 36855*	Lemma for lcfr 36874. (Contributed by NM, 24-Feb-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- .+ = ( +g ` U )   &    |- .0. = ( 0g ` U )   &    |- N = ( LSpan ` U )   &    |- A = (LSAtoms ` U )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. ( V \ { .0. } ) )   &    |- ( ph -> Y e. ( V \ { .0. } ) )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )   &    |- B = ( ( N ` { X , Y } ) i^i ( ._|_ ` { ( X .+ Y ) } ) )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- Q = ( 0g ` S )   &    |- R = ( Base ` S )   &    |- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )   &    |- ( ph -> I e. B )   &    |- L = (LKer ` U )   =>    |- ( ph -> ( ._|_ ` { X , Y } ) = ( ( L ` ( J ` X ) ) i^i ( L ` ( J ` Y ) ) ) )
Theorem	lcfrlem25 36856*	Lemma for lcfr 36874. Special case of lcfrlem35 36866 when ( ( J ` Y ) ` I ) is zero. (Contributed by NM, 11-Mar-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- .+ = ( +g ` U )   &    |- .0. = ( 0g ` U )   &    |- N = ( LSpan ` U )   &    |- A = (LSAtoms ` U )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. ( V \ { .0. } ) )   &    |- ( ph -> Y e. ( V \ { .0. } ) )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )   &    |- B = ( ( N ` { X , Y } ) i^i ( ._|_ ` { ( X .+ Y ) } ) )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- Q = ( 0g ` S )   &    |- R = ( Base ` S )   &    |- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )   &    |- ( ph -> I e. B )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- ( ph -> ( ( J ` Y ) ` I ) = Q )   &    |- ( ph -> I =/= .0. )   =>    |- ( ph -> ( ._|_ ` { ( X .+ Y ) } ) = ( L ` ( J ` Y ) ) )
Theorem	lcfrlem26 36857*	Lemma for lcfr 36874. Special case of lcfrlem36 36867 when ( ( J ` Y ) ` I ) is zero. (Contributed by NM, 11-Mar-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- .+ = ( +g ` U )   &    |- .0. = ( 0g ` U )   &    |- N = ( LSpan ` U )   &    |- A = (LSAtoms ` U )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. ( V \ { .0. } ) )   &    |- ( ph -> Y e. ( V \ { .0. } ) )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )   &    |- B = ( ( N ` { X , Y } ) i^i ( ._|_ ` { ( X .+ Y ) } ) )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- Q = ( 0g ` S )   &    |- R = ( Base ` S )   &    |- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )   &    |- ( ph -> I e. B )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- ( ph -> ( ( J ` Y ) ` I ) = Q )   &    |- ( ph -> I =/= .0. )   =>    |- ( ph -> ( X .+ Y ) e. ( ._|_ ` ( L ` ( J ` Y ) ) ) )
Theorem	lcfrlem27 36858*	Lemma for lcfr 36874. Special case of lcfrlem37 36868 when ( ( J ` Y ) ` I ) is zero. (Contributed by NM, 11-Mar-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- .+ = ( +g ` U )   &    |- .0. = ( 0g ` U )   &    |- N = ( LSpan ` U )   &    |- A = (LSAtoms ` U )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. ( V \ { .0. } ) )   &    |- ( ph -> Y e. ( V \ { .0. } ) )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )   &    |- B = ( ( N ` { X , Y } ) i^i ( ._|_ ` { ( X .+ Y ) } ) )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- Q = ( 0g ` S )   &    |- R = ( Base ` S )   &    |- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )   &    |- ( ph -> I e. B )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- ( ph -> ( ( J ` Y ) ` I ) = Q )   &    |- ( ph -> I =/= .0. )   &    |- ( ph -> G e. ( LSubSp ` D ) )   &    |- ( ph -> G C_ { f e. (LFnl ` U ) | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) } )   &    |- E = U_ g e. G ( ._|_ ` ( L ` g ) )   &    |- ( ph -> X e. E )   &    |- ( ph -> Y e. E )   =>    |- ( ph -> ( X .+ Y ) e. E )
Theorem	lcfrlem28 36859*	Lemma for lcfr 36874. TODO: This can be a hypothesis since the zero version of ( J ` Y ) ` I needs it. (Contributed by NM, 9-Mar-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- .+ = ( +g ` U )   &    |- .0. = ( 0g ` U )   &    |- N = ( LSpan ` U )   &    |- A = (LSAtoms ` U )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. ( V \ { .0. } ) )   &    |- ( ph -> Y e. ( V \ { .0. } ) )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )   &    |- B = ( ( N ` { X , Y } ) i^i ( ._|_ ` { ( X .+ Y ) } ) )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- Q = ( 0g ` S )   &    |- R = ( Base ` S )   &    |- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )   &    |- ( ph -> I e. B )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- ( ph -> ( ( J ` Y ) ` I ) =/= Q )   =>    |- ( ph -> I =/= .0. )
Theorem	lcfrlem29 36860*	Lemma for lcfr 36874. (Contributed by NM, 9-Mar-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- .+ = ( +g ` U )   &    |- .0. = ( 0g ` U )   &    |- N = ( LSpan ` U )   &    |- A = (LSAtoms ` U )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. ( V \ { .0. } ) )   &    |- ( ph -> Y e. ( V \ { .0. } ) )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )   &    |- B = ( ( N ` { X , Y } ) i^i ( ._|_ ` { ( X .+ Y ) } ) )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- Q = ( 0g ` S )   &    |- R = ( Base ` S )   &    |- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )   &    |- ( ph -> I e. B )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- ( ph -> ( ( J ` Y ) ` I ) =/= Q )   &    |- F = ( invr ` S )   =>    |- ( ph -> ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) e. R )
Theorem	lcfrlem30 36861*	Lemma for lcfr 36874. (Contributed by NM, 6-Mar-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- .+ = ( +g ` U )   &    |- .0. = ( 0g ` U )   &    |- N = ( LSpan ` U )   &    |- A = (LSAtoms ` U )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. ( V \ { .0. } ) )   &    |- ( ph -> Y e. ( V \ { .0. } ) )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )   &    |- B = ( ( N ` { X , Y } ) i^i ( ._|_ ` { ( X .+ Y ) } ) )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- Q = ( 0g ` S )   &    |- R = ( Base ` S )   &    |- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )   &    |- ( ph -> I e. B )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- ( ph -> ( ( J ` Y ) ` I ) =/= Q )   &    |- F = ( invr ` S )   &    |- .- = ( -g ` D )   &    |- C = ( ( J ` X ) .- ( ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) ( .s ` D ) ( J ` Y ) ) )   =>    |- ( ph -> C e. (LFnl ` U ) )
Theorem	lcfrlem31 36862*	Lemma for lcfr 36874. (Contributed by NM, 10-Mar-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- .+ = ( +g ` U )   &    |- .0. = ( 0g ` U )   &    |- N = ( LSpan ` U )   &    |- A = (LSAtoms ` U )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. ( V \ { .0. } ) )   &    |- ( ph -> Y e. ( V \ { .0. } ) )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )   &    |- B = ( ( N ` { X , Y } ) i^i ( ._|_ ` { ( X .+ Y ) } ) )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- Q = ( 0g ` S )   &    |- R = ( Base ` S )   &    |- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )   &    |- ( ph -> I e. B )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- ( ph -> ( ( J ` Y ) ` I ) =/= Q )   &    |- F = ( invr ` S )   &    |- .- = ( -g ` D )   &    |- C = ( ( J ` X ) .- ( ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) ( .s ` D ) ( J ` Y ) ) )   &    |- ( ph -> ( ( J ` X ) ` I ) =/= Q )   &    |- ( ph -> C = ( 0g ` D ) )   =>    |- ( ph -> ( N ` { X } ) = ( N ` { Y } ) )
Theorem	lcfrlem32 36863*	Lemma for lcfr 36874. (Contributed by NM, 10-Mar-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- .+ = ( +g ` U )   &    |- .0. = ( 0g ` U )   &    |- N = ( LSpan ` U )   &    |- A = (LSAtoms ` U )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. ( V \ { .0. } ) )   &    |- ( ph -> Y e. ( V \ { .0. } ) )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )   &    |- B = ( ( N ` { X , Y } ) i^i ( ._|_ ` { ( X .+ Y ) } ) )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- Q = ( 0g ` S )   &    |- R = ( Base ` S )   &    |- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )   &    |- ( ph -> I e. B )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- ( ph -> ( ( J ` Y ) ` I ) =/= Q )   &    |- F = ( invr ` S )   &    |- .- = ( -g ` D )   &    |- C = ( ( J ` X ) .- ( ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) ( .s ` D ) ( J ` Y ) ) )   &    |- ( ph -> ( ( J ` X ) ` I ) =/= Q )   =>    |- ( ph -> C =/= ( 0g ` D ) )
Theorem	lcfrlem33 36864*	Lemma for lcfr 36874. (Contributed by NM, 10-Mar-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- .+ = ( +g ` U )   &    |- .0. = ( 0g ` U )   &    |- N = ( LSpan ` U )   &    |- A = (LSAtoms ` U )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. ( V \ { .0. } ) )   &    |- ( ph -> Y e. ( V \ { .0. } ) )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )   &    |- B = ( ( N ` { X , Y } ) i^i ( ._|_ ` { ( X .+ Y ) } ) )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- Q = ( 0g ` S )   &    |- R = ( Base ` S )   &    |- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )   &    |- ( ph -> I e. B )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- ( ph -> ( ( J ` Y ) ` I ) =/= Q )   &    |- F = ( invr ` S )   &    |- .- = ( -g ` D )   &    |- C = ( ( J ` X ) .- ( ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) ( .s ` D ) ( J ` Y ) ) )   &    |- ( ph -> ( ( J ` X ) ` I ) = Q )   =>    |- ( ph -> C =/= ( 0g ` D ) )
Theorem	lcfrlem34 36865*	Lemma for lcfr 36874. (Contributed by NM, 10-Mar-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- .+ = ( +g ` U )   &    |- .0. = ( 0g ` U )   &    |- N = ( LSpan ` U )   &    |- A = (LSAtoms ` U )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. ( V \ { .0. } ) )   &    |- ( ph -> Y e. ( V \ { .0. } ) )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )   &    |- B = ( ( N ` { X , Y } ) i^i ( ._|_ ` { ( X .+ Y ) } ) )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- Q = ( 0g ` S )   &    |- R = ( Base ` S )   &    |- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )   &    |- ( ph -> I e. B )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- ( ph -> ( ( J ` Y ) ` I ) =/= Q )   &    |- F = ( invr ` S )   &    |- .- = ( -g ` D )   &    |- C = ( ( J ` X ) .- ( ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) ( .s ` D ) ( J ` Y ) ) )   =>    |- ( ph -> C =/= ( 0g ` D ) )
Theorem	lcfrlem35 36866*	Lemma for lcfr 36874. (Contributed by NM, 2-Mar-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- .+ = ( +g ` U )   &    |- .0. = ( 0g ` U )   &    |- N = ( LSpan ` U )   &    |- A = (LSAtoms ` U )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. ( V \ { .0. } ) )   &    |- ( ph -> Y e. ( V \ { .0. } ) )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )   &    |- B = ( ( N ` { X , Y } ) i^i ( ._|_ ` { ( X .+ Y ) } ) )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- Q = ( 0g ` S )   &    |- R = ( Base ` S )   &    |- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )   &    |- ( ph -> I e. B )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- ( ph -> ( ( J ` Y ) ` I ) =/= Q )   &    |- F = ( invr ` S )   &    |- .- = ( -g ` D )   &    |- C = ( ( J ` X ) .- ( ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) ( .s ` D ) ( J ` Y ) ) )   =>    |- ( ph -> ( ._|_ ` { ( X .+ Y ) } ) = ( L ` C ) )
Theorem	lcfrlem36 36867*	Lemma for lcfr 36874. (Contributed by NM, 6-Mar-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- .+ = ( +g ` U )   &    |- .0. = ( 0g ` U )   &    |- N = ( LSpan ` U )   &    |- A = (LSAtoms ` U )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. ( V \ { .0. } ) )   &    |- ( ph -> Y e. ( V \ { .0. } ) )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )   &    |- B = ( ( N ` { X , Y } ) i^i ( ._|_ ` { ( X .+ Y ) } ) )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- Q = ( 0g ` S )   &    |- R = ( Base ` S )   &    |- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )   &    |- ( ph -> I e. B )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- ( ph -> ( ( J ` Y ) ` I ) =/= Q )   &    |- F = ( invr ` S )   &    |- .- = ( -g ` D )   &    |- C = ( ( J ` X ) .- ( ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) ( .s ` D ) ( J ` Y ) ) )   =>    |- ( ph -> ( X .+ Y ) e. ( ._|_ ` ( L ` C ) ) )
Theorem	lcfrlem37 36868*	Lemma for lcfr 36874. (Contributed by NM, 8-Mar-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- V = ( Base ` U )   &    |- .+ = ( +g ` U )   &    |- .0. = ( 0g ` U )   &    |- N = ( LSpan ` U )   &    |- A = (LSAtoms ` U )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. ( V \ { .0. } ) )   &    |- ( ph -> Y e. ( V \ { .0. } ) )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )   &    |- B = ( ( N ` { X , Y } ) i^i ( ._|_ ` { ( X .+ Y ) } ) )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- Q = ( 0g ` S )   &    |- R = ( Base ` S )   &    |- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )   &    |- ( ph -> I e. B )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- ( ph -> ( ( J ` Y ) ` I ) =/= Q )   &    |- F = ( invr ` S )   &    |- .- = ( -g ` D )   &    |- C = ( ( J ` X ) .- ( ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) ( .s ` D ) ( J ` Y ) ) )   &    |- ( ph -> G e. ( LSubSp ` D ) )   &    |- ( ph -> G C_ { f e. (LFnl ` U ) | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) } )   &    |- E = U_ g e. G ( ._|_ ` ( L ` g ) )   &    |- ( ph -> X e. E )   &    |- ( ph -> Y e. E )   =>    |- ( ph -> ( X .+ Y ) e. E )
Theorem	lcfrlem38 36869*	Lemma for lcfr 36874. Combine lcfrlem27 36858 and lcfrlem37 36868. (Contributed by NM, 11-Mar-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- .+ = ( +g ` U )   &    |- F = (LFnl ` U )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- Q = ( LSubSp ` D )   &    |- C = { f e. (LFnl ` U ) | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }   &    |- E = U_ g e. G ( ._|_ ` ( L ` g ) )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> G e. Q )   &    |- ( ph -> G C_ C )   &    |- ( ph -> X e. E )   &    |- ( ph -> Y e. E )   &    |- .0. = ( 0g ` U )   &    |- ( ph -> X =/= .0. )   &    |- ( ph -> Y =/= .0. )   &    |- N = ( LSpan ` U )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )   &    |- B = ( ( N ` { X , Y } ) i^i ( ._|_ ` { ( X .+ Y ) } ) )   &    |- ( ph -> I e. B )   &    |- ( ph -> I =/= .0. )   &    |- V = ( Base ` U )   &    |- .x. = ( .s ` U )   &    |- S = (Scalar ` U )   &    |- R = ( Base ` S )   &    |- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )   =>    |- ( ph -> ( X .+ Y ) e. E )
Theorem	lcfrlem39 36870*	Lemma for lcfr 36874. Eliminate J. (Contributed by NM, 11-Mar-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- .+ = ( +g ` U )   &    |- F = (LFnl ` U )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- Q = ( LSubSp ` D )   &    |- C = { f e. (LFnl ` U ) | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }   &    |- E = U_ g e. G ( ._|_ ` ( L ` g ) )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> G e. Q )   &    |- ( ph -> G C_ C )   &    |- ( ph -> X e. E )   &    |- ( ph -> Y e. E )   &    |- .0. = ( 0g ` U )   &    |- ( ph -> X =/= .0. )   &    |- ( ph -> Y =/= .0. )   &    |- N = ( LSpan ` U )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )   &    |- B = ( ( N ` { X , Y } ) i^i ( ._|_ ` { ( X .+ Y ) } ) )   &    |- ( ph -> I e. B )   &    |- ( ph -> I =/= .0. )   =>    |- ( ph -> ( X .+ Y ) e. E )
Theorem	lcfrlem40 36871*	Lemma for lcfr 36874. Eliminate B and I. (Contributed by NM, 11-Mar-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- .+ = ( +g ` U )   &    |- F = (LFnl ` U )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- Q = ( LSubSp ` D )   &    |- C = { f e. (LFnl ` U ) | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }   &    |- E = U_ g e. G ( ._|_ ` ( L ` g ) )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> G e. Q )   &    |- ( ph -> G C_ C )   &    |- ( ph -> X e. E )   &    |- ( ph -> Y e. E )   &    |- .0. = ( 0g ` U )   &    |- ( ph -> X =/= .0. )   &    |- ( ph -> Y =/= .0. )   &    |- N = ( LSpan ` U )   &    |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )   =>    |- ( ph -> ( X .+ Y ) e. E )
Theorem	lcfrlem41 36872*	Lemma for lcfr 36874. Eliminate span condition. (Contributed by NM, 11-Mar-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- .+ = ( +g ` U )   &    |- F = (LFnl ` U )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- Q = ( LSubSp ` D )   &    |- C = { f e. (LFnl ` U ) | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }   &    |- E = U_ g e. G ( ._|_ ` ( L ` g ) )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> G e. Q )   &    |- ( ph -> G C_ C )   &    |- ( ph -> X e. E )   &    |- ( ph -> Y e. E )   &    |- .0. = ( 0g ` U )   &    |- ( ph -> X =/= .0. )   &    |- ( ph -> Y =/= .0. )   =>    |- ( ph -> ( X .+ Y ) e. E )
Theorem	lcfrlem42 36873*	Lemma for lcfr 36874. Eliminate nonzero condition. (Contributed by NM, 11-Mar-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- .+ = ( +g ` U )   &    |- F = (LFnl ` U )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- Q = ( LSubSp ` D )   &    |- C = { f e. (LFnl ` U ) | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }   &    |- E = U_ g e. G ( ._|_ ` ( L ` g ) )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> G e. Q )   &    |- ( ph -> G C_ C )   &    |- ( ph -> X e. E )   &    |- ( ph -> Y e. E )   =>    |- ( ph -> ( X .+ Y ) e. E )
Theorem	lcfr 36874*	Reconstruction of a subspace from a dual subspace of functionals with closed kernels. Our proof was suggested by Mario Carneiro, 20-Feb-2015. (Contributed by NM, 5-Mar-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- S = ( LSubSp ` U )   &    |- F = (LFnl ` U )   &    |- L = (LKer ` U )   &    |- D = (LDual ` U )   &    |- T = ( LSubSp ` D )   &    |- C = { f e. F | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }   &    |- Q = U_ g e. R ( ._|_ ` ( L ` g ) )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> R e. T )   &    |- ( ph -> R C_ C )   =>    |- ( ph -> Q e. S )
Syntax	clcd 36875	Extend class notation with vector space of functionals with closed kernels.
class LCDual
Definition	df-lcdual 36876*	Dual vector space of functionals with closed kernels. Note: we could also define this directly without mapd by using mapdrn 36938. TODO: see if it makes sense to go back and replace some of the LDual stuff with this. TODO: We could simplify df-mapd 36914 using ( Base ` ( (LCDual ` K ) ` W ) ). (Contributed by NM, 13-Mar-2015.)
|- LCDual = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> ( (LDual ` ( ( DVecH ` k ) ` w ) ) ↾s { f e. (LFnl ` ( ( DVecH ` k ) ` w ) ) | ( ( ( ocH ` k ) ` w ) ` ( ( ( ocH ` k ) ` w ) ` ( (LKer ` ( ( DVecH ` k ) ` w ) ) ` f ) ) ) = ( (LKer ` ( ( DVecH ` k ) ` w ) ) ` f ) } ) ) )
Theorem	lcdfval 36877*	Dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.)
|- H = ( LHyp ` K )   =>    |- ( K e. X -> (LCDual ` K ) = ( w e. H |-> ( (LDual ` ( ( DVecH ` K ) ` w ) ) ↾s { f e. (LFnl ` ( ( DVecH ` K ) ` w ) ) | ( ( ( ocH ` K ) ` w ) ` ( ( ( ocH ` K ) ` w ) ` ( (LKer ` ( ( DVecH ` K ) ` w ) ) ` f ) ) ) = ( (LKer ` ( ( DVecH ` K ) ` w ) ) ` f ) } ) ) )
Theorem	lcdval 36878*	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 )   &    |- ( ph -> ( K e. X /\ W e. H ) )   =>    |- ( ph -> C = ( D ↾s { f e. F | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) } ) )
Theorem	lcdval2 36879*	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 )   &    |- ( ph -> ( K e. X /\ W e. H ) )   &    |- B = { f e. F | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }   =>    |- ( ph -> C = ( D ↾s B ) )
Theorem	lcdlvec 36880	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 )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   =>    |- ( ph -> C e. LVec )
Theorem	lcdlmod 36881	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 )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   =>    |- ( ph -> C e. LMod )
Theorem	lcdvbase 36882*	Vector base set of a dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.)
|- H = ( LHyp ` K )   &    |- ._|_ = ( ( ocH ` K ) ` W )   &    |- C = ( (LCDual ` K ) ` W )   &    |- V = ( Base ` C )   &    |- U = ( ( DVecH ` K ) ` W )   &    |- F = (LFnl ` U )   &    |- L = (LKer ` U )   &    |- B = { f e. F | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }   &    |- ( ph -> ( K e. HL /\ W e. H ) )   =>    |- ( ph -> V = B )
Theorem	lcdvbasess 36883	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 )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   =>    |- ( ph -> V C_ F )
Theorem	lcdvbaselfl 36884	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 )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. V )   =>    |- ( ph -> X e. F )
Theorem	lcdvbasecl 36885	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 )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> F e. E )   &    |- ( ph -> X e. V )   =>    |- ( ph -> ( F ` X ) e. R )
Theorem	lcdvadd 36886	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 )   &    |- .+ = ( +g ` D )   &    |- C = ( (LCDual ` K ) ` W )   &    |- .+b = ( +g ` C )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   =>    |- ( ph -> .+b = .+ )
Theorem	lcdvaddval 36887	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 )   &    |- .+ = ( +g ` R )   &    |- C = ( (LCDual ` K ) ` W )   &    |- D = ( Base ` C )   &    |- .+b = ( +g ` C )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> F e. D )   &    |- ( ph -> G e. D )   &    |- ( ph -> X e. V )   =>    |- ( ph -> ( ( F .+b G ) ` X ) = ( ( F ` X ) .+ ( G ` X ) ) )
Theorem	lcdsca 36888	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 = (oppr ` F )   &    |- C = ( (LCDual ` K ) ` W )   &    |- R = (Scalar ` C )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   =>    |- ( ph -> R = O )
Theorem	lcdsbase 36889	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 )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   =>    |- ( ph -> R = L )
Theorem	lcdsadd 36890	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 )   &    |- .+ = ( +g ` F )   &    |- C = ( (LCDual ` K ) ` W )   &    |- S = (Scalar ` C )   &    |- .+b = ( +g ` S )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   =>    |- ( ph -> .+b = .+ )
Theorem	lcdsmul 36891	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 )   &    |- .x. = ( .r ` F )   &    |- C = ( (LCDual ` K ) ` W )   &    |- S = (Scalar ` C )   &    |- .xb = ( .r ` S )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. L )   &    |- ( ph -> Y e. L )   =>    |- ( ph -> ( X .xb Y ) = ( Y .x. X ) )
Theorem	lcdvs 36892	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 )   &    |- .x. = ( .s ` D )   &    |- C = ( (LCDual ` K ) ` W )   &    |- .xb = ( .s ` C )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   =>    |- ( ph -> .xb = .x. )
Theorem	lcdvsval 36893	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 )   &    |- .x. = ( .r ` S )   &    |- C = ( (LCDual ` K ) ` W )   &    |- F = ( Base ` C )   &    |- .xb = ( .s ` C )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. R )   &    |- ( ph -> G e. F )   &    |- ( ph -> A e. V )   =>    |- ( ph -> ( ( X .xb G ) ` A ) = ( ( G ` A ) .x. X ) )
Theorem	lcdvscl 36894	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 )   &    |- .x. = ( .s ` C )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. R )   &    |- ( ph -> G e. V )   =>    |- ( ph -> ( X .x. G ) e. V )
Theorem	lcdlssvscl 36895	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 )   &    |- .x. = ( .s ` C )   &    |- S = ( LSubSp ` C )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> L e. S )   &    |- ( ph -> X e. R )   &    |- ( ph -> Y e. L )   =>    |- ( ph -> ( X .x. Y ) e. L )
Theorem	lcdvsass 36896	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 )   &    |- .x. = ( .r ` R )   &    |- C = ( (LCDual ` K ) ` W )   &    |- F = ( Base ` C )   &    |- .xb = ( .s ` C )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   &    |- ( ph -> X e. L )   &    |- ( ph -> Y e. L )   &    |- ( ph -> G e. F )   =>    |- ( ph -> ( ( Y .x. X ) .xb G ) = ( X .xb ( Y .xb G ) ) )
Theorem	lcd0 36897	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. = ( 0g ` F )   &    |- C = ( (LCDual ` K ) ` W )   &    |- S = (Scalar ` C )   &    |- O = ( 0g ` S )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   =>    |- ( ph -> O = .0. )
Theorem	lcd1 36898	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. = ( 1r ` F )   &    |- C = ( (LCDual ` K ) ` W )   &    |- S = (Scalar ` C )   &    |- I = ( 1r ` S )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   =>    |- ( ph -> I = .1. )
Theorem	lcdneg 36899	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 = ( invg ` R )   &    |- C = ( (LCDual ` K ) ` W )   &    |- S = (Scalar ` C )   &    |- N = ( invg ` S )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   =>    |- ( ph -> N = M )
Theorem	lcd0v 36900	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. = ( 0g ` R )   &    |- C = ( (LCDual ` K ) ` W )   &    |- O = ( 0g ` C )   &    |- ( ph -> ( K e. HL /\ W e. H ) )   =>    |- ( ph -> O = ( V X. { .0. } ) )