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Theorem mapdh9a 37079
Description: Lemma for part (9) in [Baer] p. 48. TODO: why is this 50% larger than mapdh9aOLDN 37080? (Contributed by NM, 14-May-2015.)
Hypotheses
Ref Expression
mapdh8a.h |- H = ( LHyp ` K )
mapdh8a.u |- U = ( ( DVecH ` K ) ` W )
mapdh8a.v |- V = ( Base ` U )
mapdh8a.s |- .- = ( -g ` U )
mapdh8a.o |- .0. = ( 0g ` U )
mapdh8a.n |- N = ( LSpan ` U )
mapdh8a.c |- C = ( (LCDual ` K ) ` W )
mapdh8a.d |- D = ( Base ` C )
mapdh8a.r |- R = ( -g ` C )
mapdh8a.q |- Q = ( 0g ` C )
mapdh8a.j |- J = ( LSpan ` C )
mapdh8a.m |- M = ( (mapd ` K ) ` W )
mapdh8a.i |- I = ( x e. _V |-> if ( ( 2nd ` x ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) ) )
mapdh8a.k |- ( ph -> ( K e. HL /\ W e. H ) )
mapdh8h.f |- ( ph -> F e. D )
mapdh8h.mn |- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
mapdh9a.x |- ( ph -> X e. ( V \ { .0. } ) )
mapdh9a.t |- ( ph -> T e. V )
Assertion
Ref Expression
mapdh9a |- ( ph -> E! y e. D A. z e. V ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) )
Distinct variable groups:   x, h, .-   .0. , h, x   C, h   D, h, x   h, F, x   h, I   h, J, x   h, M, x   h, N, x   ph, h   R, h, x   x, Q   T, h, x   U, h   h, X, x   x, I   h, V   y, z, D   y, F, z   y, I, z   y, N, z   y, .0. , z   y, T, z   z, U   y, V, z   y, X, z   ph, y, z   z, h, x
Allowed substitution hints:   ph( x)   C( x, y, z)   Q( y, z, h)   R( y, z)   U( x, y)   H( x, y, z, h)   J( y, z)   K( x, y, z, h)   M( y, z)   .- ( y, z)   V( x)   W( x, y, z, h)
Proof of Theorem mapdh9a
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 mapdh8a.h . . . . . . 7 |- H = ( LHyp ` K )
2 mapdh8a.u . . . . . . 7 |- U = ( ( DVecH ` K ) ` W )
3 mapdh8a.v . . . . . . 7 |- V = ( Base ` U )
4 mapdh8a.s . . . . . . 7 |- .- = ( -g ` U )
5 mapdh8a.o . . . . . . 7 |- .0. = ( 0g ` U )
6 mapdh8a.n . . . . . . 7 |- N = ( LSpan ` U )
7 mapdh8a.c . . . . . . 7 |- C = ( (LCDual ` K ) ` W )
8 mapdh8a.d . . . . . . 7 |- D = ( Base ` C )
9 mapdh8a.r . . . . . . 7 |- R = ( -g ` C )
10 mapdh8a.q . . . . . . 7 |- Q = ( 0g ` C )
11 mapdh8a.j . . . . . . 7 |- J = ( LSpan ` C )
12 mapdh8a.m . . . . . . 7 |- M = ( (mapd ` K ) ` W )
13 mapdh8a.i . . . . . . 7 |- I = ( x e. _V |-> if ( ( 2nd ` x ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) ) )
14 mapdh8a.k . . . . . . . 8 |- ( ph -> ( K e. HL /\ W e. H ) )
15143ad2ant1 1082 . . . . . . 7 |- ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> ( K e. HL /\ W e. H ) )
16 mapdh8h.f . . . . . . . 8 |- ( ph -> F e. D )
17163ad2ant1 1082 . . . . . . 7 |- ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> F e. D )
18 mapdh8h.mn . . . . . . . 8 |- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
19183ad2ant1 1082 . . . . . . 7 |- ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
20 mapdh9a.x . . . . . . . 8 |- ( ph -> X e. ( V \ { .0. } ) )
21203ad2ant1 1082 . . . . . . 7 |- ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> X e. ( V \ { .0. } ) )
22 simp3ll 1132 . . . . . . 7 |- ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> z e. ( V \ { .0. } ) )
23 simp3rl 1134 . . . . . . 7 |- ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> w e. ( V \ { .0. } ) )
24 simplrl 800 . . . . . . . . 9 |- ( ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) -> ( N ` { z } ) =/= ( N ` { X } ) )
25243ad2ant3 1084 . . . . . . . 8 |- ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> ( N ` { z } ) =/= ( N ` { X } ) )
2625necomd 2849 . . . . . . 7 |- ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> ( N ` { X } ) =/= ( N ` { z } ) )
27 simprrl 804 . . . . . . . . 9 |- ( ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) -> ( N ` { w } ) =/= ( N ` { X } ) )
28273ad2ant3 1084 . . . . . . . 8 |- ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> ( N ` { w } ) =/= ( N ` { X } ) )
2928necomd 2849 . . . . . . 7 |- ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> ( N ` { X } ) =/= ( N ` { w } ) )
30 simplrr 801 . . . . . . . 8 |- ( ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) -> ( N ` { z } ) =/= ( N ` { T } ) )
31303ad2ant3 1084 . . . . . . 7 |- ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> ( N ` { z } ) =/= ( N ` { T } ) )
32 simprrr 805 . . . . . . . 8 |- ( ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) -> ( N ` { w } ) =/= ( N ` { T } ) )
33323ad2ant3 1084 . . . . . . 7 |- ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> ( N ` { w } ) =/= ( N ` { T } ) )
34 mapdh9a.t . . . . . . . 8 |- ( ph -> T e. V )
35343ad2ant1 1082 . . . . . . 7 |- ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> T e. V )
361, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 17, 19, 21, 22, 23, 26, 29, 31, 33, 35mapdh8 37078 . . . . . 6 |- ( ( ph /\ ( z e. V /\ w e. V ) /\ ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) ) -> ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) = ( I ` <. w , ( I ` <. X , F , w >. ) , T >. ) )
37363exp 1264 . . . . 5 |- ( ph -> ( ( z e. V /\ w e. V ) -> ( ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) -> ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) = ( I ` <. w , ( I ` <. X , F , w >. ) , T >. ) ) ) )
3837ralrimivv 2970 . . . 4 |- ( ph -> A. z e. V A. w e. V ( ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) -> ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) = ( I ` <. w , ( I ` <. X , F , w >. ) , T >. ) ) )
3920eldifad 3586 . . . . . . . 8 |- ( ph -> X e. V )
401, 2, 3, 6, 14, 39, 34dvh3dim 36735 . . . . . . 7 |- ( ph -> E. z e. V -. z e. ( N ` { X , T } ) )
41 eqid 2622 . . . . . . . . . . 11 |- ( LSubSp ` U ) = ( LSubSp ` U )
421, 2, 14dvhlmod 36399 . . . . . . . . . . . 12 |- ( ph -> U e. LMod )
4342ad2antrr 762 . . . . . . . . . . 11 |- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> U e. LMod )
443, 41, 6, 42, 39, 34lspprcl 18978 . . . . . . . . . . . 12 |- ( ph -> ( N ` { X , T } ) e. ( LSubSp ` U ) )
4544ad2antrr 762 . . . . . . . . . . 11 |- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> ( N ` { X , T } ) e. ( LSubSp ` U ) )
46 simplr 792 . . . . . . . . . . 11 |- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> z e. V )
47 simpr 477 . . . . . . . . . . 11 |- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> -. z e. ( N ` { X , T } ) )
483, 5, 41, 43, 45, 46, 47lssneln0 18952 . . . . . . . . . 10 |- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> z e. ( V \ { .0. } ) )
491, 2, 14dvhlvec 36398 . . . . . . . . . . . 12 |- ( ph -> U e. LVec )
5049ad2antrr 762 . . . . . . . . . . 11 |- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> U e. LVec )
5139ad2antrr 762 . . . . . . . . . . 11 |- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> X e. V )
5234ad2antrr 762 . . . . . . . . . . 11 |- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> T e. V )
533, 6, 50, 46, 51, 52, 47lspindpi 19132 . . . . . . . . . 10 |- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) )
5448, 53jca 554 . . . . . . . . 9 |- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X , T } ) ) -> ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) )
5554ex 450 . . . . . . . 8 |- ( ( ph /\ z e. V ) -> ( -. z e. ( N ` { X , T } ) -> ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) )
5655reximdva 3017 . . . . . . 7 |- ( ph -> ( E. z e. V -. z e. ( N ` { X , T } ) -> E. z e. V ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) )
5740, 56mpd 15 . . . . . 6 |- ( ph -> E. z e. V ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) )
5814ad2antrr 762 . . . . . . . . . 10 |- ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> ( K e. HL /\ W e. H ) )
5916ad2antrr 762 . . . . . . . . . . 11 |- ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> F e. D )
6018ad2antrr 762 . . . . . . . . . . 11 |- ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
6120ad2antrr 762 . . . . . . . . . . 11 |- ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> X e. ( V \ { .0. } ) )
62 simplr 792 . . . . . . . . . . 11 |- ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> z e. V )
63 simprrl 804 . . . . . . . . . . . 12 |- ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> ( N ` { z } ) =/= ( N ` { X } ) )
6463necomd 2849 . . . . . . . . . . 11 |- ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> ( N ` { X } ) =/= ( N ` { z } ) )
6510, 13, 1, 12, 2, 3, 4, 5, 6, 7, 8, 9, 11, 58, 59, 60, 61, 62, 64mapdhcl 37016 . . . . . . . . . 10 |- ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> ( I ` <. X , F , z >. ) e. D )
66 eqidd 2623 . . . . . . . . . . . 12 |- ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> ( I ` <. X , F , z >. ) = ( I ` <. X , F , z >. ) )
67 simprl 794 . . . . . . . . . . . . 13 |- ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> z e. ( V \ { .0. } ) )
6810, 13, 1, 12, 2, 3, 4, 5, 6, 7, 8, 9, 11, 58, 59, 60, 61, 67, 65, 64mapdheq 37017 . . . . . . . . . . . 12 |- ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> ( ( I ` <. X , F , z >. ) = ( I ` <. X , F , z >. ) <-> ( ( M ` ( N ` { z } ) ) = ( J ` { ( I ` <. X , F , z >. ) } ) /\ ( M ` ( N ` { ( X .- z ) } ) ) = ( J ` { ( F R ( I ` <. X , F , z >. ) ) } ) ) ) )
6966, 68mpbid 222 . . . . . . . . . . 11 |- ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> ( ( M ` ( N ` { z } ) ) = ( J ` { ( I ` <. X , F , z >. ) } ) /\ ( M ` ( N ` { ( X .- z ) } ) ) = ( J ` { ( F R ( I ` <. X , F , z >. ) ) } ) ) )
7069simpld 475 . . . . . . . . . 10 |- ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> ( M ` ( N ` { z } ) ) = ( J ` { ( I ` <. X , F , z >. ) } ) )
7134ad2antrr 762 . . . . . . . . . 10 |- ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> T e. V )
72 simprrr 805 . . . . . . . . . 10 |- ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> ( N ` { z } ) =/= ( N ` { T } ) )
7310, 13, 1, 12, 2, 3, 4, 5, 6, 7, 8, 9, 11, 58, 65, 70, 67, 71, 72mapdhcl 37016 . . . . . . . . 9 |- ( ( ( ph /\ z e. V ) /\ ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) -> ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) e. D )
7473ex 450 . . . . . . . 8 |- ( ( ph /\ z e. V ) -> ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) -> ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) e. D ) )
7574ancld 576 . . . . . . 7 |- ( ( ph /\ z e. V ) -> ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) -> ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) e. D ) ) )
7675reximdva 3017 . . . . . 6 |- ( ph -> ( E. z e. V ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) -> E. z e. V ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) e. D ) ) )
7757, 76mpd 15 . . . . 5 |- ( ph -> E. z e. V ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) e. D ) )
78 eleq1 2689 . . . . . . 7 |- ( z = w -> ( z e. ( V \ { .0. } ) <-> w e. ( V \ { .0. } ) ) )
79 sneq 4187 . . . . . . . . . 10 |- ( z = w -> { z } = { w } )
8079fveq2d 6195 . . . . . . . . 9 |- ( z = w -> ( N ` { z } ) = ( N ` { w } ) )
8180neeq1d 2853 . . . . . . . 8 |- ( z = w -> ( ( N ` { z } ) =/= ( N ` { X } ) <-> ( N ` { w } ) =/= ( N ` { X } ) ) )
8280neeq1d 2853 . . . . . . . 8 |- ( z = w -> ( ( N ` { z } ) =/= ( N ` { T } ) <-> ( N ` { w } ) =/= ( N ` { T } ) ) )
8381, 82anbi12d 747 . . . . . . 7 |- ( z = w -> ( ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) <-> ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) )
8478, 83anbi12d 747 . . . . . 6 |- ( z = w -> ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) <-> ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) )
85 oteq1 4411 . . . . . . . 8 |- ( z = w -> <. z , ( I ` <. X , F , z >. ) , T >. = <. w , ( I ` <. X , F , z >. ) , T >. )
86 oteq3 4413 . . . . . . . . . 10 |- ( z = w -> <. X , F , z >. = <. X , F , w >. )
8786fveq2d 6195 . . . . . . . . 9 |- ( z = w -> ( I ` <. X , F , z >. ) = ( I ` <. X , F , w >. ) )
8887oteq2d 4415 . . . . . . . 8 |- ( z = w -> <. w , ( I ` <. X , F , z >. ) , T >. = <. w , ( I ` <. X , F , w >. ) , T >. )
8985, 88eqtrd 2656 . . . . . . 7 |- ( z = w -> <. z , ( I ` <. X , F , z >. ) , T >. = <. w , ( I ` <. X , F , w >. ) , T >. )
9089fveq2d 6195 . . . . . 6 |- ( z = w -> ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) = ( I ` <. w , ( I ` <. X , F , w >. ) , T >. ) )
9184, 90reusv3 4876 . . . . 5 |- ( E. z e. V ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) e. D ) -> ( A. z e. V A. w e. V ( ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) -> ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) = ( I ` <. w , ( I ` <. X , F , w >. ) , T >. ) ) <-> E. y e. D A. z e. V ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) ) )
9277, 91syl 17 . . . 4 |- ( ph -> ( A. z e. V A. w e. V ( ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) /\ ( w e. ( V \ { .0. } ) /\ ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { T } ) ) ) ) -> ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) = ( I ` <. w , ( I ` <. X , F , w >. ) , T >. ) ) <-> E. y e. D A. z e. V ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) ) )
9338, 92mpbid 222 . . 3 |- ( ph -> E. y e. D A. z e. V ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) )
94 ioran 511 . . . . . . . 8 |- ( -. ( z e. ( N ` { X } ) \/ z e. ( N ` { T } ) ) <-> ( -. z e. ( N ` { X } ) /\ -. z e. ( N ` { T } ) ) )
95 elun 3753 . . . . . . . 8 |- ( z e. ( ( N ` { X } ) u. ( N ` { T } ) ) <-> ( z e. ( N ` { X } ) \/ z e. ( N ` { T } ) ) )
9694, 95xchnxbir 323 . . . . . . 7 |- ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) <-> ( -. z e. ( N ` { X } ) /\ -. z e. ( N ` { T } ) ) )
9742ad2antrr 762 . . . . . . . . . 10 |- ( ( ( ph /\ z e. V ) /\ ( -. z e. ( N ` { X } ) /\ -. z e. ( N ` { T } ) ) ) -> U e. LMod )
983, 41, 6lspsncl 18977 . . . . . . . . . . . 12 |- ( ( U e. LMod /\ X e. V ) -> ( N ` { X } ) e. ( LSubSp ` U ) )
9942, 39, 98syl2anc 693 . . . . . . . . . . 11 |- ( ph -> ( N ` { X } ) e. ( LSubSp ` U ) )
10099ad2antrr 762 . . . . . . . . . 10 |- ( ( ( ph /\ z e. V ) /\ ( -. z e. ( N ` { X } ) /\ -. z e. ( N ` { T } ) ) ) -> ( N ` { X } ) e. ( LSubSp ` U ) )
101 simplr 792 . . . . . . . . . 10 |- ( ( ( ph /\ z e. V ) /\ ( -. z e. ( N ` { X } ) /\ -. z e. ( N ` { T } ) ) ) -> z e. V )
102 simprl 794 . . . . . . . . . 10 |- ( ( ( ph /\ z e. V ) /\ ( -. z e. ( N ` { X } ) /\ -. z e. ( N ` { T } ) ) ) -> -. z e. ( N ` { X } ) )
1033, 5, 41, 97, 100, 101, 102lssneln0 18952 . . . . . . . . 9 |- ( ( ( ph /\ z e. V ) /\ ( -. z e. ( N ` { X } ) /\ -. z e. ( N ` { T } ) ) ) -> z e. ( V \ { .0. } ) )
104103ex 450 . . . . . . . 8 |- ( ( ph /\ z e. V ) -> ( ( -. z e. ( N ` { X } ) /\ -. z e. ( N ` { T } ) ) -> z e. ( V \ { .0. } ) ) )
10542ad2antrr 762 . . . . . . . . . . 11 |- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X } ) ) -> U e. LMod )
106 simplr 792 . . . . . . . . . . 11 |- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X } ) ) -> z e. V )
10739ad2antrr 762 . . . . . . . . . . 11 |- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X } ) ) -> X e. V )
108 simpr 477 . . . . . . . . . . 11 |- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X } ) ) -> -. z e. ( N ` { X } ) )
1093, 6, 105, 106, 107, 108lspsnne2 19118 . . . . . . . . . 10 |- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { X } ) ) -> ( N ` { z } ) =/= ( N ` { X } ) )
110109ex 450 . . . . . . . . 9 |- ( ( ph /\ z e. V ) -> ( -. z e. ( N ` { X } ) -> ( N ` { z } ) =/= ( N ` { X } ) ) )
11142ad2antrr 762 . . . . . . . . . . 11 |- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { T } ) ) -> U e. LMod )
112 simplr 792 . . . . . . . . . . 11 |- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { T } ) ) -> z e. V )
11334ad2antrr 762 . . . . . . . . . . 11 |- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { T } ) ) -> T e. V )
114 simpr 477 . . . . . . . . . . 11 |- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { T } ) ) -> -. z e. ( N ` { T } ) )
1153, 6, 111, 112, 113, 114lspsnne2 19118 . . . . . . . . . 10 |- ( ( ( ph /\ z e. V ) /\ -. z e. ( N ` { T } ) ) -> ( N ` { z } ) =/= ( N ` { T } ) )
116115ex 450 . . . . . . . . 9 |- ( ( ph /\ z e. V ) -> ( -. z e. ( N ` { T } ) -> ( N ` { z } ) =/= ( N ` { T } ) ) )
117110, 116anim12d 586 . . . . . . . 8 |- ( ( ph /\ z e. V ) -> ( ( -. z e. ( N ` { X } ) /\ -. z e. ( N ` { T } ) ) -> ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) )
118104, 117jcad 555 . . . . . . 7 |- ( ( ph /\ z e. V ) -> ( ( -. z e. ( N ` { X } ) /\ -. z e. ( N ` { T } ) ) -> ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) )
11996, 118syl5bi 232 . . . . . 6 |- ( ( ph /\ z e. V ) -> ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) ) )
120119imim1d 82 . . . . 5 |- ( ( ph /\ z e. V ) -> ( ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) -> ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) ) )
121120ralimdva 2962 . . . 4 |- ( ph -> ( A. z e. V ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) -> A. z e. V ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) ) )
122121reximdv 3016 . . 3 |- ( ph -> ( E. y e. D A. z e. V ( ( z e. ( V \ { .0. } ) /\ ( ( N ` { z } ) =/= ( N ` { X } ) /\ ( N ` { z } ) =/= ( N ` { T } ) ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) -> E. y e. D A. z e. V ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) ) )
12393, 122mpd 15 . 2 |- ( ph -> E. y e. D A. z e. V ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) )
1243, 6, 42, 39, 34lspprid1 18997 . . . . . . . 8 |- ( ph -> X e. ( N ` { X , T } ) )
12541, 6, 42, 44, 124lspsnel5a 18996 . . . . . . 7 |- ( ph -> ( N ` { X } ) C_ ( N ` { X , T } ) )
1263, 6, 42, 39, 34lspprid2 18998 . . . . . . . 8 |- ( ph -> T e. ( N ` { X , T } ) )
12741, 6, 42, 44, 126lspsnel5a 18996 . . . . . . 7 |- ( ph -> ( N ` { T } ) C_ ( N ` { X , T } ) )
128125, 127unssd 3789 . . . . . 6 |- ( ph -> ( ( N ` { X } ) u. ( N ` { T } ) ) C_ ( N ` { X , T } ) )
129128ssneld 3605 . . . . 5 |- ( ph -> ( -. z e. ( N ` { X , T } ) -> -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) ) )
130129reximdv 3016 . . . 4 |- ( ph -> ( E. z e. V -. z e. ( N ` { X , T } ) -> E. z e. V -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) ) )
13140, 130mpd 15 . . 3 |- ( ph -> E. z e. V -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) )
132 reusv1 4866 . . 3 |- ( E. z e. V -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> ( E! y e. D A. z e. V ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) <-> E. y e. D A. z e. V ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) ) )
133131, 132syl 17 . 2 |- ( ph -> ( E! y e. D A. z e. V ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) <-> E. y e. D A. z e. V ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) ) )
134123, 133mpbird 247 1 |- ( ph -> E! y e. D A. z e. V ( -. z e. ( ( N ` { X } ) u. ( N ` { T } ) ) -> y = ( I ` <. z , ( I ` <. X , F , z >. ) , T >. ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   E!wreu 2914   _Vcvv 3200   \ cdif 3571   u. cun 3572   ifcif 4086   {csn 4177   {cpr 4179   <.cotp 4185   |-> cmpt 4729   ` cfv 5888   iota_crio 6610  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   Basecbs 15857   0gc0g 16100   -gcsg 17424   LModclmod 18863   LSubSpclss 18932   LSpanclspn 18971   LVecclvec 19102   HLchlt 34637   LHypclh 35270   DVecHcdvh 36367  LCDualclcd 36875  mapdcmpd 36913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-riotaBAD 34239
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-ot 4186  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-undef 7399  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-0g 16102  df-mre 16246  df-mrc 16247  df-acs 16249  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-cntz 17750  df-oppg 17776  df-lsm 18051  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-drng 18749  df-lmod 18865  df-lss 18933  df-lsp 18972  df-lvec 19103  df-lsatoms 34263  df-lshyp 34264  df-lcv 34306  df-lfl 34345  df-lkr 34373  df-ldual 34411  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-llines 34784  df-lplanes 34785  df-lvols 34786  df-lines 34787  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446  df-tgrp 36031  df-tendo 36043  df-edring 36045  df-dveca 36291  df-disoa 36318  df-dvech 36368  df-dib 36428  df-dic 36462  df-dih 36518  df-doch 36637  df-djh 36684  df-lcdual 36876  df-mapd 36914
This theorem is referenced by:  hdmap1eulem  37113
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