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Theorem lcfl7lem 34499
Description: Lemma for lcfl7N 34501. If two functionals  G and  J are equal, they are determined by the same vector. (Contributed by NM, 4-Jan-2015.)
Hypotheses
Ref Expression
lcfl7lem.h  |-  H  =  ( LHyp `  K
)
lcfl7lem.o  |-  ._|_  =  ( ( ocH `  K
) `  W )
lcfl7lem.u  |-  U  =  ( ( DVecH `  K
) `  W )
lcfl7lem.v  |-  V  =  ( Base `  U
)
lcfl7lem.a  |-  .+  =  ( +g  `  U )
lcfl7lem.t  |-  .x.  =  ( .s `  U )
lcfl7lem.s  |-  S  =  (Scalar `  U )
lcfl7lem.r  |-  R  =  ( Base `  S
)
lcfl7lem.z  |-  .0.  =  ( 0g `  U )
lcfl7lem.f  |-  F  =  (LFnl `  U )
lcfl7lem.l  |-  L  =  (LKer `  U )
lcfl7lem.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
lcfl7lem.g  |-  G  =  ( v  e.  V  |->  ( iota_ k  e.  R  E. w  e.  (  ._|_  `  { X }
) v  =  ( w  .+  ( k 
.x.  X ) ) ) )
lcfl7lem.j  |-  J  =  ( v  e.  V  |->  ( iota_ k  e.  R  E. w  e.  (  ._|_  `  { Y }
) v  =  ( w  .+  ( k 
.x.  Y ) ) ) )
lcfl7lem.x  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
lcfl7lem.x2  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
lcfl7lem.gj  |-  ( ph  ->  G  =  J )
Assertion
Ref Expression
lcfl7lem  |-  ( ph  ->  X  =  Y )
Distinct variable groups:    v, k, w,  .+    ._|_ , k, v, w   
w,  .0.    R, k, v    S, k, w    v, V    .x. , k, v, w    k, X, v, w    k, Y, v, w
Allowed substitution hints:    ph( w, v, k)    R( w)    S( v)    U( w, v, k)    F( w, v, k)    G( w, v, k)    H( w, v, k)    J( w, v, k)    K( w, v, k)    L( w, v, k)    V( w, k)    W( w, v, k)    .0. ( v, k)

Proof of Theorem lcfl7lem
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 lcfl7lem.h . . . . . 6  |-  H  =  ( LHyp `  K
)
2 lcfl7lem.o . . . . . 6  |-  ._|_  =  ( ( ocH `  K
) `  W )
3 lcfl7lem.u . . . . . 6  |-  U  =  ( ( DVecH `  K
) `  W )
4 lcfl7lem.v . . . . . 6  |-  V  =  ( Base `  U
)
5 lcfl7lem.z . . . . . 6  |-  .0.  =  ( 0g `  U )
6 lcfl7lem.a . . . . . 6  |-  .+  =  ( +g  `  U )
7 lcfl7lem.t . . . . . 6  |-  .x.  =  ( .s `  U )
8 lcfl7lem.l . . . . . 6  |-  L  =  (LKer `  U )
9 lcfl7lem.s . . . . . 6  |-  S  =  (Scalar `  U )
10 lcfl7lem.r . . . . . 6  |-  R  =  ( Base `  S
)
11 lcfl7lem.g . . . . . 6  |-  G  =  ( v  e.  V  |->  ( iota_ k  e.  R  E. w  e.  (  ._|_  `  { X }
) v  =  ( w  .+  ( k 
.x.  X ) ) ) )
12 lcfl7lem.k . . . . . 6  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
13 lcfl7lem.x . . . . . 6  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
141, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13dochsnkr2cl 34474 . . . . 5  |-  ( ph  ->  X  e.  ( ( 
._|_  `  ( L `  G ) )  \  {  .0.  } ) )
1514eldifad 3425 . . . 4  |-  ( ph  ->  X  e.  (  ._|_  `  ( L `  G
) ) )
16 lcfl7lem.gj . . . . . . . 8  |-  ( ph  ->  G  =  J )
1716fveq2d 5852 . . . . . . 7  |-  ( ph  ->  ( L `  G
)  =  ( L `
 J ) )
18 lcfl7lem.j . . . . . . . 8  |-  J  =  ( v  e.  V  |->  ( iota_ k  e.  R  E. w  e.  (  ._|_  `  { Y }
) v  =  ( w  .+  ( k 
.x.  Y ) ) ) )
19 lcfl7lem.x2 . . . . . . . 8  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
201, 2, 3, 4, 5, 6, 7, 8, 9, 10, 18, 12, 19dochsnkr2 34473 . . . . . . 7  |-  ( ph  ->  ( L `  J
)  =  (  ._|_  `  { Y } ) )
2117, 20eqtrd 2443 . . . . . 6  |-  ( ph  ->  ( L `  G
)  =  (  ._|_  `  { Y } ) )
2221fveq2d 5852 . . . . 5  |-  ( ph  ->  (  ._|_  `  ( L `
 G ) )  =  (  ._|_  `  (  ._|_  `  { Y }
) ) )
23 eqid 2402 . . . . . . 7  |-  ( LSpan `  U )  =  (
LSpan `  U )
2419eldifad 3425 . . . . . . . 8  |-  ( ph  ->  Y  e.  V )
2524snssd 4116 . . . . . . 7  |-  ( ph  ->  { Y }  C_  V )
261, 3, 2, 4, 23, 12, 25dochocsp 34379 . . . . . 6  |-  ( ph  ->  (  ._|_  `  ( (
LSpan `  U ) `  { Y } ) )  =  (  ._|_  `  { Y } ) )
2726fveq2d 5852 . . . . 5  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( ( LSpan `  U
) `  { Y } ) ) )  =  (  ._|_  `  (  ._|_  `  { Y }
) ) )
28 eqid 2402 . . . . . . . 8  |-  ( (
DIsoH `  K ) `  W )  =  ( ( DIsoH `  K ) `  W )
291, 3, 4, 23, 28dihlsprn 34331 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  e.  V
)  ->  ( ( LSpan `  U ) `  { Y } )  e. 
ran  ( ( DIsoH `  K ) `  W
) )
3012, 24, 29syl2anc 659 . . . . . 6  |-  ( ph  ->  ( ( LSpan `  U
) `  { Y } )  e.  ran  ( ( DIsoH `  K
) `  W )
)
311, 28, 2dochoc 34367 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( LSpan `  U ) `  { Y } )  e.  ran  ( ( DIsoH `  K
) `  W )
)  ->  (  ._|_  `  (  ._|_  `  ( (
LSpan `  U ) `  { Y } ) ) )  =  ( (
LSpan `  U ) `  { Y } ) )
3212, 30, 31syl2anc 659 . . . . 5  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( ( LSpan `  U
) `  { Y } ) ) )  =  ( ( LSpan `  U ) `  { Y } ) )
3322, 27, 323eqtr2d 2449 . . . 4  |-  ( ph  ->  (  ._|_  `  ( L `
 G ) )  =  ( ( LSpan `  U ) `  { Y } ) )
3415, 33eleqtrd 2492 . . 3  |-  ( ph  ->  X  e.  ( (
LSpan `  U ) `  { Y } ) )
351, 3, 12dvhlmod 34110 . . . 4  |-  ( ph  ->  U  e.  LMod )
369, 10, 4, 7, 23lspsnel 17967 . . . 4  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( X  e.  ( ( LSpan `  U ) `  { Y } )  <->  E. s  e.  R  X  =  ( s  .x.  Y
) ) )
3735, 24, 36syl2anc 659 . . 3  |-  ( ph  ->  ( X  e.  ( ( LSpan `  U ) `  { Y } )  <->  E. s  e.  R  X  =  ( s  .x.  Y ) ) )
3834, 37mpbid 210 . 2  |-  ( ph  ->  E. s  e.  R  X  =  ( s  .x.  Y ) )
39 simp3 999 . . . 4  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  X  =  ( s  .x.  Y
) )
40 fveq2 5848 . . . . . . . . . 10  |-  ( X  =  ( s  .x.  Y )  ->  ( G `  X )  =  ( G `  ( s  .x.  Y
) ) )
41403ad2ant3 1020 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( G `  X )  =  ( G `  ( s 
.x.  Y ) ) )
42 eqid 2402 . . . . . . . . . . . 12  |-  ( 1r
`  S )  =  ( 1r `  S
)
431, 2, 3, 4, 6, 7, 5, 9, 10, 42, 12, 19, 18dochfl1 34476 . . . . . . . . . . 11  |-  ( ph  ->  ( J `  Y
)  =  ( 1r
`  S ) )
4416fveq1d 5850 . . . . . . . . . . 11  |-  ( ph  ->  ( G `  Y
)  =  ( J `
 Y ) )
451, 2, 3, 4, 6, 7, 5, 9, 10, 42, 12, 13, 11dochfl1 34476 . . . . . . . . . . 11  |-  ( ph  ->  ( G `  X
)  =  ( 1r
`  S ) )
4643, 44, 453eqtr4rd 2454 . . . . . . . . . 10  |-  ( ph  ->  ( G `  X
)  =  ( G `
 Y ) )
47463ad2ant1 1018 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( G `  X )  =  ( G `  Y ) )
48353ad2ant1 1018 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  U  e.  LMod )
49 lcfl7lem.f . . . . . . . . . . . 12  |-  F  =  (LFnl `  U )
501, 2, 3, 4, 5, 6, 7, 49, 9, 10, 11, 12, 13dochflcl 34475 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  F )
51503ad2ant1 1018 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  G  e.  F )
52 simp2 998 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  s  e.  R )
53243ad2ant1 1018 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  Y  e.  V )
54 eqid 2402 . . . . . . . . . . 11  |-  ( .r
`  S )  =  ( .r `  S
)
559, 10, 54, 4, 7, 49lflmul 32066 . . . . . . . . . 10  |-  ( ( U  e.  LMod  /\  G  e.  F  /\  (
s  e.  R  /\  Y  e.  V )
)  ->  ( G `  ( s  .x.  Y
) )  =  ( s ( .r `  S ) ( G `
 Y ) ) )
5648, 51, 52, 53, 55syl112anc 1234 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( G `  ( s  .x.  Y
) )  =  ( s ( .r `  S ) ( G `
 Y ) ) )
5741, 47, 563eqtr3d 2451 . . . . . . . 8  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( G `  Y )  =  ( s ( .r `  S ) ( G `
 Y ) ) )
5857oveq1d 6292 . . . . . . 7  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( ( G `  Y )
( .r `  S
) ( ( invr `  S ) `  ( G `  Y )
) )  =  ( ( s ( .r
`  S ) ( G `  Y ) ) ( .r `  S ) ( (
invr `  S ) `  ( G `  Y
) ) ) )
599lmodring 17838 . . . . . . . . . 10  |-  ( U  e.  LMod  ->  S  e. 
Ring )
6035, 59syl 17 . . . . . . . . 9  |-  ( ph  ->  S  e.  Ring )
61603ad2ant1 1018 . . . . . . . 8  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  S  e.  Ring )
629, 10, 4, 49lflcl 32062 . . . . . . . . . 10  |-  ( ( U  e.  LMod  /\  G  e.  F  /\  Y  e.  V )  ->  ( G `  Y )  e.  R )
6335, 50, 24, 62syl3anc 1230 . . . . . . . . 9  |-  ( ph  ->  ( G `  Y
)  e.  R )
64633ad2ant1 1018 . . . . . . . 8  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( G `  Y )  e.  R
)
651, 3, 12dvhlvec 34109 . . . . . . . . . . 11  |-  ( ph  ->  U  e.  LVec )
669lvecdrng 18069 . . . . . . . . . . 11  |-  ( U  e.  LVec  ->  S  e.  DivRing )
6765, 66syl 17 . . . . . . . . . 10  |-  ( ph  ->  S  e.  DivRing )
6844, 43eqtrd 2443 . . . . . . . . . . 11  |-  ( ph  ->  ( G `  Y
)  =  ( 1r
`  S ) )
69 eqid 2402 . . . . . . . . . . . . 13  |-  ( 0g
`  S )  =  ( 0g `  S
)
7069, 42drngunz 17729 . . . . . . . . . . . 12  |-  ( S  e.  DivRing  ->  ( 1r `  S )  =/=  ( 0g `  S ) )
7167, 70syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( 1r `  S
)  =/=  ( 0g
`  S ) )
7268, 71eqnetrd 2696 . . . . . . . . . 10  |-  ( ph  ->  ( G `  Y
)  =/=  ( 0g
`  S ) )
73 eqid 2402 . . . . . . . . . . 11  |-  ( invr `  S )  =  (
invr `  S )
7410, 69, 73drnginvrcl 17731 . . . . . . . . . 10  |-  ( ( S  e.  DivRing  /\  ( G `  Y )  e.  R  /\  ( G `  Y )  =/=  ( 0g `  S
) )  ->  (
( invr `  S ) `  ( G `  Y
) )  e.  R
)
7567, 63, 72, 74syl3anc 1230 . . . . . . . . 9  |-  ( ph  ->  ( ( invr `  S
) `  ( G `  Y ) )  e.  R )
76753ad2ant1 1018 . . . . . . . 8  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( ( invr `  S ) `  ( G `  Y ) )  e.  R )
7710, 54ringass 17533 . . . . . . . 8  |-  ( ( S  e.  Ring  /\  (
s  e.  R  /\  ( G `  Y )  e.  R  /\  (
( invr `  S ) `  ( G `  Y
) )  e.  R
) )  ->  (
( s ( .r
`  S ) ( G `  Y ) ) ( .r `  S ) ( (
invr `  S ) `  ( G `  Y
) ) )  =  ( s ( .r
`  S ) ( ( G `  Y
) ( .r `  S ) ( (
invr `  S ) `  ( G `  Y
) ) ) ) )
7861, 52, 64, 76, 77syl13anc 1232 . . . . . . 7  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( (
s ( .r `  S ) ( G `
 Y ) ) ( .r `  S
) ( ( invr `  S ) `  ( G `  Y )
) )  =  ( s ( .r `  S ) ( ( G `  Y ) ( .r `  S
) ( ( invr `  S ) `  ( G `  Y )
) ) ) )
7910, 69, 54, 42, 73drnginvrr 17734 . . . . . . . . . 10  |-  ( ( S  e.  DivRing  /\  ( G `  Y )  e.  R  /\  ( G `  Y )  =/=  ( 0g `  S
) )  ->  (
( G `  Y
) ( .r `  S ) ( (
invr `  S ) `  ( G `  Y
) ) )  =  ( 1r `  S
) )
8067, 63, 72, 79syl3anc 1230 . . . . . . . . 9  |-  ( ph  ->  ( ( G `  Y ) ( .r
`  S ) ( ( invr `  S
) `  ( G `  Y ) ) )  =  ( 1r `  S ) )
81803ad2ant1 1018 . . . . . . . 8  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( ( G `  Y )
( .r `  S
) ( ( invr `  S ) `  ( G `  Y )
) )  =  ( 1r `  S ) )
8281oveq2d 6293 . . . . . . 7  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( s
( .r `  S
) ( ( G `
 Y ) ( .r `  S ) ( ( invr `  S
) `  ( G `  Y ) ) ) )  =  ( s ( .r `  S
) ( 1r `  S ) ) )
8358, 78, 823eqtrrd 2448 . . . . . 6  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( s
( .r `  S
) ( 1r `  S ) )  =  ( ( G `  Y ) ( .r
`  S ) ( ( invr `  S
) `  ( G `  Y ) ) ) )
8410, 54, 42ringridm 17541 . . . . . . 7  |-  ( ( S  e.  Ring  /\  s  e.  R )  ->  (
s ( .r `  S ) ( 1r
`  S ) )  =  s )
8561, 52, 84syl2anc 659 . . . . . 6  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( s
( .r `  S
) ( 1r `  S ) )  =  s )
8683, 85, 813eqtr3d 2451 . . . . 5  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  s  =  ( 1r `  S ) )
87 oveq1 6284 . . . . . 6  |-  ( s  =  ( 1r `  S )  ->  (
s  .x.  Y )  =  ( ( 1r
`  S )  .x.  Y ) )
884, 9, 7, 42lmodvs1 17858 . . . . . . . 8  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  (
( 1r `  S
)  .x.  Y )  =  Y )
8935, 24, 88syl2anc 659 . . . . . . 7  |-  ( ph  ->  ( ( 1r `  S )  .x.  Y
)  =  Y )
90893ad2ant1 1018 . . . . . 6  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( ( 1r `  S )  .x.  Y )  =  Y )
9187, 90sylan9eqr 2465 . . . . 5  |-  ( ( ( ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y
) )  /\  s  =  ( 1r `  S ) )  -> 
( s  .x.  Y
)  =  Y )
9286, 91mpdan 666 . . . 4  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( s  .x.  Y )  =  Y )
9339, 92eqtrd 2443 . . 3  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  X  =  Y )
9493rexlimdv3a 2897 . 2  |-  ( ph  ->  ( E. s  e.  R  X  =  ( s  .x.  Y )  ->  X  =  Y ) )
9538, 94mpd 15 1  |-  ( ph  ->  X  =  Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   E.wrex 2754    \ cdif 3410   {csn 3971    |-> cmpt 4452   ran crn 4823   ` cfv 5568   iota_crio 6238  (class class class)co 6277   Basecbs 14839   +g cplusg 14907   .rcmulr 14908  Scalarcsca 14910   .scvsca 14911   0gc0g 15052   1rcur 17471   Ringcrg 17516   invrcinvr 17638   DivRingcdr 17714   LModclmod 17830   LSpanclspn 17935   LVecclvec 18066  LFnlclfn 32055  LKerclk 32083   HLchlt 32348   LHypclh 32981   DVecHcdvh 34078   DIsoHcdih 34228   ocHcoch 34347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-riotaBAD 31957
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-tpos 6957  df-undef 7004  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-map 7458  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-3 10635  df-4 10636  df-5 10637  df-6 10638  df-n0 10836  df-z 10905  df-uz 11127  df-fz 11725  df-struct 14841  df-ndx 14842  df-slot 14843  df-base 14844  df-sets 14845  df-ress 14846  df-plusg 14920  df-mulr 14921  df-sca 14923  df-vsca 14924  df-0g 15054  df-preset 15879  df-poset 15897  df-plt 15910  df-lub 15926  df-glb 15927  df-join 15928  df-meet 15929  df-p0 15991  df-p1 15992  df-lat 15998  df-clat 16060  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-submnd 16289  df-grp 16379  df-minusg 16380  df-sbg 16381  df-subg 16520  df-cntz 16677  df-lsm 16978  df-cmn 17122  df-abl 17123  df-mgp 17460  df-ur 17472  df-ring 17518  df-oppr 17590  df-dvdsr 17608  df-unit 17609  df-invr 17639  df-dvr 17650  df-drng 17716  df-lmod 17832  df-lss 17897  df-lsp 17936  df-lvec 18067  df-lsatoms 31974  df-lshyp 31975  df-lfl 32056  df-lkr 32084  df-oposet 32174  df-ol 32176  df-oml 32177  df-covers 32264  df-ats 32265  df-atl 32296  df-cvlat 32320  df-hlat 32349  df-llines 32495  df-lplanes 32496  df-lvols 32497  df-lines 32498  df-psubsp 32500  df-pmap 32501  df-padd 32793  df-lhyp 32985  df-laut 32986  df-ldil 33101  df-ltrn 33102  df-trl 33157  df-tgrp 33742  df-tendo 33754  df-edring 33756  df-dveca 34002  df-disoa 34029  df-dvech 34079  df-dib 34139  df-dic 34173  df-dih 34229  df-doch 34348  df-djh 34395
This theorem is referenced by:  lcfl7N  34501  lcfrlem9  34550
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