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Theorem lcfl7lem 36296
Description: Lemma for lcfl7N 36298. 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 36271 . . . . 5  |-  ( ph  ->  X  e.  ( ( 
._|_  `  ( L `  G ) )  \  {  .0.  } ) )
1514eldifad 3488 . . . 4  |-  ( ph  ->  X  e.  (  ._|_  `  ( L `  G
) ) )
16 lcfl7lem.gj . . . . . . . 8  |-  ( ph  ->  G  =  J )
1716fveq2d 5868 . . . . . . 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 36270 . . . . . . 7  |-  ( ph  ->  ( L `  J
)  =  (  ._|_  `  { Y } ) )
2117, 20eqtrd 2508 . . . . . 6  |-  ( ph  ->  ( L `  G
)  =  (  ._|_  `  { Y } ) )
2221fveq2d 5868 . . . . 5  |-  ( ph  ->  (  ._|_  `  ( L `
 G ) )  =  (  ._|_  `  (  ._|_  `  { Y }
) ) )
23 eqid 2467 . . . . . . 7  |-  ( LSpan `  U )  =  (
LSpan `  U )
2419eldifad 3488 . . . . . . . 8  |-  ( ph  ->  Y  e.  V )
2524snssd 4172 . . . . . . 7  |-  ( ph  ->  { Y }  C_  V )
261, 3, 2, 4, 23, 12, 25dochocsp 36176 . . . . . 6  |-  ( ph  ->  (  ._|_  `  ( (
LSpan `  U ) `  { Y } ) )  =  (  ._|_  `  { Y } ) )
2726fveq2d 5868 . . . . 5  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( ( LSpan `  U
) `  { Y } ) ) )  =  (  ._|_  `  (  ._|_  `  { Y }
) ) )
28 eqid 2467 . . . . . . . 8  |-  ( (
DIsoH `  K ) `  W )  =  ( ( DIsoH `  K ) `  W )
291, 3, 4, 23, 28dihlsprn 36128 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  e.  V
)  ->  ( ( LSpan `  U ) `  { Y } )  e. 
ran  ( ( DIsoH `  K ) `  W
) )
3012, 24, 29syl2anc 661 . . . . . 6  |-  ( ph  ->  ( ( LSpan `  U
) `  { Y } )  e.  ran  ( ( DIsoH `  K
) `  W )
)
311, 28, 2dochoc 36164 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( LSpan `  U ) `  { Y } )  e.  ran  ( ( DIsoH `  K
) `  W )
)  ->  (  ._|_  `  (  ._|_  `  ( (
LSpan `  U ) `  { Y } ) ) )  =  ( (
LSpan `  U ) `  { Y } ) )
3212, 30, 31syl2anc 661 . . . . 5  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( ( LSpan `  U
) `  { Y } ) ) )  =  ( ( LSpan `  U ) `  { Y } ) )
3322, 27, 323eqtr2d 2514 . . . 4  |-  ( ph  ->  (  ._|_  `  ( L `
 G ) )  =  ( ( LSpan `  U ) `  { Y } ) )
3415, 33eleqtrd 2557 . . 3  |-  ( ph  ->  X  e.  ( (
LSpan `  U ) `  { Y } ) )
351, 3, 12dvhlmod 35907 . . . 4  |-  ( ph  ->  U  e.  LMod )
369, 10, 4, 7, 23lspsnel 17429 . . . 4  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( X  e.  ( ( LSpan `  U ) `  { Y } )  <->  E. s  e.  R  X  =  ( s  .x.  Y
) ) )
3735, 24, 36syl2anc 661 . . 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 998 . . . 4  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  X  =  ( s  .x.  Y
) )
40 fveq2 5864 . . . . . . . . . 10  |-  ( X  =  ( s  .x.  Y )  ->  ( G `  X )  =  ( G `  ( s  .x.  Y
) ) )
41403ad2ant3 1019 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( G `  X )  =  ( G `  ( s 
.x.  Y ) ) )
42 eqid 2467 . . . . . . . . . . . 12  |-  ( 1r
`  S )  =  ( 1r `  S
)
431, 2, 3, 4, 6, 7, 5, 9, 10, 42, 12, 19, 18dochfl1 36273 . . . . . . . . . . 11  |-  ( ph  ->  ( J `  Y
)  =  ( 1r
`  S ) )
4416fveq1d 5866 . . . . . . . . . . 11  |-  ( ph  ->  ( G `  Y
)  =  ( J `
 Y ) )
451, 2, 3, 4, 6, 7, 5, 9, 10, 42, 12, 13, 11dochfl1 36273 . . . . . . . . . . 11  |-  ( ph  ->  ( G `  X
)  =  ( 1r
`  S ) )
4643, 44, 453eqtr4rd 2519 . . . . . . . . . 10  |-  ( ph  ->  ( G `  X
)  =  ( G `
 Y ) )
47463ad2ant1 1017 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( G `  X )  =  ( G `  Y ) )
48353ad2ant1 1017 . . . . . . . . . 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 36272 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  F )
51503ad2ant1 1017 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  G  e.  F )
52 simp2 997 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  s  e.  R )
53243ad2ant1 1017 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  Y  e.  V )
54 eqid 2467 . . . . . . . . . . 11  |-  ( .r
`  S )  =  ( .r `  S
)
559, 10, 54, 4, 7, 49lflmul 33865 . . . . . . . . . 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 1232 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( G `  ( s  .x.  Y
) )  =  ( s ( .r `  S ) ( G `
 Y ) ) )
5741, 47, 563eqtr3d 2516 . . . . . . . 8  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( G `  Y )  =  ( s ( .r `  S ) ( G `
 Y ) ) )
5857oveq1d 6297 . . . . . . 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
) ) ) )
599lmodrng 17300 . . . . . . . . . 10  |-  ( U  e.  LMod  ->  S  e. 
Ring )
6035, 59syl 16 . . . . . . . . 9  |-  ( ph  ->  S  e.  Ring )
61603ad2ant1 1017 . . . . . . . 8  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  S  e.  Ring )
629, 10, 4, 49lflcl 33861 . . . . . . . . . 10  |-  ( ( U  e.  LMod  /\  G  e.  F  /\  Y  e.  V )  ->  ( G `  Y )  e.  R )
6335, 50, 24, 62syl3anc 1228 . . . . . . . . 9  |-  ( ph  ->  ( G `  Y
)  e.  R )
64633ad2ant1 1017 . . . . . . . 8  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( G `  Y )  e.  R
)
651, 3, 12dvhlvec 35906 . . . . . . . . . . 11  |-  ( ph  ->  U  e.  LVec )
669lvecdrng 17531 . . . . . . . . . . 11  |-  ( U  e.  LVec  ->  S  e.  DivRing )
6765, 66syl 16 . . . . . . . . . 10  |-  ( ph  ->  S  e.  DivRing )
6844, 43eqtrd 2508 . . . . . . . . . . 11  |-  ( ph  ->  ( G `  Y
)  =  ( 1r
`  S ) )
69 eqid 2467 . . . . . . . . . . . . 13  |-  ( 0g
`  S )  =  ( 0g `  S
)
7069, 42drngunz 17191 . . . . . . . . . . . 12  |-  ( S  e.  DivRing  ->  ( 1r `  S )  =/=  ( 0g `  S ) )
7167, 70syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( 1r `  S
)  =/=  ( 0g
`  S ) )
7268, 71eqnetrd 2760 . . . . . . . . . 10  |-  ( ph  ->  ( G `  Y
)  =/=  ( 0g
`  S ) )
73 eqid 2467 . . . . . . . . . . 11  |-  ( invr `  S )  =  (
invr `  S )
7410, 69, 73drnginvrcl 17193 . . . . . . . . . 10  |-  ( ( S  e.  DivRing  /\  ( G `  Y )  e.  R  /\  ( G `  Y )  =/=  ( 0g `  S
) )  ->  (
( invr `  S ) `  ( G `  Y
) )  e.  R
)
7567, 63, 72, 74syl3anc 1228 . . . . . . . . 9  |-  ( ph  ->  ( ( invr `  S
) `  ( G `  Y ) )  e.  R )
76753ad2ant1 1017 . . . . . . . 8  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( ( invr `  S ) `  ( G `  Y ) )  e.  R )
7710, 54rngass 16999 . . . . . . . 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 1230 . . . . . . 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 17196 . . . . . . . . . 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 1228 . . . . . . . . 9  |-  ( ph  ->  ( ( G `  Y ) ( .r
`  S ) ( ( invr `  S
) `  ( G `  Y ) ) )  =  ( 1r `  S ) )
81803ad2ant1 1017 . . . . . . . 8  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( ( G `  Y )
( .r `  S
) ( ( invr `  S ) `  ( G `  Y )
) )  =  ( 1r `  S ) )
8281oveq2d 6298 . . . . . . 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 2513 . . . . . 6  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( s
( .r `  S
) ( 1r `  S ) )  =  ( ( G `  Y ) ( .r
`  S ) ( ( invr `  S
) `  ( G `  Y ) ) ) )
8410, 54, 42rngridm 17007 . . . . . . 7  |-  ( ( S  e.  Ring  /\  s  e.  R )  ->  (
s ( .r `  S ) ( 1r
`  S ) )  =  s )
8561, 52, 84syl2anc 661 . . . . . 6  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( s
( .r `  S
) ( 1r `  S ) )  =  s )
8683, 85, 813eqtr3d 2516 . . . . 5  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  s  =  ( 1r `  S ) )
87 oveq1 6289 . . . . . 6  |-  ( s  =  ( 1r `  S )  ->  (
s  .x.  Y )  =  ( ( 1r
`  S )  .x.  Y ) )
884, 9, 7, 42lmodvs1 17320 . . . . . . . 8  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  (
( 1r `  S
)  .x.  Y )  =  Y )
8935, 24, 88syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( ( 1r `  S )  .x.  Y
)  =  Y )
90893ad2ant1 1017 . . . . . 6  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( ( 1r `  S )  .x.  Y )  =  Y )
9187, 90sylan9eqr 2530 . . . . 5  |-  ( ( ( ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y
) )  /\  s  =  ( 1r `  S ) )  -> 
( s  .x.  Y
)  =  Y )
9286, 91mpdan 668 . . . 4  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  ( s  .x.  Y )  =  Y )
9339, 92eqtrd 2508 . . 3  |-  ( (
ph  /\  s  e.  R  /\  X  =  ( s  .x.  Y ) )  ->  X  =  Y )
9493rexlimdv3a 2957 . 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 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815    \ cdif 3473   {csn 4027    |-> cmpt 4505   ran crn 5000   ` cfv 5586   iota_crio 6242  (class class class)co 6282   Basecbs 14483   +g cplusg 14548   .rcmulr 14549  Scalarcsca 14551   .scvsca 14552   0gc0g 14688   1rcur 16940   Ringcrg 16983   invrcinvr 17101   DivRingcdr 17176   LModclmod 17292   LSpanclspn 17397   LVecclvec 17528  LFnlclfn 33854  LKerclk 33882   HLchlt 34147   LHypclh 34780   DVecHcdvh 35875   DIsoHcdih 36025   ocHcoch 36144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-riotaBAD 33756
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-tpos 6952  df-undef 6999  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-struct 14485  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-mulr 14562  df-sca 14564  df-vsca 14565  df-0g 14690  df-poset 15426  df-plt 15438  df-lub 15454  df-glb 15455  df-join 15456  df-meet 15457  df-p0 15519  df-p1 15520  df-lat 15526  df-clat 15588  df-mnd 15725  df-submnd 15775  df-grp 15855  df-minusg 15856  df-sbg 15857  df-subg 15990  df-cntz 16147  df-lsm 16449  df-cmn 16593  df-abl 16594  df-mgp 16929  df-ur 16941  df-rng 16985  df-oppr 17053  df-dvdsr 17071  df-unit 17072  df-invr 17102  df-dvr 17113  df-drng 17178  df-lmod 17294  df-lss 17359  df-lsp 17398  df-lvec 17529  df-lsatoms 33773  df-lshyp 33774  df-lfl 33855  df-lkr 33883  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-llines 34294  df-lplanes 34295  df-lvols 34296  df-lines 34297  df-psubsp 34299  df-pmap 34300  df-padd 34592  df-lhyp 34784  df-laut 34785  df-ldil 34900  df-ltrn 34901  df-trl 34955  df-tgrp 35539  df-tendo 35551  df-edring 35553  df-dveca 35799  df-disoa 35826  df-dvech 35876  df-dib 35936  df-dic 35970  df-dih 36026  df-doch 36145  df-djh 36192
This theorem is referenced by:  lcfl7N  36298  lcfrlem9  36347
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