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Theorem dihordlem7 30308
Description: Part of proof of Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
Hypotheses
Ref Expression
dihordlem8.b  |-  B  =  ( Base `  K
)
dihordlem8.l  |-  .<_  =  ( le `  K )
dihordlem8.a  |-  A  =  ( Atoms `  K )
dihordlem8.h  |-  H  =  ( LHyp `  K
)
dihordlem8.p  |-  P  =  ( ( oc `  K ) `  W
)
dihordlem8.o  |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )
dihordlem8.t  |-  T  =  ( ( LTrn `  K
) `  W )
dihordlem8.e  |-  E  =  ( ( TEndo `  K
) `  W )
dihordlem8.u  |-  U  =  ( ( DVecH `  K
) `  W )
dihordlem8.s  |-  .+  =  ( +g  `  U )
dihordlem8.g  |-  G  =  ( iota_ h  e.  T
( h `  P
)  =  R )
Assertion
Ref Expression
dihordlem7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T  /\  <. f ,  O >.  =  ( <. ( s `  G
) ,  s >.  .+  <. g ,  O >. ) ) )  -> 
( f  =  ( ( s `  G
)  o.  g )  /\  O  =  s ) )
Distinct variable groups:    .<_ , h    A, h    B, h    h, H   
h, K    P, h    R, h    T, h    h, W
Allowed substitution hints:    A( f, g, s)    B( f, g, s)    P( f, g, s)    .+ ( f,
g, h, s)    Q( f, g, h, s)    R( f, g, s)    T( f, g, s)    U( f, g, h, s)    E( f, g, h, s)    G( f, g, h, s)    H( f, g, s)    K( f, g, s)    .<_ ( f, g, s)    O( f, g, h, s)    W( f, g, s)

Proof of Theorem dihordlem7
StepHypRef Expression
1 simp33 998 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T  /\  <. f ,  O >.  =  ( <. ( s `  G
) ,  s >.  .+  <. g ,  O >. ) ) )  ->  <. f ,  O >.  =  ( <. ( s `  G ) ,  s
>.  .+  <. g ,  O >. ) )
2 simp1 960 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T  /\  <. f ,  O >.  =  ( <. ( s `  G
) ,  s >.  .+  <. g ,  O >. ) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
3 simp2l 986 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T  /\  <. f ,  O >.  =  ( <. ( s `  G
) ,  s >.  .+  <. g ,  O >. ) ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
4 simp2r 987 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T  /\  <. f ,  O >.  =  ( <. ( s `  G
) ,  s >.  .+  <. g ,  O >. ) ) )  -> 
( R  e.  A  /\  -.  R  .<_  W ) )
5 simp31 996 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T  /\  <. f ,  O >.  =  ( <. ( s `  G
) ,  s >.  .+  <. g ,  O >. ) ) )  -> 
s  e.  E )
6 simp32 997 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T  /\  <. f ,  O >.  =  ( <. ( s `  G
) ,  s >.  .+  <. g ,  O >. ) ) )  -> 
g  e.  T )
7 dihordlem8.b . . . . 5  |-  B  =  ( Base `  K
)
8 dihordlem8.l . . . . 5  |-  .<_  =  ( le `  K )
9 dihordlem8.a . . . . 5  |-  A  =  ( Atoms `  K )
10 dihordlem8.h . . . . 5  |-  H  =  ( LHyp `  K
)
11 dihordlem8.p . . . . 5  |-  P  =  ( ( oc `  K ) `  W
)
12 dihordlem8.o . . . . 5  |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )
13 dihordlem8.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
14 dihordlem8.e . . . . 5  |-  E  =  ( ( TEndo `  K
) `  W )
15 dihordlem8.u . . . . 5  |-  U  =  ( ( DVecH `  K
) `  W )
16 dihordlem8.s . . . . 5  |-  .+  =  ( +g  `  U )
17 dihordlem8.g . . . . 5  |-  G  =  ( iota_ h  e.  T
( h `  P
)  =  R )
187, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17dihordlem6 30307 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T ) )  -> 
( <. ( s `  G ) ,  s
>.  .+  <. g ,  O >. )  =  <. (
( s `  G
)  o.  g ) ,  s >. )
192, 3, 4, 5, 6, 18syl122anc 1196 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T  /\  <. f ,  O >.  =  ( <. ( s `  G
) ,  s >.  .+  <. g ,  O >. ) ) )  -> 
( <. ( s `  G ) ,  s
>.  .+  <. g ,  O >. )  =  <. (
( s `  G
)  o.  g ) ,  s >. )
201, 19eqtrd 2285 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T  /\  <. f ,  O >.  =  ( <. ( s `  G
) ,  s >.  .+  <. g ,  O >. ) ) )  ->  <. f ,  O >.  = 
<. ( ( s `  G )  o.  g
) ,  s >.
)
21 fvex 5391 . . . 4  |-  ( s `
 G )  e. 
_V
22 vex 2730 . . . 4  |-  g  e. 
_V
2321, 22coex 5122 . . 3  |-  ( ( s `  G )  o.  g )  e. 
_V
24 vex 2730 . . 3  |-  s  e. 
_V
2523, 24opth2 4141 . 2  |-  ( <.
f ,  O >.  = 
<. ( ( s `  G )  o.  g
) ,  s >.  <->  ( f  =  ( ( s `  G )  o.  g )  /\  O  =  s )
)
2620, 25sylib 190 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( s  e.  E  /\  g  e.  T  /\  <. f ,  O >.  =  ( <. ( s `  G
) ,  s >.  .+  <. g ,  O >. ) ) )  -> 
( f  =  ( ( s `  G
)  o.  g )  /\  O  =  s ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   <.cop 3547   class class class wbr 3920    e. cmpt 3974    _I cid 4197    |` cres 4582    o. ccom 4584   ` cfv 4592  (class class class)co 5710   iota_crio 6181   Basecbs 13022   +g cplusg 13082   lecple 13089   occoc 13090   Atomscatm 28357   HLchlt 28444   LHypclh 29077   LTrncltrn 29194   TEndoctendo 29845   DVecHcdvh 30172
This theorem is referenced by:  dihordlem7b  30309
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-er 6546  df-map 6660  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-n 9627  df-2 9684  df-3 9685  df-4 9686  df-5 9687  df-6 9688  df-n0 9845  df-z 9904  df-uz 10110  df-fz 10661  df-struct 13024  df-ndx 13025  df-slot 13026  df-base 13027  df-plusg 13095  df-mulr 13096  df-sca 13098  df-vsca 13099  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28270  df-ol 28272  df-oml 28273  df-covers 28360  df-ats 28361  df-atl 28392  df-cvlat 28416  df-hlat 28445  df-llines 28591  df-lplanes 28592  df-lvols 28593  df-lines 28594  df-psubsp 28596  df-pmap 28597  df-padd 28889  df-lhyp 29081  df-laut 29082  df-ldil 29197  df-ltrn 29198  df-trl 29252  df-tendo 29848  df-edring 29850  df-dvech 30173
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