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Theorem dihordlem7 31851
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 995 . . 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 957 . . . 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 983 . . . 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 984 . . . 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 993 . . . 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 994 . . . 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 31850 . . . 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 1193 . . 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 2467 . 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 5733 . . . 4  |-  ( s `
 G )  e. 
_V
22 vex 2951 . . . 4  |-  g  e. 
_V
2321, 22coex 5404 . . 3  |-  ( ( s `  G )  o.  g )  e. 
_V
24 vex 2951 . . 3  |-  s  e. 
_V
2523, 24opth2 4430 . 2  |-  ( <.
f ,  O >.  = 
<. ( ( s `  G )  o.  g
) ,  s >.  <->  ( f  =  ( ( s `  G )  o.  g )  /\  O  =  s )
)
2620, 25sylib 189 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 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   <.cop 3809   class class class wbr 4204    e. cmpt 4258    _I cid 4485    |` cres 4871    o. ccom 4873   ` cfv 5445  (class class class)co 6072   iota_crio 6533   Basecbs 13457   +g cplusg 13517   lecple 13524   occoc 13525   Atomscatm 29900   HLchlt 29987   LHypclh 30620   LTrncltrn 30737   TEndoctendo 31388   DVecHcdvh 31715
This theorem is referenced by:  dihordlem7b  31852
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-undef 6534  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-oadd 6719  df-er 6896  df-map 7011  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-nn 9990  df-2 10047  df-3 10048  df-4 10049  df-5 10050  df-6 10051  df-n0 10211  df-z 10272  df-uz 10478  df-fz 11033  df-struct 13459  df-ndx 13460  df-slot 13461  df-base 13462  df-plusg 13530  df-mulr 13531  df-sca 13533  df-vsca 13534  df-poset 14391  df-plt 14403  df-lub 14419  df-glb 14420  df-join 14421  df-meet 14422  df-p0 14456  df-p1 14457  df-lat 14463  df-clat 14525  df-oposet 29813  df-ol 29815  df-oml 29816  df-covers 29903  df-ats 29904  df-atl 29935  df-cvlat 29959  df-hlat 29988  df-llines 30134  df-lplanes 30135  df-lvols 30136  df-lines 30137  df-psubsp 30139  df-pmap 30140  df-padd 30432  df-lhyp 30624  df-laut 30625  df-ldil 30740  df-ltrn 30741  df-trl 30795  df-tendo 31391  df-edring 31393  df-dvech 31716
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