Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dihordlem7b Unicode version

Theorem dihordlem7b 31405
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
dihordlem7b  |-  ( ( ( 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  =  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 dihordlem7b
StepHypRef Expression
1 dihordlem8.b . . . . 5  |-  B  =  ( Base `  K
)
2 dihordlem8.l . . . . 5  |-  .<_  =  ( le `  K )
3 dihordlem8.a . . . . 5  |-  A  =  ( Atoms `  K )
4 dihordlem8.h . . . . 5  |-  H  =  ( LHyp `  K
)
5 dihordlem8.p . . . . 5  |-  P  =  ( ( oc `  K ) `  W
)
6 dihordlem8.o . . . . 5  |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )
7 dihordlem8.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
8 dihordlem8.e . . . . 5  |-  E  =  ( ( TEndo `  K
) `  W )
9 dihordlem8.u . . . . 5  |-  U  =  ( ( DVecH `  K
) `  W )
10 dihordlem8.s . . . . 5  |-  .+  =  ( +g  `  U )
11 dihordlem8.g . . . . 5  |-  G  =  ( iota_ h  e.  T
( h `  P
)  =  R )
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11dihordlem7 31404 . . . 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 >. ) ) )  -> 
( f  =  ( ( s `  G
)  o.  g )  /\  O  =  s ) )
1312simpld 445 . . 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  =  ( ( s `  G )  o.  g ) )
1412simprd 449 . . . . . 6  |-  ( ( ( 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 >. ) ) )  ->  O  =  s )
1514fveq1d 5527 . . . . 5  |-  ( ( ( 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 >. ) ) )  -> 
( O `  G
)  =  ( s `
 G ) )
16 simp1 955 . . . . . . 7  |-  ( ( ( 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 ) )
172, 3, 4, 5lhpocnel2 30208 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
18173ad2ant1 976 . . . . . . 7  |-  ( ( ( 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 >. ) ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
19 simp2r 982 . . . . . . 7  |-  ( ( ( 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 ) )
202, 3, 4, 7, 11ltrniotacl 30768 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  G  e.  T )
2116, 18, 19, 20syl3anc 1182 . . . . . 6  |-  ( ( ( 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 )
226, 1tendo02 30976 . . . . . 6  |-  ( G  e.  T  ->  ( O `  G )  =  (  _I  |`  B ) )
2321, 22syl 15 . . . . 5  |-  ( ( ( 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 >. ) ) )  -> 
( O `  G
)  =  (  _I  |`  B ) )
2415, 23eqtr3d 2317 . . . 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 `  G
)  =  (  _I  |`  B ) )
2524coeq1d 4845 . . 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 )  o.  g
)  =  ( (  _I  |`  B )  o.  g ) )
26 simp32 992 . . . . 5  |-  ( ( ( 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 )
271, 4, 7ltrn1o 30313 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  g  e.  T
)  ->  g : B
-1-1-onto-> B )
2816, 26, 27syl2anc 642 . . . 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 : B -1-1-onto-> B )
29 f1of 5472 . . . 4  |-  ( g : B -1-1-onto-> B  ->  g : B
--> B )
30 fcoi2 5416 . . . 4  |-  ( g : B --> B  -> 
( (  _I  |`  B )  o.  g )  =  g )
3128, 29, 303syl 18 . . 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 >. ) ) )  -> 
( (  _I  |`  B )  o.  g )  =  g )
3213, 25, 313eqtrd 2319 . 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  =  g )
3332, 14jca 518 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  =  g  /\  O  =  s ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   <.cop 3643   class class class wbr 4023    e. cmpt 4077    _I cid 4304    |` cres 4691    o. ccom 4693   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148   +g cplusg 13208   lecple 13215   occoc 13216   Atomscatm 29453   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   TEndoctendo 30941   DVecHcdvh 31268
This theorem is referenced by:  dihord10  31413
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348  df-tendo 30944  df-edring 30946  df-dvech 31269
  Copyright terms: Public domain W3C validator