Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemj2 Unicode version

Theorem cdlemj2 30141
Description: Part of proof of Lemma J of [Crawley] p. 118. Eliminate  p. (Contributed by NM, 20-Jun-2013.)
Hypotheses
Ref Expression
cdlemj.b  |-  B  =  ( Base `  K
)
cdlemj.h  |-  H  =  ( LHyp `  K
)
cdlemj.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemj.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemj.e  |-  E  =  ( ( TEndo `  K
) `  W )
Assertion
Ref Expression
cdlemj2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )
) )  ->  ( U `  h )  =  ( V `  h ) )

Proof of Theorem cdlemj2
StepHypRef Expression
1 simpl1 963 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )
) )  /\  (
p  e.  ( Atoms `  K )  /\  -.  p ( le `  K ) W ) )  ->  ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
) )
2 simpl2 964 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )
) )  /\  (
p  e.  ( Atoms `  K )  /\  -.  p ( le `  K ) W ) )  ->  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) ) )
3 simpl3l 1015 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )
) )  /\  (
p  e.  ( Atoms `  K )  /\  -.  p ( le `  K ) W ) )  ->  ( R `  F )  =/=  ( R `  g )
)
4 simpl3r 1016 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )
) )  /\  (
p  e.  ( Atoms `  K )  /\  -.  p ( le `  K ) W ) )  ->  ( R `  g )  =/=  ( R `  h )
)
5 simpr 449 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )
) )  /\  (
p  e.  ( Atoms `  K )  /\  -.  p ( le `  K ) W ) )  ->  ( p  e.  ( Atoms `  K )  /\  -.  p ( le
`  K ) W ) )
6 cdlemj.b . . . . . 6  |-  B  =  ( Base `  K
)
7 cdlemj.h . . . . . 6  |-  H  =  ( LHyp `  K
)
8 cdlemj.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
9 cdlemj.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
10 cdlemj.e . . . . . 6  |-  E  =  ( ( TEndo `  K
) `  W )
11 eqid 2256 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
12 eqid 2256 . . . . . 6  |-  ( Atoms `  K )  =  (
Atoms `  K )
136, 7, 8, 9, 10, 11, 12cdlemj1 30140 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )  /\  ( p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W ) ) )  ->  (
( U `  h
) `  p )  =  ( ( V `
 h ) `  p ) )
141, 2, 3, 4, 5, 13syl113anc 1199 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )
) )  /\  (
p  e.  ( Atoms `  K )  /\  -.  p ( le `  K ) W ) )  ->  ( ( U `  h ) `  p )  =  ( ( V `  h
) `  p )
)
1514exp32 591 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )
) )  ->  (
p  e.  ( Atoms `  K )  ->  ( -.  p ( le `  K ) W  -> 
( ( U `  h ) `  p
)  =  ( ( V `  h ) `
 p ) ) ) )
1615ralrimiv 2596 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )
) )  ->  A. p  e.  ( Atoms `  K )
( -.  p ( le `  K ) W  ->  ( ( U `  h ) `  p )  =  ( ( V `  h
) `  p )
) )
17 simp11 990 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
18 simp121 1092 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )
) )  ->  U  e.  E )
19 simp133 1097 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )
) )  ->  h  e.  T )
207, 8, 10tendocl 30086 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  h  e.  T
)  ->  ( U `  h )  e.  T
)
2117, 18, 19, 20syl3anc 1187 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )
) )  ->  ( U `  h )  e.  T )
22 simp122 1093 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )
) )  ->  V  e.  E )
237, 8, 10tendocl 30086 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E  /\  h  e.  T
)  ->  ( V `  h )  e.  T
)
2417, 22, 19, 23syl3anc 1187 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )
) )  ->  ( V `  h )  e.  T )
2511, 12, 7, 8ltrneq 29468 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U `  h )  e.  T  /\  ( V `  h
)  e.  T )  ->  ( A. p  e.  ( Atoms `  K )
( -.  p ( le `  K ) W  ->  ( ( U `  h ) `  p )  =  ( ( V `  h
) `  p )
)  <->  ( U `  h )  =  ( V `  h ) ) )
2617, 21, 24, 25syl3anc 1187 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )
) )  ->  ( A. p  e.  ( Atoms `  K ) ( -.  p ( le
`  K ) W  ->  ( ( U `
 h ) `  p )  =  ( ( V `  h
) `  p )
)  <->  ( U `  h )  =  ( V `  h ) ) )
2716, 26mpbid 203 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )
) )  ->  ( U `  h )  =  ( V `  h ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2419   A.wral 2516   class class class wbr 3963    _I cid 4241    |` cres 4628   ` cfv 4638   Basecbs 13075   lecple 13142   Atomscatm 28583   HLchlt 28670   LHypclh 29303   LTrncltrn 29420   trLctrl 29477   TEndoctendo 30071
This theorem is referenced by:  cdlemj3  30142
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 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-fal 1316  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-undef 6229  df-riota 6237  df-map 6707  df-poset 14007  df-plt 14019  df-lub 14035  df-glb 14036  df-join 14037  df-meet 14038  df-p0 14072  df-p1 14073  df-lat 14079  df-clat 14141  df-oposet 28496  df-ol 28498  df-oml 28499  df-covers 28586  df-ats 28587  df-atl 28618  df-cvlat 28642  df-hlat 28671  df-llines 28817  df-lplanes 28818  df-lvols 28819  df-lines 28820  df-psubsp 28822  df-pmap 28823  df-padd 29115  df-lhyp 29307  df-laut 29308  df-ldil 29423  df-ltrn 29424  df-trl 29478  df-tendo 30074
  Copyright terms: Public domain W3C validator