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Theorem cdlemj2 34097
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
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 simpl1 1008 . . . . 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 1009 . . . . 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 1060 . . . . 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 1061 . . . . 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 462 . . . . 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 2429 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
12 eqid 2429 . . . . . 6  |-  ( Atoms `  K )  =  (
Atoms `  K )
136, 7, 8, 9, 10, 11, 12cdlemj1 34096 . . . . 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 1276 . . . 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 608 . . 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 2844 . 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 1035 . . 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 1137 . . . 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 1142 . . . 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 34042 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  h  e.  T
)  ->  ( U `  h )  e.  T
)
2117, 18, 19, 20syl3anc 1264 . . 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 1138 . . . 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 34042 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E  /\  h  e.  T
)  ->  ( V `  h )  e.  T
)
2417, 22, 19, 23syl3anc 1264 . . 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 33422 . . 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 1264 . 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 213 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782   class class class wbr 4426    _I cid 4764    |` cres 4856   ` cfv 5601   Basecbs 15084   lecple 15159   Atomscatm 32537   HLchlt 32624   LHypclh 33257   LTrncltrn 33374   trLctrl 33432   TEndoctendo 34027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-riotaBAD 32233
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-undef 7028  df-map 7482  df-preset 16124  df-poset 16142  df-plt 16155  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-p0 16236  df-p1 16237  df-lat 16243  df-clat 16305  df-oposet 32450  df-ol 32452  df-oml 32453  df-covers 32540  df-ats 32541  df-atl 32572  df-cvlat 32596  df-hlat 32625  df-llines 32771  df-lplanes 32772  df-lvols 32773  df-lines 32774  df-psubsp 32776  df-pmap 32777  df-padd 33069  df-lhyp 33261  df-laut 33262  df-ldil 33377  df-ltrn 33378  df-trl 33433  df-tendo 34030
This theorem is referenced by:  cdlemj3  34098
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