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Theorem cdlemj2 31556
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 960 . . . . 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 961 . . . . 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 1012 . . . . 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 1013 . . . . 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 448 . . . . 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 2435 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
12 eqid 2435 . . . . . 6  |-  ( Atoms `  K )  =  (
Atoms `  K )
136, 7, 8, 9, 10, 11, 12cdlemj1 31555 . . . . 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 1196 . . . 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 589 . . 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 2780 . 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 987 . . 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 1089 . . . 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 1094 . . . 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 31501 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  h  e.  T
)  ->  ( U `  h )  e.  T
)
2117, 18, 19, 20syl3anc 1184 . . 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 1090 . . . 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 31501 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E  /\  h  e.  T
)  ->  ( V `  h )  e.  T
)
2417, 22, 19, 23syl3anc 1184 . . 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 30883 . . 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 1184 . 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 202 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 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   class class class wbr 4204    _I cid 4485    |` cres 4872   ` cfv 5446   Basecbs 13461   lecple 13528   Atomscatm 29998   HLchlt 30085   LHypclh 30718   LTrncltrn 30835   trLctrl 30892   TEndoctendo 31486
This theorem is referenced by:  cdlemj3  31557
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 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-fal 1329  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-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-map 7012  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-llines 30232  df-lplanes 30233  df-lvols 30234  df-lines 30235  df-psubsp 30237  df-pmap 30238  df-padd 30530  df-lhyp 30722  df-laut 30723  df-ldil 30838  df-ltrn 30839  df-trl 30893  df-tendo 31489
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