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Theorem cdlemj2 31621
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 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 ) )  ->  ( ( 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 962 . . . . 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 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 `  F )  =/=  ( R `  g )
)
4 simpl3r 1014 . . . . 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 2438 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
12 eqid 2438 . . . . . 6  |-  ( Atoms `  K )  =  (
Atoms `  K )
136, 7, 8, 9, 10, 11, 12cdlemj1 31620 . . . . 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 1197 . . . 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 590 . . 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 2790 . 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 988 . . 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 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 )
) )  ->  U  e.  E )
19 simp133 1095 . . . 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 31566 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  h  e.  T
)  ->  ( U `  h )  e.  T
)
2117, 18, 19, 20syl3anc 1185 . . 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 1091 . . . 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 31566 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  V  e.  E  /\  h  e.  T
)  ->  ( V `  h )  e.  T
)
2417, 22, 19, 23syl3anc 1185 . . 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 30948 . . 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 1185 . 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 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   class class class wbr 4214    _I cid 4495    |` cres 4882   ` cfv 5456   Basecbs 13471   lecple 13538   Atomscatm 30063   HLchlt 30150   LHypclh 30783   LTrncltrn 30900   trLctrl 30957   TEndoctendo 31551
This theorem is referenced by:  cdlemj3  31622
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-fal 1330  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-map 7022  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-p1 14471  df-lat 14477  df-clat 14539  df-oposet 29976  df-ol 29978  df-oml 29979  df-covers 30066  df-ats 30067  df-atl 30098  df-cvlat 30122  df-hlat 30151  df-llines 30297  df-lplanes 30298  df-lvols 30299  df-lines 30300  df-psubsp 30302  df-pmap 30303  df-padd 30595  df-lhyp 30787  df-laut 30788  df-ldil 30903  df-ltrn 30904  df-trl 30958  df-tendo 31554
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