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Theorem cdlemj3 29813
Description: Part of proof of Lemma J of [Crawley] p. 118. Eliminate  g. (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
cdlemj3  |-  ( ( ( ( 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 ) )  ->  ( U `  h )  =  ( V `  h ) )

Proof of Theorem cdlemj3
StepHypRef Expression
1 simpl1 963 . . 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 ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 eqid 2253 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
3 eqid 2253 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 cdlemj.h . . . 4  |-  H  =  ( LHyp `  K
)
52, 3, 4lhpexle2 29000 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. u  e.  (
Atoms `  K ) ( u ( le `  K ) W  /\  u  =/=  ( R `  F )  /\  u  =/=  ( R `  h
) ) )
61, 5syl 17 . 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 ) )  ->  E. u  e.  ( Atoms `  K )
( u ( le
`  K ) W  /\  u  =/=  ( R `  F )  /\  u  =/=  ( R `  h )
) )
7 simpl1l 1011 . . . . . . 7  |-  ( ( ( ( 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 ) )  ->  K  e.  HL )
87adantr 453 . . . . . 6  |-  ( ( ( ( ( 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 ) )  /\  ( u  e.  ( Atoms `  K
)  /\  ( u
( le `  K
) W  /\  u  =/=  ( R `  F
)  /\  u  =/=  ( R `  h ) ) ) )  ->  K  e.  HL )
9 simpl1r 1012 . . . . . . 7  |-  ( ( ( ( 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 ) )  ->  W  e.  H )
109adantr 453 . . . . . 6  |-  ( ( ( ( ( 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 ) )  /\  ( u  e.  ( Atoms `  K
)  /\  ( u
( le `  K
) W  /\  u  =/=  ( R `  F
)  /\  u  =/=  ( R `  h ) ) ) )  ->  W  e.  H )
11 simprl 735 . . . . . 6  |-  ( ( ( ( ( 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 ) )  /\  ( u  e.  ( Atoms `  K
)  /\  ( u
( le `  K
) W  /\  u  =/=  ( R `  F
)  /\  u  =/=  ( R `  h ) ) ) )  ->  u  e.  ( Atoms `  K ) )
12 simprr1 1008 . . . . . 6  |-  ( ( ( ( ( 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 ) )  /\  ( u  e.  ( Atoms `  K
)  /\  ( u
( le `  K
) W  /\  u  =/=  ( R `  F
)  /\  u  =/=  ( R `  h ) ) ) )  ->  u ( le `  K ) W )
13 cdlemj.b . . . . . . 7  |-  B  =  ( Base `  K
)
14 cdlemj.t . . . . . . 7  |-  T  =  ( ( LTrn `  K
) `  W )
15 cdlemj.r . . . . . . 7  |-  R  =  ( ( trL `  K
) `  W )
1613, 2, 3, 4, 14, 15cdlemfnid 29554 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( u  e.  ( Atoms `  K )  /\  u ( le `  K ) W ) )  ->  E. g  e.  T  ( ( R `  g )  =  u  /\  g  =/=  (  _I  |`  B ) ) )
178, 10, 11, 12, 16syl22anc 1188 . . . . 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 ) )  /\  ( u  e.  ( Atoms `  K
)  /\  ( u
( le `  K
) W  /\  u  =/=  ( R `  F
)  /\  u  =/=  ( R `  h ) ) ) )  ->  E. g  e.  T  ( ( R `  g )  =  u  /\  g  =/=  (  _I  |`  B ) ) )
18 simp1l 984 . . . . . . . . 9  |-  ( ( ( ( ( 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 ) )  /\  ( u  e.  ( Atoms `  K
)  /\  ( u
( le `  K
) W  /\  u  =/=  ( R `  F
)  /\  u  =/=  ( R `  h ) ) )  /\  (
g  e.  T  /\  ( ( R `  g )  =  u  /\  g  =/=  (  _I  |`  B ) ) ) )  ->  (
( 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
) ) )
19 simp1r 985 . . . . . . . . 9  |-  ( ( ( ( ( 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 ) )  /\  ( u  e.  ( Atoms `  K
)  /\  ( u
( le `  K
) W  /\  u  =/=  ( R `  F
)  /\  u  =/=  ( R `  h ) ) )  /\  (
g  e.  T  /\  ( ( R `  g )  =  u  /\  g  =/=  (  _I  |`  B ) ) ) )  ->  h  =/=  (  _I  |`  B ) )
20 simp3l 988 . . . . . . . . 9  |-  ( ( ( ( ( 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 ) )  /\  ( u  e.  ( Atoms `  K
)  /\  ( u
( le `  K
) W  /\  u  =/=  ( R `  F
)  /\  u  =/=  ( R `  h ) ) )  /\  (
g  e.  T  /\  ( ( R `  g )  =  u  /\  g  =/=  (  _I  |`  B ) ) ) )  ->  g  e.  T )
21 simp3rr 1034 . . . . . . . . 9  |-  ( ( ( ( ( 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 ) )  /\  ( u  e.  ( Atoms `  K
)  /\  ( u
( le `  K
) W  /\  u  =/=  ( R `  F
)  /\  u  =/=  ( R `  h ) ) )  /\  (
g  e.  T  /\  ( ( R `  g )  =  u  /\  g  =/=  (  _I  |`  B ) ) ) )  ->  g  =/=  (  _I  |`  B ) )
22 simp2r2 1063 . . . . . . . . . . 11  |-  ( ( ( ( ( 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 ) )  /\  ( u  e.  ( Atoms `  K
)  /\  ( u
( le `  K
) W  /\  u  =/=  ( R `  F
)  /\  u  =/=  ( R `  h ) ) )  /\  (
g  e.  T  /\  ( ( R `  g )  =  u  /\  g  =/=  (  _I  |`  B ) ) ) )  ->  u  =/=  ( R `  F
) )
2322necomd 2495 . . . . . . . . . 10  |-  ( ( ( ( ( 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 ) )  /\  ( u  e.  ( Atoms `  K
)  /\  ( u
( le `  K
) W  /\  u  =/=  ( R `  F
)  /\  u  =/=  ( R `  h ) ) )  /\  (
g  e.  T  /\  ( ( R `  g )  =  u  /\  g  =/=  (  _I  |`  B ) ) ) )  ->  ( R `  F )  =/=  u )
24 simp3rl 1033 . . . . . . . . . 10  |-  ( ( ( ( ( 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 ) )  /\  ( u  e.  ( Atoms `  K
)  /\  ( u
( le `  K
) W  /\  u  =/=  ( R `  F
)  /\  u  =/=  ( R `  h ) ) )  /\  (
g  e.  T  /\  ( ( R `  g )  =  u  /\  g  =/=  (  _I  |`  B ) ) ) )  ->  ( R `  g )  =  u )
2523, 24neeqtrrd 2436 . . . . . . . . 9  |-  ( ( ( ( ( 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 ) )  /\  ( u  e.  ( Atoms `  K
)  /\  ( u
( le `  K
) W  /\  u  =/=  ( R `  F
)  /\  u  =/=  ( R `  h ) ) )  /\  (
g  e.  T  /\  ( ( R `  g )  =  u  /\  g  =/=  (  _I  |`  B ) ) ) )  ->  ( R `  F )  =/=  ( R `  g
) )
26 simp2r3 1064 . . . . . . . . . 10  |-  ( ( ( ( ( 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 ) )  /\  ( u  e.  ( Atoms `  K
)  /\  ( u
( le `  K
) W  /\  u  =/=  ( R `  F
)  /\  u  =/=  ( R `  h ) ) )  /\  (
g  e.  T  /\  ( ( R `  g )  =  u  /\  g  =/=  (  _I  |`  B ) ) ) )  ->  u  =/=  ( R `  h
) )
2724, 26eqnetrd 2430 . . . . . . . . 9  |-  ( ( ( ( ( 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 ) )  /\  ( u  e.  ( Atoms `  K
)  /\  ( u
( le `  K
) W  /\  u  =/=  ( R `  F
)  /\  u  =/=  ( R `  h ) ) )  /\  (
g  e.  T  /\  ( ( R `  g )  =  u  /\  g  =/=  (  _I  |`  B ) ) ) )  ->  ( R `  g )  =/=  ( R `  h
) )
28 cdlemj.e . . . . . . . . . 10  |-  E  =  ( ( TEndo `  K
) `  W )
2913, 4, 14, 15, 28cdlemj2 29812 . . . . . . . . 9  |-  ( ( ( ( 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 ) )
3018, 19, 20, 21, 25, 27, 29syl132anc 1205 . . . . . . . 8  |-  ( ( ( ( ( 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 ) )  /\  ( u  e.  ( Atoms `  K
)  /\  ( u
( le `  K
) W  /\  u  =/=  ( R `  F
)  /\  u  =/=  ( R `  h ) ) )  /\  (
g  e.  T  /\  ( ( R `  g )  =  u  /\  g  =/=  (  _I  |`  B ) ) ) )  ->  ( U `  h )  =  ( V `  h ) )
31303expia 1158 . . . . . . 7  |-  ( ( ( ( ( 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 ) )  /\  ( u  e.  ( Atoms `  K
)  /\  ( u
( le `  K
) W  /\  u  =/=  ( R `  F
)  /\  u  =/=  ( R `  h ) ) ) )  -> 
( ( g  e.  T  /\  ( ( R `  g )  =  u  /\  g  =/=  (  _I  |`  B ) ) )  ->  ( U `  h )  =  ( V `  h ) ) )
3231exp3a 427 . . . . . 6  |-  ( ( ( ( ( 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 ) )  /\  ( u  e.  ( Atoms `  K
)  /\  ( u
( le `  K
) W  /\  u  =/=  ( R `  F
)  /\  u  =/=  ( R `  h ) ) ) )  -> 
( g  e.  T  ->  ( ( ( R `
 g )  =  u  /\  g  =/=  (  _I  |`  B ) )  ->  ( U `  h )  =  ( V `  h ) ) ) )
3332rexlimdv 2628 . . . . 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 ) )  /\  ( u  e.  ( Atoms `  K
)  /\  ( u
( le `  K
) W  /\  u  =/=  ( R `  F
)  /\  u  =/=  ( R `  h ) ) ) )  -> 
( E. g  e.  T  ( ( R `
 g )  =  u  /\  g  =/=  (  _I  |`  B ) )  ->  ( U `  h )  =  ( V `  h ) ) )
3417, 33mpd 16 . . . 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 ) )  /\  ( u  e.  ( Atoms `  K
)  /\  ( u
( le `  K
) W  /\  u  =/=  ( R `  F
)  /\  u  =/=  ( R `  h ) ) ) )  -> 
( U `  h
)  =  ( V `
 h ) )
3534exp32 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 ) )  ->  ( u  e.  ( Atoms `  K )  ->  ( ( u ( le `  K ) W  /\  u  =/=  ( R `  F
)  /\  u  =/=  ( R `  h ) )  ->  ( U `  h )  =  ( V `  h ) ) ) )
3635rexlimdv 2628 . 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 ) )  ->  ( E. u  e.  ( Atoms `  K ) ( u ( le `  K
) W  /\  u  =/=  ( R `  F
)  /\  u  =/=  ( R `  h ) )  ->  ( U `  h )  =  ( V `  h ) ) )
376, 36mpd 16 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 ) )  ->  ( U `  h )  =  ( V `  h ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   E.wrex 2510   class class class wbr 3920    _I cid 4197    |` cres 4582   ` cfv 4592   Basecbs 13022   lecple 13089   Atomscatm 28254   HLchlt 28341   LHypclh 28974   LTrncltrn 29091   trLctrl 29148   TEndoctendo 29742
This theorem is referenced by:  tendocan  29814
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-map 6660  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28167  df-ol 28169  df-oml 28170  df-covers 28257  df-ats 28258  df-atl 28289  df-cvlat 28313  df-hlat 28342  df-llines 28488  df-lplanes 28489  df-lvols 28490  df-lines 28491  df-psubsp 28493  df-pmap 28494  df-padd 28786  df-lhyp 28978  df-laut 28979  df-ldil 29094  df-ltrn 29095  df-trl 29149  df-tendo 29745
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