Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemj3 Unicode version

Theorem cdlemj3 31064
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
Dummy variables  g  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl1 958 . . 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 2358 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
3 eqid 2358 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 cdlemj.h . . . 4  |-  H  =  ( LHyp `  K
)
52, 3, 4lhpexle2 30251 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. u  e.  (
Atoms `  K ) ( u ( le `  K ) W  /\  u  =/=  ( R `  F )  /\  u  =/=  ( R `  h
) ) )
61, 5syl 15 . 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 1006 . . . . . . 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 451 . . . . . 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 1007 . . . . . . 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 451 . . . . . 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 732 . . . . . 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 1003 . . . . . 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 30805 . . . . . 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 1183 . . . . 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 979 . . . . . . . . 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 980 . . . . . . . . 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 983 . . . . . . . . 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 1029 . . . . . . . . 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 1058 . . . . . . . . . . 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 2604 . . . . . . . . . 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 1028 . . . . . . . . . 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 2545 . . . . . . . . 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 1059 . . . . . . . . . 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 2539 . . . . . . . . 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 31063 . . . . . . . . 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 1200 . . . . . . . 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 1153 . . . . . . 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 425 . . . . . 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 2742 . . . . 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 14 . . . 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 588 . . 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 2742 . 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 14 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 4    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710    =/= wne 2521   E.wrex 2620   class class class wbr 4102    _I cid 4383    |` cres 4770   ` cfv 5334   Basecbs 13239   lecple 13306   Atomscatm 29505   HLchlt 29592   LHypclh 30225   LTrncltrn 30342   trLctrl 30399   TEndoctendo 30993
This theorem is referenced by:  tendocan  31065
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-fal 1320  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-iin 3987  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-undef 6382  df-riota 6388  df-map 6859  df-poset 14173  df-plt 14185  df-lub 14201  df-glb 14202  df-join 14203  df-meet 14204  df-p0 14238  df-p1 14239  df-lat 14245  df-clat 14307  df-oposet 29418  df-ol 29420  df-oml 29421  df-covers 29508  df-ats 29509  df-atl 29540  df-cvlat 29564  df-hlat 29593  df-llines 29739  df-lplanes 29740  df-lvols 29741  df-lines 29742  df-psubsp 29744  df-pmap 29745  df-padd 30037  df-lhyp 30229  df-laut 30230  df-ldil 30345  df-ltrn 30346  df-trl 30400  df-tendo 30996
  Copyright terms: Public domain W3C validator