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

Theorem cdlemj3 34472
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 991 . . 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 2443 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
3 eqid 2443 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 cdlemj.h . . . 4  |-  H  =  ( LHyp `  K
)
52, 3, 4lhpexle2 33659 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. u  e.  (
Atoms `  K ) ( u ( le `  K ) W  /\  u  =/=  ( R `  F )  /\  u  =/=  ( R `  h
) ) )
61, 5syl 16 . 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 1039 . . . . 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 ) )  ->  K  e.  HL )
87adantr 465 . . . 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 ) ) ) )  ->  K  e.  HL )
9 simpl1r 1040 . . . . 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 ) )  ->  W  e.  H )
109adantr 465 . . . 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 ) ) ) )  ->  W  e.  H )
11 simprl 755 . . . 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  e.  ( Atoms `  K ) )
12 simprr1 1036 . . . 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 ( le `  K ) W )
13 cdlemj.b . . . . 5  |-  B  =  ( Base `  K
)
14 cdlemj.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
15 cdlemj.r . . . . 5  |-  R  =  ( ( trL `  K
) `  W )
1613, 2, 3, 4, 14, 15cdlemfnid 34213 . . . 4  |-  ( ( ( 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 1219 . . 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 ) ) ) )  ->  E. g  e.  T  ( ( R `  g )  =  u  /\  g  =/=  (  _I  |`  B ) ) )
18 simp1l 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 ) )  /\  ( 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 1013 . . . . . . 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 ) ) ) )  ->  h  =/=  (  _I  |`  B ) )
20 simp3l 1016 . . . . . . 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 ) ) ) )  ->  g  e.  T )
21 simp3rr 1062 . . . . . . 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 ) ) ) )  ->  g  =/=  (  _I  |`  B ) )
22 simp2r2 1091 . . . . . . . . 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 ) ) ) )  ->  u  =/=  ( R `  F
) )
2322necomd 2700 . . . . . . . 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 ) ) ) )  ->  ( R `  F )  =/=  u )
24 simp3rl 1061 . . . . . . . 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 ) ) ) )  ->  ( R `  g )  =  u )
2523, 24neeqtrrd 2637 . . . . . . 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 ) ) ) )  ->  ( R `  F )  =/=  ( R `  g
) )
26 simp2r3 1092 . . . . . . . 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  =/=  ( R `  h
) )
2724, 26eqnetrd 2631 . . . . . . 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 ) ) ) )  ->  ( R `  g )  =/=  ( R `  h
) )
28 cdlemj.e . . . . . . . 8  |-  E  =  ( ( TEndo `  K
) `  W )
2913, 4, 14, 15, 28cdlemj2 34471 . . . . . . 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 )  /\  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 1236 . . . . . 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 ) )
31303expia 1189 . . . . 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 ) ) ) )  -> 
( ( g  e.  T  /\  ( ( R `  g )  =  u  /\  g  =/=  (  _I  |`  B ) ) )  ->  ( U `  h )  =  ( V `  h ) ) )
3231expd 436 . . . 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 ) ) ) )  -> 
( g  e.  T  ->  ( ( ( R `
 g )  =  u  /\  g  =/=  (  _I  |`  B ) )  ->  ( U `  h )  =  ( V `  h ) ) ) )
3332rexlimdv 2845 . . 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 ) ) ) )  -> 
( E. g  e.  T  ( ( R `
 g )  =  u  /\  g  =/=  (  _I  |`  B ) )  ->  ( U `  h )  =  ( V `  h ) ) )
3417, 33mpd 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 ) )  /\  ( u  e.  ( Atoms `  K
)  /\  ( u
( le `  K
) W  /\  u  =/=  ( R `  F
)  /\  u  =/=  ( R `  h ) ) ) )  -> 
( U `  h
)  =  ( V `
 h ) )
356, 34rexlimddv 2850 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   E.wrex 2721   class class class wbr 4297    _I cid 4636    |` cres 4847   ` cfv 5423   Basecbs 14179   lecple 14250   Atomscatm 32913   HLchlt 33000   LHypclh 33633   LTrncltrn 33750   trLctrl 33807   TEndoctendo 34401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-riotaBAD 32609
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-undef 6797  df-map 7221  df-poset 15121  df-plt 15133  df-lub 15149  df-glb 15150  df-join 15151  df-meet 15152  df-p0 15214  df-p1 15215  df-lat 15221  df-clat 15283  df-oposet 32826  df-ol 32828  df-oml 32829  df-covers 32916  df-ats 32917  df-atl 32948  df-cvlat 32972  df-hlat 33001  df-llines 33147  df-lplanes 33148  df-lvols 33149  df-lines 33150  df-psubsp 33152  df-pmap 33153  df-padd 33445  df-lhyp 33637  df-laut 33638  df-ldil 33753  df-ltrn 33754  df-trl 33808  df-tendo 34404
This theorem is referenced by:  tendocan  34473
  Copyright terms: Public domain W3C validator