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Theorem cdlemj3 31351
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 960 . . 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 2430 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
3 eqid 2430 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 cdlemj.h . . . 4  |-  H  =  ( LHyp `  K
)
52, 3, 4lhpexle2 30538 . . 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 1008 . . . . 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 452 . . . 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 1009 . . . . 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 452 . . . 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 733 . . . 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 1005 . . . 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 31092 . . . 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 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 ) )  /\  ( 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 981 . . . . . . 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 982 . . . . . . 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 985 . . . . . . 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 1031 . . . . . . 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 1060 . . . . . . . . 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 2676 . . . . . . . 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 1030 . . . . . . . 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 2617 . . . . . . 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 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 ) ) ) )  ->  u  =/=  ( R `  h
) )
2724, 26eqnetrd 2611 . . . . . . 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 31350 . . . . . . 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 1202 . . . . . 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 1155 . . . . 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 ) ) )
3231exp3a 426 . . . 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 2816 . . 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 2821 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2593   E.wrex 2693   class class class wbr 4199    _I cid 4480    |` cres 4866   ` cfv 5440   Basecbs 13452   lecple 13519   Atomscatm 29792   HLchlt 29879   LHypclh 30512   LTrncltrn 30629   trLctrl 30686   TEndoctendo 31280
This theorem is referenced by:  tendocan  31352
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 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687
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 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-reu 2699  df-rmo 2700  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-iun 4082  df-iin 4083  df-br 4200  df-opab 4254  df-mpt 4255  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-1st 6335  df-2nd 6336  df-undef 6529  df-riota 6535  df-map 7006  df-poset 14386  df-plt 14398  df-lub 14414  df-glb 14415  df-join 14416  df-meet 14417  df-p0 14451  df-p1 14452  df-lat 14458  df-clat 14520  df-oposet 29705  df-ol 29707  df-oml 29708  df-covers 29795  df-ats 29796  df-atl 29827  df-cvlat 29851  df-hlat 29880  df-llines 30026  df-lplanes 30027  df-lvols 30028  df-lines 30029  df-psubsp 30031  df-pmap 30032  df-padd 30324  df-lhyp 30516  df-laut 30517  df-ldil 30632  df-ltrn 30633  df-trl 30687  df-tendo 31283
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