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Theorem cdlemj3 36946
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 997 . . 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 2454 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
3 eqid 2454 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 cdlemj.h . . . 4  |-  H  =  ( LHyp `  K
)
52, 3, 4lhpexle2 36131 . . 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 1045 . . . . 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 463 . . . 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 1046 . . . . 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 463 . . . 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 754 . . . 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 1042 . . . 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 36687 . . . 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 1227 . . 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 1018 . . . . . . 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 1019 . . . . . . 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 1022 . . . . . . 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 1068 . . . . . . 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 1097 . . . . . . . . 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 2725 . . . . . . . 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 1067 . . . . . . . 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 2754 . . . . . . 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 1098 . . . . . . . 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 2747 . . . . . . 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 36945 . . . . . . 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 1244 . . . . . 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 1196 . . . . 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 434 . . . 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 2944 . . 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 2950 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   E.wrex 2805   class class class wbr 4439    _I cid 4779    |` cres 4990   ` cfv 5570   Basecbs 14716   lecple 14791   Atomscatm 35385   HLchlt 35472   LHypclh 36105   LTrncltrn 36222   trLctrl 36280   TEndoctendo 36875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-riotaBAD 35081
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-undef 6994  df-map 7414  df-preset 15756  df-poset 15774  df-plt 15787  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-p0 15868  df-p1 15869  df-lat 15875  df-clat 15937  df-oposet 35298  df-ol 35300  df-oml 35301  df-covers 35388  df-ats 35389  df-atl 35420  df-cvlat 35444  df-hlat 35473  df-llines 35619  df-lplanes 35620  df-lvols 35621  df-lines 35622  df-psubsp 35624  df-pmap 35625  df-padd 35917  df-lhyp 36109  df-laut 36110  df-ldil 36225  df-ltrn 36226  df-trl 36281  df-tendo 36878
This theorem is referenced by:  tendocan  36947
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