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Theorem cdlemd8 30733
Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 1-Jun-2012.)
Hypotheses
Ref Expression
cdlemd4.l  |-  .<_  =  ( le `  K )
cdlemd4.j  |-  .\/  =  ( join `  K )
cdlemd4.a  |-  A  =  ( Atoms `  K )
cdlemd4.h  |-  H  =  ( LHyp `  K
)
cdlemd4.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
cdlemd8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  P )  =  P ) )  ->  ( F `  R )  =  ( G `  R ) )

Proof of Theorem cdlemd8
StepHypRef Expression
1 simp3r 986 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  P )  =  P ) )  ->  ( F `  P )  =  P )
2 simp11 987 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  P )  =  P ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
3 simp12l 1070 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  P )  =  P ) )  ->  F  e.  T
)
4 simp2 958 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  P )  =  P ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
5 eqid 2430 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
6 cdlemd4.l . . . . . 6  |-  .<_  =  ( le `  K )
7 cdlemd4.a . . . . . 6  |-  A  =  ( Atoms `  K )
8 cdlemd4.h . . . . . 6  |-  H  =  ( LHyp `  K
)
9 cdlemd4.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
105, 6, 7, 8, 9ltrnideq 30703 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F  =  (  _I  |`  ( Base `  K ) )  <-> 
( F `  P
)  =  P ) )
112, 3, 4, 10syl3anc 1184 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  P )  =  P ) )  ->  ( F  =  (  _I  |`  ( Base `  K ) )  <-> 
( F `  P
)  =  P ) )
121, 11mpbird 224 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  P )  =  P ) )  ->  F  =  (  _I  |`  ( Base `  K ) ) )
1312fveq1d 5716 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  P )  =  P ) )  ->  ( F `  R )  =  ( (  _I  |`  ( Base `  K ) ) `
 R ) )
14 simp3l 985 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  P )  =  P ) )  ->  ( F `  P )  =  ( G `  P ) )
1514, 1eqtr3d 2464 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  P )  =  P ) )  ->  ( G `  P )  =  P )
16 simp12r 1071 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  P )  =  P ) )  ->  G  e.  T
)
175, 6, 7, 8, 9ltrnideq 30703 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( G  =  (  _I  |`  ( Base `  K ) )  <-> 
( G `  P
)  =  P ) )
182, 16, 4, 17syl3anc 1184 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  P )  =  P ) )  ->  ( G  =  (  _I  |`  ( Base `  K ) )  <-> 
( G `  P
)  =  P ) )
1915, 18mpbird 224 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  P )  =  P ) )  ->  G  =  (  _I  |`  ( Base `  K ) ) )
2019fveq1d 5716 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  P )  =  P ) )  ->  ( G `  R )  =  ( (  _I  |`  ( Base `  K ) ) `
 R ) )
2113, 20eqtr4d 2465 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  P )  =  P ) )  ->  ( F `  R )  =  ( G `  R ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4199    _I cid 4480    |` cres 4866   ` cfv 5440   Basecbs 13452   lecple 13519   joincjn 14384   Atomscatm 29792   HLchlt 29879   LHypclh 30512   LTrncltrn 30629
This theorem is referenced by:  cdlemd9  30734
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-3an 938  df-tru 1328  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-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-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-lhyp 30516  df-laut 30517  df-ldil 30632  df-ltrn 30633  df-trl 30687
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