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Theorem cdlemg4b2 30617
Description: TODO: FIX COMMENT (Contributed by NM, 24-Apr-2013.)
Hypotheses
Ref Expression
cdlemg4.l  |-  .<_  =  ( le `  K )
cdlemg4.a  |-  A  =  ( Atoms `  K )
cdlemg4.h  |-  H  =  ( LHyp `  K
)
cdlemg4.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg4.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemg4.j  |-  .\/  =  ( join `  K )
cdlemg4b.v  |-  V  =  ( R `  G
)
Assertion
Ref Expression
cdlemg4b2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  G  e.  T )  ->  (
( G `  P
)  .\/  V )  =  ( P  .\/  ( G `  P ) ) )

Proof of Theorem cdlemg4b2
StepHypRef Expression
1 cdlemg4b.v . . . 4  |-  V  =  ( R `  G
)
2 cdlemg4.l . . . . . 6  |-  .<_  =  ( le `  K )
3 cdlemg4.j . . . . . 6  |-  .\/  =  ( join `  K )
4 eqid 2316 . . . . . 6  |-  ( meet `  K )  =  (
meet `  K )
5 cdlemg4.a . . . . . 6  |-  A  =  ( Atoms `  K )
6 cdlemg4.h . . . . . 6  |-  H  =  ( LHyp `  K
)
7 cdlemg4.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
8 cdlemg4.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
92, 3, 4, 5, 6, 7, 8trlval2 30170 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  G )  =  ( ( P  .\/  ( G `  P )
) ( meet `  K
) W ) )
1093com23 1157 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  G  e.  T )  ->  ( R `  G )  =  ( ( P 
.\/  ( G `  P ) ) (
meet `  K ) W ) )
111, 10syl5eq 2360 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  G  e.  T )  ->  V  =  ( ( P 
.\/  ( G `  P ) ) (
meet `  K ) W ) )
1211oveq2d 5916 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  G  e.  T )  ->  (
( G `  P
)  .\/  V )  =  ( ( G `
 P )  .\/  ( ( P  .\/  ( G `  P ) ) ( meet `  K
) W ) ) )
13 simp1 955 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  G  e.  T )  ->  ( K  e.  HL  /\  W  e.  H ) )
14 simp2l 981 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  G  e.  T )  ->  P  e.  A )
152, 5, 6, 7ltrnel 30146 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( G `  P )  e.  A  /\  -.  ( G `  P )  .<_  W ) )
16153com23 1157 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  G  e.  T )  ->  (
( G `  P
)  e.  A  /\  -.  ( G `  P
)  .<_  W ) )
17 eqid 2316 . . . 4  |-  ( ( P  .\/  ( G `
 P ) ) ( meet `  K
) W )  =  ( ( P  .\/  ( G `  P ) ) ( meet `  K
) W )
182, 3, 4, 5, 6, 17cdleme0cq 30222 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( ( G `  P )  e.  A  /\  -.  ( G `  P ) 
.<_  W ) ) )  ->  ( ( G `
 P )  .\/  ( ( P  .\/  ( G `  P ) ) ( meet `  K
) W ) )  =  ( P  .\/  ( G `  P ) ) )
1913, 14, 16, 18syl12anc 1180 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  G  e.  T )  ->  (
( G `  P
)  .\/  ( ( P  .\/  ( G `  P ) ) (
meet `  K ) W ) )  =  ( P  .\/  ( G `  P )
) )
2012, 19eqtrd 2348 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  G  e.  T )  ->  (
( G `  P
)  .\/  V )  =  ( P  .\/  ( G `  P ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701   class class class wbr 4060   ` cfv 5292  (class class class)co 5900   lecple 13262   joincjn 14127   meetcmee 14128   Atomscatm 29271   HLchlt 29358   LHypclh 29991   LTrncltrn 30108   trLctrl 30165
This theorem is referenced by:  cdlemg4b12  30618  cdlemg4c  30619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-iin 3945  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-undef 6340  df-riota 6346  df-map 6817  df-poset 14129  df-plt 14141  df-lub 14157  df-glb 14158  df-join 14159  df-meet 14160  df-p0 14194  df-p1 14195  df-lat 14201  df-clat 14263  df-oposet 29184  df-ol 29186  df-oml 29187  df-covers 29274  df-ats 29275  df-atl 29306  df-cvlat 29330  df-hlat 29359  df-psubsp 29510  df-pmap 29511  df-padd 29803  df-lhyp 29995  df-laut 29996  df-ldil 30111  df-ltrn 30112  df-trl 30166
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