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Theorem cdlemg4a 31405
Description: TODO: FIX COMMENT If fg(p) = p, then tr f = tr g. (Contributed by NM, 23-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 )
Assertion
Ref Expression
cdlemg4a  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( F `  ( G `  P )
)  =  P )  ->  ( R `  F )  =  ( R `  G ) )

Proof of Theorem cdlemg4a
StepHypRef Expression
1 simp3 959 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( F `  ( G `  P )
)  =  P )  ->  ( F `  ( G `  P ) )  =  P )
21oveq2d 6097 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( F `  ( G `  P )
)  =  P )  ->  ( ( G `
 P ) (
join `  K )
( F `  ( G `  P )
) )  =  ( ( G `  P
) ( join `  K
) P ) )
3 simp1l 981 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( F `  ( G `  P )
)  =  P )  ->  K  e.  HL )
4 simp1 957 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( F `  ( G `  P )
)  =  P )  ->  ( K  e.  HL  /\  W  e.  H ) )
5 simp23 992 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( F `  ( G `  P )
)  =  P )  ->  G  e.  T
)
6 simp21 990 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( F `  ( G `  P )
)  =  P )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
7 cdlemg4.l . . . . . . . 8  |-  .<_  =  ( le `  K )
8 cdlemg4.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
9 cdlemg4.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
10 cdlemg4.t . . . . . . . 8  |-  T  =  ( ( LTrn `  K
) `  W )
117, 8, 9, 10ltrnel 30936 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( G `  P )  e.  A  /\  -.  ( G `  P )  .<_  W ) )
1211simpld 446 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( G `  P )  e.  A
)
134, 5, 6, 12syl3anc 1184 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( F `  ( G `  P )
)  =  P )  ->  ( G `  P )  e.  A
)
14 simp21l 1074 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( F `  ( G `  P )
)  =  P )  ->  P  e.  A
)
15 eqid 2436 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
1615, 8hlatjcom 30165 . . . . 5  |-  ( ( K  e.  HL  /\  ( G `  P )  e.  A  /\  P  e.  A )  ->  (
( G `  P
) ( join `  K
) P )  =  ( P ( join `  K ) ( G `
 P ) ) )
173, 13, 14, 16syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( F `  ( G `  P )
)  =  P )  ->  ( ( G `
 P ) (
join `  K ) P )  =  ( P ( join `  K
) ( G `  P ) ) )
182, 17eqtrd 2468 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( F `  ( G `  P )
)  =  P )  ->  ( ( G `
 P ) (
join `  K )
( F `  ( G `  P )
) )  =  ( P ( join `  K
) ( G `  P ) ) )
1918oveq1d 6096 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( F `  ( G `  P )
)  =  P )  ->  ( ( ( G `  P ) ( join `  K
) ( F `  ( G `  P ) ) ) ( meet `  K ) W )  =  ( ( P ( join `  K
) ( G `  P ) ) (
meet `  K ) W ) )
20 simp22 991 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( F `  ( G `  P )
)  =  P )  ->  F  e.  T
)
214, 5, 6, 11syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( F `  ( G `  P )
)  =  P )  ->  ( ( G `
 P )  e.  A  /\  -.  ( G `  P )  .<_  W ) )
22 eqid 2436 . . . 4  |-  ( meet `  K )  =  (
meet `  K )
23 cdlemg4.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
247, 15, 22, 8, 9, 10, 23trlval2 30960 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( G `  P )  e.  A  /\  -.  ( G `  P )  .<_  W ) )  ->  ( R `  F )  =  ( ( ( G `  P ) ( join `  K ) ( F `
 ( G `  P ) ) ) ( meet `  K
) W ) )
254, 20, 21, 24syl3anc 1184 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( F `  ( G `  P )
)  =  P )  ->  ( R `  F )  =  ( ( ( G `  P ) ( join `  K ) ( F `
 ( G `  P ) ) ) ( meet `  K
) W ) )
267, 15, 22, 8, 9, 10, 23trlval2 30960 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  G )  =  ( ( P ( join `  K ) ( G `
 P ) ) ( meet `  K
) W ) )
274, 5, 6, 26syl3anc 1184 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( F `  ( G `  P )
)  =  P )  ->  ( R `  G )  =  ( ( P ( join `  K ) ( G `
 P ) ) ( meet `  K
) W ) )
2819, 25, 273eqtr4d 2478 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( F `  ( G `  P )
)  =  P )  ->  ( R `  F )  =  ( R `  G ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   lecple 13536   joincjn 14401   meetcmee 14402   Atomscatm 30061   HLchlt 30148   LHypclh 30781   LTrncltrn 30898   trLctrl 30955
This theorem is referenced by:  cdlemg4f  31412
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-map 7020  df-poset 14403  df-plt 14415  df-glb 14432  df-join 14433  df-p0 14468  df-lat 14475  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-lhyp 30785  df-laut 30786  df-ldil 30901  df-ltrn 30902  df-trl 30956
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