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Theorem cdlemg4a 36477
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 998 . . . . 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 6312 . . . 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 1020 . . . . 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 996 . . . . . 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 1031 . . . . . 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 1029 . . . . . 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 36006 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( G `  P )  e.  A  /\  -.  ( G `  P )  .<_  W ) )
1211simpld 459 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( G `  P )  e.  A
)
134, 5, 6, 12syl3anc 1228 . . . . 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 1113 . . . . 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 2457 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
1615, 8hlatjcom 35235 . . . . 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 1228 . . . 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 2498 . . 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 6311 . 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 1030 . . 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 1228 . . 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 2457 . . . 4  |-  ( meet `  K )  =  (
meet `  K )
23 cdlemg4.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
247, 15, 22, 8, 9, 10, 23trlval2 36031 . . 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 1228 . 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 36031 . . 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 1228 . 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 2508 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   lecple 14719   joincjn 15700   meetcmee 15701   Atomscatm 35131   HLchlt 35218   LHypclh 35851   LTrncltrn 35968   trLctrl 36026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-map 7440  df-preset 15684  df-poset 15702  df-plt 15715  df-lub 15731  df-glb 15732  df-join 15733  df-meet 15734  df-p0 15796  df-lat 15803  df-oposet 35044  df-ol 35046  df-oml 35047  df-covers 35134  df-ats 35135  df-atl 35166  df-cvlat 35190  df-hlat 35219  df-lhyp 35855  df-laut 35856  df-ldil 35971  df-ltrn 35972  df-trl 36027
This theorem is referenced by:  cdlemg4f  36484
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