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Theorem cdlemg4a 35404
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 6298 . . . 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 34935 . . . . . . 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 2467 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
1615, 8hlatjcom 34164 . . . . 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 2508 . . 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 6297 . 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 2467 . . . 4  |-  ( meet `  K )  =  (
meet `  K )
23 cdlemg4.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
247, 15, 22, 8, 9, 10, 23trlval2 34959 . . 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 34959 . . 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 2518 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 1379    e. wcel 1767   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   lecple 14555   joincjn 15424   meetcmee 15425   Atomscatm 34060   HLchlt 34147   LHypclh 34780   LTrncltrn 34897   trLctrl 34954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-map 7419  df-poset 15426  df-plt 15438  df-lub 15454  df-glb 15455  df-join 15456  df-meet 15457  df-p0 15519  df-lat 15526  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-lhyp 34784  df-laut 34785  df-ldil 34900  df-ltrn 34901  df-trl 34955
This theorem is referenced by:  cdlemg4f  35411
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