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Theorem cdlemg4a 34591
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 990 . . . . 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 6217 . . . 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 1012 . . . . 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 988 . . . . . 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 1023 . . . . . 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 1021 . . . . . 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 34122 . . . . . . 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 1219 . . . . 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 1105 . . . . 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 2454 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
1615, 8hlatjcom 33351 . . . . 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 1219 . . . 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 2495 . . 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 6216 . 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 1022 . . 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 1219 . . 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 2454 . . . 4  |-  ( meet `  K )  =  (
meet `  K )
23 cdlemg4.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
247, 15, 22, 8, 9, 10, 23trlval2 34146 . . 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 1219 . 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 34146 . . 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 1219 . 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 2505 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 965    = wceq 1370    e. wcel 1758   class class class wbr 4401   ` cfv 5527  (class class class)co 6201   lecple 14365   joincjn 15234   meetcmee 15235   Atomscatm 33247   HLchlt 33334   LHypclh 33967   LTrncltrn 34084   trLctrl 34141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-map 7327  df-poset 15236  df-plt 15248  df-lub 15264  df-glb 15265  df-join 15266  df-meet 15267  df-p0 15329  df-lat 15336  df-oposet 33160  df-ol 33162  df-oml 33163  df-covers 33250  df-ats 33251  df-atl 33282  df-cvlat 33306  df-hlat 33335  df-lhyp 33971  df-laut 33972  df-ldil 34087  df-ltrn 34088  df-trl 34142
This theorem is referenced by:  cdlemg4f  34598
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