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Theorem cdlemg44b 31591
Description: Eliminate  ( F `
 P )  =/= 
P,  ( G `  P )  =/=  P from cdlemg44a 31590. (Contributed by NM, 3-Jun-2013.)
Hypotheses
Ref Expression
cdlemg44.h  |-  H  =  ( LHyp `  K
)
cdlemg44.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg44.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemg44.l  |-  .<_  =  ( le `  K )
cdlemg44.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
cdlemg44b  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `  F )  =/=  ( R `  G )
)  ->  ( F `  ( G `  P
) )  =  ( G `  ( F `
 P ) ) )

Proof of Theorem cdlemg44b
StepHypRef Expression
1 simpl1 961 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `
 F )  =/=  ( R `  G
) )  /\  ( F `  P )  =  P )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simpl21 1036 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `
 F )  =/=  ( R `  G
) )  /\  ( F `  P )  =  P )  ->  F  e.  T )
3 simpl23 1038 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `
 F )  =/=  ( R `  G
) )  /\  ( F `  P )  =  P )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
4 simpl22 1037 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `
 F )  =/=  ( R `  G
) )  /\  ( F `  P )  =  P )  ->  G  e.  T )
5 cdlemg44.l . . . . . 6  |-  .<_  =  ( le `  K )
6 cdlemg44.a . . . . . 6  |-  A  =  ( Atoms `  K )
7 cdlemg44.h . . . . . 6  |-  H  =  ( LHyp `  K
)
8 cdlemg44.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
95, 6, 7, 8ltrnel 30998 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( G `  P )  e.  A  /\  -.  ( G `  P )  .<_  W ) )
101, 4, 3, 9syl3anc 1185 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `
 F )  =/=  ( R `  G
) )  /\  ( F `  P )  =  P )  ->  (
( G `  P
)  e.  A  /\  -.  ( G `  P
)  .<_  W ) )
11 simpr 449 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `
 F )  =/=  ( R `  G
) )  /\  ( F `  P )  =  P )  ->  ( F `  P )  =  P )
125, 6, 7, 8ltrnateq 31040 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( G `  P )  e.  A  /\  -.  ( G `  P )  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( F `  ( G `  P ) )  =  ( G `  P
) )
131, 2, 3, 10, 11, 12syl131anc 1198 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `
 F )  =/=  ( R `  G
) )  /\  ( F `  P )  =  P )  ->  ( F `  ( G `  P ) )  =  ( G `  P
) )
1411fveq2d 5734 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `
 F )  =/=  ( R `  G
) )  /\  ( F `  P )  =  P )  ->  ( G `  ( F `  P ) )  =  ( G `  P
) )
1513, 14eqtr4d 2473 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `
 F )  =/=  ( R `  G
) )  /\  ( F `  P )  =  P )  ->  ( F `  ( G `  P ) )  =  ( G `  ( F `  P )
) )
16 simpr 449 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `
 F )  =/=  ( R `  G
) )  /\  ( G `  P )  =  P )  ->  ( G `  P )  =  P )
1716fveq2d 5734 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `
 F )  =/=  ( R `  G
) )  /\  ( G `  P )  =  P )  ->  ( F `  ( G `  P ) )  =  ( F `  P
) )
18 simpl1 961 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `
 F )  =/=  ( R `  G
) )  /\  ( G `  P )  =  P )  ->  ( K  e.  HL  /\  W  e.  H ) )
19 simpl22 1037 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `
 F )  =/=  ( R `  G
) )  /\  ( G `  P )  =  P )  ->  G  e.  T )
20 simpl23 1038 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `
 F )  =/=  ( R `  G
) )  /\  ( G `  P )  =  P )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
21 simpl21 1036 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `
 F )  =/=  ( R `  G
) )  /\  ( G `  P )  =  P )  ->  F  e.  T )
225, 6, 7, 8ltrnel 30998 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )
2318, 21, 20, 22syl3anc 1185 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `
 F )  =/=  ( R `  G
) )  /\  ( G `  P )  =  P )  ->  (
( F `  P
)  e.  A  /\  -.  ( F `  P
)  .<_  W ) )
245, 6, 7, 8ltrnateq 31040 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( F `  P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )  /\  ( G `
 P )  =  P )  ->  ( G `  ( F `  P ) )  =  ( F `  P
) )
2518, 19, 20, 23, 16, 24syl131anc 1198 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `
 F )  =/=  ( R `  G
) )  /\  ( G `  P )  =  P )  ->  ( G `  ( F `  P ) )  =  ( F `  P
) )
2617, 25eqtr4d 2473 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `
 F )  =/=  ( R `  G
) )  /\  ( G `  P )  =  P )  ->  ( F `  ( G `  P ) )  =  ( G `  ( F `  P )
) )
27 simpl1 961 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `
 F )  =/=  ( R `  G
) )  /\  (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
28 simpl2 962 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `
 F )  =/=  ( R `  G
) )  /\  (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P ) )  ->  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )
29 simprl 734 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `
 F )  =/=  ( R `  G
) )  /\  (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P ) )  ->  ( F `  P )  =/=  P
)
30 simprr 735 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `
 F )  =/=  ( R `  G
) )  /\  (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P ) )  ->  ( G `  P )  =/=  P
)
31 simpl3 963 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `
 F )  =/=  ( R `  G
) )  /\  (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P ) )  ->  ( R `  F )  =/=  ( R `  G )
)
32 cdlemg44.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
337, 8, 32, 5, 6cdlemg44a 31590 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( ( F `
 P )  =/= 
P  /\  ( G `  P )  =/=  P  /\  ( R `  F
)  =/=  ( R `
 G ) ) )  ->  ( F `  ( G `  P
) )  =  ( G `  ( F `
 P ) ) )
3427, 28, 29, 30, 31, 33syl113anc 1197 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `
 F )  =/=  ( R `  G
) )  /\  (
( F `  P
)  =/=  P  /\  ( G `  P )  =/=  P ) )  ->  ( F `  ( G `  P ) )  =  ( G `
 ( F `  P ) ) )
3515, 26, 34pm2.61da2ne 2685 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( R `  F )  =/=  ( R `  G )
)  ->  ( F `  ( G `  P
) )  =  ( G `  ( F `
 P ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4214   ` cfv 5456   lecple 13538   Atomscatm 30123   HLchlt 30210   LHypclh 30843   LTrncltrn 30960   trLctrl 31017
This theorem is referenced by:  cdlemg44  31592
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-map 7022  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-p1 14471  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-llines 30357  df-psubsp 30362  df-pmap 30363  df-padd 30655  df-lhyp 30847  df-laut 30848  df-ldil 30963  df-ltrn 30964  df-trl 31018
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