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Theorem cdlemg44b 35403
Description: Eliminate  ( F `
 P )  =/= 
P,  ( G `  P )  =/=  P from cdlemg44a 35402. (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 994 . . . 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 1069 . . . 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 1071 . . . 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 1070 . . . . 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 34810 . . . . 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 1223 . . . 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 461 . . . 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 34852 . . . 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 1236 . . 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 5861 . . 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 2504 . 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 461 . . . 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 5861 . . 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 994 . . . 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 1070 . . . 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 1071 . . . 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 1069 . . . . 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 34810 . . . . 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 1223 . . . 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 34852 . . . 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 1236 . . 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 2504 . 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 994 . . 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 995 . . 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 755 . . 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 756 . . 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 996 . . 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 35402 . . 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 1235 . 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 2779 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   class class class wbr 4440   ` cfv 5579   lecple 14551   Atomscatm 33935   HLchlt 34022   LHypclh 34655   LTrncltrn 34772   trLctrl 34829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-map 7412  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-p1 15516  df-lat 15522  df-clat 15584  df-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023  df-llines 34169  df-psubsp 34174  df-pmap 34175  df-padd 34467  df-lhyp 34659  df-laut 34660  df-ldil 34775  df-ltrn 34776  df-trl 34830
This theorem is referenced by:  cdlemg44  35404
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