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Theorem cdlemg44b 30921
Description: Eliminate  ( F `
 P )  =/= 
P,  ( G `  P )  =/=  P from cdlemg44a 30920. (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 958 . . . 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 1033 . . . 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 1035 . . . 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 1034 . . . . 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 30328 . . . . 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 1182 . . . 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 447 . . . 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 30370 . . . 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 1195 . . 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 5529 . . 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 2318 . 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 447 . . . 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 5529 . . 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 958 . . . 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 1034 . . . 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 1035 . . . 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 1033 . . . . 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 30328 . . . . 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 1182 . . . 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 30370 . . . 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 1195 . . 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 2318 . 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 958 . . 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 959 . . 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 732 . . 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 733 . . 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 960 . . 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 30920 . . 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 1194 . 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 2525 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255   lecple 13215   Atomscatm 29453   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   trLctrl 30347
This theorem is referenced by:  cdlemg44  30922
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348
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