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Theorem cdlemg44b 33751
Description: Eliminate  ( F `
 P )  =/= 
P,  ( G `  P )  =/=  P from cdlemg44a 33750. (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 1000 . . . 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 1075 . . . 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 1077 . . . 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 1076 . . . . 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 33156 . . . . 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 1230 . . . 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 459 . . . 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 33199 . . . 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 1243 . . 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 5853 . . 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 2446 . 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 459 . . . 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 5853 . . 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 1000 . . . 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 1076 . . . 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 1077 . . . 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 1075 . . . . 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 33156 . . . . 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 1230 . . . 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 33199 . . . 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 1243 . . 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 2446 . 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 1000 . . 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 1001 . . 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 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 ) )  ->  ( F `  P )  =/=  P
)
30 simprr 758 . . 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 1002 . . 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 33750 . . 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 1242 . 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 2722 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   class class class wbr 4395   ` cfv 5569   lecple 14916   Atomscatm 32281   HLchlt 32368   LHypclh 33001   LTrncltrn 33118   trLctrl 33176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-map 7459  df-preset 15881  df-poset 15899  df-plt 15912  df-lub 15928  df-glb 15929  df-join 15930  df-meet 15931  df-p0 15993  df-p1 15994  df-lat 16000  df-clat 16062  df-oposet 32194  df-ol 32196  df-oml 32197  df-covers 32284  df-ats 32285  df-atl 32316  df-cvlat 32340  df-hlat 32369  df-llines 32515  df-psubsp 32520  df-pmap 32521  df-padd 32813  df-lhyp 33005  df-laut 33006  df-ldil 33121  df-ltrn 33122  df-trl 33177
This theorem is referenced by:  cdlemg44  33752
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