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Theorem cdlemg17dN 34626
Description: TODO: fix comment. (Contributed by NM, 9-May-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemg12.l  |-  .<_  =  ( le `  K )
cdlemg12.j  |-  .\/  =  ( join `  K )
cdlemg12.m  |-  ./\  =  ( meet `  K )
cdlemg12.a  |-  A  =  ( Atoms `  K )
cdlemg12.h  |-  H  =  ( LHyp `  K
)
cdlemg12.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg12b.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
cdlemg17dN  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) )  /\  ( G `  P )  =/=  P ) )  -> 
( R `  G
)  =  ( ( P  .\/  Q ) 
./\  W ) )
Distinct variable groups:    A, r    G, r    .\/ , r    .<_ , r    P, r    Q, r    W, r
Allowed substitution hints:    R( r)    T( r)    H( r)    K( r)    ./\ ( r)

Proof of Theorem cdlemg17dN
StepHypRef Expression
1 simp1 988 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) )  /\  ( G `  P )  =/=  P ) )  -> 
( K  e.  HL  /\  W  e.  H  /\  G  e.  T )
)
2 simp21 1021 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) )  /\  ( G `  P )  =/=  P ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
3 simpl1 991 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  HL )
4 simpl2 992 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  e.  H )
5 simpl3 993 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  G  e.  T )
6 simpr 461 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
7 cdlemg12.l . . . . 5  |-  .<_  =  ( le `  K )
8 cdlemg12.j . . . . 5  |-  .\/  =  ( join `  K )
9 cdlemg12.m . . . . 5  |-  ./\  =  ( meet `  K )
10 cdlemg12.a . . . . 5  |-  A  =  ( Atoms `  K )
11 cdlemg12.h . . . . 5  |-  H  =  ( LHyp `  K
)
12 cdlemg12.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
13 cdlemg12b.r . . . . 5  |-  R  =  ( ( trL `  K
) `  W )
147, 8, 9, 10, 11, 12, 13trlval2 34126 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  G )  =  ( ( P  .\/  ( G `  P )
)  ./\  W )
)
153, 4, 5, 6, 14syl211anc 1225 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  G )  =  ( ( P  .\/  ( G `  P )
)  ./\  W )
)
161, 2, 15syl2anc 661 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) )  /\  ( G `  P )  =/=  P ) )  -> 
( R `  G
)  =  ( ( P  .\/  ( G `
 P ) ) 
./\  W ) )
17 simp11 1018 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) )  /\  ( G `  P )  =/=  P ) )  ->  K  e.  HL )
18 simp12 1019 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) )  /\  ( G `  P )  =/=  P ) )  ->  W  e.  H )
1917, 18jca 532 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) )  /\  ( G `  P )  =/=  P ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
20 simp22 1022 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) )  /\  ( G `  P )  =/=  P ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
21 simp13 1020 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) )  /\  ( G `  P )  =/=  P ) )  ->  G  e.  T )
22 simp23 1023 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) )  /\  ( G `  P )  =/=  P ) )  ->  P  =/=  Q )
23 simp33 1026 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) )  /\  ( G `  P )  =/=  P ) )  -> 
( G `  P
)  =/=  P )
24 simp31 1024 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) )  /\  ( G `  P )  =/=  P ) )  -> 
( R `  G
)  .<_  ( P  .\/  Q ) )
25 simp32 1025 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) )  /\  ( G `  P )  =/=  P ) )  ->  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )
267, 8, 9, 10, 11, 12, 13cdlemg17b 34625 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( G  e.  T  /\  P  =/=  Q
)  /\  ( ( G `  P )  =/=  P  /\  ( R `
 G )  .<_  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( G `  P )  =  Q )
2719, 2, 20, 21, 22, 23, 24, 25, 26syl323anc 1249 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) )  /\  ( G `  P )  =/=  P ) )  -> 
( G `  P
)  =  Q )
2827oveq2d 6211 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) )  /\  ( G `  P )  =/=  P ) )  -> 
( P  .\/  ( G `  P )
)  =  ( P 
.\/  Q ) )
2928oveq1d 6210 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) )  /\  ( G `  P )  =/=  P ) )  -> 
( ( P  .\/  ( G `  P ) )  ./\  W )  =  ( ( P 
.\/  Q )  ./\  W ) )
3016, 29eqtrd 2493 1  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( ( R `  G )  .<_  ( P  .\/  Q
)  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) )  /\  ( G `  P )  =/=  P ) )  -> 
( R `  G
)  =  ( ( P  .\/  Q ) 
./\  W ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2645   E.wrex 2797   class class class wbr 4395   ` cfv 5521  (class class class)co 6195   lecple 14359   joincjn 15228   meetcmee 15229   Atomscatm 33227   HLchlt 33314   LHypclh 33947   LTrncltrn 34064   trLctrl 34121
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 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-iin 4277  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-1st 6682  df-2nd 6683  df-map 7321  df-poset 15230  df-plt 15242  df-lub 15258  df-glb 15259  df-join 15260  df-meet 15261  df-p0 15323  df-p1 15324  df-lat 15330  df-clat 15392  df-oposet 33140  df-ol 33142  df-oml 33143  df-covers 33230  df-ats 33231  df-atl 33262  df-cvlat 33286  df-hlat 33315  df-psubsp 33466  df-pmap 33467  df-padd 33759  df-lhyp 33951  df-laut 33952  df-ldil 34067  df-ltrn 34068  df-trl 34122
This theorem is referenced by: (None)
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