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Theorem cdlemg17iqN 30789
Description: cdlemg17i 30784 with  P and  Q swapped. (Contributed by NM, 13-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
cdlemg17iqN  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  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 `  ( F `  Q ) )  =  ( F `
 P ) )
Distinct variable groups:    A, r    G, r    .\/ , r    .<_ , r    P, r    Q, r    W, r    F, r
Allowed substitution hints:    R( r)    T( r)    H( r)    K( r)    ./\ ( r)

Proof of Theorem cdlemg17iqN
StepHypRef Expression
1 simp11 987 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  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 )
2 simp12 988 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  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
)
31, 2jca 519 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  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 ) )
4 simp21 990 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  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 ) )
5 simp22 991 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  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 ) )
6 simp13l 1072 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  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 ) )  ->  F  e.  T
)
7 simp13r 1073 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  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
)
8 simp23 992 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  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
)
9 simp33 995 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  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
)
10 simp31 993 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  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 ) )
11 simp32 994 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  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
) ) )
12 cdlemg12.l . . . 4  |-  .<_  =  ( le `  K )
13 cdlemg12.j . . . 4  |-  .\/  =  ( join `  K )
14 cdlemg12.m . . . 4  |-  ./\  =  ( meet `  K )
15 cdlemg12.a . . . 4  |-  A  =  ( Atoms `  K )
16 cdlemg12.h . . . 4  |-  H  =  ( LHyp `  K
)
17 cdlemg12.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
18 cdlemg12b.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
1912, 13, 14, 15, 16, 17, 18cdlemg17pq 30787 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q
)  /\  ( ( G `  P )  =/=  P  /\  ( R `
 G )  .<_  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  Q  =/=  P
)  /\  ( ( G `  Q )  =/=  Q  /\  ( R `
 G )  .<_  ( Q  .\/  P )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( Q  .\/  r )  =  ( P  .\/  r
) ) ) ) )
203, 4, 5, 6, 7, 8, 9, 10, 11, 19syl333anc 1216 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  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 )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  Q  =/=  P
)  /\  ( ( G `  Q )  =/=  Q  /\  ( R `
 G )  .<_  ( Q  .\/  P )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( Q  .\/  r )  =  ( P  .\/  r
) ) ) ) )
2112, 13, 14, 15, 16, 17, 18cdlemg17i 30784 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  Q  =/=  P
)  /\  ( ( G `  Q )  =/=  Q  /\  ( R `
 G )  .<_  ( Q  .\/  P )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( Q  .\/  r )  =  ( P  .\/  r
) ) ) )  ->  ( G `  ( F `  Q ) )  =  ( F `
 P ) )
2220, 21syl 16 1  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  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 `  ( F `  Q ) )  =  ( F `
 P ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2551   E.wrex 2651   class class class wbr 4154   ` cfv 5395  (class class class)co 6021   lecple 13464   joincjn 14329   meetcmee 14330   Atomscatm 29379   HLchlt 29466   LHypclh 30099   LTrncltrn 30216   trLctrl 30273
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-undef 6480  df-riota 6486  df-map 6957  df-poset 14331  df-plt 14343  df-lub 14359  df-glb 14360  df-join 14361  df-meet 14362  df-p0 14396  df-p1 14397  df-lat 14403  df-clat 14465  df-oposet 29292  df-ol 29294  df-oml 29295  df-covers 29382  df-ats 29383  df-atl 29414  df-cvlat 29438  df-hlat 29467  df-llines 29613  df-lplanes 29614  df-lvols 29615  df-lines 29616  df-psubsp 29618  df-pmap 29619  df-padd 29911  df-lhyp 30103  df-laut 30104  df-ldil 30219  df-ltrn 30220  df-trl 30274
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