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Theorem cdleme35fnpq 29917
Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT (Contributed by NM, 19-Mar-2013.)
Hypotheses
Ref Expression
cdleme35.l  |-  .<_  =  ( le `  K )
cdleme35.j  |-  .\/  =  ( join `  K )
cdleme35.m  |-  ./\  =  ( meet `  K )
cdleme35.a  |-  A  =  ( Atoms `  K )
cdleme35.h  |-  H  =  ( LHyp `  K
)
cdleme35.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme35.f  |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
Assertion
Ref Expression
cdleme35fnpq  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  -.  F  .<_  ( P  .\/  Q ) )

Proof of Theorem cdleme35fnpq
StepHypRef Expression
1 simp3 957 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  -.  R  .<_  ( P  .\/  Q ) )
2 simp11 985 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
3 simp12l 1068 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  P  e.  A )
4 simp13l 1070 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  Q  e.  A )
5 cdleme35.l . . . . . 6  |-  .<_  =  ( le `  K )
6 cdleme35.j . . . . . 6  |-  .\/  =  ( join `  K )
7 cdleme35.m . . . . . 6  |-  ./\  =  ( meet `  K )
8 cdleme35.a . . . . . 6  |-  A  =  ( Atoms `  K )
9 cdleme35.h . . . . . 6  |-  H  =  ( LHyp `  K
)
10 cdleme35.u . . . . . 6  |-  U  =  ( ( P  .\/  Q )  ./\  W )
115, 6, 7, 8, 9, 10cdlemeulpq 29688 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A ) )  ->  U  .<_  ( P  .\/  Q ) )
122, 3, 4, 11syl12anc 1180 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  U  .<_  ( P  .\/  Q
) )
13 simp11l 1066 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  K  e.  HL )
14 hllat 28832 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
1513, 14syl 15 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  K  e.  Lat )
16 simp2rl 1024 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  R  e.  A )
17 cdleme35.f . . . . . . . 8  |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
18 eqid 2284 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
195, 6, 7, 8, 9, 10, 17, 18cdleme1b 29694 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  F  e.  ( Base `  K ) )
202, 3, 4, 16, 19syl13anc 1184 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  F  e.  ( Base `  K
) )
215, 6, 7, 8, 9, 10, 18cdleme0aa 29678 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A
)  ->  U  e.  ( Base `  K )
)
222, 3, 4, 21syl3anc 1182 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  U  e.  ( Base `  K
) )
2318, 6, 8hlatjcl 28835 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
2413, 3, 4, 23syl3anc 1182 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
2518, 5, 6latjle12 14164 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( F  e.  ( Base `  K )  /\  U  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
) ) )  -> 
( ( F  .<_  ( P  .\/  Q )  /\  U  .<_  ( P 
.\/  Q ) )  <-> 
( F  .\/  U
)  .<_  ( P  .\/  Q ) ) )
2615, 20, 22, 24, 25syl13anc 1184 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  (
( F  .<_  ( P 
.\/  Q )  /\  U  .<_  ( P  .\/  Q ) )  <->  ( F  .\/  U )  .<_  ( P 
.\/  Q ) ) )
2726biimpd 198 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  (
( F  .<_  ( P 
.\/  Q )  /\  U  .<_  ( P  .\/  Q ) )  ->  ( F  .\/  U )  .<_  ( P  .\/  Q ) ) )
2812, 27mpan2d 655 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  ( F  .<_  ( P  .\/  Q )  ->  ( F  .\/  U )  .<_  ( P 
.\/  Q ) ) )
2918, 8atbase 28758 . . . . . . 7  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
3016, 29syl 15 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  R  e.  ( Base `  K
) )
3118, 5, 6latlej1 14162 . . . . . 6  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  U  e.  ( Base `  K
) )  ->  R  .<_  ( R  .\/  U
) )
3215, 30, 22, 31syl3anc 1182 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  R  .<_  ( R  .\/  U
) )
335, 6, 7, 8, 9, 10, 17cdleme35a 29916 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  ( F  .\/  U )  =  ( R  .\/  U
) )
3432, 33breqtrrd 4050 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  R  .<_  ( F  .\/  U
) )
3518, 6latjcl 14152 . . . . . 6  |-  ( ( K  e.  Lat  /\  F  e.  ( Base `  K )  /\  U  e.  ( Base `  K
) )  ->  ( F  .\/  U )  e.  ( Base `  K
) )
3615, 20, 22, 35syl3anc 1182 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  ( F  .\/  U )  e.  ( Base `  K
) )
3718, 5lattr 14158 . . . . 5  |-  ( ( K  e.  Lat  /\  ( R  e.  ( Base `  K )  /\  ( F  .\/  U )  e.  ( Base `  K
)  /\  ( P  .\/  Q )  e.  (
Base `  K )
) )  ->  (
( R  .<_  ( F 
.\/  U )  /\  ( F  .\/  U ) 
.<_  ( P  .\/  Q
) )  ->  R  .<_  ( P  .\/  Q
) ) )
3815, 30, 36, 24, 37syl13anc 1184 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  (
( R  .<_  ( F 
.\/  U )  /\  ( F  .\/  U ) 
.<_  ( P  .\/  Q
) )  ->  R  .<_  ( P  .\/  Q
) ) )
3934, 38mpand 656 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  (
( F  .\/  U
)  .<_  ( P  .\/  Q )  ->  R  .<_  ( P  .\/  Q ) ) )
4028, 39syld 40 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  ( F  .<_  ( P  .\/  Q )  ->  R  .<_  ( P  .\/  Q ) ) )
411, 40mtod 168 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q
) )  ->  -.  F  .<_  ( P  .\/  Q ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1685    =/= wne 2447   class class class wbr 4024   ` cfv 5221  (class class class)co 5820   Basecbs 13144   lecple 13211   joincjn 14074   meetcmee 14075   Latclat 14147   Atomscatm 28732   HLchlt 28819   LHypclh 29452
This theorem is referenced by:  cdleme35sn3a  29927  cdleme46frvlpq  29972
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-undef 6292  df-riota 6300  df-poset 14076  df-plt 14088  df-lub 14104  df-glb 14105  df-join 14106  df-meet 14107  df-p0 14141  df-p1 14142  df-lat 14148  df-clat 14210  df-oposet 28645  df-ol 28647  df-oml 28648  df-covers 28735  df-ats 28736  df-atl 28767  df-cvlat 28791  df-hlat 28820  df-lines 28969  df-psubsp 28971  df-pmap 28972  df-padd 29264  df-lhyp 29456
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