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Theorem cdlemd1 29554
Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 29-May-2012.)
Hypotheses
Ref Expression
cdlemd1.l  |-  .<_  =  ( le `  K )
cdlemd1.j  |-  .\/  =  ( join `  K )
cdlemd1.m  |-  ./\  =  ( meet `  K )
cdlemd1.a  |-  A  =  ( Atoms `  K )
cdlemd1.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
cdlemd1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  R  =  ( ( P  .\/  (
( P  .\/  R
)  ./\  W )
)  ./\  ( Q  .\/  ( ( Q  .\/  R )  ./\  W )
) ) )

Proof of Theorem cdlemd1
StepHypRef Expression
1 simpll 733 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  K  e.  HL )
2 simpr1l 1017 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  P  e.  A
)
3 simpr2l 1019 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  Q  e.  A
)
4 simpr31 1050 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  R  e.  A
)
5 simpr32 1051 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  P  =/=  Q
)
6 simpr33 1052 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  -.  R  .<_  ( P  .\/  Q ) )
7 cdlemd1.l . . . 4  |-  .<_  =  ( le `  K )
8 cdlemd1.j . . . 4  |-  .\/  =  ( join `  K )
9 cdlemd1.m . . . 4  |-  ./\  =  ( meet `  K )
10 cdlemd1.a . . . 4  |-  A  =  ( Atoms `  K )
117, 8, 9, 102llnma2 29145 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
) ) )  -> 
( ( R  .\/  P )  ./\  ( R  .\/  Q ) )  =  R )
121, 2, 3, 4, 5, 6, 11syl132anc 1205 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  ( ( R 
.\/  P )  ./\  ( R  .\/  Q ) )  =  R )
13 hllat 28720 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
1413ad2antrr 709 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  K  e.  Lat )
15 eqid 2258 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1615, 10atbase 28646 . . . . . 6  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
174, 16syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  R  e.  (
Base `  K )
)
1815, 10atbase 28646 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
192, 18syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  P  e.  (
Base `  K )
)
2015, 8latjcom 14127 . . . . 5  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  P  e.  ( Base `  K
) )  ->  ( R  .\/  P )  =  ( P  .\/  R
) )
2114, 17, 19, 20syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  ( R  .\/  P )  =  ( P 
.\/  R ) )
22 simpl 445 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
23 simpr1 966 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
24 cdlemd1.h . . . . . 6  |-  H  =  ( LHyp `  K
)
2515, 7, 8, 9, 10, 24cdlemc1 29547 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  (
Base `  K )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( ( P  .\/  R )  ./\  W )
)  =  ( P 
.\/  R ) )
2622, 17, 23, 25syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  ( P  .\/  ( ( P  .\/  R )  ./\  W )
)  =  ( P 
.\/  R ) )
2721, 26eqtr4d 2293 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  ( R  .\/  P )  =  ( P 
.\/  ( ( P 
.\/  R )  ./\  W ) ) )
2815, 10atbase 28646 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
293, 28syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  Q  e.  (
Base `  K )
)
3015, 8latjcom 14127 . . . . 5  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( R  .\/  Q )  =  ( Q  .\/  R
) )
3114, 17, 29, 30syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  ( R  .\/  Q )  =  ( Q 
.\/  R ) )
32 simpr2 967 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
3315, 7, 8, 9, 10, 24cdlemc1 29547 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  (
Base `  K )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( Q  .\/  ( ( Q  .\/  R )  ./\  W )
)  =  ( Q 
.\/  R ) )
3422, 17, 32, 33syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  ( Q  .\/  ( ( Q  .\/  R )  ./\  W )
)  =  ( Q 
.\/  R ) )
3531, 34eqtr4d 2293 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  ( R  .\/  Q )  =  ( Q 
.\/  ( ( Q 
.\/  R )  ./\  W ) ) )
3627, 35oveq12d 5810 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  ( ( R 
.\/  P )  ./\  ( R  .\/  Q ) )  =  ( ( P  .\/  ( ( P  .\/  R ) 
./\  W ) ) 
./\  ( Q  .\/  ( ( Q  .\/  R )  ./\  W )
) ) )
3712, 36eqtr3d 2292 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  R  =  ( ( P  .\/  (
( P  .\/  R
)  ./\  W )
)  ./\  ( Q  .\/  ( ( Q  .\/  R )  ./\  W )
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2421   class class class wbr 3997   ` cfv 4673  (class class class)co 5792   Basecbs 13110   lecple 13177   joincjn 14040   meetcmee 14041   Latclat 14113   Atomscatm 28620   HLchlt 28707   LHypclh 29340
This theorem is referenced by:  cdlemd2  29555
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-poset 14042  df-plt 14054  df-lub 14070  df-glb 14071  df-join 14072  df-meet 14073  df-p0 14107  df-p1 14108  df-lat 14114  df-clat 14176  df-oposet 28533  df-ol 28535  df-oml 28536  df-covers 28623  df-ats 28624  df-atl 28655  df-cvlat 28679  df-hlat 28708  df-psubsp 28859  df-pmap 28860  df-padd 29152  df-lhyp 29344
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