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Theorem cdleme0cq 33489
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 25-Apr-2013.)
Hypotheses
Ref Expression
cdleme0.l  |-  .<_  =  ( le `  K )
cdleme0.j  |-  .\/  =  ( join `  K )
cdleme0.m  |-  ./\  =  ( meet `  K )
cdleme0.a  |-  A  =  ( Atoms `  K )
cdleme0.h  |-  H  =  ( LHyp `  K
)
cdleme0.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
Assertion
Ref Expression
cdleme0cq  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( Q  .\/  U )  =  ( P  .\/  Q ) )

Proof of Theorem cdleme0cq
StepHypRef Expression
1 cdleme0.u . . 3  |-  U  =  ( ( P  .\/  Q )  ./\  W )
21oveq2i 6316 . 2  |-  ( Q 
.\/  U )  =  ( Q  .\/  (
( P  .\/  Q
)  ./\  W )
)
3 simpll 758 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  K  e.  HL )
4 simprrl 772 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  Q  e.  A )
5 hllat 32637 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
65ad2antrr 730 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  K  e.  Lat )
7 eqid 2429 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
8 cdleme0.a . . . . . . 7  |-  A  =  ( Atoms `  K )
97, 8atbase 32563 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
109ad2antrl 732 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  P  e.  ( Base `  K )
)
117, 8atbase 32563 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
124, 11syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  Q  e.  ( Base `  K )
)
13 cdleme0.j . . . . . 6  |-  .\/  =  ( join `  K )
147, 13latjcl 16248 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
156, 10, 12, 14syl3anc 1264 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( P  .\/  Q )  e.  (
Base `  K )
)
16 cdleme0.h . . . . . 6  |-  H  =  ( LHyp `  K
)
177, 16lhpbase 33271 . . . . 5  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1817ad2antlr 731 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  W  e.  ( Base `  K )
)
19 cdleme0.l . . . . . 6  |-  .<_  =  ( le `  K )
207, 19, 13latlej2 16258 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  Q  .<_  ( P  .\/  Q
) )
216, 10, 12, 20syl3anc 1264 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  Q  .<_  ( P  .\/  Q ) )
22 cdleme0.m . . . . 5  |-  ./\  =  ( meet `  K )
237, 19, 13, 22, 8atmod3i1 33137 . . . 4  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  ( P  .\/  Q
)  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  /\  Q  .<_  ( P  .\/  Q
) )  ->  ( Q  .\/  ( ( P 
.\/  Q )  ./\  W ) )  =  ( ( P  .\/  Q
)  ./\  ( Q  .\/  W ) ) )
243, 4, 15, 18, 21, 23syl131anc 1277 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( Q  .\/  ( ( P  .\/  Q )  ./\  W )
)  =  ( ( P  .\/  Q ) 
./\  ( Q  .\/  W ) ) )
25 eqid 2429 . . . . . 6  |-  ( 1.
`  K )  =  ( 1. `  K
)
2619, 13, 25, 8, 16lhpjat2 33294 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Q  .\/  W
)  =  ( 1.
`  K ) )
2726adantrl 720 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( Q  .\/  W )  =  ( 1. `  K ) )
2827oveq2d 6321 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( ( P  .\/  Q )  ./\  ( Q  .\/  W ) )  =  ( ( P  .\/  Q ) 
./\  ( 1. `  K ) ) )
29 hlol 32635 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OL )
3029ad2antrr 730 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  K  e.  OL )
317, 22, 25olm11 32501 . . . 4  |-  ( ( K  e.  OL  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  (
( P  .\/  Q
)  ./\  ( 1. `  K ) )  =  ( P  .\/  Q
) )
3230, 15, 31syl2anc 665 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( ( P  .\/  Q )  ./\  ( 1. `  K ) )  =  ( P 
.\/  Q ) )
3324, 28, 323eqtrd 2474 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( Q  .\/  ( ( P  .\/  Q )  ./\  W )
)  =  ( P 
.\/  Q ) )
342, 33syl5eq 2482 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( Q  .\/  U )  =  ( P  .\/  Q ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Basecbs 15084   lecple 15159   joincjn 16140   meetcmee 16141   1.cp1 16235   Latclat 16242   OLcol 32448   Atomscatm 32537   HLchlt 32624   LHypclh 33257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-preset 16124  df-poset 16142  df-plt 16155  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-p0 16236  df-p1 16237  df-lat 16243  df-clat 16305  df-oposet 32450  df-ol 32452  df-oml 32453  df-covers 32540  df-ats 32541  df-atl 32572  df-cvlat 32596  df-hlat 32625  df-psubsp 32776  df-pmap 32777  df-padd 33069  df-lhyp 33261
This theorem is referenced by:  cdleme11g  33539  cdlemg4b2  33885  cdlemg13a  33926
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