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Theorem cdleme0cp 29092
Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: Reformat as in cdlemg3a 29475- swap consequent equality; make antecedent use df-3an 941. (Contributed by NM, 13-Jun-2012.)
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
cdleme0cp  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  ( P  .\/  U )  =  ( P  .\/  Q ) )

Proof of Theorem cdleme0cp
StepHypRef Expression
1 cdleme0.u . . 3  |-  U  =  ( ( P  .\/  Q )  ./\  W )
21oveq2i 5721 . 2  |-  ( P 
.\/  U )  =  ( P  .\/  (
( P  .\/  Q
)  ./\  W )
)
3 simpll 733 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  K  e.  HL )
4 simprll 741 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  P  e.  A )
5 hllat 28242 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
65ad2antrr 709 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  K  e.  Lat )
7 eqid 2253 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
8 cdleme0.a . . . . . . 7  |-  A  =  ( Atoms `  K )
97, 8atbase 28168 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
104, 9syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  P  e.  ( Base `  K )
)
11 simprr 736 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  Q  e.  A )
127, 8atbase 28168 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
1311, 12syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  Q  e.  ( Base `  K )
)
14 cdleme0.j . . . . . 6  |-  .\/  =  ( join `  K )
157, 14latjcl 14000 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
166, 10, 13, 15syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  ( P  .\/  Q )  e.  (
Base `  K )
)
17 cdleme0.h . . . . . 6  |-  H  =  ( LHyp `  K
)
187, 17lhpbase 28876 . . . . 5  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1918ad2antlr 710 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  W  e.  ( Base `  K )
)
20 cdleme0.l . . . . . 6  |-  .<_  =  ( le `  K )
2120, 14, 8hlatlej1 28253 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  P  .<_  ( P  .\/  Q ) )
223, 4, 11, 21syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  P  .<_  ( P  .\/  Q ) )
23 cdleme0.m . . . . 5  |-  ./\  =  ( meet `  K )
247, 20, 14, 23, 8atmod3i1 28742 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  ( P  .\/  Q
)  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  /\  P  .<_  ( P  .\/  Q
) )  ->  ( P  .\/  ( ( P 
.\/  Q )  ./\  W ) )  =  ( ( P  .\/  Q
)  ./\  ( P  .\/  W ) ) )
253, 4, 16, 19, 22, 24syl131anc 1200 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  ( P  .\/  ( ( P  .\/  Q )  ./\  W )
)  =  ( ( P  .\/  Q ) 
./\  ( P  .\/  W ) ) )
26 eqid 2253 . . . . . 6  |-  ( 1.
`  K )  =  ( 1. `  K
)
2720, 14, 26, 8, 17lhpjat2 28899 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( P  .\/  W
)  =  ( 1.
`  K ) )
2827adantrr 700 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  ( P  .\/  W )  =  ( 1. `  K ) )
2928oveq2d 5726 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  ( ( P  .\/  Q )  ./\  ( P  .\/  W ) )  =  ( ( P  .\/  Q ) 
./\  ( 1. `  K ) ) )
30 hlol 28240 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OL )
3130ad2antrr 709 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  K  e.  OL )
327, 23, 26olm11 28106 . . . 4  |-  ( ( K  e.  OL  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  (
( P  .\/  Q
)  ./\  ( 1. `  K ) )  =  ( P  .\/  Q
) )
3331, 16, 32syl2anc 645 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  ( ( P  .\/  Q )  ./\  ( 1. `  K ) )  =  ( P 
.\/  Q ) )
3425, 29, 333eqtrd 2289 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  ( P  .\/  ( ( P  .\/  Q )  ./\  W )
)  =  ( P 
.\/  Q ) )
352, 34syl5eq 2297 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  ( P  .\/  U )  =  ( P  .\/  Q ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   Basecbs 13022   lecple 13089   joincjn 13922   meetcmee 13923   1.cp1 13988   Latclat 13995   OLcol 28053   Atomscatm 28142   HLchlt 28229   LHypclh 28862
This theorem is referenced by:  cdleme11c  29139  cdlemg4b1  29487  cdlemg4g  29494  cdlemg13a  29529  cdlemg17a  29539  cdlemg17f  29544  cdlemg18b  29557  cdlemg18c  29558
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28055  df-ol 28057  df-oml 28058  df-covers 28145  df-ats 28146  df-atl 28177  df-cvlat 28201  df-hlat 28230  df-psubsp 28381  df-pmap 28382  df-padd 28674  df-lhyp 28866
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