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Theorem cdleme0cp 34197
Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: Reformat as in cdlemg3a 34580- swap consequent equality; make antecedent use df-3an 967. (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 6212 . 2  |-  ( P 
.\/  U )  =  ( P  .\/  (
( P  .\/  Q
)  ./\  W )
)
3 simpll 753 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  K  e.  HL )
4 simprll 761 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  P  e.  A )
5 hllat 33347 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
65ad2antrr 725 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  K  e.  Lat )
7 eqid 2454 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
8 cdleme0.a . . . . . . 7  |-  A  =  ( Atoms `  K )
97, 8atbase 33273 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
104, 9syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  P  e.  ( Base `  K )
)
11 simprr 756 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  Q  e.  A )
127, 8atbase 33273 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
1311, 12syl 16 . . . . 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 15341 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
166, 10, 13, 15syl3anc 1219 . . . 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 33981 . . . . 5  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1918ad2antlr 726 . . . 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 33358 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  P  .<_  ( P  .\/  Q ) )
223, 4, 11, 21syl3anc 1219 . . . 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 33847 . . . 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 1232 . . 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 2454 . . . . . 6  |-  ( 1.
`  K )  =  ( 1. `  K
)
2720, 14, 26, 8, 17lhpjat2 34004 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( P  .\/  W
)  =  ( 1.
`  K ) )
2827adantrr 716 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  ( P  .\/  W )  =  ( 1. `  K ) )
2928oveq2d 6217 . . 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 33345 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OL )
3130ad2antrr 725 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  K  e.  OL )
327, 23, 26olm11 33211 . . . 4  |-  ( ( K  e.  OL  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  (
( P  .\/  Q
)  ./\  ( 1. `  K ) )  =  ( P  .\/  Q
) )
3331, 16, 32syl2anc 661 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  ( ( P  .\/  Q )  ./\  ( 1. `  K ) )  =  ( P 
.\/  Q ) )
3425, 29, 333eqtrd 2499 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  ( P  .\/  ( ( P  .\/  Q )  ./\  W )
)  =  ( P 
.\/  Q ) )
352, 34syl5eq 2507 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  ( P  .\/  U )  =  ( P  .\/  Q ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   class class class wbr 4401   ` cfv 5527  (class class class)co 6201   Basecbs 14293   lecple 14365   joincjn 15234   meetcmee 15235   1.cp1 15328   Latclat 15335   OLcol 33158   Atomscatm 33247   HLchlt 33334   LHypclh 33967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-iin 4283  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689  df-poset 15236  df-plt 15248  df-lub 15264  df-glb 15265  df-join 15266  df-meet 15267  df-p0 15329  df-p1 15330  df-lat 15336  df-clat 15398  df-oposet 33160  df-ol 33162  df-oml 33163  df-covers 33250  df-ats 33251  df-atl 33282  df-cvlat 33306  df-hlat 33335  df-psubsp 33486  df-pmap 33487  df-padd 33779  df-lhyp 33971
This theorem is referenced by:  cdleme11c  34244  cdlemg4b1  34592  cdlemg4g  34599  cdlemg13a  34634  cdlemg17a  34644  cdlemg17f  34649  cdlemg18b  34662  cdlemg18c  34663
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