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Theorem cdleme8 30985
Description: Part of proof of Lemma E in [Crawley] p. 113, 2nd paragraph on p. 114.  C represents s1. In their notation, we prove p  \/ s1 = p  \/ s. (Contributed by NM, 9-Jun-2012.)
Hypotheses
Ref Expression
cdleme8.l  |-  .<_  =  ( le `  K )
cdleme8.j  |-  .\/  =  ( join `  K )
cdleme8.m  |-  ./\  =  ( meet `  K )
cdleme8.a  |-  A  =  ( Atoms `  K )
cdleme8.h  |-  H  =  ( LHyp `  K
)
cdleme8.4  |-  C  =  ( ( P  .\/  S )  ./\  W )
Assertion
Ref Expression
cdleme8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  ( P  .\/  C )  =  ( P  .\/  S
) )

Proof of Theorem cdleme8
StepHypRef Expression
1 cdleme8.4 . . 3  |-  C  =  ( ( P  .\/  S )  ./\  W )
21oveq2i 6085 . 2  |-  ( P 
.\/  C )  =  ( P  .\/  (
( P  .\/  S
)  ./\  W )
)
3 simp1l 981 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  K  e.  HL )
4 simp2l 983 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  P  e.  A )
5 hllat 30099 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
63, 5syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  K  e.  Lat )
7 eqid 2436 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
8 cdleme8.a . . . . . . 7  |-  A  =  ( Atoms `  K )
97, 8atbase 30025 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
104, 9syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  P  e.  ( Base `  K
) )
117, 8atbase 30025 . . . . . 6  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
12113ad2ant3 980 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  S  e.  ( Base `  K
) )
13 cdleme8.j . . . . . 6  |-  .\/  =  ( join `  K )
147, 13latjcl 14472 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  S  e.  ( Base `  K
) )  ->  ( P  .\/  S )  e.  ( Base `  K
) )
156, 10, 12, 14syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  ( P  .\/  S )  e.  ( Base `  K
) )
16 simp1r 982 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  W  e.  H )
17 cdleme8.h . . . . . 6  |-  H  =  ( LHyp `  K
)
187, 17lhpbase 30733 . . . . 5  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1916, 18syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  W  e.  ( Base `  K
) )
20 cdleme8.l . . . . . 6  |-  .<_  =  ( le `  K )
217, 20, 13latlej1 14482 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  S  e.  ( Base `  K
) )  ->  P  .<_  ( P  .\/  S
) )
226, 10, 12, 21syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  P  .<_  ( P  .\/  S
) )
23 cdleme8.m . . . . 5  |-  ./\  =  ( meet `  K )
247, 20, 13, 23, 8atmod3i1 30599 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  ( P  .\/  S
)  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  /\  P  .<_  ( P  .\/  S
) )  ->  ( P  .\/  ( ( P 
.\/  S )  ./\  W ) )  =  ( ( P  .\/  S
)  ./\  ( P  .\/  W ) ) )
253, 4, 15, 19, 22, 24syl131anc 1197 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  ( P  .\/  ( ( P 
.\/  S )  ./\  W ) )  =  ( ( P  .\/  S
)  ./\  ( P  .\/  W ) ) )
26 eqid 2436 . . . . . 6  |-  ( 1.
`  K )  =  ( 1. `  K
)
2720, 13, 26, 8, 17lhpjat2 30756 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( P  .\/  W
)  =  ( 1.
`  K ) )
28273adant3 977 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  ( P  .\/  W )  =  ( 1. `  K
) )
2928oveq2d 6090 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  (
( P  .\/  S
)  ./\  ( P  .\/  W ) )  =  ( ( P  .\/  S )  ./\  ( 1. `  K ) ) )
30 hlol 30097 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OL )
313, 30syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  K  e.  OL )
327, 23, 26olm11 29963 . . . 4  |-  ( ( K  e.  OL  /\  ( P  .\/  S )  e.  ( Base `  K
) )  ->  (
( P  .\/  S
)  ./\  ( 1. `  K ) )  =  ( P  .\/  S
) )
3331, 15, 32syl2anc 643 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  (
( P  .\/  S
)  ./\  ( 1. `  K ) )  =  ( P  .\/  S
) )
3425, 29, 333eqtrd 2472 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  ( P  .\/  ( ( P 
.\/  S )  ./\  W ) )  =  ( P  .\/  S ) )
352, 34syl5eq 2480 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  ( P  .\/  C )  =  ( P  .\/  S
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4205   ` cfv 5447  (class class class)co 6074   Basecbs 13462   lecple 13529   joincjn 14394   meetcmee 14395   1.cp1 14460   Latclat 14467   OLcol 29910   Atomscatm 29999   HLchlt 30086   LHypclh 30719
This theorem is referenced by:  cdleme8tN  30990  cdleme15a  31009  cdleme17b  31022  cdlemg3a  31332
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-iin 4089  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-undef 6536  df-riota 6542  df-poset 14396  df-plt 14408  df-lub 14424  df-glb 14425  df-join 14426  df-meet 14427  df-p0 14461  df-p1 14462  df-lat 14468  df-clat 14530  df-oposet 29912  df-ol 29914  df-oml 29915  df-covers 30002  df-ats 30003  df-atl 30034  df-cvlat 30058  df-hlat 30087  df-psubsp 30238  df-pmap 30239  df-padd 30531  df-lhyp 30723
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