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Theorem cdleme10 29610
Description: Part of proof of Lemma E in [Crawley] p. 113, 2nd paragraph on p. 114.  D represents s2. In their notation, we prove s  \/ s2 = s  \/ r. (Contributed by NM, 9-Jun-2012.)
Hypotheses
Ref Expression
cdleme10.l  |-  .<_  =  ( le `  K )
cdleme10.j  |-  .\/  =  ( join `  K )
cdleme10.m  |-  ./\  =  ( meet `  K )
cdleme10.a  |-  A  =  ( Atoms `  K )
cdleme10.h  |-  H  =  ( LHyp `  K
)
cdleme10.d  |-  D  =  ( ( R  .\/  S )  ./\  W )
Assertion
Ref Expression
cdleme10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( S  .\/  D )  =  ( S  .\/  R ) )

Proof of Theorem cdleme10
StepHypRef Expression
1 cdleme10.d . . 3  |-  D  =  ( ( R  .\/  S )  ./\  W )
21oveq2i 5803 . 2  |-  ( S 
.\/  D )  =  ( S  .\/  (
( R  .\/  S
)  ./\  W )
)
3 simp1l 984 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  K  e.  HL )
4 simp3l 988 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  S  e.  A )
5 simp2 961 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  R  e.  A )
6 eqid 2258 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
7 cdleme10.j . . . . . 6  |-  .\/  =  ( join `  K )
8 cdleme10.a . . . . . 6  |-  A  =  ( Atoms `  K )
96, 7, 8hlatjcl 28723 . . . . 5  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  ( Base `  K ) )
103, 5, 4, 9syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( R  .\/  S )  e.  (
Base `  K )
)
11 simp1r 985 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  W  e.  H )
12 cdleme10.h . . . . . 6  |-  H  =  ( LHyp `  K
)
136, 12lhpbase 29354 . . . . 5  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1411, 13syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  W  e.  ( Base `  K )
)
15 hllat 28720 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
163, 15syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  K  e.  Lat )
176, 8atbase 28646 . . . . . 6  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
18173ad2ant2 982 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  R  e.  ( Base `  K )
)
196, 8atbase 28646 . . . . . 6  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
204, 19syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  S  e.  ( Base `  K )
)
21 cdleme10.l . . . . . 6  |-  .<_  =  ( le `  K )
226, 21, 7latlej2 14129 . . . . 5  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  S  e.  ( Base `  K
) )  ->  S  .<_  ( R  .\/  S
) )
2316, 18, 20, 22syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  S  .<_  ( R  .\/  S ) )
24 cdleme10.m . . . . 5  |-  ./\  =  ( meet `  K )
256, 21, 7, 24, 8atmod3i1 29220 . . . 4  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  ( R  .\/  S
)  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  /\  S  .<_  ( R  .\/  S
) )  ->  ( S  .\/  ( ( R 
.\/  S )  ./\  W ) )  =  ( ( R  .\/  S
)  ./\  ( S  .\/  W ) ) )
263, 4, 10, 14, 23, 25syl131anc 1200 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( S  .\/  ( ( R  .\/  S )  ./\  W )
)  =  ( ( R  .\/  S ) 
./\  ( S  .\/  W ) ) )
276, 7latjcom 14127 . . . . 5  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  S  e.  ( Base `  K
) )  ->  ( R  .\/  S )  =  ( S  .\/  R
) )
2816, 18, 20, 27syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( R  .\/  S )  =  ( S  .\/  R ) )
29 eqid 2258 . . . . . 6  |-  ( 1.
`  K )  =  ( 1. `  K
)
3021, 7, 29, 8, 12lhpjat2 29377 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  -> 
( S  .\/  W
)  =  ( 1.
`  K ) )
31303adant2 979 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( S  .\/  W )  =  ( 1. `  K ) )
3228, 31oveq12d 5810 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( ( R  .\/  S )  ./\  ( S  .\/  W ) )  =  ( ( S  .\/  R ) 
./\  ( 1. `  K ) ) )
33 hlol 28718 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OL )
343, 33syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  K  e.  OL )
356, 7latjcl 14118 . . . . 5  |-  ( ( K  e.  Lat  /\  S  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  ( S  .\/  R )  e.  ( Base `  K
) )
3616, 20, 18, 35syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( S  .\/  R )  e.  (
Base `  K )
)
376, 24, 29olm11 28584 . . . 4  |-  ( ( K  e.  OL  /\  ( S  .\/  R )  e.  ( Base `  K
) )  ->  (
( S  .\/  R
)  ./\  ( 1. `  K ) )  =  ( S  .\/  R
) )
3834, 36, 37syl2anc 645 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( ( S  .\/  R )  ./\  ( 1. `  K ) )  =  ( S 
.\/  R ) )
3926, 32, 383eqtrd 2294 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( S  .\/  ( ( R  .\/  S )  ./\  W )
)  =  ( S 
.\/  R ) )
402, 39syl5eq 2302 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( S  .\/  D )  =  ( S  .\/  R ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   class class class wbr 3997   ` cfv 4673  (class class class)co 5792   Basecbs 13110   lecple 13177   joincjn 14040   meetcmee 14041   1.cp1 14106   Latclat 14113   OLcol 28531   Atomscatm 28620   HLchlt 28707   LHypclh 29340
This theorem is referenced by:  cdleme10tN  29614  cdleme20aN  29665  cdleme20g  29671
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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