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Theorem omlspjN 32906
Description: Contraction of a Sasaki projection. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
omlspj.b  |-  B  =  ( Base `  K
)
omlspj.l  |-  .<_  =  ( le `  K )
omlspj.j  |-  .\/  =  ( join `  K )
omlspj.m  |-  ./\  =  ( meet `  K )
omlspj.o  |-  ._|_  =  ( oc `  K )
Assertion
Ref Expression
omlspjN  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( ( X  .\/  (  ._|_  `  Y
) )  ./\  Y
)  =  X )

Proof of Theorem omlspjN
StepHypRef Expression
1 omllat 32887 . . . . . 6  |-  ( K  e.  OML  ->  K  e.  Lat )
213ad2ant1 1009 . . . . 5  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  K  e.  Lat )
3 omlop 32886 . . . . . . 7  |-  ( K  e.  OML  ->  K  e.  OP )
433ad2ant1 1009 . . . . . 6  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  K  e.  OP )
5 simp2r 1015 . . . . . 6  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  Y  e.  B )
6 omlspj.b . . . . . . 7  |-  B  =  ( Base `  K
)
7 omlspj.o . . . . . . 7  |-  ._|_  =  ( oc `  K )
86, 7opoccl 32839 . . . . . 6  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  (  ._|_  `  Y )  e.  B )
94, 5, 8syl2anc 661 . . . . 5  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  (  ._|_  `  Y )  e.  B
)
10 omlspj.m . . . . . 6  |-  ./\  =  ( meet `  K )
116, 10latmcom 15245 . . . . 5  |-  ( ( K  e.  Lat  /\  (  ._|_  `  Y )  e.  B  /\  Y  e.  B )  ->  (
(  ._|_  `  Y )  ./\  Y )  =  ( Y  ./\  (  ._|_  `  Y ) ) )
122, 9, 5, 11syl3anc 1218 . . . 4  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( (  ._|_  `  Y )  ./\  Y )  =  ( Y 
./\  (  ._|_  `  Y
) ) )
13 eqid 2443 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
146, 7, 10, 13opnoncon 32853 . . . . 5  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  ( Y  ./\  (  ._|_  `  Y ) )  =  ( 0. `  K ) )
154, 5, 14syl2anc 661 . . . 4  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( Y  ./\  (  ._|_  `  Y ) )  =  ( 0.
`  K ) )
1612, 15eqtrd 2475 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( (  ._|_  `  Y )  ./\  Y )  =  ( 0.
`  K ) )
1716oveq2d 6107 . 2  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( X  .\/  ( (  ._|_  `  Y
)  ./\  Y )
)  =  ( X 
.\/  ( 0. `  K ) ) )
18 simp1 988 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  K  e.  OML )
19 simp2l 1014 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  X  e.  B )
20 simp3 990 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  X  .<_  Y )
21 eqid 2443 . . . . . 6  |-  ( cm
`  K )  =  ( cm `  K
)
226, 21cmtidN 32902 . . . . 5  |-  ( ( K  e.  OML  /\  Y  e.  B )  ->  Y ( cm `  K ) Y )
2318, 5, 22syl2anc 661 . . . 4  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  Y ( cm `  K ) Y )
246, 7, 21cmt3N 32896 . . . . 5  |-  ( ( K  e.  OML  /\  Y  e.  B  /\  Y  e.  B )  ->  ( Y ( cm
`  K ) Y  <-> 
(  ._|_  `  Y )
( cm `  K
) Y ) )
2518, 5, 5, 24syl3anc 1218 . . . 4  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( Y
( cm `  K
) Y  <->  (  ._|_  `  Y ) ( cm
`  K ) Y ) )
2623, 25mpbid 210 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  (  ._|_  `  Y ) ( cm
`  K ) Y )
27 omlspj.l . . . 4  |-  .<_  =  ( le `  K )
28 omlspj.j . . . 4  |-  .\/  =  ( join `  K )
296, 27, 28, 10, 21omlmod1i2N 32905 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  (  ._|_  `  Y
)  e.  B  /\  Y  e.  B )  /\  ( X  .<_  Y  /\  (  ._|_  `  Y )
( cm `  K
) Y ) )  ->  ( X  .\/  ( (  ._|_  `  Y
)  ./\  Y )
)  =  ( ( X  .\/  (  ._|_  `  Y ) )  ./\  Y ) )
3018, 19, 9, 5, 20, 26, 29syl132anc 1236 . 2  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( X  .\/  ( (  ._|_  `  Y
)  ./\  Y )
)  =  ( ( X  .\/  (  ._|_  `  Y ) )  ./\  Y ) )
31 omlol 32885 . . . 4  |-  ( K  e.  OML  ->  K  e.  OL )
32313ad2ant1 1009 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  K  e.  OL )
336, 28, 13olj01 32870 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( X  .\/  ( 0. `  K ) )  =  X )
3432, 19, 33syl2anc 661 . 2  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( X  .\/  ( 0. `  K
) )  =  X )
3517, 30, 343eqtr3d 2483 1  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( ( X  .\/  (  ._|_  `  Y
) )  ./\  Y
)  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   Basecbs 14174   lecple 14245   occoc 14246   joincjn 15114   meetcmee 15115   0.cp0 15207   Latclat 15215   OPcops 32817   cmccmtN 32818   OLcol 32819   OMLcoml 32820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-poset 15116  df-lub 15144  df-glb 15145  df-join 15146  df-meet 15147  df-p0 15209  df-lat 15216  df-oposet 32821  df-cmtN 32822  df-ol 32823  df-oml 32824
This theorem is referenced by:  doca2N  34771  djajN  34782
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