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Theorem omlspjN 30073
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 30054 . . . . . 6  |-  ( K  e.  OML  ->  K  e.  Lat )
213ad2ant1 976 . . . . 5  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  K  e.  Lat )
3 omlop 30053 . . . . . . 7  |-  ( K  e.  OML  ->  K  e.  OP )
433ad2ant1 976 . . . . . 6  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  K  e.  OP )
5 simp2r 982 . . . . . 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 30006 . . . . . 6  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  (  ._|_  `  Y )  e.  B )
94, 5, 8syl2anc 642 . . . . 5  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  (  ._|_  `  Y )  e.  B
)
10 omlspj.m . . . . . 6  |-  ./\  =  ( meet `  K )
116, 10latmcom 14197 . . . . 5  |-  ( ( K  e.  Lat  /\  (  ._|_  `  Y )  e.  B  /\  Y  e.  B )  ->  (
(  ._|_  `  Y )  ./\  Y )  =  ( Y  ./\  (  ._|_  `  Y ) ) )
122, 9, 5, 11syl3anc 1182 . . . 4  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( (  ._|_  `  Y )  ./\  Y )  =  ( Y 
./\  (  ._|_  `  Y
) ) )
13 eqid 2296 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
146, 7, 10, 13opnoncon 30020 . . . . 5  |-  ( ( K  e.  OP  /\  Y  e.  B )  ->  ( Y  ./\  (  ._|_  `  Y ) )  =  ( 0. `  K ) )
154, 5, 14syl2anc 642 . . . 4  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( Y  ./\  (  ._|_  `  Y ) )  =  ( 0.
`  K ) )
1612, 15eqtrd 2328 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( (  ._|_  `  Y )  ./\  Y )  =  ( 0.
`  K ) )
1716oveq2d 5890 . 2  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( X  .\/  ( (  ._|_  `  Y
)  ./\  Y )
)  =  ( X 
.\/  ( 0. `  K ) ) )
18 simp1 955 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  K  e.  OML )
19 simp2l 981 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  X  e.  B )
20 simp3 957 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  X  .<_  Y )
21 eqid 2296 . . . . . 6  |-  ( cm
`  K )  =  ( cm `  K
)
226, 21cmtidN 30069 . . . . 5  |-  ( ( K  e.  OML  /\  Y  e.  B )  ->  Y ( cm `  K ) Y )
2318, 5, 22syl2anc 642 . . . 4  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  Y ( cm `  K ) Y )
246, 7, 21cmt3N 30063 . . . . 5  |-  ( ( K  e.  OML  /\  Y  e.  B  /\  Y  e.  B )  ->  ( Y ( cm
`  K ) Y  <-> 
(  ._|_  `  Y )
( cm `  K
) Y ) )
2518, 5, 5, 24syl3anc 1182 . . . 4  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( Y
( cm `  K
) Y  <->  (  ._|_  `  Y ) ( cm
`  K ) Y ) )
2623, 25mpbid 201 . . 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 30072 . . 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 1200 . 2  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( X  .\/  ( (  ._|_  `  Y
)  ./\  Y )
)  =  ( ( X  .\/  (  ._|_  `  Y ) )  ./\  Y ) )
31 omlol 30052 . . . 4  |-  ( K  e.  OML  ->  K  e.  OL )
32313ad2ant1 976 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  K  e.  OL )
336, 28, 13olj01 30037 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( X  .\/  ( 0. `  K ) )  =  X )
3432, 19, 33syl2anc 642 . 2  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( X  .\/  ( 0. `  K
) )  =  X )
3517, 30, 343eqtr3d 2336 1  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( ( X  .\/  (  ._|_  `  Y
) )  ./\  Y
)  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   occoc 13232   joincjn 14094   meetcmee 14095   0.cp0 14159   Latclat 14167   OPcops 29984   cmccmtN 29985   OLcol 29986   OMLcoml 29987
This theorem is referenced by:  doca2N  31938  djajN  31949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-lat 14168  df-oposet 29988  df-cmtN 29989  df-ol 29990  df-oml 29991
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