Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  omlspjN Structured version   Unicode version

Theorem omlspjN 35087
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 35068 . . . . . 6  |-  ( K  e.  OML  ->  K  e.  Lat )
213ad2ant1 1017 . . . . 5  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  K  e.  Lat )
3 omlop 35067 . . . . . . 7  |-  ( K  e.  OML  ->  K  e.  OP )
433ad2ant1 1017 . . . . . 6  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  K  e.  OP )
5 simp2r 1023 . . . . . 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 35020 . . . . . 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 15831 . . . . 5  |-  ( ( K  e.  Lat  /\  (  ._|_  `  Y )  e.  B  /\  Y  e.  B )  ->  (
(  ._|_  `  Y )  ./\  Y )  =  ( Y  ./\  (  ._|_  `  Y ) ) )
122, 9, 5, 11syl3anc 1228 . . . 4  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( (  ._|_  `  Y )  ./\  Y )  =  ( Y 
./\  (  ._|_  `  Y
) ) )
13 eqid 2457 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
146, 7, 10, 13opnoncon 35034 . . . . 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 2498 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( (  ._|_  `  Y )  ./\  Y )  =  ( 0.
`  K ) )
1716oveq2d 6312 . 2  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( X  .\/  ( (  ._|_  `  Y
)  ./\  Y )
)  =  ( X 
.\/  ( 0. `  K ) ) )
18 simp1 996 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  K  e.  OML )
19 simp2l 1022 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  X  e.  B )
20 simp3 998 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  X  .<_  Y )
21 eqid 2457 . . . . . 6  |-  ( cm
`  K )  =  ( cm `  K
)
226, 21cmtidN 35083 . . . . 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 35077 . . . . 5  |-  ( ( K  e.  OML  /\  Y  e.  B  /\  Y  e.  B )  ->  ( Y ( cm
`  K ) Y  <-> 
(  ._|_  `  Y )
( cm `  K
) Y ) )
2518, 5, 5, 24syl3anc 1228 . . . 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 35086 . . 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 1246 . 2  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( X  .\/  ( (  ._|_  `  Y
)  ./\  Y )
)  =  ( ( X  .\/  (  ._|_  `  Y ) )  ./\  Y ) )
31 omlol 35066 . . . 4  |-  ( K  e.  OML  ->  K  e.  OL )
32313ad2ant1 1017 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  K  e.  OL )
336, 28, 13olj01 35051 . . 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 2506 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 973    = wceq 1395    e. wcel 1819   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   Basecbs 14643   lecple 14718   occoc 14719   joincjn 15699   meetcmee 15700   0.cp0 15793   Latclat 15801   OPcops 34998   cmccmtN 34999   OLcol 35000   OMLcoml 35001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-preset 15683  df-poset 15701  df-lub 15730  df-glb 15731  df-join 15732  df-meet 15733  df-p0 15795  df-lat 15802  df-oposet 35002  df-cmtN 35003  df-ol 35004  df-oml 35005
This theorem is referenced by:  doca2N  36954  djajN  36965
  Copyright terms: Public domain W3C validator