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Theorem omlmod1i2N 29985
Description: Analog of modular law atmod1i2 30583 that holds in any OML. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
omlmod.b  |-  B  =  ( Base `  K
)
omlmod.l  |-  .<_  =  ( le `  K )
omlmod.j  |-  .\/  =  ( join `  K )
omlmod.m  |-  ./\  =  ( meet `  K )
omlmod.c  |-  C  =  ( cm `  K
)
Assertion
Ref Expression
omlmod1i2N  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  -> 
( X  .\/  ( Y  ./\  Z ) )  =  ( ( X 
.\/  Y )  ./\  Z ) )

Proof of Theorem omlmod1i2N
StepHypRef Expression
1 simp1 957 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  K  e.  OML )
2 simp23 992 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  Z  e.  B )
3 simp21 990 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  X  e.  B )
4 simp22 991 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  Y  e.  B )
5 simp3l 985 . . . . 5  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  X  .<_  Z )
6 omlmod.b . . . . . . 7  |-  B  =  ( Base `  K
)
7 omlmod.l . . . . . . 7  |-  .<_  =  ( le `  K )
8 omlmod.c . . . . . . 7  |-  C  =  ( cm `  K
)
96, 7, 8lecmtN 29981 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .<_  Z  ->  X C Z ) )
101, 3, 2, 9syl3anc 1184 . . . . 5  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  -> 
( X  .<_  Z  ->  X C Z ) )
115, 10mpd 15 . . . 4  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  X C Z )
126, 8cmtcomN 29974 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Z  e.  B )  ->  ( X C Z  <-> 
Z C X ) )
131, 3, 2, 12syl3anc 1184 . . . 4  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  -> 
( X C Z  <-> 
Z C X ) )
1411, 13mpbid 202 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  Z C X )
15 simp3r 986 . . . 4  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  Y C Z )
166, 8cmtcomN 29974 . . . . 5  |-  ( ( K  e.  OML  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y C Z  <-> 
Z C Y ) )
171, 4, 2, 16syl3anc 1184 . . . 4  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  -> 
( Y C Z  <-> 
Z C Y ) )
1815, 17mpbid 202 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  Z C Y )
19 omlmod.j . . . 4  |-  .\/  =  ( join `  K )
20 omlmod.m . . . 4  |-  ./\  =  ( meet `  K )
216, 19, 20, 8omlfh1N 29983 . . 3  |-  ( ( K  e.  OML  /\  ( Z  e.  B  /\  X  e.  B  /\  Y  e.  B
)  /\  ( Z C X  /\  Z C Y ) )  -> 
( Z  ./\  ( X  .\/  Y ) )  =  ( ( Z 
./\  X )  .\/  ( Z  ./\  Y ) ) )
221, 2, 3, 4, 14, 18, 21syl132anc 1202 . 2  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  -> 
( Z  ./\  ( X  .\/  Y ) )  =  ( ( Z 
./\  X )  .\/  ( Z  ./\  Y ) ) )
23 omllat 29967 . . . 4  |-  ( K  e.  OML  ->  K  e.  Lat )
24233ad2ant1 978 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  K  e.  Lat )
256, 19latjcl 14471 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
2624, 3, 4, 25syl3anc 1184 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  -> 
( X  .\/  Y
)  e.  B )
276, 20latmcom 14496 . . 3  |-  ( ( K  e.  Lat  /\  Z  e.  B  /\  ( X  .\/  Y )  e.  B )  -> 
( Z  ./\  ( X  .\/  Y ) )  =  ( ( X 
.\/  Y )  ./\  Z ) )
2824, 2, 26, 27syl3anc 1184 . 2  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  -> 
( Z  ./\  ( X  .\/  Y ) )  =  ( ( X 
.\/  Y )  ./\  Z ) )
296, 7, 20latleeqm2 14501 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .<_  Z  <->  ( Z  ./\ 
X )  =  X ) )
3024, 3, 2, 29syl3anc 1184 . . . 4  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  -> 
( X  .<_  Z  <->  ( Z  ./\ 
X )  =  X ) )
315, 30mpbid 202 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  -> 
( Z  ./\  X
)  =  X )
326, 20latmcom 14496 . . . 4  |-  ( ( K  e.  Lat  /\  Z  e.  B  /\  Y  e.  B )  ->  ( Z  ./\  Y
)  =  ( Y 
./\  Z ) )
3324, 2, 4, 32syl3anc 1184 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  -> 
( Z  ./\  Y
)  =  ( Y 
./\  Z ) )
3431, 33oveq12d 6091 . 2  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  -> 
( ( Z  ./\  X )  .\/  ( Z 
./\  Y ) )  =  ( X  .\/  ( Y  ./\  Z ) ) )
3522, 28, 343eqtr3rd 2476 1  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  -> 
( X  .\/  ( Y  ./\  Z ) )  =  ( ( X 
.\/  Y )  ./\  Z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   joincjn 14393   meetcmee 14394   Latclat 14466   cmccmtN 29898   OMLcoml 29900
This theorem is referenced by:  omlspjN  29986
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-lat 14467  df-oposet 29901  df-cmtN 29902  df-ol 29903  df-oml 29904
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