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Theorem omlmod1i2N 33211
Description: Analog of modular law atmod1i2 33809 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 988 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  K  e.  OML )
2 simp23 1023 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  Z  e.  B )
3 simp21 1021 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  X  e.  B )
4 simp22 1022 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  Y  e.  B )
5 simp3l 1016 . . . . 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 33207 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .<_  Z  ->  X C Z ) )
101, 3, 2, 9syl3anc 1219 . . . . 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 33200 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Z  e.  B )  ->  ( X C Z  <-> 
Z C X ) )
131, 3, 2, 12syl3anc 1219 . . . 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 210 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  Z C X )
15 simp3r 1017 . . . 4  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  Y C Z )
166, 8cmtcomN 33200 . . . . 5  |-  ( ( K  e.  OML  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y C Z  <-> 
Z C Y ) )
171, 4, 2, 16syl3anc 1219 . . . 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 210 . . 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 33209 . . 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 1237 . 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 33193 . . . 4  |-  ( K  e.  OML  ->  K  e.  Lat )
24233ad2ant1 1009 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  K  e.  Lat )
256, 19latjcl 15323 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
2624, 3, 4, 25syl3anc 1219 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  -> 
( X  .\/  Y
)  e.  B )
276, 20latmcom 15347 . . 3  |-  ( ( K  e.  Lat  /\  Z  e.  B  /\  ( X  .\/  Y )  e.  B )  -> 
( Z  ./\  ( X  .\/  Y ) )  =  ( ( X 
.\/  Y )  ./\  Z ) )
2824, 2, 26, 27syl3anc 1219 . 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 15352 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .<_  Z  <->  ( Z  ./\ 
X )  =  X ) )
3024, 3, 2, 29syl3anc 1219 . . . 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 210 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  -> 
( Z  ./\  X
)  =  X )
326, 20latmcom 15347 . . . 4  |-  ( ( K  e.  Lat  /\  Z  e.  B  /\  Y  e.  B )  ->  ( Z  ./\  Y
)  =  ( Y 
./\  Z ) )
3324, 2, 4, 32syl3anc 1219 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  -> 
( Z  ./\  Y
)  =  ( Y 
./\  Z ) )
3431, 33oveq12d 6208 . 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 2501 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   class class class wbr 4390   ` cfv 5516  (class class class)co 6190   Basecbs 14276   lecple 14347   joincjn 15216   meetcmee 15217   Latclat 15317   cmccmtN 33124   OMLcoml 33126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-poset 15218  df-lub 15246  df-glb 15247  df-join 15248  df-meet 15249  df-p0 15311  df-lat 15318  df-oposet 33127  df-cmtN 33128  df-ol 33129  df-oml 33130
This theorem is referenced by:  omlspjN  33212
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