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Theorem omlmod1i2N 35382
Description: Analog of modular law atmod1i2 35980 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 994 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  K  e.  OML )
2 simp23 1029 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  Z  e.  B )
3 simp21 1027 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  X  e.  B )
4 simp22 1028 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  Y  e.  B )
5 simp3l 1022 . . . . 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 35378 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .<_  Z  ->  X C Z ) )
101, 3, 2, 9syl3anc 1226 . . . . 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 35371 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Z  e.  B )  ->  ( X C Z  <-> 
Z C X ) )
131, 3, 2, 12syl3anc 1226 . . . 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 1023 . . . 4  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  Y C Z )
166, 8cmtcomN 35371 . . . . 5  |-  ( ( K  e.  OML  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y C Z  <-> 
Z C Y ) )
171, 4, 2, 16syl3anc 1226 . . . 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 35380 . . 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 1244 . 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 35364 . . . 4  |-  ( K  e.  OML  ->  K  e.  Lat )
24233ad2ant1 1015 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  K  e.  Lat )
256, 19latjcl 15880 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
2624, 3, 4, 25syl3anc 1226 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  -> 
( X  .\/  Y
)  e.  B )
276, 20latmcom 15904 . . 3  |-  ( ( K  e.  Lat  /\  Z  e.  B  /\  ( X  .\/  Y )  e.  B )  -> 
( Z  ./\  ( X  .\/  Y ) )  =  ( ( X 
.\/  Y )  ./\  Z ) )
2824, 2, 26, 27syl3anc 1226 . 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 15909 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .<_  Z  <->  ( Z  ./\ 
X )  =  X ) )
3024, 3, 2, 29syl3anc 1226 . . . 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 15904 . . . 4  |-  ( ( K  e.  Lat  /\  Z  e.  B  /\  Y  e.  B )  ->  ( Z  ./\  Y
)  =  ( Y 
./\  Z ) )
3324, 2, 4, 32syl3anc 1226 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  -> 
( Z  ./\  Y
)  =  ( Y 
./\  Z ) )
3431, 33oveq12d 6288 . 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 2504 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14716   lecple 14791   joincjn 15772   meetcmee 15773   Latclat 15874   cmccmtN 35295   OMLcoml 35297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-preset 15756  df-poset 15774  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-p0 15868  df-lat 15875  df-oposet 35298  df-cmtN 35299  df-ol 35300  df-oml 35301
This theorem is referenced by:  omlspjN  35383
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