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Theorem omlmod1i2N 29450
Description: Analog of modular law atmod1i2 30048 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 955 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  K  e.  OML )
2 simp23 990 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  Z  e.  B )
3 simp21 988 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  X  e.  B )
4 simp22 989 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  Y  e.  B )
5 simp3l 983 . . . . 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 29446 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .<_  Z  ->  X C Z ) )
101, 3, 2, 9syl3anc 1182 . . . . 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 14 . . . 4  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  X C Z )
126, 8cmtcomN 29439 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Z  e.  B )  ->  ( X C Z  <-> 
Z C X ) )
131, 3, 2, 12syl3anc 1182 . . . 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 201 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  Z C X )
15 simp3r 984 . . . 4  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  Y C Z )
166, 8cmtcomN 29439 . . . . 5  |-  ( ( K  e.  OML  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y C Z  <-> 
Z C Y ) )
171, 4, 2, 16syl3anc 1182 . . . 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 201 . . 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 29448 . . 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 1200 . 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 29432 . . . 4  |-  ( K  e.  OML  ->  K  e.  Lat )
24233ad2ant1 976 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  ->  K  e.  Lat )
256, 19latjcl 14156 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
2624, 3, 4, 25syl3anc 1182 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  -> 
( X  .\/  Y
)  e.  B )
276, 20latmcom 14181 . . 3  |-  ( ( K  e.  Lat  /\  Z  e.  B  /\  ( X  .\/  Y )  e.  B )  -> 
( Z  ./\  ( X  .\/  Y ) )  =  ( ( X 
.\/  Y )  ./\  Z ) )
2824, 2, 26, 27syl3anc 1182 . 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 14186 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .<_  Z  <->  ( Z  ./\ 
X )  =  X ) )
3024, 3, 2, 29syl3anc 1182 . . . 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 201 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  -> 
( Z  ./\  X
)  =  X )
326, 20latmcom 14181 . . . 4  |-  ( ( K  e.  Lat  /\  Z  e.  B  /\  Y  e.  B )  ->  ( Z  ./\  Y
)  =  ( Y 
./\  Z ) )
3324, 2, 4, 32syl3anc 1182 . . 3  |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  ( X  .<_  Z  /\  Y C Z ) )  -> 
( Z  ./\  Y
)  =  ( Y 
./\  Z ) )
3431, 33oveq12d 5876 . 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 2324 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Latclat 14151   cmccmtN 29363   OMLcoml 29365
This theorem is referenced by:  omlspjN  29451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-oposet 29366  df-cmtN 29367  df-ol 29368  df-oml 29369
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