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Theorem atmod1i1m 29972
Description: Version of modular law pmod1i 29962 that holds in a Hilbert lattice, when an element meets an atom. (Contributed by NM, 2-Sep-2012.) (Revised by Mario Carneiro, 10-May-2013.)
Hypotheses
Ref Expression
atmod.b  |-  B  =  ( Base `  K
)
atmod.l  |-  .<_  =  ( le `  K )
atmod.j  |-  .\/  =  ( join `  K )
atmod.m  |-  ./\  =  ( meet `  K )
atmod.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
atmod1i1m  |-  ( ( ( K  e.  HL  /\  P  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  ./\  P )  .<_  Z )  ->  (
( X  ./\  P
)  .\/  ( Y  ./\ 
Z ) )  =  ( ( ( X 
./\  P )  .\/  Y )  ./\  Z )
)

Proof of Theorem atmod1i1m
StepHypRef Expression
1 simpl1l 1008 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  ./\  P
)  .<_  Z )  /\  ( X  ./\  P )  e.  A )  ->  K  e.  HL )
2 simpr 448 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  ./\  P
)  .<_  Z )  /\  ( X  ./\  P )  e.  A )  -> 
( X  ./\  P
)  e.  A )
3 simpl22 1036 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  ./\  P
)  .<_  Z )  /\  ( X  ./\  P )  e.  A )  ->  Y  e.  B )
4 simpl23 1037 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  ./\  P
)  .<_  Z )  /\  ( X  ./\  P )  e.  A )  ->  Z  e.  B )
5 simpl3 962 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  ./\  P
)  .<_  Z )  /\  ( X  ./\  P )  e.  A )  -> 
( X  ./\  P
)  .<_  Z )
6 atmod.b . . . 4  |-  B  =  ( Base `  K
)
7 atmod.l . . . 4  |-  .<_  =  ( le `  K )
8 atmod.j . . . 4  |-  .\/  =  ( join `  K )
9 atmod.m . . . 4  |-  ./\  =  ( meet `  K )
10 atmod.a . . . 4  |-  A  =  ( Atoms `  K )
116, 7, 8, 9, 10atmod1i1 29971 . . 3  |-  ( ( K  e.  HL  /\  ( ( X  ./\  P )  e.  A  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  ./\  P
)  .<_  Z )  -> 
( ( X  ./\  P )  .\/  ( Y 
./\  Z ) )  =  ( ( ( X  ./\  P )  .\/  Y )  ./\  Z
) )
121, 2, 3, 4, 5, 11syl131anc 1197 . 2  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  ./\  P
)  .<_  Z )  /\  ( X  ./\  P )  e.  A )  -> 
( ( X  ./\  P )  .\/  ( Y 
./\  Z ) )  =  ( ( ( X  ./\  P )  .\/  Y )  ./\  Z
) )
13 simp1l 981 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  ./\  P )  .<_  Z )  ->  K  e.  HL )
14 hlol 29476 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OL )
1513, 14syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  ./\  P )  .<_  Z )  ->  K  e.  OL )
1615adantr 452 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  ./\  P
)  .<_  Z )  /\  ( X  ./\  P )  =  ( 0. `  K ) )  ->  K  e.  OL )
17 hllat 29478 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
1813, 17syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  ./\  P )  .<_  Z )  ->  K  e.  Lat )
1918adantr 452 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  ./\  P
)  .<_  Z )  /\  ( X  ./\  P )  =  ( 0. `  K ) )  ->  K  e.  Lat )
20 simpl22 1036 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  ./\  P
)  .<_  Z )  /\  ( X  ./\  P )  =  ( 0. `  K ) )  ->  Y  e.  B )
21 simpl23 1037 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  ./\  P
)  .<_  Z )  /\  ( X  ./\  P )  =  ( 0. `  K ) )  ->  Z  e.  B )
226, 9latmcl 14407 . . . . 5  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  ./\  Z
)  e.  B )
2319, 20, 21, 22syl3anc 1184 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  ./\  P
)  .<_  Z )  /\  ( X  ./\  P )  =  ( 0. `  K ) )  -> 
( Y  ./\  Z
)  e.  B )
24 eqid 2387 . . . . 5  |-  ( 0.
`  K )  =  ( 0. `  K
)
256, 8, 24olj02 29341 . . . 4  |-  ( ( K  e.  OL  /\  ( Y  ./\  Z )  e.  B )  -> 
( ( 0. `  K )  .\/  ( Y  ./\  Z ) )  =  ( Y  ./\  Z ) )
2616, 23, 25syl2anc 643 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  ./\  P
)  .<_  Z )  /\  ( X  ./\  P )  =  ( 0. `  K ) )  -> 
( ( 0. `  K )  .\/  ( Y  ./\  Z ) )  =  ( Y  ./\  Z ) )
27 oveq1 6027 . . . 4  |-  ( ( X  ./\  P )  =  ( 0. `  K )  ->  (
( X  ./\  P
)  .\/  ( Y  ./\ 
Z ) )  =  ( ( 0. `  K )  .\/  ( Y  ./\  Z ) ) )
2827adantl 453 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  ./\  P
)  .<_  Z )  /\  ( X  ./\  P )  =  ( 0. `  K ) )  -> 
( ( X  ./\  P )  .\/  ( Y 
./\  Z ) )  =  ( ( 0.
`  K )  .\/  ( Y  ./\  Z ) ) )
29 oveq1 6027 . . . . . 6  |-  ( ( X  ./\  P )  =  ( 0. `  K )  ->  (
( X  ./\  P
)  .\/  Y )  =  ( ( 0.
`  K )  .\/  Y ) )
3029adantl 453 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  ./\  P
)  .<_  Z )  /\  ( X  ./\  P )  =  ( 0. `  K ) )  -> 
( ( X  ./\  P )  .\/  Y )  =  ( ( 0.
`  K )  .\/  Y ) )
316, 8, 24olj02 29341 . . . . . 6  |-  ( ( K  e.  OL  /\  Y  e.  B )  ->  ( ( 0. `  K )  .\/  Y
)  =  Y )
3216, 20, 31syl2anc 643 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  ./\  P
)  .<_  Z )  /\  ( X  ./\  P )  =  ( 0. `  K ) )  -> 
( ( 0. `  K )  .\/  Y
)  =  Y )
3330, 32eqtrd 2419 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  ./\  P
)  .<_  Z )  /\  ( X  ./\  P )  =  ( 0. `  K ) )  -> 
( ( X  ./\  P )  .\/  Y )  =  Y )
3433oveq1d 6035 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  ./\  P
)  .<_  Z )  /\  ( X  ./\  P )  =  ( 0. `  K ) )  -> 
( ( ( X 
./\  P )  .\/  Y )  ./\  Z )  =  ( Y  ./\  Z ) )
3526, 28, 343eqtr4d 2429 . 2  |-  ( ( ( ( K  e.  HL  /\  P  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  ./\  P
)  .<_  Z )  /\  ( X  ./\  P )  =  ( 0. `  K ) )  -> 
( ( X  ./\  P )  .\/  ( Y 
./\  Z ) )  =  ( ( ( X  ./\  P )  .\/  Y )  ./\  Z
) )
36 simp21 990 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  ./\  P )  .<_  Z )  ->  X  e.  B )
37 simp1r 982 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  ./\  P )  .<_  Z )  ->  P  e.  A )
386, 9, 24, 10meetat2 29412 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  P  e.  A )  ->  ( ( X  ./\  P )  e.  A  \/  ( X  ./\  P )  =  ( 0. `  K ) ) )
3915, 36, 37, 38syl3anc 1184 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  ./\  P )  .<_  Z )  ->  (
( X  ./\  P
)  e.  A  \/  ( X  ./\  P )  =  ( 0. `  K ) ) )
4012, 35, 39mpjaodan 762 1  |-  ( ( ( K  e.  HL  /\  P  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  ./\  P )  .<_  Z )  ->  (
( X  ./\  P
)  .\/  ( Y  ./\ 
Z ) )  =  ( ( ( X 
./\  P )  .\/  Y )  ./\  Z )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   Basecbs 13396   lecple 13463   joincjn 14328   meetcmee 14329   0.cp0 14393   Latclat 14401   OLcol 29289   Atomscatm 29378   HLchlt 29465
This theorem is referenced by:  dalawlem3  29987  dalawlem6  29990
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-undef 6479  df-riota 6485  df-poset 14330  df-plt 14342  df-lub 14358  df-glb 14359  df-join 14360  df-meet 14361  df-p0 14395  df-lat 14402  df-clat 14464  df-oposet 29291  df-ol 29293  df-oml 29294  df-covers 29381  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466  df-psubsp 29617  df-pmap 29618  df-padd 29910
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