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Theorem atmod1i1m 30592
Description: Version of modular law pmod1i 30582 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 30591 . . 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 30096 . . . . . 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 30098 . . . . . . 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 14472 . . . . 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 2435 . . . . 5  |-  ( 0.
`  K )  =  ( 0. `  K
)
256, 8, 24olj02 29961 . . . 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 6080 . . . 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 6080 . . . . . 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 29961 . . . . . 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 2467 . . . 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 6088 . . 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 2477 . 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 30032 . . 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 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   joincjn 14393   meetcmee 14394   0.cp0 14458   Latclat 14466   OLcol 29909   Atomscatm 29998   HLchlt 30085
This theorem is referenced by:  dalawlem3  30607  dalawlem6  30610
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-iin 4088  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-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-psubsp 30237  df-pmap 30238  df-padd 30530
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