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Theorem atmod1i1m 30669
Description: Version of modular law pmod1i 30659 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 1006 . . 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 447 . . 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 1034 . . 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 1035 . . 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 960 . . 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 30668 . . 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 1195 . 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 979 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  ./\  P )  .<_  Z )  ->  K  e.  HL )
14 hlol 30173 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OL )
1513, 14syl 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  ./\  P )  .<_  Z )  ->  K  e.  OL )
1615adantr 451 . . . 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 30175 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
1813, 17syl 15 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  ./\  P )  .<_  Z )  ->  K  e.  Lat )
1918adantr 451 . . . . 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 1034 . . . . 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 1035 . . . . 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 14173 . . . . 5  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  ./\  Z
)  e.  B )
2319, 20, 21, 22syl3anc 1182 . . . 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 2296 . . . . 5  |-  ( 0.
`  K )  =  ( 0. `  K
)
256, 8, 24olj02 30038 . . . 4  |-  ( ( K  e.  OL  /\  ( Y  ./\  Z )  e.  B )  -> 
( ( 0. `  K )  .\/  ( Y  ./\  Z ) )  =  ( Y  ./\  Z ) )
2616, 23, 25syl2anc 642 . . 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 5881 . . . 4  |-  ( ( X  ./\  P )  =  ( 0. `  K )  ->  (
( X  ./\  P
)  .\/  ( Y  ./\ 
Z ) )  =  ( ( 0. `  K )  .\/  ( Y  ./\  Z ) ) )
2827adantl 452 . . 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 5881 . . . . . 6  |-  ( ( X  ./\  P )  =  ( 0. `  K )  ->  (
( X  ./\  P
)  .\/  Y )  =  ( ( 0.
`  K )  .\/  Y ) )
3029adantl 452 . . . . 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 30038 . . . . . 6  |-  ( ( K  e.  OL  /\  Y  e.  B )  ->  ( ( 0. `  K )  .\/  Y
)  =  Y )
3216, 20, 31syl2anc 642 . . . . 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 2328 . . . 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 5889 . . 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 2338 . 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 988 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  ./\  P )  .<_  Z )  ->  X  e.  B )
37 simp1r 980 . . 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 30109 . . 3  |-  ( ( K  e.  OL  /\  X  e.  B  /\  P  e.  A )  ->  ( ( X  ./\  P )  e.  A  \/  ( X  ./\  P )  =  ( 0. `  K ) ) )
3915, 36, 37, 38syl3anc 1182 . 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 761 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 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   0.cp0 14159   Latclat 14167   OLcol 29986   Atomscatm 30075   HLchlt 30162
This theorem is referenced by:  dalawlem3  30684  dalawlem6  30687
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-psubsp 30314  df-pmap 30315  df-padd 30607
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