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Theorem llnmod2i2 30121
Description: Version of modular law pmod1i 30106 that holds in a Hilbert lattice, when one element is a lattice line (expressed as the join  P  .\/  Q). (Contributed by NM, 16-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
llnmod2i2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  ( ( X  ./\  ( P  .\/  Q ) )  .\/  Y
)  =  ( X 
./\  ( ( P 
.\/  Q )  .\/  Y ) ) )

Proof of Theorem llnmod2i2
StepHypRef Expression
1 simp11 985 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  K  e.  HL )
2 hllat 29622 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 15 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  K  e.  Lat )
4 simp13 987 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  Y  e.  B )
5 simp2l 981 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  P  e.  A )
6 simp2r 982 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  Q  e.  A )
7 atmod.b . . . . . 6  |-  B  =  ( Base `  K
)
8 atmod.j . . . . . 6  |-  .\/  =  ( join `  K )
9 atmod.a . . . . . 6  |-  A  =  ( Atoms `  K )
107, 8, 9hlatjcl 29625 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  B )
111, 5, 6, 10syl3anc 1182 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  ( P  .\/  Q )  e.  B
)
12 simp12 986 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  X  e.  B )
13 atmod.m . . . . 5  |-  ./\  =  ( meet `  K )
147, 13latmcl 14256 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  B  /\  X  e.  B )  ->  (
( P  .\/  Q
)  ./\  X )  e.  B )
153, 11, 12, 14syl3anc 1182 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  ( ( P  .\/  Q )  ./\  X )  e.  B )
167, 8latjcom 14264 . . 3  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  ( ( P  .\/  Q )  ./\  X )  e.  B )  ->  ( Y  .\/  ( ( P 
.\/  Q )  ./\  X ) )  =  ( ( ( P  .\/  Q )  ./\  X )  .\/  Y ) )
173, 4, 15, 16syl3anc 1182 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  ( Y  .\/  ( ( P  .\/  Q )  ./\  X )
)  =  ( ( ( P  .\/  Q
)  ./\  X )  .\/  Y ) )
187, 8latjcl 14255 . . . . 5  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  ( P  .\/  Q )  e.  B )  -> 
( Y  .\/  ( P  .\/  Q ) )  e.  B )
193, 4, 11, 18syl3anc 1182 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  ( Y  .\/  ( P  .\/  Q
) )  e.  B
)
207, 13latmcom 14280 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( Y  .\/  ( P 
.\/  Q ) )  e.  B )  -> 
( X  ./\  ( Y  .\/  ( P  .\/  Q ) ) )  =  ( ( Y  .\/  ( P  .\/  Q ) )  ./\  X )
)
213, 12, 19, 20syl3anc 1182 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  ( X  ./\  ( Y  .\/  ( P  .\/  Q ) ) )  =  ( ( Y  .\/  ( P 
.\/  Q ) ) 
./\  X ) )
227, 8latjcom 14264 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  B  /\  Y  e.  B )  ->  (
( P  .\/  Q
)  .\/  Y )  =  ( Y  .\/  ( P  .\/  Q ) ) )
233, 11, 4, 22syl3anc 1182 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  ( ( P  .\/  Q )  .\/  Y )  =  ( Y 
.\/  ( P  .\/  Q ) ) )
2423oveq2d 5961 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  ( X  ./\  ( ( P  .\/  Q )  .\/  Y ) )  =  ( X 
./\  ( Y  .\/  ( P  .\/  Q ) ) ) )
25 simp3 957 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  Y  .<_  X )
26 atmod.l . . . . 5  |-  .<_  =  ( le `  K )
277, 26, 8, 13, 9llnmod1i2 30118 . . . 4  |-  ( ( ( K  e.  HL  /\  Y  e.  B  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  ( Y  .\/  ( ( P  .\/  Q )  ./\  X )
)  =  ( ( Y  .\/  ( P 
.\/  Q ) ) 
./\  X ) )
281, 4, 12, 5, 6, 25, 27syl321anc 1204 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  ( Y  .\/  ( ( P  .\/  Q )  ./\  X )
)  =  ( ( Y  .\/  ( P 
.\/  Q ) ) 
./\  X ) )
2921, 24, 283eqtr4d 2400 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  ( X  ./\  ( ( P  .\/  Q )  .\/  Y ) )  =  ( Y 
.\/  ( ( P 
.\/  Q )  ./\  X ) ) )
307, 13latmcom 14280 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( P  .\/  Q )  e.  B )  -> 
( X  ./\  ( P  .\/  Q ) )  =  ( ( P 
.\/  Q )  ./\  X ) )
313, 12, 11, 30syl3anc 1182 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  ( X  ./\  ( P  .\/  Q
) )  =  ( ( P  .\/  Q
)  ./\  X )
)
3231oveq1d 5960 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  ( ( X  ./\  ( P  .\/  Q ) )  .\/  Y
)  =  ( ( ( P  .\/  Q
)  ./\  X )  .\/  Y ) )
3317, 29, 323eqtr4rd 2401 1  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  ( ( X  ./\  ( P  .\/  Q ) )  .\/  Y
)  =  ( X 
./\  ( ( P 
.\/  Q )  .\/  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   class class class wbr 4104   ` cfv 5337  (class class class)co 5945   Basecbs 13245   lecple 13312   joincjn 14177   meetcmee 14178   Latclat 14250   Atomscatm 29522   HLchlt 29609
This theorem is referenced by:  dalawlem11  30139
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-iin 3989  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-undef 6385  df-riota 6391  df-poset 14179  df-plt 14191  df-lub 14207  df-glb 14208  df-join 14209  df-meet 14210  df-p0 14244  df-lat 14251  df-clat 14313  df-oposet 29435  df-ol 29437  df-oml 29438  df-covers 29525  df-ats 29526  df-atl 29557  df-cvlat 29581  df-hlat 29610  df-psubsp 29761  df-pmap 29762  df-padd 30054
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