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Theorem atmod1i2 30658
Description: Version of modular law pmod1i 30647 that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 14-May-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
atmod1i2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( X  .\/  ( P  ./\  Y
) )  =  ( ( X  .\/  P
)  ./\  Y )
)

Proof of Theorem atmod1i2
StepHypRef Expression
1 simpl 445 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
) )  ->  K  e.  HL )
2 simpr2 965 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
) )  ->  X  e.  B )
3 simpr1 964 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
) )  ->  P  e.  A )
4 atmod.b . . . . . 6  |-  B  =  ( Base `  K
)
5 atmod.j . . . . . 6  |-  .\/  =  ( join `  K )
6 atmod.a . . . . . 6  |-  A  =  ( Atoms `  K )
7 eqid 2438 . . . . . 6  |-  ( pmap `  K )  =  (
pmap `  K )
8 eqid 2438 . . . . . 6  |-  ( + P `  K )  =  ( + P `  K )
94, 5, 6, 7, 8pmapjat1 30652 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  ->  ( ( pmap `  K
) `  ( X  .\/  P ) )  =  ( ( ( pmap `  K ) `  X
) ( + P `  K ) ( (
pmap `  K ) `  P ) ) )
101, 2, 3, 9syl3anc 1185 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
) )  ->  (
( pmap `  K ) `  ( X  .\/  P
) )  =  ( ( ( pmap `  K
) `  X )
( + P `  K ) ( (
pmap `  K ) `  P ) ) )
114, 6atbase 30089 . . . . . 6  |-  ( P  e.  A  ->  P  e.  B )
123, 11syl 16 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
) )  ->  P  e.  B )
13 simpr3 966 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
) )  ->  Y  e.  B )
14 atmod.l . . . . . 6  |-  .<_  =  ( le `  K )
15 atmod.m . . . . . 6  |-  ./\  =  ( meet `  K )
164, 14, 5, 15, 7, 8hlmod1i 30655 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  B  /\  Y  e.  B
) )  ->  (
( X  .<_  Y  /\  ( ( pmap `  K
) `  ( X  .\/  P ) )  =  ( ( ( pmap `  K ) `  X
) ( + P `  K ) ( (
pmap `  K ) `  P ) ) )  ->  ( ( X 
.\/  P )  ./\  Y )  =  ( X 
.\/  ( P  ./\  Y ) ) ) )
171, 2, 12, 13, 16syl13anc 1187 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
) )  ->  (
( X  .<_  Y  /\  ( ( pmap `  K
) `  ( X  .\/  P ) )  =  ( ( ( pmap `  K ) `  X
) ( + P `  K ) ( (
pmap `  K ) `  P ) ) )  ->  ( ( X 
.\/  P )  ./\  Y )  =  ( X 
.\/  ( P  ./\  Y ) ) ) )
1810, 17mpan2d 657 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
) )  ->  ( X  .<_  Y  ->  (
( X  .\/  P
)  ./\  Y )  =  ( X  .\/  ( P  ./\  Y ) ) ) )
19183impia 1151 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( ( X  .\/  P )  ./\  Y )  =  ( X 
.\/  ( P  ./\  Y ) ) )
2019eqcomd 2443 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( X  .\/  ( P  ./\  Y
) )  =  ( ( X  .\/  P
)  ./\  Y )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   joincjn 14403   meetcmee 14404   Atomscatm 30063   HLchlt 30150   pmapcpmap 30296   + Pcpadd 30594
This theorem is referenced by:  atmod2i2  30661  atmod3i2  30664  atmod4i2  30666  lhpmod2i2  30837  dihmeetlem7N  32110  dihjatc1  32111
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-lat 14477  df-clat 14539  df-oposet 29976  df-ol 29978  df-oml 29979  df-covers 30066  df-ats 30067  df-atl 30098  df-cvlat 30122  df-hlat 30151  df-psubsp 30302  df-pmap 30303  df-padd 30595
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