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Theorem 4atex2-0aOLDN 30877
Description: Same as 4atex2 30876 except that  S is zero. (Contributed by NM, 27-May-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
4that.l  |-  .<_  =  ( le `  K )
4that.j  |-  .\/  =  ( join `  K )
4that.a  |-  A  =  ( Atoms `  K )
4that.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
4atex2-0aOLDN  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  =  ( 0. `  K ) )  /\  ( P  =/=  Q  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( S  .\/  z )  =  ( T  .\/  z ) ) )
Distinct variable groups:    z, r, A    H, r    .\/ , r,
z    K, r, z    .<_ , r, z    P, r, z    Q, r, z    S, r, z    W, r, z    T, r, z
Allowed substitution hint:    H( z)

Proof of Theorem 4atex2-0aOLDN
StepHypRef Expression
1 simp32l 1083 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  =  ( 0. `  K ) )  /\  ( P  =/=  Q  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  T  e.  A
)
2 simp32r 1084 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  =  ( 0. `  K ) )  /\  ( P  =/=  Q  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  -.  T  .<_  W )
3 simp1l 982 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  =  ( 0. `  K ) )  /\  ( P  =/=  Q  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  K  e.  HL )
4 hlol 30161 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OL )
53, 4syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  =  ( 0. `  K ) )  /\  ( P  =/=  Q  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  K  e.  OL )
6 eqid 2438 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
7 4that.a . . . . . 6  |-  A  =  ( Atoms `  K )
86, 7atbase 30089 . . . . 5  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
91, 8syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  =  ( 0. `  K ) )  /\  ( P  =/=  Q  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  T  e.  (
Base `  K )
)
10 4that.j . . . . 5  |-  .\/  =  ( join `  K )
11 eqid 2438 . . . . 5  |-  ( 0.
`  K )  =  ( 0. `  K
)
126, 10, 11olj02 30026 . . . 4  |-  ( ( K  e.  OL  /\  T  e.  ( Base `  K ) )  -> 
( ( 0. `  K )  .\/  T
)  =  T )
135, 9, 12syl2anc 644 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  =  ( 0. `  K ) )  /\  ( P  =/=  Q  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( ( 0.
`  K )  .\/  T )  =  T )
14 simp23 993 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  =  ( 0. `  K ) )  /\  ( P  =/=  Q  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  S  =  ( 0. `  K ) )
1514oveq1d 6098 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  =  ( 0. `  K ) )  /\  ( P  =/=  Q  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( S  .\/  T )  =  ( ( 0. `  K ) 
.\/  T ) )
1610, 7hlatjidm 30168 . . . 4  |-  ( ( K  e.  HL  /\  T  e.  A )  ->  ( T  .\/  T
)  =  T )
173, 1, 16syl2anc 644 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  =  ( 0. `  K ) )  /\  ( P  =/=  Q  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( T  .\/  T )  =  T )
1813, 15, 173eqtr4d 2480 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  =  ( 0. `  K ) )  /\  ( P  =/=  Q  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( S  .\/  T )  =  ( T 
.\/  T ) )
19 breq1 4217 . . . . 5  |-  ( z  =  T  ->  (
z  .<_  W  <->  T  .<_  W ) )
2019notbid 287 . . . 4  |-  ( z  =  T  ->  ( -.  z  .<_  W  <->  -.  T  .<_  W ) )
21 oveq2 6091 . . . . 5  |-  ( z  =  T  ->  ( S  .\/  z )  =  ( S  .\/  T
) )
22 oveq2 6091 . . . . 5  |-  ( z  =  T  ->  ( T  .\/  z )  =  ( T  .\/  T
) )
2321, 22eqeq12d 2452 . . . 4  |-  ( z  =  T  ->  (
( S  .\/  z
)  =  ( T 
.\/  z )  <->  ( S  .\/  T )  =  ( T  .\/  T ) ) )
2420, 23anbi12d 693 . . 3  |-  ( z  =  T  ->  (
( -.  z  .<_  W  /\  ( S  .\/  z )  =  ( T  .\/  z ) )  <->  ( -.  T  .<_  W  /\  ( S 
.\/  T )  =  ( T  .\/  T
) ) ) )
2524rspcev 3054 . 2  |-  ( ( T  e.  A  /\  ( -.  T  .<_  W  /\  ( S  .\/  T )  =  ( T 
.\/  T ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( S  .\/  z )  =  ( T  .\/  z
) ) )
261, 2, 18, 25syl12anc 1183 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  =  ( 0. `  K ) )  /\  ( P  =/=  Q  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( S  .\/  z )  =  ( T  .\/  z ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   joincjn 14403   0.cp0 14468   OLcol 29974   Atomscatm 30063   HLchlt 30150   LHypclh 30783
This theorem is referenced by:  4atex2-0bOLDN  30878
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-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-lub 14433  df-glb 14434  df-join 14435  df-p0 14470  df-lat 14477  df-oposet 29976  df-ol 29978  df-oml 29979  df-ats 30067  df-atl 30098  df-cvlat 30122  df-hlat 30151
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