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Theorem cdlemd3 29078
Description: Part of proof of Lemma D in [Crawley] p. 113. The  R  =/=  P requirement is not mentioned in their proof. (Contributed by NM, 29-May-2012.)
Hypotheses
Ref Expression
cdlemd3.l  |-  .<_  =  ( le `  K )
cdlemd3.j  |-  .\/  =  ( join `  K )
cdlemd3.a  |-  A  =  ( Atoms `  K )
cdlemd3.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
cdlemd3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  -.  R  .<_  ( P  .\/  S
) )

Proof of Theorem cdlemd3
StepHypRef Expression
1 simp33 998 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  -.  S  .<_  ( P  .\/  Q
) )
2 simp1l 984 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  K  e.  HL )
3 simp31 996 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  R  e.  A )
4 simp32 997 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  S  e.  A )
5 simp21l 1077 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  P  e.  A )
6 simp233 1106 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  R  =/=  P )
7 cdlemd3.l . . . . 5  |-  .<_  =  ( le `  K )
8 cdlemd3.j . . . . 5  |-  .\/  =  ( join `  K )
9 cdlemd3.a . . . . 5  |-  A  =  ( Atoms `  K )
107, 8, 9hlatexch1 28273 . . . 4  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  P  e.  A
)  /\  R  =/=  P )  ->  ( R  .<_  ( P  .\/  S
)  ->  S  .<_  ( P  .\/  R ) ) )
112, 3, 4, 5, 6, 10syl131anc 1200 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( R  .<_  ( P  .\/  S
)  ->  S  .<_  ( P  .\/  R ) ) )
12 simp22l 1079 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  Q  e.  A )
137, 8, 9hlatlej1 28253 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  P  .<_  ( P  .\/  Q ) )
142, 5, 12, 13syl3anc 1187 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  P  .<_  ( P  .\/  Q ) )
15 simp232 1105 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  R  .<_  ( P  .\/  Q ) )
16 hllat 28242 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
172, 16syl 17 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  K  e.  Lat )
18 eqid 2253 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
1918, 9atbase 28168 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
205, 19syl 17 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  P  e.  ( Base `  K )
)
2118, 9atbase 28168 . . . . . . 7  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
223, 21syl 17 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  R  e.  ( Base `  K )
)
2318, 9atbase 28168 . . . . . . . 8  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
2412, 23syl 17 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  Q  e.  ( Base `  K )
)
2518, 8latjcl 14000 . . . . . . 7  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
2617, 20, 24, 25syl3anc 1187 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( P  .\/  Q )  e.  (
Base `  K )
)
2718, 7, 8latjle12 14012 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  R  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
) ) )  -> 
( ( P  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) )  <-> 
( P  .\/  R
)  .<_  ( P  .\/  Q ) ) )
2817, 20, 22, 26, 27syl13anc 1189 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( ( P  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) )  <->  ( P  .\/  R )  .<_  ( P  .\/  Q ) ) )
2914, 15, 28mpbi2and 892 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( P  .\/  R )  .<_  ( P 
.\/  Q ) )
3018, 9atbase 28168 . . . . . 6  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
314, 30syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  S  e.  ( Base `  K )
)
3218, 8latjcl 14000 . . . . . 6  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  ( P  .\/  R )  e.  ( Base `  K
) )
3317, 20, 22, 32syl3anc 1187 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( P  .\/  R )  e.  (
Base `  K )
)
3418, 7lattr 14006 . . . . 5  |-  ( ( K  e.  Lat  /\  ( S  e.  ( Base `  K )  /\  ( P  .\/  R )  e.  ( Base `  K
)  /\  ( P  .\/  Q )  e.  (
Base `  K )
) )  ->  (
( S  .<_  ( P 
.\/  R )  /\  ( P  .\/  R ) 
.<_  ( P  .\/  Q
) )  ->  S  .<_  ( P  .\/  Q
) ) )
3517, 31, 33, 26, 34syl13anc 1189 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( ( S  .<_  ( P  .\/  R )  /\  ( P 
.\/  R )  .<_  ( P  .\/  Q ) )  ->  S  .<_  ( P  .\/  Q ) ) )
3629, 35mpan2d 658 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( S  .<_  ( P  .\/  R
)  ->  S  .<_  ( P  .\/  Q ) ) )
3711, 36syld 42 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( R  .<_  ( P  .\/  S
)  ->  S  .<_  ( P  .\/  Q ) ) )
381, 37mtod 170 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  -.  R  .<_  ( P  .\/  S
) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   Basecbs 13022   lecple 13089   joincjn 13922   Latclat 13995   Atomscatm 28142   HLchlt 28229   LHypclh 28862
This theorem is referenced by:  cdlemd4  29079
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-poset 13924  df-plt 13936  df-lub 13952  df-join 13954  df-lat 13996  df-covers 28145  df-ats 28146  df-atl 28177  df-cvlat 28201  df-hlat 28230
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