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Theorem cdlemd3 29656
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 995 . 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 981 . . . 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 993 . . . 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 994 . . . 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 1074 . . . 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 1103 . . . 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 28851 . . . 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 1197 . . 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 1076 . . . . . 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 28831 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  P  .<_  ( P  .\/  Q ) )
142, 5, 12, 13syl3anc 1184 . . . . 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 1102 . . . . 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 28820 . . . . . . 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 2284 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
1918, 9atbase 28746 . . . . . . 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 28746 . . . . . . 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 28746 . . . . . . . 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 14150 . . . . . . 7  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
2617, 20, 24, 25syl3anc 1184 . . . . . 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 14162 . . . . . 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 1186 . . . . 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 889 . . . 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 28746 . . . . . 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 14150 . . . . . 6  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  ( P  .\/  R )  e.  ( Base `  K
) )
3317, 20, 22, 32syl3anc 1184 . . . . 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 14156 . . . . 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 1186 . . . 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 657 . . 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 936    = wceq 1624    e. wcel 1685    =/= wne 2447   class class class wbr 4024   ` cfv 5221  (class class class)co 5819   Basecbs 13142   lecple 13209   joincjn 14072   Latclat 14145   Atomscatm 28720   HLchlt 28807   LHypclh 29440
This theorem is referenced by:  cdlemd4  29657
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-undef 6291  df-riota 6299  df-poset 14074  df-plt 14086  df-lub 14102  df-join 14104  df-lat 14146  df-covers 28723  df-ats 28724  df-atl 28755  df-cvlat 28779  df-hlat 28808
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